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PAGE 1 1 EXPERIMENTAL CHARACT ERIZATION AND APPLIC ATION OF THERMODYNAMICALLY CO NSISTENT VISCOELASTI C MODELS TO DESCRIBE THE BEHAVIOR OF COLL AGEN DERMAL GRAFTS By MULUGETA A HAILE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 Mulugeta A Haile PAGE 3 3 To the memory of Ab iy e e PAGE 4 4 ACKNOWLEDGMENTS My deepest gratitude is due to my advisor and mentor Dr. Peter G Ifju for his support, patience, and encouragement. I am always indebted for the research, teaching and other countless professional opportunitie s I was given while working with him. I would also like to thank Dr. Bhavani Sa nkar, Dr. Ghatu Subhash, and Dr. Jack Mecholsky for serving on my PhD committee and for their very useful feedback on the contents of this thesis. Finally, I would like to acknowledge the love and support I have received from my f amily and friends througho ut my doctoral studies I am so blessed to have them in my life. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF ABBREVIATIONS ................................ ................................ ........................... 15 ABSTRACT ................................ ................................ ................................ ................... 18 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 21 Background ................................ ................................ ................................ ............. 24 Dissertation Outline ................................ ................................ ................................ 32 Main Contribution of the Thesis ................................ ................................ .............. 34 2 LITERATURE REVIEW ................................ ................................ .......................... 36 Introduction to Time Dependent Material ................................ ................................ 36 Phenomenological Models Differential Forms ................................ ...................... 37 Prony Series ................................ ................................ ................................ ........... 39 Integral Forms ................................ ................................ ................................ ......... 43 Hyperelasticity and Pseudoelasticity ................................ ................................ ....... 45 Time Dependent Elasticity: Viscoelasticity ................................ .............................. 48 Linear and Quasilinear Theories of Viscoelasticity ................................ ................. 50 Nonlinear Viscoelasticity ................................ ................................ ......................... 52 Strong and Weak Principles of Fading Memory ................................ ............... 54 Recent Studies ................................ ................................ ................................ 55 3 EXPERIMENTAL CHARACTERIZATION ................................ ............................... 58 Introduction ................................ ................................ ................................ ............. 58 Uniaxial Creep and Relaxation Tests ................................ ................................ ...... 58 Dynamic Test ................................ ................................ ................................ .......... 60 Experimental Procedures ................................ ................................ ........................ 60 Forming Speckle Pattern ................................ ................................ .................. 61 Hydration of Samples ................................ ................................ ....................... 63 Force Measur ement ................................ ................................ ......................... 64 Strain Measurement using DIC ................................ ................................ ........ 66 Mechanical Testing ................................ ................................ ................................ 69 Response to Step Strain ................................ ................................ ................... 69 Elastic response to step stretch ................................ ................................ 71 Relaxation response to step strain ................................ ............................. 73 Elastic Response and Isochronous Stress Strain Curves ................................ ....... 76 PAGE 6 6 Response to Ramp Strain ................................ ................................ ................ 76 Isochron ous Stress Strain ................................ ................................ ................ 79 Stress Relaxation at Multiple Strain Levels ................................ ...................... 79 Concluding Remarks ................................ ................................ ............................... 82 4 ELASTIC IMAGE REGISTRATION AND REFRACTION CORRECTION FOR STRAIN MEASUREMENTS IN FLUID MEDIUM ................................ .................... 84 Background ................................ ................................ ................................ ............. 84 Introduction ................................ ................................ ................................ ............. 86 Refraction Induced Image Distortion ................................ ................................ ....... 87 Registration of Point Sets ................................ ................................ ....................... 90 Local Weighted Mean Transform ................................ ................................ ..... 90 Experimental Procedures ................................ ................................ ........................ 93 The Digital Image Correlation (DIC) Test Setup ................................ ............... 95 Acquiring DIC images of the template ................................ .............................. 95 Acquiring DIC Images of the Test Specimen ................................ .................... 96 Results and discussions ................................ ................................ ......................... 97 Recovering the Transformation ................................ ................................ ........ 98 Transforming images of the test specimen ................................ ..................... 100 Concluding Remarks ................................ ................................ ............................. 106 5 APPLICATION OF THERMODYNAMICALLY CONSISTENT MODELS .............. 108 Introduction ................................ ................................ ................................ ........... 108 Analysis ................................ ................................ ................................ ................ 109 The Quasilinear Model (QLV) ................................ ................................ ............... 110 Implementation of the Quasilinear Viscoelastic Theory ................................ .. 111 The Elastic Response Function, e ( ) ................................ ............................ 112 The Reduced Relaxation Function, G(t) ................................ ......................... 117 Non Equilibrium Thermodynamic Model ................................ ............................... 122 Thermoviscoelastic Relaxation ................................ ................................ ....... 124 Thermoviscoelastic Creep ................................ ................................ .............. 125 Schape ................................ ................................ ................ 127 Introduction ................................ ................................ ................................ ..... 127 Application of Schapery's Theory ................................ ................................ ... 130 Concluding Remarks ................................ ................................ ............................. 131 6 EQUIBIAXIAL CHARACTERIZATION USING BUBBLE INFLATION TEST ......... 133 Introduction ................................ ................................ ................................ ........... 133 Literature Survey on Equibiaxial Testing ................................ ............................... 136 Bubble Inflation Test of Dermal Grafts ................................ ................................ .. 137 Materials and Methods ................................ ................................ .......................... 139 Sample Preparation ................................ ................................ ........................ 143 Dealing with Membrane Porosity ................................ ................................ .... 144 Curvature Modeling Geometrical Formulation ................................ .............. 146 PAGE 7 7 Biaxial Stress Strain Relations ................................ ................................ .............. 147 Planar verses Bubble Inflation Equivalence ................................ .......................... 149 DIC Based Detection of Anisotropy ................................ ................................ ...... 151 Concluding Remarks ................................ ................................ ............................. 153 7 THREE DIMENSIONAL CONSTITUTIVE MODELS BASED ON BUBBLE INFLATION TEST ................................ ................................ ................................ 155 Introduction ................................ ................................ ................................ ........... 155 Three Dimensional Thermoviscoelastic Model ................................ ..................... 155 Application to Inflated Membrane ................................ ................................ ......... 158 Three Dimensional QLV Theory ................................ ................................ ........... 163 Three dimensional Generalization of Schapery's Theory ................................ ..... 166 Application of 3D Schapery's Theory to Bubble inflation Test Data ................ 169 Finding Material Parameters ................................ ................................ .......... 172 Concluding Remar ks ................................ ................................ ............................. 175 8 CONCLUSIONS AND FUTURE WORK ................................ ............................... 177 APPENDIX A THERMODYNAMICALLY CONSISTENT CONSTITUTIVE FORMULATION ....... 183 LIST OF REFERENCES ................................ ................................ ............................. 191 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 202 PAGE 8 8 LIST OF TABLES Table page 2 1 Viscoelastic creep and relaxation functions based on phenomenological models ................................ ................................ ................................ ................ 39 2 2 Prony coefficients of a three and five arm Weichert models, coefficients with 95% confidence bounds, R 2 = 0.97 ................................ ................................ .... 42 3 1 Dimensions of dry and wet dermal collagen grafts used in step stretch and relaxation tests ................................ ................................ ................................ ... 69 3 2 Step stretch and hold test parameters ................................ ................................ 6 9 3 3 Experimental control parameter and results of the stress relaxation test ........... 75 3 4 Dimensions of collagen grafts in dry and wet conditions ................................ .... 77 3 5 Ramp stretch and hold test parameters ................................ .............................. 77 5 1 Materials constants for the elastic response characterization of collage grafts, strained to 40% at a stretch rate of 500mm/min. Constants determined with 9 5% confidents bound, R 2 =0.92 ................................ ................................ ....... 115 5 2 Materials constants for the elastic response characterization of collage grafts, strained to 4 0% at a stretch rate of 10mm/min. Constants determined with 95% Confidents bound, R 2 =0.92 ................................ ................................ ...... 116 5 3 Coefficients of the reduced relaxation function based on a step stretch test (500 mm/min) estimated with 95% confidence bound, R 2 =0.97 ....................... 120 5 4 Experimental mat erial constants used to nonlinear viscoelastic model, coefficients with 95% confidence bound, R 2 = 0.92 ................................ .......... 131 6 1 Stress Strain equiva lence between uniaxial, bubble inflation and planar tests (Reuge et al.2001) ................................ ................................ ............................ 149 7 1 Viscoelastic material parameters used with Eq. ( 7 47 ) and Schapery type thermodynamic model adapted for bubble inflation experiments ...................... 173 PAGE 9 9 LIST OF FIGURES Figure page 1 1 Polypropylene mesh that is still in clinical use for hernia repair is composed of a single stranded polymer in a crystalline molecular structure (Venaporta TM LTD; Silva et al. 2001 ) ................................ ................................ ........................ 21 1 2 An allograft is used in various type of reconstructive surgery, it serves as a scaffold for the host tissue to grow and proliferate (PubMed Central) ................ 22 1 3 A section of the Type I collagen (McKeon et al. 2009 ) ................................ ....... 23 1 4 Molecular structure of the most prevalent GAG known as chondroitin sulfate (Wiki) ................................ ................................ ................................ .................. 26 1 5 Light image of Type I collagen fibrillar (left) and a saturated gel showing fibrillar network embedded in aqueous matrix (Knapp et. al 1997) ..................... 26 1 6 Response to a ramp stretch of collagen dermal grafts, (A) Input strain, and (B) Ramp and relaxation stresses. ................................ ................................ ..... 28 1 7 A typical uniaxial stress strain response of collagen dermal graft pulled at a strain rate of 0.0087strain unit per sec. ................................ .............................. 29 1 8 Uniaxial stress relaxation responses of collagen dermal grafts tested at strain levels of 35% and 40% evidently the relaxation rates depend on the strain level. ................................ ................................ ................................ ................... 29 1 9 Collagen grafts are hydrated with saline solution before mechanical testing. ..... 30 1 10 A typical uniaxial strain filed of collagen grafts using digital image correlation technique ................................ ................................ ................................ ............ 31 2 1 Maxwell model is only accurate for steady state (Meyers et al. 1999) ................ 37 2 2 The Voigt model is only accurate for a solid undergoing reversible strain .......... 38 2 3 The standard linear solid model combing a Maxwell arm with a linear spring .... 38 2 4 Schematic of Maxwell Weichert viscoelastic model (Roylance 2001) ................ 40 2 5 A Five arm prony series approximation of dermal collagen grafts ...................... 41 2 6 A three arm prony series approximation of dermal collagen grafts ......................... 42 2 7 A five arm Weichert model of dermal collage grafts ................................ ........... 42 2 8 Viscoelastic material spectrum ................................ ................................ ........... 49 PAGE 10 10 2 9 Stress relaxation of collagen grafts at various strain level. Contrary to Fung's prediction the relaxation rate is not strain independent ................................ ...... 51 3 1 Test accessories used in the mechanical characterization of dermal collagen samples ................................ ................................ ................................ .............. 61 3 2 Speckled specimen prior to rehydration no flaking of the speck le pattern was observed after loading. ................................ ................................ ....................... 63 3 3 Rehydration of dermal grafts before mechanical testing, the speckled specimen is hydrated for 30 min before being tested ................................ ......... 64 3 4 Custom made grip used in the mechanical testing of dermal grafts ................... 65 3 5 Experimental setup showing DIC arrangement and LabView VI ........................ 66 3 6 DIC uses an affine mapping between deformed and undeformed image space ................................ ................................ ................................ .................. 67 3 7 Typical DIC result obtained from VIC 3D software (correlated solution) ............. 68 3 8 DIC full field displacements and strain measurements superimposed on the speckle image ................................ ................................ ................................ ..... 70 3 9 Experimental Setup for a uniaxial creep and relaxation tests ............................. 71 3 10 Instantaneous elastic response of dermal grafts to a step strain of 40% tested at the maximum stretch rate of 500mm/min ................................ ............. 72 3 11 Relaxation stress and modulus of dermal grafts at 40% ste p strain ................... 74 3 12 Reduced modulus of dermal grafts at 40% step strain for three different samples (Top); the reduced relaxation funct ion with error bars of 1 standard deviation above and below the average value ................................ .................... 76 3 13 Stress strain response of collagen dermal grafts tested at multiple strain rates ................................ ................................ ................................ ................... 78 3 14 Isochronous stress strain response of hydrated dermal collagen grafts ............. 79 3 15 Tensile test samples were excised in a random direction from a large collagen dermis piece ................................ ................................ ......................... 80 3 16 Relaxation response of dermal collagen samples tested at multiple strain levels using samples randomly cut out of a large dermis ................................ ... 81 3 17 Relaxation modulus and the reduced relaxation forms of dermal collagen samples tested at multiple strains ................................ ................................ ....... 82 PAGE 11 11 4 1 Distortion of a rectangular grid due to light refraction and geometrical imperfection of a viewing glass window, Left: Undistorted image, Right: pincushion distortion of rectangular grid ................................ ............................. 85 4 2 A simplified ray optics analysis establishing relationship between experimental setup paramete rs and angle of refraction is often fixed by the design of the chamber and can be varied ................................ ........................ 88 4 3 Refracted and non refracted points of a rectangular grid submerged under water ................................ ................................ ................................ ................... 89 4 4 A soft tissue hydration chamber used for underwater mechanical test of human dermis, the chamber is manufactured MEA A Machine Shop ................ 94 4 5 Schematic representation of the test setup, the twin CCD cameras & VIC 3D comprise the DIC system; MTS machine is controlled by TestWorks software .. 94 4 6 DIC setup for acquiring: (a) reference, and (b) refracted images of the template ................................ ................................ ................................ .............. 96 4 7 Uniaxial test setup inside (a) water filled, and (b) an empty hydration chamber ................................ ................................ ................................ ............. 97 4 8 Template images acquired inside (a) an empty, and (b) water filled chamber .... 98 4 9 Control point selection window (a) refracted (b) non refracted template images ................................ ................................ ................................ ................ 99 4 10 Mapping functions u and v recovered using control points selected in Figure 4 9 ................................ ................................ ................................ ...................... 99 4 11 Template panel (a) reference and, (b) underwater image after transform ........ 100 4 12 Images of the test specimen at 35% strain (a) reference, (b) underwater and, (c) reconstructed ................................ ................................ ............................... 101 4 13 Strain and displacement fields obtained from reference images at 35% strain 102 4 14 Strain and displacement fields obtained from underwater images at 35% strain ................................ ................................ ................................ ................. 102 4 15 Strain and displacement fields obtained from the reconstructed images at 35% strain ................................ ................................ ................................ ......... 103 4 16 In a typical uniaxial stress strain test strain values are sought along lines L 1 and L 2 and the errors of strain measures along these line is shown in Figure 4 17 and Figure 4 18 ................................ ................................ ........................ 104 PAGE 12 12 4 17 Strain along line L 1 of the reference, reconstructed and underwater images .... 104 4 18 Strain along line L 2 of the reference, reconstructed and underwater images .... 105 4 19 Strain errors on underwater and reconstructed images along line L 1 ............... 106 5 1 Stress strain curves of collagen graft at a constant stretch rate of 500 "instantaneous elastic stiffness" of the material. ................................ ............... 111 5 2 Light image of Type I collagen fibrillar showing fibrillar network embedded in aqueous matrix, the first part of the uniaxial stress strain curve is a mere reorientation of fibers (Knapp et al. 1997) ................................ ........................ 112 5 3 The variation of Young's modulus w ith stress at a strain rate of 500mm/min, determination of material constants lines were required to fit the experimental data. ................................ ............... 113 5 4 The variation of Young's modulus with stress at a strain rate of 10mm/min, determination of material constants lines were required to fit the experimental data. ................................ ............... 114 5 5 Comaprison of the stiff' Young's modulus of two samples tested at a stretch rate of 500mm/min. ................................ ................................ ........................... 114 5 6 Comaprison of the stiff' Young's modulus of two samples tested at a stretch rate of 10mm/min. ................................ ................................ ............................. 115 5 7 Comparison of the viscoelastic material constants of four samples tested at multiple strain rates. ................................ ................................ ......................... 116 5 8 Comparison between experimental data and mathematical expression, ................................ ................................ ............................... 117 5 9 Reduced relaxation functions for two specimens tested at a stretch rate ......... 120 5 10 Estimated reduced relaxation functions for two specimens tested at a stretch rate of 500mm/min. ................................ ................................ ........................... 121 5 11 Application of the quasilinear constitutive theory, Eq. ( 5 16 ) to predict strain dependent relaxation behavior of dermal collagen grafts ................................ 122 5 12 Stress relaxation predicted by the energy method. Experimental constants = 0.15, G( ) = 0.471, =1.29, = 1.342, T* = 0.89 .............. 125 PAGE 13 13 5 13 Creep prediction based on experimental constants obtained from uniaxial = 0.15, G( ) = 0.471, =1.29, = 1.29, T* = 0.89 ................................ ................................ ................................ .................. 126 5 14 Nonlinear theory of Schapery applied to describe the relaxation behavior of dermal collagen grafts ................................ ................................ ...................... 130 6 1 Light image of saturated gel showing fibrillar network embedded in aqueous matrix (Knapp 1997). ................................ ................................ ........................ 133 6 2 Schematic drawing of circumferentially sutured graft with the microstructure section view showing randomly arranged fibers ins ide aqueous matrix. .......... 134 6 3 A mechanical test frame used for equibiaxial extension of elastomer (Axel TM physical testing services Inc.) ................................ ................................ ........... 137 6 4 Pictorial view of bubble inflation apparatus use for testing the behavior of collagen dermal grafts ................................ ................................ ...................... 139 6 5 Schematic view of the experimental setup showing section view of the bubble inflation test equipment and the arrangement of digital image correlation system ................................ ................................ ................................ .............. 142 6 6 Sample preparation for the equibiaxial test ................................ ...................... 143 6 7 Comparison of pressure vs. strain curves of an Extra thin Thera Band Latex with a thin brown latex. The Former was used as a backing material during inflati on testing. The reinforcement of the extra thin band on brown thin shown as 'combined' is insignificant. ................................ ................................ 145 6 8 Inflated porous ma trix with and without latex backing. Without a backing material the maximum pressure achieved was 39.6 psi whereas with latex the pressure buildup was unlimited. The samples were speckled using Sharpie marker. ................................ ................................ ................................ 145 6 9 A state of equibiaxial deformation exists at the pole of an inflated bubble ........ 146 6 10 Measured and approximated contours of a hyperelastic membrane, for h/d < 0.5 the bubble shape is spherical and for h/d > 0.5 it becomes ellipsoid (Reuge et al. 2001) ................................ ................................ ........................... 147 6 11 Dimension of inflated specimen showing the change in thickness during test .. 148 6 12 Comparison of equibiaxial and Uniaxial stress strain behavior of collagen grafts ................................ ................................ ................................ ................ 150 6 13 Average equibiaxial stress strain response of collagen grafts (error bars at one standard deviation) ................................ ................................ .................... 151 PAGE 14 14 6 14 DIC strain field of inflated Thera Band latex, elliptic pattern indicates bidirectionality of the membrane. ................................ ................................ ...... 152 6 15 DIC pattern showing the w field and principal strain on the surface of an inflated Thera Band latex. ................................ ................................ ................ 152 6 16 DIC strain field of inflated collagen graft, random patterns indicates anisotropy of the material ................................ ................................ ................. 153 6 17 DIC w field and principal strain of inflated collagen scaffold ............................. 153 7 1 Formulation of thermoviscoelastic model using Green strain components E 2 and E 3 ................................ ................................ ................................ ............... 156 7 2 Para meters describing the creep behavior of inflated bubble, A is the initial radius, and a(t) radius at time t of a circular region enclosing the pole ............. 159 7 3 Thin walled isotropic spherical bubble for which Eq. ( 7 10 ) is formulated ......... 160 7 4 Typical strain field on the surface of an inflated dermal graft ............................ 161 7 5 The equibiaxial viscoelastic creep response of collagen grafts to multiple pressure inputs. Thin inflated membrane theory is used with Haslach's (2004) evolution equation to formulate the creep law. MATLAB's stiff integrator ode15s wa s used to solve the problem. ................................ ........... 163 7 6 Constant inflation pressure is maintained during inflation creep test of dermal grafts ................................ ................................ ................................ ................ 169 7 7 Creep responses at multiple inflation pressure (linear and log log scales) ....... 1 72 A 1 Generalized Voigt model (Schapery 1997) ................................ ....................... 186 PAGE 15 15 LIST OF ABBREVIATION S a point on Fung type stress stretch curve deformation gradient displacement fields in x, y and z directions elastic potential function, strain energy function Fung type material constants Gibbs energy green strain components Helmholtz free energy influence function, ablivator internal state variables internal state variables moisture content right Cauchy stress tensor second Piola Kirchhoff stress tensor steady state relaxation modulus, G infinity temperature thermodynamic force time independent material constants of thermoviscoelasticity constant semi definite positive matrices energy functions instantaneous elastic response function invariants of the right Cauchy Green deformatio n tensor PAGE 16 16 materials constants of the non equilibrium formulation coordinate points, pixel location in the reference or unrefracted image coordinate points, pixel location in the refracted or distorted image creep compliance dashpot constants, damping constant exponent of the creep power law, slope of the log log plot green strain rate, material derivative of the lagrangian strain tensor initial or elastic creep compliance instantaneous radius of a control area at the bubble pole; initial radius local weighted mean transform mapping functions normalized relaxation function, reduced modulus quasilinear viscoelasticity relaxation function relaxation time for constant strain relaxation time for constant stress spring constant at infinity spring constant, modulus of elasticity, green strain step pressure, instantaneous pressure step strain s tep s tress PAGE 17 17 strain dependent material parameters in the irreversible thermodynamic s theory strain shift factor which modulates the time scale s train, c reep strain stress dependent material constants in the irreversible thermodynamics theory stress shift factor which modulates the time scale stress, relaxation stress stretch stretch acceleration temperature shift factor which modulates the time scale the distance of point (x i y i ) from (n 1) th nearest control point on the reference image time scale, variable of integration time stress superposition principle time temperature superposition principle transient creep compliance transient elastic modulus unit step function variable of integration weight function PAGE 18 18 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EXPERIMENTAL CHARACT ERIZATION AND APPLIC ATION OF THERMODYNAMICALLY CO NSISTENT VISCOELASTI C MODELS TO DESCRIBE THE BEHAVIOR OF COLL AGEN DERMAL GRAFTS By Mulugeta A Haile May 2010 Chair: Peter G Ifju Major: Mechanical Engineering The mechanical behavior of c ollagen dermal grafts is nonlinear viscoelastic rather than elastic and a full description of the respon se to load requires a viscoelastic material model which is built on firm theoretical and experimental foundations. Experimentally, the mechanical behavior of dermal grafts has been studied using uniaxial and equibiaxial viscoelastic tests. In a uniaxial te st, the instantaneous elastic response of the material and the subsequent relaxation behavior have been investigated by applying a step like stretch fast e nough to preclude the simultaneous stress relaxation and thereby allow separation of elastic and visc ous responses of the material. The elastic response of dermal grafts is nonlinear with a highly compliant region followed by a stiff linear region. In the highly compliant region the deformation is the result of a progressive reorientation of collagen fi bers inside the fluid saturated gl ycosaminoglycan (GAG) matrix, w hereas in the stiff linear region, collagen fibril extension is the principal mode of deformation. In the stress relaxation phase of the uniaxial test, the specimen was held at a constant st rain while simultaneously measuring the change in stress. It was observed PAGE 19 19 that rapid stress relaxation in dermal graft s generally occur within 10 to 12 seconds of the hold strain and a steady state stress is reached in less than 3 minutes. In addition, the relaxation modulus and relaxation rates depend on strain level implying the presence of considerable viscoelastic nonlinearity that cannot be described by linear ad hoc type constitutive theories This work has considered three thermodynamically consiste nt constitutive theories to describe the experimentally observed behavior of dermal grafts. The quasilinear viscoelastic model (or Q LV) has been shown to describe the strain dependent relaxation stress fairly well. However, the relaxation rate predicted by the QLV theory is significantly slower than what was actually observed during relaxation tests. The n on equilibrium thermoviscoelastic model predicted the relaxation and creep behaviors of dermal grafts from the same set of experimental data. The resultin g model prediction agrees reasonably well with the long term relaxation and creep behaviors, but falls short in predicting the transient response. We have pointed out that such disparity in the transient response prediction can be alleviated by making the constants k and k to vary with time rather than keeping them constant. Schapery's nonlinear constitutive theory which is founded on the principle of irreversible thermodynamics was able to describe the viscoelastic nonlinearity of dermal grafts in a much superior way than both thermoviscoelastic and quasilinear theories. The predictions are accurate in both stress and stress relaxation rate. The thermodynamic consistenc y of these models arise s from the fact that, given a set of data, they can be extended to include such stat e variables as temperature and humidity which generally play a considerable role in defining the mechanical behavior PAGE 20 20 of dermal grafts. Ad hoc spring and dashpot models are not readily adaptable to provide such a robust mechani cal description. Finally, a bubble inflation test (or bulge test) has been carried out to investigate the equibiaxial stress strain behavior of dermal grafts. In equibiaxial loading, the range of the highly compliant region is considerably short er compared to a uniaxial test. However, the stiffness of the dermal grafts in equibiaxial loading is almost the same as the stiffness obtained from a comparable uniaxial test. This very important observation allows the use of a constant multiplicative factor to rela te the stress strain data from uniaxial tests to that of an equibiaxial test. PAGE 21 21 CHAPTER 1 INTRODUCTION Synthetic materials such as silicone, Teflon, Dacron mersilene, and polypropylene mesh have been used as soft tissue replacement in various types of surgical reconstructions (Yanna s et al. 1980; Matsunda et al. 1993) In some patients, synthetic materials support the body's natural wound healing process however complications that arise following surgical repair with synthetic implants lead to chronic inflammation that cause infection. In some surgical application s a secondary surgery is required to remove the synthetic implant, thus increasing patient morbidity (Yanna s et al. 1993) To avoid the limitations of synthetic materials, surgeons began implanting autologous tissue (or autografts) harvested from one part of the body for transplantation to another part However the increa sed pain, risk and operative time associated with managing two surgical sites has put stringent limitation on the use of autologous tissue in surgical operations. Figure 1 1 Polypropylene mesh that is still in clinical use for hernia repair is composed of a si ngle stranded polymer in a crystalline molecular structure (Venaporta TM LTD; Silva et al. 2001 ) PAGE 22 22 The use of allotransplantation, the transplantation of body parts harvested from genetically non identical members of the same species, has long been reported in the biomechanics literature (Veis 19 75 ; Elsdale et al. 1972; Knapp et al. 1997; Holmes et al. 2001; Tranquillo 1999). Human sourced allograft, a transfer from a human donor to a recipient, or bovine (and porcine) sourced xenografts offer biocompatible matrix for remodeling damaged soft tissue in a way that eschews the potential complications associated with both synthetic and autologous implants( Hollister et al. 2002; Kleinman et al. 1987 ; Miller et al. 19 82 ). Dermal collagen grafts (such brands as surgical repairs including abdominal wall reconstruction, burn repairs, tissue replacement after cancer resection and many others. These grafts are g enerally made from type I collagen derived from donated human dermis (RTI Biologics). Figure 1 2 An allograft is used in various type of reconstructive surgery, it serves as a scaffold for the host tissue to grow and proliferate (PubMed Central) PAGE 23 23 Collagen is a natural protein and the predominant biomolecule found in dermis, tendon, ligament, cartilage, and other parts of the body (Bateman et al. 1996) It is the main load carrying element in skin and the basic structural component that gives mechanical integrity and strength to animal's body (Fung 1993; Banga 1966; Bauer et al. 1971). In molecular form collagen is a protein containing sizable domains of triple helical conformation and functioning primarily as supporting elements in an extracellular matrix. The arrangement of amino acids in the collagen molecules is shown in Figure 1 3 E very third residue is glycine and proline and OH proline follow each other relatively frequently (Fung 1993) The individual chains are left handed helices with approximately three residues per turn. The chains are in turn coiled around each other following a right handed twist with a pitch of about 8.6 nm. The three slight longitudinal displacements. T he amino acids within each chain are displaced by a distance of 0.291 nm, with a relative twist of 110 0 making the distance between each third glycine 0.873 nm. An excellent article abo ut the molecular structure of collagen i s available in Freeman et al (2004; 2005), McKeon (2009) Fung (1993). Figure 1 3 A section of the Type I collagen ( McKeon et al. 2009 ) PAGE 24 24 To dat e 20 types of structurally different collagen molecules have been identified ( McKeon et al. 2009 ) Dermis, the material from which surgical grafts are made, contains mainly type I collagen molecule. Structurally, type I is a triple helix composed of three left handed polyproline II helices in tertwined in a right handed manner (McKeon et al. 2009; Silver et al. 2001; Fung 1993). In structural form fiber forming collagen molecules of type I pack together through entropy driven self assembly process forming a network of fibrils with a diameter of 50 200 nm (Veis 1982). The initially weak entropy driven interaction is reinforced by covalent cros s linking and the small fibrilla r segments polymer ize into a highly interconnected randomly arranged long and continuous fibrils via an end to end fusion (Veis 1982; Kadler et al. 1996). In fibrils adjacent collagen molecules are displaced from one another by 67 nm. Purified interstitial collagen scaffolds are reconstituted in vitro u sing a series of controlled chemical processes. Often the extraction process involves dissolving the dermis (or a connective tissue) in acetic acid (of known pH and ionic strength) and bringing the solution into perception at a predetermined temperature (H olmes et al. 1985). Reconstituted Type I collagen grafts are simple and highly biomimetic and are widely used to repair skin wound injuries resulting from trauma and hernia. The s e material s optimize the healing process through a series of complex biochem ical and mechanical interactions with the host tissue. Background In biomechanics the so called the contractile theory of wound healing ( and the consequent formation scar tissue ) explain s the response of an injured tissue in adult humans (Galko et al. 20 04). In a mother's womb, a wounded mammalian fetus PAGE 25 25 undergoes regenerative healing whereby the damaged tissue is fully repaired by the body without a trace. In contrast, wound healing in adult mammals is characterized by contraction of the wound and the sub sequent formation of scar tissue at the wound site (Yannas 2001). The complete biochemical pathway that brings about the formation of scar tissue are not well known (Martin 1997; Singer et al. 1999), but empirical studies have led to the development of wou nd closure rule which states that the initial wound area will completely be filled by some combination of contracted scar tissue, and regenerated tissue. In adult mammals, no regeneration occurs and thus would naturally heal through contraction followed by the proliferation of scar tissue (Yannas 2001). Such healing, however, has several undesirable consequences ; first, scar tissue is stiff and can cause a severe physical limitation and is cosmetically unpleasant ; second, a body may not be able to overcome a loss of large amount of tissue in which case the entire healing will be disrupted resulting in chronic wound (Birk et al. 1991) It is believed that the provision of collagen implants inhibit wound contraction and disrupt scar formation and promote rege neration (Yannas 2001; Colwell et al. 2003; Galko et.al 2004). Therefore, collagen grafts serve the following purposes: 1. Biological purpose; the graft facilitate s cell attachment, distribution and growth of regenerative tissue and facilitate s the transport of nutrients and signals, 2. Biophysical purpose; the graft provide s structural sup port at the site of replacement, According to the contractile theory, the interaction of the host cells and collagen dermal grafts at the wound site is primary mechanical, and as such the micro and macro mechanical properties of the graft plays a prominent role in the regeneration process. Therefore, knowledge of the mechanical property of this material is important PAGE 26 26 in developing theoretical models that could be used to design and manufacture superior grade graft s (Freyman et al. 2001; Roy et al. 1997). In order to develop a mechanical material model, an understanding of the geometrical arrangement of collagen fibers within the matrix is helpful. Reconstituted c ollagen is a porous matrix of collagen fibers with glycosaminoglycans (GAG) bonded to the collagen fibers (Voytik et al. 2001; Parry et al. 1988) At the molecular level the GAGs are long unbranched polysaccharides consisting of a repeating disaccharide un it as shown in Figure 1 4 Figure 1 4 Molecular structure of the most prevalent GAG known as c hondroitin sulfate (Wiki) Figure 1 5 Light image of Type I collagen fibrillar (left) and a saturated gel showing fibrillar network embedded in aqueous matrix (Knapp et. al 1997) The mechanical behavior of collagen grafts is viscoelastic rather than elastic and a full description of the mechanical response to load requires a nonlinear viscoelastic PAGE 27 27 model that is built on a robust analytical foundation consistent with the theory of continuum mechanics (the Clausius Duhem inequality, 2 nd law thermodynamics). Most au thors discuss soft tissue behavior in the framework of the linear theory of viscoelasticity relating stress and strain using phenomenological models of Maxwell, Voigt, Kelvin, Weichret, and others (Fung 1993) These models not only are limited to represent ation of a small subset of the material property but also have no thermodynamic basis; lack of which effects of temperature and humidity cannot be accounted for. The development of thermodynamically consistent viscoelastic constitutive laws generally follo ws two approaches. These are the rational (or functional) thermodynamic approach (Truesdell 1984; Coleman and Noll 1962; Haslach 2004) and the irreversible (non equilibrium) thermodynamic approach (S c hapery 1966, 1969, Lou and S c hapery 1971). In the ration al thermodynamics, the well known Helmholtz free energy is represented in terms of conjugate thermodynamic variables such as strain and stress or temperature and entropy, etc, and then a constitutive equation is formed by taking the derivatives of the free energy with respect the appropriate conjugate variable (Zeng et al. 2002) The irreversible thermodynamic formulation on the other hand, uses certain internal variables in order to describe the internal state of a material. Analytical equations that desc ribe the evolution of the internal state variables then become the governing constitutive equations (Schapery 1969) To be of any practical use, however, the constitutive models must be substantiated with experimental and empirical studies. A casual glanc e at the stress strain time responses of collagen grafts shown in Figure 1 6 through Figure 1 8 reveals that the PAGE 28 28 rate of stress relaxation is strain dependent and the rate of creep is stress dependent. S uch behavior cannot be considered linear (Christensen 1980; Dafermos 1968; Dehoff 1978) and requires a more general description than the ubiquitous phenomenological model that is often used by many authors. Figure 1 6 Response to a ramp stretch of collagen dermal grafts, (A) Input strain, and (B) R amp and relaxati on stresses. PAGE 29 29 Figure 1 7 A typical uniaxial stress strain response of collagen dermal graft pulled at a strain rate of 0.0087strain unit per sec. Figure 1 8 Uniaxial stress relaxation responses of collagen dermal grafts tested at strain levels of 35% and 40% evidently the relaxation rates depend on the strain level. Analytical and experimental study on collagen dermal grafts is being pursued by a team of researchers f rom across a spectrum of disciplines. A series of microstructural studies regarding collagen fibril morphology and correlation of specific molecular PAGE 30 30 features has been undertaken mainly by biomechanics and soft tissue researchers (Osborne et al. 1998; Vies 1982; Knapp 1997; Chandran et al. 2004; Hsu et al. 1994). A second thread of published work on purely mechanical behavior of these materials have focused on the development of viscoelastic constitutive theories and mathematical models (Fung 1971; Ozerdem e t.al 1995; Roeder et al. 2002; Holzapfel 2000; Haslach 2004; Decraemer et al. 1980; S c hapery 1969). Figure 1 9 Collagen grafts are hydrated with saline solution before mechanical testing Owing to the difficult y associated with conducting mechanical tests on the moist, soft and slippery specimens of collagen implants, most of the mechanical studies however, are based on a limited and sometimes inaccurate tensile test data available in the literature. The critic al issue associated with tensile testing of wet collagen implants was the problem associated with measuring local strain on the gage area of the specimen. Traditional strain or stretch measurement techniques such as strain gage, photoelasticity, mechanical extensometer, laser extensometer etc. are not readily adaptable for soft and slippery material s Strain approximation via crosshead motion is accurate only if the material is homogenous and the deformation zone is well defined. PAGE 31 31 Collagen grafts, on the con trary exhibit significant material inhomogeneity with a not so well defined deformation therefore strain measurement based on crosshead motion would result in a grossly erroneous result. To circumvent this issue, a test setup involving digital image corr elation (DIC) technique was adopted in the current research to not only measure local axial and transverse strains, but also to obtain full field strain and displacement data with high accuracy. DIC is a non contact optical method that derives the displac ement and strain field by tracking speckle patterns on the surface of a deforming specimen ( using digital images). The underlying principle of DIC is to calculate the displacement field by tracking the deformation of a subset of a random speckle pattern ap plied to the specimen surface. Details of the experimental setup and procedures are provided in Chapters 3 and a typical DIC strain field is shown in Figure 1 10 Figure 1 10 A typical uniaxial strain filed of collagen grafts using digital image correlation technique PAGE 32 32 Dissertation Outline The outline of this dissertation is as follows. Chapter 2 presents a detailed literature review on linea r viscoelastic models (Mechanical models, Prony series, Boltzmann integral), nonlinear elasticity models (hyperelasticity, pseudoelasticity), the quasilinear theory of Y.C Fung, and the nonlinear single integral viscoelasticity models of S c hapery, and Hasl ach. Emphasis will be given to models that are most adaptable to the behavior of collagen grafts. Chapter 3 describes the experimental setup and procedures I used to characterize collagen graft s A concise description of the materials and methods includin g the test setup for studying the uniaxial viscoelastic relaxation of graft s exposed to step ( or jump) and ramp deformations are presented A detail explanation is given on the adaptations I have carried out to use an experimental facility primar i ly meant for testing traditional structural materials to testing of biological soft tissue. Information on specimen handling and preparation is also given. Chapter 4 presents a new technique for cor recting the refraction induced image distortion that occurs durin g underwater application of the digital image correlation system. Refraction causes nonlinear and non uniform image distortion and current camera calibration algorithms (used with commercial DIC) cannot compensate for the refraction distortion and result s in large reconstruction error s In this chapter, I introduce an elastic image registration technique to correct the distortion while being able to use the same DIC system It is shown that once an image is corrected with elastic image registration techni que, it can be passed onto a DIC algorithm to obtain the deformation and strain fields using the same correlation technique. I also reviewed the PAGE 33 33 nature of refraction distortion and presented an assessment of experimental setups that interacts with the refr action process In C hapter 5 I develop nonlinear constitutive laws describing the behavior of collagen scaffolds based on the experimental data obtained in C hapter 3. As mentioned earlier, the primary focus will be on existing single integral mathematical models that are an outgrowth of the Clausius Duhem inequality and that lead to thermodynamically consistent description of the experimentally observed nonlinear behavior. Emphasis will be on one dimensional characterization but these can be readily extend ed to three dimensions using deviatoric and dilatational stresses and strains as later discussed in Chapters 7 Chapter 6 describes the experimental setup and procedures developed to study the equibiaxial rheology of dermal grafts I introduce an integrate d bubble inflation and digital image correlation (DIC) technique to investigate the multiaxial elastic and viscoelastic creep behavior of a porous dermal matrix The test was performed on a custom made material test rig that I machined in MAE A shop. A sim ple method for resolving the problem associated with testing a porous media using a pneumatic system is documented. Chapter 7 presents a mathematical formulation of the three dimensional constitutive model based on the bubble inflation experimental data. The model formulation will focus on the evolution equation (non equilibrium thermodynamic) approach and the quasilinear (3d QL V) model s Finally, Chapter 8 presents a summary of the main results and conclusion of this work. Some recommendations for future work are also included in this final chapter. PAGE 34 34 E ach of the C hapters 3 through 6 are organized with their own brief introductor y notes and a chapter summary. The latter is particularly aimed at helping the reader identify specific sections within each chapter for easy reference of detailed discussions of results. Main Contribution of the Thesis The main contribution of this thesi s is in the development of a nonlinear material model to describe the behavior of dermal collagen grafts. Experimentally the material is characterized using non contact optical technique (digital image correlation) where it is shown that accurate full fie ld strain and deformation measurements can be acquired by applying minimally invasive speckle patterns on the surface of a dermal graft. Analytical studies are also conducted to represent the nonlinear mechanical behavior of dermal collagen grafts using th ermodynamically consistent constitutive models. The ultimate aim of the mechanical studies is to support the effort of designing functional non biological substitute to these materials for clinical use. All in all, this dissertation is presented as a contr ibution in: 1. The development of experimental techniques to investigate the mechanical property of dermal collagen grafts using the non contact optical method of digital image correlation, 2. The development of the most accurate experimental technique for measuring the viscoelastic stress relaxation and creep responses of collagen grafts using step and ramp loadings, 3. The development of a computational framework for obtaining the most appropriate constitutive model to describe the behavior of collagen grafts and 4. The development of an experimental setup to i nvestigate the response of collagen grafts to biaxial (in situ like) loading where a three dimensional computational model is established, PAGE 35 35 This work is different from related work in the literature on derm al grafts (reviewed in C hapter 2) in that, the experimental approaches are clearly outlined, and model validation results are presented and thermodynamically consistent formulations are experimentally verified. PAGE 36 36 CHAP TER 2 LITERATURE REVIEW Introduction to Time Dependent Material A number of experimental and analytical studies addressing various characteristics of amorphous polymers, semicrystalline solids and biological soft tissues, collectively known as material s with time dependent property have been pursued over the years (Coleman & Noll 1961; Wang 1965; Pipkin & Roger 196 8 ; Green & Riviln 1957; Dafermos 196 8 ; Fung 1993). The goal of the experimental study was to characterizing the behavior of these materials in terms their temporal response using stress relaxations and creep tests. The e xperimental data so obtained was used to develop analytical models having either an integral or differential forms. Early integral forms, such as the hereditary model, are based on Boltzmann's superposition principle and attempt to explain the time depend ent response as a linear combination of loading history (Gurtin et al. 1965; Gross 1953; Christensen 1971) Being a linear model the integral form is only applicable in a very narrow st r ain regime near the equilibrium response of a material. The differenti al forms, on the other hand, are phenomenological models which involve the ad hoc systems of springs and dashpots to predict the short term linear response of time dependent materials (Wineman et al. 2000; Smith 1953) Differential models are often easier to acquire and helpful in visualizing the observed material behavior using simple mechanical analogy as springs and dampers. For this reason, differential models in the form of mechanical springs and dashpots are frequently used by experimentalists (or exp erimenters) who are engaged more in experimental activity than in the theoretical modeling of viscoelastic material (Soussou et al. 1970; Bischoff 2000) PAGE 37 37 Phenomenological Models Diffe rential F orms Often, the property of soft tissue material is discussed within the framework of linear viscoelasticity relating stress and strain on the basis of the ad hoc spring and dashpot models of Voigt, Maxwell, and Kelvin ( Bland 1960; Mendis et al. 1 995 ). The linear springs and dashpots (that these models consist) are intended to mimic the elastic and viscous parts of the material. The differential models, despite their relative ease of use, are often accurate in describing a small subset of the obse rved material response. The Maxwell model shown in Figure 2 1 is only accurate in predicting the steady state creep of an uncross linked polymer and is not accurate in predicting stress relaxation since it assumes a zeros stress after sometime. Equation ( 2 1 ) shows the Maxwell creep model and the initial condit ion for a suddenly applied load (Green 1960; Fung 1993): ( 2 1 ) Figure 2 1 Maxwell model is only accurate for steady state ( Meyers et al. 1999 ) In Eq. ( 2 1 ) is stress, modulus of elasticity. The Kelvin Voigt ( or the Voigt) model, shown in Figure 2 2 is only good for a solid undergoing reversible viscoelastic strain such as a cross linked polymer under uniaxial creep loading. However, the model shows poor agreement with the stress relaxa tion response of many viscoelastic materials (Ferry 1961; Egen 1987) E PAGE 38 38 ( 2 2 ) Figure 2 2 The Voigt model is only accurate for a solid undergoing reversible strain The standard linear solid ( or Kelvin) model combines the Maxwell Model with a Hookean spring in parallel, Figure 2 3 The governing differential equation and initial condition are (Fung 1993) : ( 2 3 ) where ( 2 4 ) Figure 2 3 The standard linear solid model combing a Maxwell arm with a linear spring The factor is called the relaxat ion time for constant strain, is called the relaxation time for constant stress and, E R is the relaxation elastic modulus. If the differential Eqs. ( 2 2 ) ( 2 3 ) and ( 2 4 ) are solved for (t), assuming a step loading (t) = 0 U(t) where U(t) is Hea viside step function, one would obtain the time dependent creep compliance c(t). Similarly, if the roles of stress and strain are interchanged and the above equations are re solved, one would arrive at the relaxation function g ( t). E 1 E 0 E PAGE 39 39 Ideally, the time dependent relaxation function g ( t) corresponds to a force that must be applied in order to produce an elongation that changes, at t = 0 from zero to unity and remains unity thereafter (Hoang et al. 1987) Table 2 1 summarizes the creep and relaxation functions of phenomenological models. Table 2 1 Viscoelastic c reep and relaxation functions based on phenomenolo gical models Creep Function Relaxation Function Maxwell solid Kelvin Voigt solid Standard linear solid In Table 2 1 the unit impulse function (t) is the Dirac's delta which is defined as a function with singularity at the origin, (t) = 0 (for t < 0 and t > 0). The unit step function u ( t) is defined as ( Koeller 1986) : ( 2 5 ) The ratio /E with dimension of time is often referred to as the relaxation time and it characterizes the speed at which the stress approaches the steady state during viscoelastic step and hold experiment ( Roylance 2001 ). Prony Series The Kelvin Voigt, Maxwell and the Standard three parameter solid models discussed in the preceding section are often too simple to give a good quantitative fit to PAGE 40 40 experimentally observed time dependent data of a soft tissue material. In such cases experimenters use a more general and versatile version known as Maxwell W eichert (or Weichert) model. Weichert (Emil Weichert) observed that in cross linked polymers the relaxation does not occur at a single time, but rather at a distribution of time and that shorter molecules (or fibers) contribute less to the relaxation or c reep response than longer ones. He takes into account this variation by providing as many Maxwell elements ( or Maxwell arms) to the mechanical model as are necessary to fit the experimental data. Consequently, the stress relaxation of a Weichert model to a step strain is given by ( Roylance 2001 ): ( 2 6 ) where E is the steady state stiffness of the lone parallel spring, and i E i are the time constants and stiffness of the Maxwell arms. Figure 2 4 Schematic of Maxwell Weichert viscoelastic model ( Roylance 2001) General speaking, the Weich e rt model contains enough constants to accurately describe very complex viscoelastic relaxa tion behavior compared to the three ad hoc models. Dividing Eq. ( 2 6 ) by the step strain, one obtains the time dependent relaxation modulus as: E E 1 E 2 E N 1 2 N PAGE 41 41 ( 2 7 ) The sum of the exponential terms in Eq. ( 2 7 ) is the well known Prony series and is built into algorithm in many finite element software (ABAQUS manual), and the E i 's are referred to as the Prony coefficients. Using the convolution integral theorem, Eq. ( 2 6 ) can be rewritten as: ( 2 8 ) where E is replaced by G to be consistent with the notation used in most of the ( 2 8 ) is the well k nown hereditary integral for viscoelastic modeling of polymers and some biological materials (ABAQUS manual). The general class of materials in which stress depends on strain history is sometimes referred to as materials with memory. A Prony series approxi mation of a dermal collagen graft using three and five arm Weichert model s is shown in Figure 2 5 and Figure 2 6 respectively. Figure 2 7 shows the arrangement of springs and dashpots for the five arm model. Figure 2 5 A Five arm prony series approximation of dermal collagen grafts PAGE 42 42 Figure 2 6 A three arm prony series approximation of dermal collagen grafts Table 2 2 Prony c oefficients of a three and five arm Weicher t models, c oefficients with 95% confidence bounds, R 2 = 0.97 Coefficient Five arm model Three arm model E 0.799 0.803 E 1 0.119 0.092 E 2 0.122 0.146 E 3 0.698 E 4 0.799 1 16.67 0.347 2 253.9 192.3 3 0.147 4 2.098 Figure 2 7 A five arm Weichert model of dermal collage graft s E E 1 E 2 E 4 1 2 4 E 3 3 PAGE 43 43 While using the prony series approximation, it should be noted that the sum of the Prony coefficients should add up to one that is ( Fung 1993): ( 2 9 ) The Prony series approximation has found a wide spread application in modeling linear time dependent materials. However, the method can sometimes be impractical as it requires large number of elements to get a better fit to observed experimental data. The generalized Maxwell model (a large number of sim ple Maxwell elements in parallel) and the generalized Voigt model (a large number of simple Voigt elements in series) also involve the identification of a large number of material parameters ( McCrum et al. 2001) Integral Forms A general integral formulatio n under the assumption of linearity is due to Ludwig Boltzmann (1844 1906). For a linear viscoelastic material, the strain (t) at anytime can be obtained by adding the small step strains (t) caused by the total history of loading up to a time, say t Fo r a uniaxial loading, if the loading history (t) is continuous and differentiable (C1 differentiable), then for a small time interval d in the neighborhood of t the incremental strain becomes: ( 2 10 ) The proportionality constant c depends on the time interval (t ) at which the incremental strain is observed (Fung 1993). Hence, the Boltzmann superposition integral for creep is written as: PAGE 44 44 ( 2 11 ) Interchanging the role of stress and strain in Eq. ( 2 11 ) one would obtain the Boltzmann superposition integral for stress relaxation as: ( 2 12 ) The above integral forms are linear with the proportionality functions c and k The functions c ( t ) and k ( t ) are the creep and relaxation functions. In a multiaxial loading, stress and strain tensors replace the scalar quantities such that, the position vector becomes x (x 1 x 2 x 3 ), ij ( x t) and ij ( x t). Then, the te nsorial relations for creep and relaxation become (Malvern 1969): Relaxation ( 2 13 ) or ( 2 14 ) Creep ( 2 15 ) and are the tensorial relaxation and creep functions. It is interesting to note that the mechanical models discussed in the previous sections are special cases of the integral formulation and can be retrieved by considering the d iscrete relaxation form: ( 2 16 ) PAGE 45 45 where n is called the relaxation spectrum and v n is characteristic frequency associated with a frequency axis (Fung 1993) Hyperelasticity and Pseudoelasticity The theory of hyperelasticity plays a vital role in understanding viscoelastic materials. Occasionally, viscoelastic materials are studied using hyperelastic constitutive laws either by totally disregarding the time dependent response or by preconditioning the material. Fung (1972 1993 ) proposed that, if a viscoelastic material is subjected to a cyclic loa ding, such as uniaxial tension up to certain strain followed by unloading, after certain number of loading and unloading cycles the viscous effects will fade out and the material behaves like a simple elastic material (Graf et al. 1994; Carew et al. 2000) The load unload process is called preconditioning and the viscoelastic material that is preconditioned to behave like an elastic material is called p seudoelastic (Fung 1993). A p seduelastic material that exhibits a large recoverable deformation can be rep resented by a hyperelastic model. Hyperelasticity defines the stress strain behavior of nonlinear elastic material undergoing large deformations such as rubber or biological soft tissue. The stress in a hyperelastic material is not directly related by st rain as it does for Hookean elastic material s instead it is derived from the principle of virtual work using an elastic potential function (or strain energy density function) W. An elastic potential function is expressed in terms of invariants of either le ft or right Cauchy deformation tensors. Generally speaking, a material can be classified as hyperelastic if there exists a scalar function W of the deformation tensors, such that when this function is differentiated with respect to a strain component, it yields the corresponding stress component. Often the formulation of the elastic potential function originates from the PAGE 46 46 local isothermal Clausius Duhem inequality (which is the continuum mechanics version of the second law of thermodynamics). For a thermody namically allowed process (one which obeys the balance of mass and momentum) strain energy potential function W can be express by (Ogden 1972; Rivlin 1955; Spencer et al. 1958): ( 2 17 ) where S is the second Piola is the material derivative of the Lagrangian strain tensor and C is the rig ht Cauchy deformation tensor. The second Piola Kirchhoff stress tensor S is given by: ( 2 18 ) The right Cauchy tensor C can be expressed in terms of the Lagrangian strain E Eq. ( 2 17 ) to obtain: ( 2 19 ) The elastic potential function W = W ( 1 2 3 ) is a power series in the strain invariants. Hence Eq. ( 2 19 ) becomes: ( 2 20 ) where 1 = tr(C), 2 = ( 1 2 tr(C)), 3 = det(C) are invariants of the right Cauchy Green deformation tensor. Differentiating the invariants ( 1 2 3 ) with respect to C: ( 2 21 ) Consequently Eq. ( 2 20 ) becomes, PAGE 47 47 ( 2 22 ) The second Piola Kirchhoff stress S given by Eq. ( 2 22 ) is transformed into the well known Cauchy stress ten ( 2 23 ) where J = det(F). Hence, ( 2 24 ) The strain energy potential function W usually has a polynomial form, ( 2 25 ) where C ij are material constants and N = 1 2, etc For N = 1, Eq ( 2 25 ) becomes the Mooney Rivlin strain energy function, Often, t he reduced form of Eq. ( 2 25 ) with j = 0, is used to model hyperelastic materials: ( 2 26 ) The first order reduced polynomial is called a Neo Hookean solid and is a special case of a Mooney Rivlin model (with C 01 = 0) and is give by, ( 2 27 ) The second order reduced quadratic form is, ( 2 28 ) PAGE 48 48 The cubic form (the third order) reduced polynomial form is called the Yeoh model a nd is given by, ( 2 29 ) All the material constants, C ij in the above equations are obtained by non linear regression or curve fitting of experimental data. In general, the Neo Hookean form can fit test data accurately up to about 50% strain. The Moony Rivlin form is more accurate than the Neo Hookean and it can fit test dat a up to about 90% strain. Higher order forms are usually preferred if the measured strain is greater than 100% ( Bonet 2001 ). It should be noted, however, that strain invariant based potential energy functions do not represent the strain hardening character istics typically exhibited by biological soft tissue (Fung 1993), hence a more robust model is required. Time Dependent Elasticity: Viscoelasticity Elastic constitutive laws, linear (Hookean) and nonlinear (hyperelastic), are applicable for solids whose ti me dependent behavior, such as relaxation, creep and rate dependence are small However in materials such as biological tissue (and amorphous solids) the time effects cannot be ignored and hence the elasticity models are not applicable. Viscoelastic materials, exhibit the combined characteristics of an elastic solid and a viscous fluid. In a purely elastic material all the energy stored during loading is conserved and as a result the relation between load and deformation is reversible. In purely viscous fluids, the shear stress depends on strain rate and that all of the energy stored during loading is dissipated. Viscoelastic materials have both elements of these properties as such they conserve part of the elastic energy imparted during d eformation and dissipate the other. At the molecular level, elasticity is usually the result of bond PAGE 49 49 stretching along crystallographic planes in an ordered solid (Lakes 1998; Baeurle et al. 2006; Rosato 2001; Fung 1981 ) in contrast viscoelasticity is the r esult of the diffusion of atoms or molecules in solids (mainly amorphous ) Viscoelastic phenomenon can be explained as a molecular rearrangement during deformation. For instance, when str ess is applied to a viscoelastic material such as a biological tissue parts of the long fibrous chain changes its orientation with time and this movement or rearrangement manifest itself as an increase in deformation at a constant stress (or a decrease in stress at a constant strain). Materials that exhibit hysteresis creep or stress relaxation can be considered as viscoelastic. As a matter of fact, any material can be treated as viscoelastic since in reality all materials exhibit some sort of strain aging or stress relaxation with time and as such viscous fluids and elastic solids are special cases of viscoelastic materials occupying extreme end s of the viscoelastic spectrum as shown in Figure 2 8 Figure 2 8 Viscoelastic material spectrum In mathematical terms, viscoelastic materials are represented using integral equations (history integral), or a differential equations (evolution equati on) as discussed in previous sections and most literatures, classify viscoelastic materials into linear and/or nonlinear. A linear viscoelastic material is one for which the isochronous stress strain curve is linear and a nonlinear material has nonlinear i sochronous curve. Elastics S olids Viscous Fluids Viscoelastic Materials PAGE 50 50 Linear and Quasilinear Theories of Viscoelasticity describe the time dependent behavior biological materials in a small deformation regime. The theory can be represented by a separable Volterra equation of the second type where the integral function is separable in both creep and relaxation responses as: ( 2 30 ) or ( 2 31 ) (t) is the strain, E(t) and C(t) are the relaxations and creep functions, and is the variable of integration. The lower limit of integration is such that all previous history of loading is included. At small perturbation near equilibrium the linear theory is often adequate, however, for finite deformation problems involving biologi cal soft tissue the nonlinear elastic response must be accounted for. Therefore, a more robust formulation is required for nonlinear viscoelastic materials such as collagen sheets. The most complete published report on the nonlinear viscoelastic behavior of biological materials comes from Y.C Fung (1972). The Fung model, also known as the quasi linear viscoelasticity (QLV) model accounts for the elastic nonlinearity of the stress strain response: ( 2 32 ) PAGE 51 51 H ere the relaxation function is separable into a pure function of time and a pure function dependent elastic Eq. ( 2 32 ) ). If a step strain evolution is considered, which is 0 Eq. ( 2 32 ) 0 0 0 ). This stress is strain depend ent but the speed of relaxation and the time depende nt portion of the modulus are independent of the initial strain level. A similar formulation can be obtained for creep response however Fung (1992) himself s aid that the quasilinear theory does not apply well in describing creep behavior of biological material s Figure 2 9 shows the stress relaxation behavior of dermal gr afts at multiple strain levels. Figure 2 9 Stress relaxation of collagen grafts at various strain level. Contrary to Fung's prediction the relaxation rate is not strain independent The QLV theory is based on the separability of the elastic and viscous response of a material and such separation is only achieved with the application of a step strain to a PAGE 52 52 test specimen. Though strict laboratory application of step strain is not practica l, the model is still accurate when a test is run at sufficiently high stretch rate (Haile et al.2008; Fung et al. 1972). current length divided by the reference length) ob serving the fact that in a certain range = stress is: ( 2 33 ) where ( rate regardless of the applied strain level. Hence, a more robust formulation is required to explain the strain dependent relaxation rate behavior observed in human source d collagen grafts. Nonlinear V iscoelasticity Nonlinear viscoelastic theories are based on the axiom that: for a body undergoing large deformation the stress depends on the entire strain history (Findley 1976, Pipkin 1964) The nonlinearity may be of both m aterial and geometric origins. The material nonlinearity is evidenced by the fact that significant nonlinear behavior is observed at very small strains and is strongly related to the molecular structure of the material (Coleman 1964). Geometric nonlinearit ies arise when displacements and strains become large and the linearized definitions of stress and strain become inadequate. Elastic nonlinearity and time dependence can be differentiated by plotting the stress PAGE 53 53 relaxation data on a log log plot where the n on strain dependent behaviors will have the same slope. Figure 2 10 Stress time response of dermal collagen grafts at multiple strain rates When dealing with metals, it is common to plot the stress strain curve for a constant st rain rate test and regard any deviation of that curve from linearity as an indication of the onset of material nonlinearity. Because of the dependence on time of viscoelastic response, the stress strain and stress time curves, Figure 2 10 from different strain rates of even linear viscoelastic materials are not linear. As time increases during a test, relaxation occurs simultaneously with increasing strains. Thus, we must examine other methods to establish linearity for soft tissues, such as isochronous stress strain plots at different times or modulus plots at different stress levels. PAGE 54 54 Strong and Weak Principles of F ading M emory Most nonlinear viscoelastic constitutive models were formulated with the simplifying assumption of a fading memory. The principle of fading memory states that: deformations that occur in the distant past should have less influence in determining the presen t stress than those that occur in the recent past. In fact, it's natural to relate the fading memory assumption with stress relaxation. Coleman and Noll (19 60 ) first introduced what is known as the strong and weak principles of fading memory ( Colman et al. 1958; Drapaca 2007) They introduced an influence function to characterize the rapidity with which the memory is fading (rate of creep or relaxation). The recollection of strain history or creep is then the L 2 norm ( or magnitude) of the product of the in itial strain and the influence function. The stress relaxation model of Coleman Noll, Eq. ( 2 34 ) assumes that in a simple material with fading memory, the str ess will decay (or relax) to its equilibrium value given by f(C(t)) as the constitutive equation of an elastic simple solid. The Coleman Noll relaxation and creep models are therefore (Drapaca 2007): ( 2 34 ) ( 2 35 ) where is the magnitude of the symmetric tensor The Colman Noll and Wang theories are based on weak principle of fading memory obtained from very slow motion and thus the constitutive law is approximated by an elastic material. Observing that viscoelastic behavior is a combination of elastic behavior (w ith stress dependent only on the rate of deformation vector), Pipkin and Roger (1968) developed a multiple integral constitutive model arranged in series whose PAGE 55 55 first term represents the result of a one step test (stress due to jump strain), and whose n th t erm represents a correction due to the n th step. Recent S tudies The literature on the mathematical modeling of nonlinear viscoelastic materials is replete with formulations which are the outgrowths of Green Rivlin (1957) Col man Noll (1960) and Pipkin Ro gers (1968) The majority of these formulation s are originally intended to model the behavior of polymers (Veis 1975; Dillard et al. 1987; Ferry 1961) Until recently constitutive equations of biosolids and biological tissue are not well known (Lanir et a l. 1983) Lack of this knowledge was a handicap to the development of tissue engineering and biomechanics, because without constitutive equations boundary value problems cannot be formulated, de tailed analysis cannot be made and, finite element methods and predictions cannot be tested or evaluated. Direct adaptation of the existing continuum mechanics literature may not produce an accurate result, as the best known materials like metals and plastics seem to have few counterparts in biosolids and living tiss ues as far as the mechanical properties are concerned (Fung 1993). In his 1972 publication Fung stated that if a jump strain is imposed on a uniaxial specimen, the stress induced became a separable function of time and stretch. By theorizing so, Fung was able to write the relaxation stress as a linear convolution of the very successfully fits the stress relaxation data; however, it does not fit creep data well. Decraemer et al. (1980) small range of frequencies in hysteresis tests. PAGE 56 56 A non phenomenological approach of formulating a nonlinear viscoelastic model has been to write the Helmholtz energy using conjugate thermodynamic variables. The constitutive equations are then formed by taking the derivatives of the free energy with respect the appropriate conjugate variable. Haslach and Zeng (1 999; Haslach 2004) first proposed a system of evolution differential equations to determine the long term behavior of viscoelastic materials using non equilibrium thermodynamic formulation. They defined a generalized energy function that satisfies the Onsa ger reciprocal relationship for state and control variables. The construction of the model requires a pair of conjugate thermodynamic variables defining the long term manifold so that later long term behavior is described by a time independent energy funct ion. On the experimental front, studies o f the behavior of collagen grafts abound. Chandran and Barocas (2004) studied the microstructural mechanics of collagen gels in confined compression in an effort to identify the effect of network porosity on viscoelastic response. Ozerdem and Tozeren (1995) examined the physical response of type I collagen by imposing a quick stretch and noted that the graft tension decays towards a steady state value within several seconds. The instantaneous gel stiffness increased and the relaxed stiffness decreased with the extent of stretching. Anot he r series of experimenta l study on the tensile mechanical properties of three dime nsional collagen matrixes uses c onfocal reflection microscopy to correlate the observed viscoelastic response with specific micro structural features (e.g., diameter and length). Roeder et.al (2002) reported that a change in collagen concentration has an effect on fibril density while pH of the polymerization reaction determines the diameter and length of fibril. PAGE 57 57 The study of viscoelasticity, experimentally and analytically, is at the height of continuum mechanics and has never been neglected by researchers and there is a backlog of literature online for the avid reader. PAGE 58 58 CHAPTER 3 EXPERIMENTAL CHARACT ERIZATION Introduction From the point of view of computa tional mechanics, the property of a material is known if its constitutive equation is known. The constitutive equation of a material can only be determined by experiment. Experimenting with biological materials does not differ much from industrial materials except possibly for the reasons that: (1) large samples and quantities of biological materials, particularly human s ourced tissue is rarely available; (2) controlling the test environment (such as temperature and humidity) and maintaining the condition of a biological sample as close to the in vivo state during mechanical testing requires specialized equipment procedu res and test setups; and (3) biological samples are intrinsically inhomogeneous and exhibit pronounced anisotropy (at least transversely orthotropic), and as such when preparing samples they must be cut out (excised) from the same site of the host tissue a nd in the same direction. Even then there is a significant variability of the experimental data among biological samples tested at the same test conditions. Experimentally, the mechanics of time dependence can be studied using uniaxial tensile creep, relaxation or dynamic (hysteresis) tests. Uniaxial Creep and Relaxation Tests Creep is the tendency of a material to slowly deform under the influence of stresse s It happens as a result of prolonged exposure to levels of stress that are well below the strength of the material. Unlike metals, dermal collagen grafts exhibit considerable creep deformation at room temperature, and the rate of this creep PAGE 59 59 deformation c ontains significant information about the constituents of the material. Viscoelastic creep data can be analyzed by plotting strain time curves of the graft for various input stress levels. Alternatively, creep can be presented with a single curve of the vi scoelastic creep compliance by plotting the creep modulus ( strain divided by to the constant input stress, c(t) = 0 ). It has been observed that below a certain critical stress, the viscoelastic creep modulus is independent of the applied stress level (Findley et al. 1976; Anthony et al. 1942) The relaxation test is almost the converse of a creep test where a uniaxial test specimen is held at a constant strain ( 0 spectrum of stress time curves representing th e relaxation behavior of collagen grafts at different strain levels can be used to represent the property of the graft material. Alternatively a single curve of the viscoelastic relaxation modulus, which is the ratio of stress to the constant input strain 0 ) may be used to describe the general relaxation behavior of this materials especially at lower strains (Heymans et al. 2004) The molecular mechanisms of creep and relaxation in dermal grafts is not well known, however it ha s been observed that with a prolonged application of stretch collagen molecule's periodicity increases contributing to the stress relaxation phenomena (Riedl et al. 1980; Nemetschek et al. 1980). Whereas the re alignment and straightening of collagen molecules in which th e amino acids are placed, contribute to most of the creep strain ( Oakley et al. 19 98 ; Fung 1993). For most biological materials the relaxation response reaches the steady state value faster than the creep response. PAGE 60 60 Dynamic Test Creep and stress relaxation tests are static tests and describe the long term (minutes to days) behavior of viscoelastic materials, but are less accurate at shorter times (such as seconds and less). Some biological materials, such as artery wall, are exposed to cyclic loading whose s hort term dynamic response has more clinical relevance than their long term or static property. In such cases, dynamic tests are capable of determining the short term time transient frequency response of the material. When a viscoelastic material is subje cted to a sinusoidal stress, the resulting strain lags the applied strain by some finite angle It was believed that ( Lockett 1972; Zang 2005) this phase lag is a unique property of a given viscoelastic material and contains enough information to fully characterize the creep and relaxation behavior. However, dynamic tests require more sophisticated equipment and test setups and such f acility was not available in the experimental stress analysis lab at the time of writing of this manuscript and hence the experimental characterizations presented in this chapter are based on relaxation and creep tests. Experimental Procedure s A series of stress relaxation experiments were performed on human dermal collagen graft samples provided by RTI Biologics, Gainesville Florida. The samples were surgical grade implants reconstituted from type I collagen derived from donated dermis. All the tensile sa mples were 10x40 mm rectangular grafts individually wrapped in a sterile plastic packet and labeled with tissue numbers. No special storage and handling was required throughout the test, and the grafts didn't pose biological hazard. For extra precaution, however, samples were carefully PAGE 61 61 disposed in a waste bin specifically designated for biological materials. Some of the simple accessories used to throughout the test are shown in Figure 3 1 Figure 3 1 Test accessories used in the mechanical characterization of dermal collagen samples Forming S peckle P attern As will be presented later, a digital image correlation technique was used to measure the deformation and strain field during loading. DIC, as in most optical techniques, relies on target features (or speckles) for tracking the motion of a test specimen. The graft surface, in this test, is densely speckled (manually tattooed) with a felt point water proof marker with a nominal diameter of 1mm (or about 5 pixels) to provide a full field of random visual targets for intensity correlation system. Aerosol spra y paint which was often the method of choice for speckling digital image correlation samples does not work well with the fluid saturated specimen. In an initial attempt with spray PAGE 62 62 paint (having a modified nozzle), we observed that paint diffuses through th e wet surface of the specimen making individual dots to merge and coalesce together resulting in very large and scattered dots. Moreover, the pain of the traditionally used aerosol peeled off the specimen surface during test ing causing a loss of contrast. Marking individual speckles manually, on the other hand, works very well. To expedite the manual speckling process, we stack a number of markers (four or five in most cases) and pack them together to form a bundle whose felt tip can then be used to form multiple random dots on the surface of the specimen. Often a single marker speckling may need to follow, as a touch up, to improve the density and randomness of the overall pattern. The size and randomness of the speckle patte rn are the two most important issues while preparing all DIC test samples. The speckle size which is measured in terms of the pixel dimensions on the CCD array depends on the size of the pixel subset used in the sampling process. A common practice is to si ze the speckle so that it covers most of the area of a 3 x 3 pixel array as shown in Figure 3 2 Speckles must also be random as otherwise the pattern matching techniq ue upon which the correlation algorithm is based cannot distinguish one target from another, and as such Intensity variations useful to establish the location of targets before and after deformation is not available. Manually speckled test samples before h ydration and after the application of stretch are shown in Figure 3 2 All test samples were speckled right before the thirty minutes hydration period to minimize poss ible chemical interaction between the ink and the graft material ( glycosaminoglycan matrix and amino acid fibers). PAGE 63 63 Figure 3 2 Speckled specimen prior to rehydration no flaking of the speckle pattern was observed after loading. Hydration of Samples All samples used in this test were hydrated with 0.9% NaCl ir rigation solution for 30 min per the standard test protocol of these materials (RTI Biologics). Samples cut to size and speckled, were placed in a plastic tray and the irrigation solution was added around the edge of the scaffold to allow the matrix to slo wly soak up the liquid. It was observed that, pouring the liquid directly on the top of the scaffold or dropping the scaffold in a liquid filled tray would result in large local deformation that would cause the scaffold to w a rp and lose its porous structur e. To avoid such mishap, the tray was first filled with the solution to at least 6mm without ever adding the solution directly on top of the scaffold so as to avoid crushing it. The samples were then allowed to sit undisturbed for at least 30 min prior to testing. Figure 3 3 shows the rehydration process used to irrigate test samples throughout the test. S peckle size PAGE 64 64 Figure 3 3 Rehydration of dermal graft s before mechanical testing, the speckled specimen is hydrated for 30 min before being tested In general, the scaffolds could remain intact for several hour even days before testing however they began to disintegrate overtime and become too fragile to hand le. Samples rehydrated for a longer than 30min have shown a much lower elastic modulus and breaking strength. So to maintain consistence among tests, all samples were tested after equal hydration time ( 1min) at most. Force M easurement The uniaxial force is measured using a 200 N load cell built on an MTS Alliance R/T 30 electromechanical tensile testing machine. The load cell has a resolution of 0.1 N. The TestWork 4 software on the control frame is programmed to record and store force vs. time or force v s A ll tests are displacement controlled with pre defined stretch or strain rates. PAGE 65 65 A problem that often encountered during tensile testing of soft slippery materials such as hydrated collagen grafts was the grip p ing of specimen during testing. The knurled jaws of most conventional grips tend to crush the collagen fiber at the site of the grip and result in premature failure at the grip. Attempting to avoid failure at the grip by less tightening of jaw will of ten result in the slippage of the wet and slippery specimen during test. Such mishap is particularly common when rectangular specimen (not a dogbone) is used. To circumvent this issue, a custom made grip shown in Figure 3 4 having a wave type jaw is used to hold the specimen on the testing machine. Figure 3 4 Custom made grip used in the mechanical testing of dermal grafts PAGE 66 66 Strain M easurement u sing DIC A digital image correlation (DIC) technique is used to measure the local incremental strain field on the surface of all specimens. In a DIC technique, a random gray and white speckle pattern is first formed on the surface of a stress free specime n ( Figure 3 2 ). When a load is applied and the specimen is deforming, successive images are acquired by two pre calibrated stereoscopic digital cameras. The relative d isplacement of subsets of the spackle pattern on the deformed specimen is then compared to match subsets with the reference image (or the zero load image) and the deformation and surface strain field are estimated. Figure 3 5 Experimental setup showing DIC arrangement and LabView VI DIC is predicated on the maximization of a correlation coefficient that is determined by examining pixel intensity array on a series of images and extracting the deformation mapping function that relates the images ( Figure 3 6 ). An iterative approach is used to minimize the 2D correlation coefficient using nonlinear optimization techniques. PAGE 67 67 Figure 3 6 DIC uses an affine mapping between deformed and undeformed image space The cross correlation coefficient r ij is defined as: ( 3 1 ) where u and v are translations of the center of the sub image in the X and Y directions, F(x i y j ) is the pixel intensity (or the gray scale value) at a point (x i y j ) in the reference image, G(x i *, y j *) is the intensity at point (x i *, y j *) in the deformed image. and are the mean values of the intensity matrices F and G. The coordinates or grid points (x i y j ) and (x i *, y j *) are related by a 2D affine transformation as: ( 3 2 ) Typical results obtained from a DIC system consist of geometry of the un deformed surface in discrete coordinates (x, y, z) and the corresponding displacement vector s (u, v, w). A final post processing option involves calculating the in plane strains xx yy xy ), curvature (C xx C yy and C xy ), and strain xx yy xy )). User supplied post processing parameters include step size N (which the D IC system uses to locate grid points), subset size in pixel units. The user may also chose the filter F(x,y) G (x ,y ) y y x x Reference image Deformed image Affine Mapping PAGE 68 68 size (in pixels) that the system later uses for data averaging and smoothing purposes. Default values of the post processing parameters provide accurate r esults for most mechanical and material testing including dermal grafts. The step size for this work is fixed at 5 pixels, with a subset size of 19 pixels. Strain calculations are done with a curvilinear 90% decay filter, using a filter size of 15 pixels for data averaging and smoothing. DIC was first developed at the University of South Carolina and has been optimized and improved in recent years Typical DIC window that was obtained from correlated solutions inc. is shown in Figure 3 7 Figure 3 7 Typical DIC result obtained from VIC 3D software (correlated solution) PAGE 69 69 Mechanical Testing Response to Step Strain We'll describe the mechanical experiment on three nominally identical samples of collagen sheets. The properties of the dry and wet samples before test ing were measured and are shown in Table 3 1. Table 3 1. Dimensions of dry and wet dermal collagen graft s used in step stretch and relaxation tests Dry Samples Hydrated Samples Sample ID Thickness mm Width mm Length mm Weight grams Thickness mm Width mm Length mm Weight (grams) KV3 784 1.1 10.69 40.11 0.18 1.44 11.36 41.14 0.58 KV4 296 1.05 11 40.12 0.09 1.09 10.84 41 0.4 KV4 5335 0.51 10.44 39.86 0.09 0.76 10.92 46.55 0.3 Table 3 2. Step stretch and hold test parameters Sample ID Hydration Time (min) Strain Rate (mm/min) Ramp Time (min) Hold Duration (min) Max Hold Strain (mm/mm) Max Braking Load KV3 784 29 500 NA 30 0.4 NA KV4 296 29 500 NA 30 0.4 NA KV4 5335 29 500 NA 30 0.4 NA The three samples were subjected to a single step strain of 40% and held for 30 minutes while continuously reco r ding the relaxation force. According to the quasilinear theory, a sudden application of strain would give little time for the viscous part to manifest itself during the step deformation. However, since strict application of sudden strain is not possible wi th the MTS machine, we set the crosshead speed to its PAGE 70 70 maximum value of 500 mm/min. According to Fung (1972 ; 1993 ) this speed should give a fairly accurate result. The amplitude of the step strain is set at 40% in order to impose a strain large enough to pr oduce enough deformation on the collagen febrile without tearing the specimen. Figure 3 8 DIC full field displacements and strain measurements superimposed on the speckle image As mentioned earlier, application of a small strain would only probe the tensile behavior of the network and will do very little to reveal the intrinsic property of the fibrillar matrix. Since the fiber is responsible for carrying most of the load the applied strain must be able to induce enoug h stretch in it. I tore two samples one at 50% and a second at 45% strain before realizing that the hydrated samples cannot take a step strain more than 40%. With the analytical explanations to follow, we seek to answer the following primary questions wit h the stress relaxation experiment: PAGE 71 71 1. What does the instantaneous stress relaxation response of collagen sheets look like and how significant is the material nonlinearity, 2. What are the values of the viscoelastic parameters of a collagen sheet, such as the relaxation modulus, instantaneous elastic response and time constants, 3. How fast will the relaxation stress approaches its steady state value, and 4. What is the isochronous curve of the dermal grafts, Figure 3 9 Experimental Setup for a uniaxial creep and relaxation tests Elastic r esponse to s tep s tretch The instantaneous elastic response is the res ulting stress strain curve of the dermal grafts when tested at a very fast stretch rate. The Lagrangian stress strain curve for the three test samples is shown in Figure 3 10 Specimen Camera Left Hydration bath Camera Right PAGE 72 72 Figure 3 10 Instantaneous elastic response of dermal grafts to a step strain of 40% tested at the maximum stretch rate of 500mm/min The stress strain response of collagen grafts in the 'linear' region can be described using neo Hookean law of finite deformation. Johnson et al. (1992) however used the Winemans' (1972) representation of Mooney Rivlin materials for the linear regions: ( 3 3 ) here is the Lagrange stress, is stretch ratio ( ), C 0 lies between 1 and the PAGE 73 73 Hookean materials. Mooney Rivlin material is isotropic. Collagen grafts are transversely isotropic. Let x 3 axis to be the axis of the fiber, then the strain energy function of transversely orthotropic materials is a polynomial of the strain invariants 1 2 3 and the Greens' strain E ij (Green and Adkins 1960): ( 3 4 ) T his fact can be used in the generalization of the unaxial formula to three dimensional equation which is needed in treating compos ite materials of which is collage n grafts are a part. Collagen grafts are viscoelastic rather than elastic. If the stress given by Eq. is considered to be the elastic response and maintained constant afterwards t he induced stress will decrease with tim e. This phenomenon is called stress relaxation and the relaxation behavior of the material is an important property of collagen grafts. Relaxation r esponse to s tep s train The relaxation stress describes how the dermal grafts reli e ve stress under constant strain. Figure 3 11 shows the relaxation responses and the relaxation modulus of the three specimens. The viscoelastic relaxation modulus ( vi scoelastic counterpart of the Young's modulus ) is calculated by dividing the stresses by the initial step strain: ( 3 5 ) where (t) is the relaxation stress, of the 0 is the initial step strain applied at the end of the step loading. PAGE 74 74 Figure 3 11 Relaxation stress and modulus of dermal grafts at 4 0% step strain As shown in the figures, the grafts attained the stead state relaxation stress in less than 10 seconds. Some authors use the form of the relaxation modulus as an indicator of the extent of viscoelasticity nonlinearity in polymers. If dermal grafts were linearly viscoelastic all the curves shown in the figure would have overlapped regardless of the initial hold strain. Since the grafts are nonlinear viscoelastic, however, their relaxation modulus has a form other than Eq. ( 3 5 ) PAGE 75 75 For nonlinear viscoelastic materials, it is customary to normalize the relaxation modulus by its initial value so that G(0) = 1, at t =0. Table 3 3 sh ows test control data and corresponding experimental results of the relaxation test. Table 3 3. Experimental control parameter and results of the stress relaxation test Sample 1 Sample 2 Sample 3 Step stretch Input duration 2.1 sec 2.1 sec 2.1 sec Peak Stress (Lagrange Piola) 1.54 MPa 1.40 MPa 1.44 MPa Reduced relaxation G( ) 0.452 0.45 0.45 Strain rate 21% per sec 20.4% per sec 20.4% per sec The normalized relaxation is termed as the reduced relaxation function, and the later is an important property describing the mechanical behavior of dermal grafts. Several mathematical forms describing the normalized relation function are available in the literature (Mandis et al. 1995; Fung 1993; Pipkin 1986; Nigul et al. 1987). Some of the mathematical expressions are borrowed from linear viscoelasticity (superposition), and others are simple curve fitting. Generally speaking, the relaxation function is n ot unique, and as such any function that satisfies the fading memory assumption (Drapaca et al. 2006; Pipkin 1986) can be used to describe the relaxation response of the collagen grafts. One such expression is a power function of the form: ( 3 6 ) where a and b are experimentally obtained material constants. To obtain these constants we perform a nonlinear regression on the G(t) vs. Log(Time) data as will be described in C hapter 4. PAGE 76 76 Figure 3 12 Reduced modulus of dermal grafts at 40% step strain for three different samples (Top); the reduced relaxation function with error bars of 1 standard deviation above and below the average value Elastic R esponse and I sochronous S tress S train Curves Response to Ramp S train As said earlier, the vali dity range of experimental data obtained using step stretch (500 mm/sec) is questionable, because it is very unlikely that these materials will be exposed to such a high deformation rate during service (operation). Hence it is necessary to examine the char acteristic of the materials in a ramp type loading condition using different strain rates. Following the usual specimen preparation (speckling and rehydration) four nominally identical samples were subjected to ramp deformation at various strain rates. PAGE 77 77 Fi gure 3 11 shows the stress strain curve of four specimens tested at four different strain rates; 0.007, 0.005, 0.003, and 0.002 strain units/sec. As shown in the figure, the stress strain response of collagen grafts is sensitive to strain rate. Samples tes ted at lower strain rates exhibit a lower elastic modulus due to the fact that the specimen underwent significant viscoelastic relaxation during the prolonged ramp time. Samples dimensions and experimental control parameters are shown in Tables 3 4 and Ta ble 3 5. Table 3 4. Dimensions of collagen grafts in dry and wet conditions Dry Samples Hydrated Samples Sample ID Thickness mm Width mm Length mm Weight grams Thickness mm Width mm Length mm Weight (grams) KV4 5285 0.45 10.61 39.76 0.1 0.65 11.89 41.25 0.33 KV4 3996 0.56 10.53 39.85 0.1 0.78 10.72 42.1 0.32 KV3 827 0.53 10.66 39.67 0.1 0.72 11.34 41.81 0.31 KV4 1328 0.48 10.43 39.65 0.1 0.69 11.12 41.87 0.33 Table 3 5. Ramp stretch and hold test parameters Sample ID Wt. NaCl Absorbed (grams) Wet Volume (mm 3 ) Vol. NaCl Absorbed (mm 3 ) Porosity (percent) KV4 5285 0.23 318.80 227.72 71.43 KV4 3996 0.22 352.10 217.82 61.87 KV3 827 0.21 345.04 220.83 67.57 KV4 1328 0.24 320.17 228.72 70.44 Different stress strain curves will be generated based on the strain rate at which the deformation is taking place and such experimental data doesn't easily lend itself to a tractable mathematical formulation. Moreover, the constitutive property of the gra fts PAGE 78 78 Figure 3 13 Stress strain response of collagen dermal grafts tested at multiple strain rates One way of obtained a general and unique constitutive property was through exposing the materials to step stretch. A step stretch, by effectively minimizing the simultaneou s relaxation, was able to produce a data that is separable to elastic and PAGE 79 79 viscous response of the material. A second approach which is applicable to ramp type loading is to construct the isochronous stress strain curve of the grafts. Isochronous Stress S t rain The isochronous curve shows the stress strain relations of a viscoelastic material at constant times. For the dermal grafts, we construct the isochronous curve shown in Figure 3 14 by combining the stress strain curves of the four samples shown in Figure 3 13 Figure 3 14 Isochronous stress strain re sponse of hydrated dermal collagen grafts Stress R elaxation at M ultiple S train L evels The stress relaxation behavior of collagen grafts and the relaxation modulus as well as the reduced modulus are investigated at multiple strain initial strain levels. The samples used this time are slightly different from the rectangular samples provided with in the sterile packet. Three rectangular samples are cut out of a large dermal matrix shown in Figure 3 15 using pocket knife and accessories shown in ( Figure 3 1 ). Even PAGE 80 80 though we are aware that this material has directional property the cut was made randomly without making any reference (no reference direction was available) and hence we assume isotro pic behavior and proceed with the development of characterization methodology. Figure 3 15 Tensile test samples were excised in a random dire ction from a large collagen dermis piece The three samples were then manually speckled using a sharpie marker; rehydrated for 30 minutes and tested in a similar manner as presented in previous sections. We apply a step strain of 35%, 20%, and 12% at the m aximum stretch rate that is possible with the servo motor of the MTS testing machine. The relaxation stress was then measured for about 5 minutes. In general, the clinically important mechanical response is the one that happens within the first few minutes in most cases 60 seconds. Moreover, I have observed that the relaxation stress approaches it s stead state value in the neighborhood of 10 sec. Hence a 5 minute test would provide enough information regarding the viscoelastic behavior of the materials. Figure 3 16 shows the relaxation stress response of the three samples. Cut Directions PAGE 81 81 Figure 3 16 Relaxation response of dermal collagen samples tested at multiple strain levels using sample s randomly cut out of a large dermis Similarly, the relaxation modulus, which is the stress divided by the initial strain, and the reduced modulus, relaxation modulus normalized by its initial value, for each of the three samples is shown in Figure 3 17 From Figure 3 16 and Figure 3 17 it can be observed that; (1) samples tested at low initial strain relax to their steady state quickly compared to the one tested at high strain level, hence the dermal grafts exhibit strain dependent relaxation rate, characteristic of nonlinear viscoelasticity; (2) the relaxation modulus of the three samples is different as evidenced by the fact that the three curves of g(t) didn't overlap, here this implies material nonlinearity (same analogy applies to Young's modulus) ;(3) the reduced relaxation function appears to overlap on each other satisfying possible Fung type behavior of the material. PAGE 82 82 Figure 3 17 Relaxation modulus and the reduced relaxation forms of dermal collagen samples t ested at multiple strains Concluding Remarks This chapter has been devoted to the experimental study of dermal collagen grafts using various modalities of viscoelastic relaxation tests. Rectangular samples were prepared and stretched using a displacement control testing machine. The load and elongations are recorded for prescribed loading, strain level and hold time. From these records the following observations and conclusions were made: 1. The instantaneous response of dermal grafts to a step stretch (500 m m/min) has been examined and is shown in Figure 3 10 It is believed that a single stress strain curve can be used to describe the elastic behavior of the grafts if a stretch is applied at fast enough speed to preclude simultaneous relaxation during loading. In general, the elastic behavior can be represented by Neo Hookean or Moony Rivlin hyperelastic strain energy function. PAGE 83 83 2. The response to ramp load of dermal grafts depends on both strain and stretch rate. For the same strain amplitude a higher stretch rate of strain results in a higher peak stress. 3. Typically the stress strain curve of the grafts is nonlinear, Figure 3 13 with a highly compliant region followed by a stif f linear region. In the apparently high compliant region, the stretch simply resulted in progressive orientation of collagen fibers in the direction the princi pal stretch axis. In the linear region the collagen fibril extension is the principal mode of deformation. The compaction of the sample led to considerable resistance to further fibril l ar reorientation has resulted in a stiff region in the stress response. 4. Ramp stress strain curves at different strain rates shown in Figure 3 13 are not an efficient way of describing the mechanical response as the response drastically ch anges with strain rate of the deformation. 5. Isochronous curves shown in Figure 3 14 are better suited to describe the stress strain response of dermal grafts, it is sh own that the isochronous curve can be represented by a simple power law whose material coefficients are uniquely determined experimentally. 6. The stress relaxation response of collagen grafts to step stretch is shown in Figure 3 11 Figure 3 16 In general, rapid stress relaxation occurs within 10 to 15 seconds and a steady state valu e of the stress is reached in less than 3 minutes. The fact the relaxation modulus and relaxation rates depend on strain level imply that dermal collagen grafts exhibit pronounced material nonlinearity and as such this behavior cannot be described by linea r constitutive theories. 7. The reduced relaxation modulus, shown in Figure 3 12 and Figure 3 17 is about 0.7 (G = 0.7) and the reduced relaxation behavior of all samples can be desc ribed by a single mathematical function. Biological materials which exhibit such type of relaxation behavior (aka Fung material) can be described using the quasilinear viscoelastic theory or QLV. There has been significant sample to sample variatio n even among samples tested in identical conditions. The source of these variations can't be readily explained without having enough information about the reconstitution process. Generally biomechanics literature report s data of preconditioned specimen s p erhaps t o reduce the inherent inconsistency between identical experiments. Our goal here has been to characterize the dermal collagen grafts and explain their mechanical behavior in the final form instead of modifying them to fit some ubiquitous theory. PAGE 84 84 CH APTER 4 ELASTIC IMAGE REGIST RATION AND REFRACTIO N CORRECTION FOR STR AIN MEASUREMENTS IN FLUI D MEDIUM Background One of the requirements for mechanical testing of dermal grafts is that the sample s be tested within one minute after the thirty minute rehydrat ion period. The one minute test time was set as a requirement by RTI (RTI Biologics) in part because the specimen dehydrates if it stays out of the irrigation solution for a prolonged time. This requirement, however, imposes several restrictions to the ent ire mechanical testing and modeling process. First, the one minute duration is too short to conduct a test as it takes more than sixty seconds to clamp a specimen load on the MTS machine and run a displacement control test at a finite crosshead speed (which is often in the order of 10 mm/min if dynamic effects are to be avoided) Second a viscoelastic test cut short will result in erroneous relaxation modulus (G ) and leads t o an inaccurate constitutive model For the dermal grafts it takes at least 1 0 13min before the relaxation stress settles to its steady state value. Testing the re hydrated specimen out of the irrigation solution will cause a loss of moisture which means that the specimen may undergo significant structural change while the experime nt is being conducted. That i s, if a 30min stress relaxation test is conducted on dermal graft outside a hydration bath its moisture level will change with time with the graft be ing completely wet and moist at the early stage of the test and almost dry at the end. In separate test conducted on a dry specimen, we have observed that the stress strain response as well as the stress relaxation behavior is significantly different from an identical specimen in moist or re hydrated state. Therefore if a prolonged t est is conducted outside a rehydration bath (as a matter of fact stress PAGE 85 85 relaxation and creep tests require longer test times), the dermal graft may undergo considerable structural or microstructural change while the experiment is being run. This however, violates the assumption of non aging materials that is used in the development of most constitutive laws including the ones presented in this work. If a material experiences significant change during an experiment then the material is called an aging or ti me hardening material and as such it does not obey the principle s of frame indifference and is a different class by its own (Noll 1959; Truesdell and Noll 1965). For these reasons i.e. to keep the material property the same throughout the test and to comply with Noll and Truesdell's non aging assumption, we require to run the mechanical test on specimen s while the latter are completely submerged in a tissue bath ( or rehydration bath) filled with saline solution. Submerged tests, however, proved to be tricky as digital images acquired from the underwater s pecimen contains refraction distortion which cannot be corrected by existing DIC systems. Figure 4 1 shows a pincushion type distortion that typically occurs on under water images acquired through a glass window. There is significant change in size and geometry as well as non uniformity in the distorted image when compared to its undistorted counterpart. Figure 4 1 Distortion of a rectangular grid due to light refraction and geometrical imperfection of a viewing glass window, Left: Undistorted image Right: pincushion distortion of rectangular grid h h PAGE 86 86 A digital image correlation (DIC) system as the name implies, depends on digital images and if the images are distorted then the strain and deformation measurements will be in accurate If the distorted image shown in Figure 4 1 is input instead of the undistorted then there is going to be correlation error. As it turns out, this issue was in fact the most serious obstacle that hampers the use of non contact optical techniques such a s DIC in fluid medium as nonuniform image distortion s are difficult to correct To the authors knowledge there is no literature reported on the correction of refraction distortion as related to DIC. A phone conversation with one of the major DIC providers (Correlated Solution Inc.) has proved to be so. Introduction Measuring the deformation and strain field of a liquid submerged body using the digital image correlation (DIC) poses several obstacles. These include the difficulty associated with placing the c alibration grid in the liquid, insufficient illumination due to suspended particles, and image distortion by refraction. In a standard DIC procedure a rectangular calibration grid of known dimensions is placed close to the object being deformed and a serie s of perspective images are acquired. The system is then required to extract control points from the calibration grid and match them with known grid dimensions. The accuracy of a DIC depends heavily on the accurac y of the calibration process and the latte r while the calibration grid is submerged inside a liquid is not trivial for several reasons: (a) in cloudy liquids, such as a soft tissue nutrient bath, the dispersion of light makes extraction of control points very difficult ; (b) often it is not desirab le to immerse a calibration grid inside fluid filled containers due to test related restrictions PAGE 87 87 such as a sterile and sealed organ bath (c) finally there is always image distortion due to refraction of light. These drawbacks prohibit calibration of a DIC system with a submerged calibration grid limiting the use of non contact image based techniques in many areas of experimental mechanics, particularly in biomechanics and soft tissue engineering where strain measurement using crosshead motion is grossly ina ccurate and the use of strain gage and extensometer is unlikely. In a typical laboratory setting, however, it is possible to mitigate some of the above problems by manipulating experimental conditions and using image processing techniques. The approach p resented herein makes full use of MATLAB's open source code (The MathWorks Inc.) to perform elastic image registration, and VIC 3D (Correlated Solutions Inc.) correlation algorithm to obtain full field strain and deformation measurement from a submerged sp ecimen. As said earlier, t he study presented in this C hapter is part of a parallel research that I have taken in order to investigate the viscoelastic stress st rain behavior of dermal grafts while the latter is submerged in saline solution. Refraction Indu ced Image Distortion Refraction occurs when a light passes from one medium to another. The extent of refraction depends on the property of the media involved and the angle of incidence of the light ray. For a given test setup defined by the relative locat ion of the camera, interface plane and an object point P as shown in Figure 4 2 Snell's law provides (Kwon 1999) : ( 4 1 ) PAGE 88 88 where and to interface distance, is the interface to object distance, is the object to normal axis distance, k is the distance from the interface point to the normal axis normalized by and c is the relative refractive index of the first medium to the second medium Figure 4 2 A simplified ray optics analysis establishing relationship between experimental setup parameters and angle of refraction is often fixe d by the design of the chamber and can be varied The Plexiglas interface does merely shift the ray's point of exit and has negligible effect on the angle of refraction. The relative refractive index is not a constant and varies with the ambient temperature and particle content (and densit y) of the liquid. For a given experiment al setup defined by parameters and c the ratio k is obtained by solving (Kwon 2006): ( 4 2 ) Parameter k determines the degree of image distortion induced by r efraction in underwater imaging. In general, smaller value of k with being constant, implies a less distorted image and vice versa (Kwon 2006). Figure 4 3 shows the refracted and non refracted points of an arbitrary rectangular grid submerged underwater. The dark o k o P c c Liquid Plexiglas Air R A R L P R PAGE 89 89 squares are the actual l ocation of grid points on the un refracted image and the white squares indicate the apparent position of the same grid points on the refracted image. As shown in the figure, the refracted image is larger than the non refracted image and the degree of nonlinearity increases with the angle Figure 4 3 Refracted and non refracted points of a rectangular grid submerged under water Using the quantities defined in Figure 4 3 the Lagrangian normal strain y introduced by the apparent refraction distortion can be defined as: ( 4 3 ) where y o and y 1 are the initiation and final length of the rectangular grid before and after immersion in the liquid. The pseudo st rain given by Eq. ( 4 3 ) is the result of refraction of light ray and has nothing to do with mechanical loading of the specimen. Also note that, the pseudo strain is not uniform since points far from the center exhibit a lager pseudo displacement than points closer to the center of the image. A similar expression can be written for strains x and xy y 1 y 0 x y z Refracted PAGE 90 90 Registration of Point Sets Image registration is the process of aligning two images of the same scene through control point matching and spatial transformation. A spatial transforma tion modifies the spatial relationship between pixels in a given input image to new locations in an output image. The refraction distortion can be treated as a shift in the spatial location of control points as shown in Figure 4 3 For a given experimental setup characterized by and c of Figure 4 3 control points on the re fracted image can be brought into alignment with the unrefracted counterparts using a local weighted mean (LWM) transform. LWM is a non rigid mapping which allows a local elastic deformation of the refracted image to match with the reference image. The map ping functions in the local weighted mean transform are combinations of many local transformations each describing local geometric differences between the two images. Local Weighted Mean Transform Given N non collinear set of control points (x i y i ) on the non refracted (or reference image) and (u i v i ) on the refracted image as shown in Figure 4 3 it is possible to construct an approximate mapping between these points using polynomial functions f and g such that: ( 4 4 ) The problem of finding functions f and g that would overlay the N control points in the two images can be simplified by redefining the problem as follows: given measurements of two sets of 3D points (x i y i u i ) and (x i y i v i ) determine functions f and g that will fit the two sets of data points. There are three different options available to solve this problem; (1) fitting polynomials which treat every measurement equally, (2) PAGE 91 91 fitting polynomials that are more sensitive to local measurement but still tak e into account global data; and (3) fitting polynomials that uses only local measurement to determine the mapping functions. In this work we used the third option because it has been observed that the refraction distorted image exhibit significant non unif ormity and as such a more accurate interpolation could be achieved only in small localized subsets of the control points. Now, consider fitting a polynomial with n parameters to each measurement (x i y i u i ) and (x i y i v i ) such that the polynomial fits that measurement and n 1 of its nearest measurements. Let (x i y i ) be the coordinates of the i th control point in the reference image and u i be the u component of the same control point in the refracted image. The nearest measurement refers to the data that corresponds to the control point closest to (x i y i ) in the reference image. These polynomials can be determined by: ( 4 5 ) Then the n parameters of the polynomial can be determined by substituting the N measurements into Eq. ( 4 5 ) and solving the system linear equations obtained. ( 4 6 ) If a point in the reference image is near control point i then the point corresponding to (x, y) in the refracted image is also near control point i therefore, the u value of a point (x, y) can be determined from u values of the nearest control points wi th appropriate weights as: PAGE 92 92 ( 4 7 ) Most authors choose the weight functions as the inverse distance of point (x, y) from control point i Such approach may provide interpolation sensitive to local measurements however it also takes into account global data. As said earlier, we attempt to f it images onto one another using a set of polynomials function that perform transformation in a certain local region of the image only. That is the zone of applicability of polynomials f and g is limited to a finite local region around control point i The condition of locality is enforced by defining the weight function as (Goshtasby 1988): ( 4 8 ) where L = [(x x i ) 2 + (y y i ) 2 ] 1/2 / l n and l n is the distance of point (x i y i ) from (n 1) th nearest control point of the reference image. The local weight function ensures that polynomial i will have no influen ce on points in the reference image whose distance from control point is greater than l n The weighted mean of all polynomials passing through an arbitrary point (x, y) and having non zero weight are: ( 4 9 ) PAGE 93 93 ( 4 10 ) w here P i (x, y) and Q i (x, y) are polynomials passing through points (x i y i u i ) and (x i y i v i ) for all (n 1) control points nearest to point i Polynomials P i and Q i are often chosen to be of order two however higher degrees can also be applied as the weighted sum obtained will be always continuous and smooth all over the image. Experimental Procedures To verify the validity of the elastic image registration scheme presented in the previous section s a series of underwater tests were performed on a thin latex rubber specimen with a nominal thickness of 0.75mm. The latex surface is densely speckled (manually tattooed) with a felt point water proof marker with a nominal diameter of 1mm (or about 5 pixels). Before the start of the underwater test, the specimen was preconditioned to ensure that repeated tests performed to the same crosshead displacement in water and in air result in identical deformat ion and strain measures. The uniaxial test was carried out inside a soft tissue hydration chamber of the type shown in Figure 4 4 Figure 4 5 shows the schematic arrangements of the DIC system in relation to the MTS tensile testing machine. PAGE 94 94 Figure 4 4 A soft tissue hydration chamber used for underwater mechanical test of human dermis, the chamber is manufactured MEA A Machine Shop Figure 4 5 Schematic representation of the test setup the twin CCD cameras & VIC 3D comprise the DIC system; MTS machine is controlled by TestWorks software MTS Actuator Speckled Specimen Twin CCD Cameras DIC System MTS Machine Control Load Cell Templat e Grip Plexiglass Water PAGE 95 95 The template panel shown inside the hydration chamber on Figure 4 4 is a uniformly gridded rectangular panel of 250mm long and 120mm wide with 120x50 control points (CPs) with 0.85mm in diameter uniformly spaced apart. The CPs are printed in black on a white background similar to the calibration board of commercial digital image correlation systems. The Digital Image Correlation (DIC) Test Setup The DIC system used in this test is shown in Figure 4 6 and Figure 4 7 The sy stem is composed two stereo cameras (RETIGA 1300) connected to a PC via an IEEE 1394 firewire cable and a specialized camera trigger synchronization device. The twin cameras are calibrated by acquiring perspective images of 13x10 (offsets x=3, y =5) calibr ation grid. A bundle adjustment software purchased from correlated solutions Inc. installed in the computer calculates the in trinsic (focal length, principal point and distortion parameters) and extrinsic (translation vector and rotation matrix) for each c amera. The later is used to compute the 3 D location of points based on digitized images obtained from the two cameras during the stereo triangulation process. Acquiring DIC images of the template The template was positioned at the middle of the empty cha mber and is tightly clamped with its front face parallel to the viewing side of the chamber and its edges aligned with the axis of the testing machine as shown in Figure 4 6 a. A pair of digital images is captured and stored in the default data format (*.tif). PAGE 96 96 Figure 4 6 DIC setup for acquiring: (a) reference, and (b) refracted images of the template The chamber is then filled with running water without disturbing the template and a second pair of di gital images is acquired and stored in the same default data format, Figure 4 6 b. These two pairs of images are later used to recover the transformation functions as w ill be illustrated in the next section. Acquiring DIC Images of the Test Specimen The speckled latex specimen is securely clamped inside the empty chamber and a uniaxial stretch test was performed. Reference and deformed images of the specimen are capture d at four different strain levels (13, 20, 27, and 35%) using the pre calibrated DIC setup. The cross head displacement of the test frame is automatically controlled by Testworks software installed on separate computer. Since this set of images were captur ed while the chamber is empty as shown in Figure 4 7 b, they can be directly input into the VIC 3D correlation algorithm to calculate the displacement or strain fields on the surface of the specimen. In a repeated test shown in Figure 4 7 a, the chamber is filled with a tap water and the same uniaxial test was performed on the same specimen, i.e. in the repeat test nothing has changed except that the chamber is filled with water. Camera 1 Camera 2 (a) Material Testing Frame Empty chamber Water filled chamber Template (b) PAGE 97 97 Figure 4 7 Uniaxial test setup inside (a) water filled, and (b) an empty hydration chamber Reference and deformed images of the test specimen are captured at the four strain level s as before. Since th i s set of images were captured while the specimen is deforming underwater, the images are bound to contain refraction induced distortion and need to be corrected before being input into the DIC. The refraction correction process involvi ng the elastic image registration scheme is presented in previous section. Results and discussions Figure 4 8 shows the non refracted and the underwater images of the template panel captured by camera 2. Only the elastic image registration scheme that determines the mapping function for camera 2 will be presented. The mapping for camera 1 can be determined in the same manner. As shown in the figure, control points on t he underwater image are shifted to the left and the image is larger in size than its non refracted counterpart. The non uniformity of the distortion is evident from the variation of distances between the larger control points marked on the images. Camera 2 Camera 1 Empty chambe r Specimen Material Testing Frame Water filled chamber (a) (b) PAGE 98 98 Figure 4 8 Template images acquired inside (a) an empty, and (b) water filled chamber Recovering the Transformation The localized weighted mean transform that maps the refracted image into the non refracted image space is recovered by registering the two images of the template and comparing the spatial positions of the control points as follows : 1. Load the two images into MATLAB works pace and perform image segmentation and control point matching as shown in Figure 4 9 2. Extract coordinates of all matching control points (x, y) and (u, v) from both the underwater and reference images, 3. Find local mapping functions u i and v i per the procedure discussed in the previous section s using Equations ( 4 9 ) and ( 4 10 ) (u, v) (x, y) (a) (b) PAGE 99 99 Figure 4 9 Control point selection window (a) refracted (b) non refracted template images The local mapping functions recovered using the three steps listed above are plotted in Figure 4 10 The 3 D surface functions shown in the figure transform all underwater images acquired by the same DIC setup to register with their non refracted counterpart. Figure 4 11 compares the transformed and the reference images of the template. As can be seen on the figures the transformation has achieved a reasonable registration of the two images. Figure 4 10 Mapping functions u and v recovered using control points selected in Figure 4 9 Mat ched pair of CPs (x,y) ( u v ) PAGE 100 100 Figure 4 11 Template panel (a) reference and, (b) underwater image after transform Transforming images of the test specimen For a give n experimental set up, the u i and v i mapping functions that enable reconstruction of the underwater image are generated via control point matching of the template image Underwater images of the test specimen are swept by the parameters of the mapping functions to register the refracted image space with the refere nce image space. Figure 4 12 shows the reference image, the underwater image and the reconstructed image of the test specimen at 35% strain level. As shown in the figures, the reconstructed image registers well with the reference image. (u, v) ( x y ) (a ) (b) PAGE 101 101 Figure 4 12 Images of the test specimen at 35% strain (a) reference, (b) underwater and, (c) reconstructed As a final step, these images were directly input into VIC 3D correlation algorithm to calculate the in plane displacement and strain filed in x and y directions. Figure 4 13 through Figure 4 15 show the full field strain and displacement on the surface of the deformed specimen for the three cases at 35% strain level. The step size for this correlation is fixed at 5 pixels, with a subset size of 19 pixels. Strain cal culations are done with a curvilinear 90% decay filter, using a filter size of 15 pixels for data averaging and smoothing. (a) (c) (b) PAGE 102 102 Figure 4 13 Strain and displacement fi elds obtained from reference images at 35% strain Figure 4 14 Strain and displacement fields obtained from underwater images at 35% strain PAGE 103 103 Figure 4 15 Strain and displacement fields obtained from the reconstructed images at 35% strain Figure 4 17 shows the variation of strain along arbitrary lines L 1 and L 2 drawn in the neighborhood of the mid section of the test specimen. The plots show that, the strain field in the reconstructed image closely follows the reference i mage with a maximum error on the order of 2%. Since the transform functions u i and v i are derived from a finite number of discrete control points some mismatch error is inevitable and the registration can't be exact. The mismatch error depends on the numbe r of control point pairs used to construct the transformation mappings. The larger the number of control points used in the registration, the smaller the mismatch error would be. However a large number of control points increases the computational complexi ty of the method. PAGE 104 104 Figure 4 16 In a typical uniaxial stress strain test s train values are sought along lines L 1 and L 2 and the errors of strain measures along these line is shown in Figure 4 17 and Figure 4 18 Figure 4 17 Strain along line L 1 of the reference, reconstructed and underwater images L 2 L 1 PAGE 105 105 Figure 4 18 Strain along line L 2 of the reference, reconstructed and underwater images Figure 4 19 shows the measured strain error from underwater and reconstructed images in comparison with the reference image at the four strain levels along lines L 1 an d L 2 of Figure 4 16 Owing to the large refraction angle, points far from the center of the specimen exhibit a larger pseudo displacement than points closer to the center. The error magnitude of the reconstructed image is on the order of 1% whereas the underwater image exhib it errors in excess of 5% at the 35% strain level. The fact that the error is a function of the distance from the center of the specimen indicates that large errors would result on the DIC strain of large specimens tested underwater. It should also be not ed that for tests performed in fluids with higher refractive index such as sodium chloride irrigation solution (a common sterile tissue cleanser) or other PAGE 106 106 electrolyte solutions the DIC strain error will be higher than the same test performed in distilled w ater. Figure 4 19 Strain errors on underwater and reconstructed images along line L 1 Concluding Remarks This chapter has proposed an elastic image registration technique to undo the refraction induced distortion through control point matching and spatial transformation. The image registration technique uses a local weighted mean (LWM) transform that relies on different mathematical functions at different regions of an image to have a more accurate local alignment of curved regio ns a refracted image with its non refracted counterpart. To obtain the transformation functions, refracted and non refracted images of a template panel were compared and brought into alignment and the mapping functions that achieved the registration with t he smallest error are retrieved. The same PAGE 107 107 transformation is then applied to underwater images of a test specimen, which are acquired in similar experimental conditions as the template image, to correct the refraction distortion. Once specimen images are co rrected, they are passed onto a digital image correlation algorithm to obtain the displacement and strain field following the usual correlation technique. Strain and deformation measurements of a case study have shown good agreement between the reconstruct ed and the reference DIC patterns of an underwater test sample. The elastic image registration technique presented here is an approximate method and as such the registered images are bound to have certain mismatch error, however the strain and deformation errors incurred due to the elastic image registration process is often contained within the uncertainty bounds of most commercially available non contact techniques. In any case, the purpose of this paper has been to illustrate the use of elastic image reg istration technique that extends the use of commercially available non contact strain measuring techniques in fluid environment rather than to develop an all new system exclusive for such applications. PAGE 108 108 CHAPTER 5 APPLICATION OF THERMODYNAMICALLY CONSISTENT MODELS Introduction This chapter presents nonlinear thermodynamically con sistent viscoelastic constitutive models that describe the mechanical behavior collagen grafts. All the mathematical formulations presented here are applied to the stress relaxation data obtained in Chapter 4, where the experimental methods were fully desc ribed. Most authors describe soft tissue experiments in the framework of the linear theory of viscoelasticity relating stress and strain using the phenomenological models of Maxwell, Voigt, Kelvin, Weichert, and others ( Chandran 2006; Humphrey 1999, Fung 1993 Phenomenological models, albeit easy to formulate, have the following limitations: 1. They are only accurate in capturing a small subset of the observed viscoelastic behavior, that is, a given phenomenological model (a particular arrangement of springs and dashpots) built for describing, say, stress relaxation can't explain creep or hysteresis behaviors, 2. Coefficients of mathematical functions describing the phenomenology are very sensitive to variation in experimental data. A large variability of const ants has been observed while attempting to calibrate models using similar experimental data, 3. Phenomenological models are not born out of thermodynamic theories, as such there are no clear provisions to account for thermodynamic factors, such as change in t emperature, humidity etc. and without which mechanical description of collagen grafts is not accurate, As was discussed in the previous chapters, collagen grafts are fluid saturated fibrous matrix which exhibit pronounced material and geometric nonlinearit y. Such complex mechanical behavior requires consideration of thermodynamic variables which also govern the mechanics of the material. Therefore, it is essential to focus mainly on thermodynamically consistent constitutive models rather than springs and da shpots so PAGE 109 109 that, given a set of data; the mechanical description would be extended to include temperature and humidity as additional independent control variables. In the context of continuum mechanics, thermodynamics is expressed in terms of the Clausius Duhem inequality which determines whether a constitutive model of a material is thermodynamically consistent (or permissible) (Schapery 1992). The Clausius Duhem inequality is a mathematical formalization of the irreversibility of natural processes and th e dissipation of energy in real materials. De tailed derivation of the inequality is available in books written on classical mechanics ( Truesdell 1952; Truesdel et al. 1960; Fremond 2004 ). The thermodynamic state of a material is assumed to be fully defined by stress, three different sets of internal state variables as well as absolute temperature and moisture content (Schapery 1997). Analysis As mentioned earlier, one of the primary goals of this study is to investigate existing viscoelastic constitutive th eories that are thermodynamically consistent and adaptable to describe the experimentally observed behavior of collagen grafts. Out of a plethora of mathematical theories, we choose the following constitutive models to describe the graft behavior: 1. The quasi linear viscoelastic (QLV) theory of Fung (1972), which is based on separability of the elastic and viscous responses, 2. The Nonlinear thermoviscoelastic theory of Haslach (2004), an outgrowth of non equilibrium thermodynamics, and 3. Schapery's single int egral nonlinear theory which is the outgrowth of irreversible thermodynamics (Schapery 1969), The QLV theory is chosen because it is widely used for predicting the viscoelastic response of a broad range of soft tissue materials including skin, tendon and a rtery PAGE 110 110 (Fung 1972,1993). It is also the most accurate method for predicting the viscoelastic stress relaxation of most biological materials. The thermoviscoelastic theory (Haslach 2004; Haslach et al. 1999, Knauss 1987) has garnered a lot of momentum in rec ent times due to the fact that it allows interchangeability of material data obtained from one set of test to be used in other modes of loading (Haslach 2004). The material property data that the thermoviscoelastic formulation uses is time independent and relatively easier to formulate. Schapery's single integral theory (Schapery 1966, 1971; Lou and Schapery 1971) the main theory in polymer viscoelasticity, has been shown to be accurate and adaptable for many rubber like materials (Provenzano 2002). To my knowledge, none of these models have been used to describe the behavior of dermal collagen grafts. The Quasilinear Model (QLV) Fung (1972) developed an expression for tissue stress, ( ), as a function of stretch using the fact that in a certain range th e experimental data could be fit by a straight line d /d = ( + ). The stress is obtained under displacement control by applying a step stretch to a uniaxial specimen. The stress, defined in the sense of Lagrange and Piola is given by (Fung 1992, 1972): ( 5 1 ) where ( stretch curve, and and are material constants. According to Fung, the tensile stress at any time t is the instantaneous stress decreased by an amount depending on the past history of the material, hence theory reduces to a single expression: PAGE 111 111 ( 5 2 ) where is the instantaneous elastic response which is a pure function of stretch (or strain ), and G(t) is the reduced or normalized relaxation function which is a pure function of time. Implementation of the Quasilinear Viscoelastic Theory To Implement t he QLV theory to the collagen graft, we need to obtain the elastic response function and the reduced relaxation modulus from the step stretch data presented in the previous chapter. Figure 5 1 Stress strain curves of collage n graft at a constant stretch rate of 500 "instantaneous elastic stiffness" of the material. We have seen that the elastic response of the material is obtained by running a test at high enough stretch rate that eliminates the simultaneous relaxation of the PAGE 112 112 material. The relationship between load and deformation (stress strain curve) for a step loading is shown in Figure 5 1 Looking at Figure 5 1 the stress strain curve of can be divide d into a highly compliant region (0 highly compliant region the strain is mainly the result of orientation of collagen struts in the direction the stretch axis, Figure 5 2 where as the stiff linear region indicates the actual fibril deformation and compaction of the network as a whole. Figure 5 2 Light image of Type I col lagen fibrillar showing fibrillar network embedded in aqueous matrix, the first part of the uniaxial stress strain curve is a mere reorientation of fibers (Knapp et al. 1997) Analogues to elastic material, the stiff region can be approximated by a linear function whose slope is regarded as the instantaneous elastic stiffness of the material. Fung (1972) however uses a different approach to obtain the elastic response function. The Elastic Response Function, e ( ) The instantaneous elastic response function is a pure function of strain and is obtained by curve fitting of the slope of the stress stretch curve (d /d ) vs. stress as shown in Figure 5 2 In order to get a more refined approximation, the experimental data is fitted using two straight lines defined by the equation; PAGE 113 11 3 ( 5 3 ) Integrating Eq.. ( 5 3 ) gives ( 5 4 ) The integration constant c can be determined by considering an arbitrary point on the stretch stress curve ( *, ), then: ( 5 5 ) Figure 5 3 and Figure 5 4 show the methods used to find the elastic response functions for the dermal grafts. Figure 5 3 The variation of Young's modulus with stress at a strain rate of 500mm/min, determination of material constants lines were required to fit the experimental data. PAGE 114 114 Figure 5 4 The variation of Young's modulus with stress at a strain rate of 10mm/min, determi nation of material constants lines were required to fit the experimental data. Figure 5 5 Comaprison of the stiff' Young's modulus of two samples tested at a stretch rate of 500mm/min. PAGE 115 115 Fi gure 5 6 Comaprison of the stiff' Young's modulus of two samples tested at a stretch rate of 10mm/min. The QLV parameters are found using linear regression on the d /d vs. curve as 1 = 59.12, 2 = 5.3, 1 = 0, 2 = 0.34 (40% and 500mm/min stretch rate). Table 5 1 lists materials constants obtained using the above approach. Table 5 1. Materials constants for the elastic response characterization of collage grafts, strained to 40% at a stretch rate of 500mm/min. Con stants determined with 95% confidents bound, R 2 =0.92 KV3 784 KV3 296 Average 43.79 74.45 59.12 0.01 0.01 0 6.36 4.24 5.3 0.19 0.49 0.34 PAGE 116 116 Table 5 2. Materials constants for the elastic response characterization of collage grafts, strained to 40% at a stretch rate of 10mm/min. Constants determined with 95% Confidents bound, R 2 =0.92 KV4 3996 KV4 5285 Average 43.17 24.96 34.07 0.01 0.03 0.01 7.43 6.49 6.96 0.62 0.66 0.64 Figure 5 7 Comparison of the viscoelastic material constants of four samples tested at multiple strain rates. According to Fung (1972) parameters describe an intrinsic elastic property of the collagen grafts as such these can be used to uniquely categorize one set of grafts from another. Figure 5 7 shows the gene ral trend of the elastic material constants for PAGE 117 117 four different samples two of which are tested at 10mm/min and the other two at 500mm/min. Substituting the experimentally obtained material constants into Eq. ( 5 5 ) yields the following elastic response function for all dermal grafts, ( 5 6 ) A comparison between the experimental data and the mathematical prediction is shown in Figure 5 8 The fit is good for the entire curve. Although this fitting is not surprising because the constants and are determined from these two sets of experimental data, the ability of the mathematical expression to fit the entire data using a single randomly selected point on the curve is nontrivial. Figure 5 8 Comparison between experimental data and mathematical expression, The Reduced Relaxation Function, G(t) According to the quasilinear theory, if a step stretch (from = 1 to 0 ) is imposed on a viscoelastic specimen, the stress develop ed will be a function of time as well as of PAGE 118 118 the initial stretch 0 The history of the stress response, called relaxation function, will have the form (Fung 1993) ( 5 7 ) where G(t) is called the reduced relaxation function and (e) (t) is the elastic response. In order to express the reduced relaxation function, many authors use relaxation functions borrowed from linear viscoelasticity, and one of these forms is (Mandis et al. 1995; Pipkin 1986): ( 5 8 ) i / i ) is the discrete relaxation spectrum, or a continuous one of the form (Fung 1993; Nigul et al. 1987): ( 5 9 ) The discrete relaxation function of Eq. ( 5 8 ) is not only not unique but may lead to erroneous limiting value of G( ) particularly when an experiment is cut off prematurely. In addition a discrete relaxation spectrum cannot reconcile experimental data obtained from creep, hysteresis and relaxation tests. For this reason, Fung concluded that for at least biological materials a continuous spectrum is desirable. The most common and successfully used form of the continuous relaxation function spectrum shaped spectrum (Fung 1993; Nigul et al. 1987; Rousseau et al. 1983; Sauren et al. 1983): PAGE 119 119 ( 5 10 ) where c is a dimensionless positive constant and 1 and 2 are characteristic relaxation times. If we substitute Eq. ( 5 10 ) into Eq. ( 5 9 ) the following expression is obtained (Drapaca et al. 2006; Fung 1993): ( 5 11 ) Ei(t/ 1 Ei(t/ 2 ( 5 12 ) and the exponential integral, ( 5 13 ) From a physical point of view 1 governs fast relaxation process, 2 governs slow relaxation process, and c determines the degree to which viscous effects are present (c = 0 means there is only an elastic response) (Carew et al. 1999; Mendis et al. 1995; Drapaca et al. 2006). The form of the relaxation function G(t) is not unique, and any function that satisfies the minimum conditions required by the fading memory assumption of continuity, positivity and decreasing monotonicity could be used to model relaxation modulus (Drapaca et al. 2006). Generally, such function satisfi es thermodynamically consistent decay of the stored elastic energy (Dortmans 1984; Sauren 1983; Drapaca et al. 2006 ). PAGE 120 120 Figure 5 9 Reduced relaxation functions for two specimens tested at a stretch rate A simpler, yet accurate mo notone function that describes the normalized relaxation response of collagen grafts ( Figure 5 9 ) can be a power function of the form: ( 5 14 ) where a and b are experimentally obtained material constants. To obtain this constants we perform a nonlinear regression on the G(t) vs. Log(Time) data as shown Figure 5 10 Table 5 3. Coefficients of the reduced relaxation function based on a step stretch test (500 mm/min) estimated with 95% confidence bound, R 2 =0.97 Coefficients KV3 784 KV3 296 Average Coefficient, a 0.6413 0.7138 0.6775 Coefficient, b 0.0449 0.0667 0.0559 Figure 5 11 shows the reduced relaxation function for two specimens tested at the maximum cross head speed of 500 mm/min. For Fung type materials, the reduced modulus curves of all specimen overlap on on e another and hence finding a single mathematical equation was trivial. PAGE 121 121 Figure 5 10 Estimated reduced relaxation functions for two specimens tested at a stretch rate of 500mm/min. Finally, the reduced relaxation function has the form: ( 5 15 ) Therefore, relaxation stress of dermal collagen grafts can be expressed by the following constitutive law: ( 5 16 ) Equation ( 5 16 ) implies that the tensile stress at any time t is equal to the instantaneous stress response decreased by an amount depending on the past history, because is generally a negative value. Equation ( 5 16 ) has been solved using MATLAB's symbolic toolbox and the relaxation stress responses are plotted o n semi log arithmic scale in Figure 5 11 PAGE 122 122 Figure 5 11 Application of the quasilinear constitutive theory, Eq. ( 5 16 ) to predict strain dependent relaxation behavior of dermal collagen grafts As can be seen in Figure 5 11 the Fung model fit s the relaxation stress reasonably well however the rate of relaxation predicted by the QLV theory is significantly different from that ex hibited by the grafts. Therefore, the QLV model needs to be used with certain precautions, or a better predictive model has to be developed. Non Equilibrium Thermodynamic Model The thermodynamic, nonlinear thermoviscoelastic model is based on the Onsager relations, the local reversibility of time, and the Curie principle forbidding coupling of variables with different tensorial types (Haslach 2004, Haslach and Zeng 1999). In this model a generalized energy function is defined through conjugate relationshi ps between state and control variables and for which generalized, nonlinear Onsager type relations can be defined. The construction of the model requires a set of pairs of conjugate thermodynamic variables and relations defining the long term behavior. PAGE 123 123 Th e long term behavior is described by a time independent energy function of the control variables. The variables, such as stress, strain, and temperature are separated into pairs of state variables, x i and the control variables, y i so that pairs are conju gates in the sense that the derivative of the energy function with respect of the control variables yields the state variable at thermodynamic equilibrium (Haslach 2004). For example, temperature and entropy or stress and strain are conjugate pairs. In cre ep, stress is the control and strain is the state variable and their role is reversed for the case of stress relaxation. The simplest form of a generalized energy function is given [38] : ( 5 17 ) Where is obtained from the Legendre transform of The energy function for biological soft tissue could be one of the hyperelastic models proposed in C hapter 2. By a calculation given in Haslach and Zeng (1999), a thermoviscoelastic maximum dissipation process is represented by a system of first order nonlinear ordinary differential equations in terms of the state variables, x i and the control variables, y i as, ( 5 18 ) Since (t, (e) term response can be assumed to be ) (e) ( ) to satisfy: PAGE 124 124 ( 5 19 ) With these formalisms we'll develop the thermoviscoelastic relaxation and creep behaviors of collagen dermal grafts. Thermoviscoelastic Relaxation The stress relaxa tion thermoviscoelastic model is obtained by assuming stress as the state variable and strain as the corresponding conjugate control variable. To formulate the equation, we assume an energy function such that it satisfies the equa tion: ( 5 20 ) Then, ( 5 21 ) Hence Eq. ( 5 18 ) simplifies to: ( 5 22 ) where k is stress dependent material constant, G( ) is the final value of the relaxation function, ( *, *) is an arbitrary point on the stress strain curve ( Figure 5 1 ). Equation ( 5 22 ) is numerically solved using MATLAB's stiff integrator ode23 s that implement s an explicit Runge Kutta pair of Bogacki Shampine method. Figure 5 12 shows the long term relaxation response of collagen dermal implants. It is apparent that the thermoviscoelastic model agrees well with the experimentally observed data in the long term when a steady state response is prevailed. PAGE 125 125 Figure 5 12 Stress relaxation predicted by the energy method. Experimental constants = 0.15, G( ) = 0.471, =1.29, = 1.342, T* = 0.89 Thermoviscoelastic Creep The nonlinear thermoviscoelastic evolution equation describing the creep res ponse (strain as a state variable and stress as a control variable) follows the QLV theory where the long term stress is calculated as a product of the elastic response and the reduced relaxation function (Haslach 2004) ( 5 23 ) The limiting value G( ) can be determined from the QLV model as, ( 5 24 ) where 1 and 2 are characteristic time constants of the relaxation process. PAGE 126 126 The nonlinear viscoelastic evolution Eq. ( 5 18 ) for the stretch, as the state variable and the stress, as the control variable becomes a first order dif ferential equation with creep constant k : ( 5 25 ) To generate numerical results, Eq. ( 5 25 ) is solved using MATLAB's ode23s and results are plotted in Figure 5 13 Figure 5 13 Creep prediction based on experimental constants obtained from uniaxial = 0.15, G( ) = 0.471, =1.29, = 1.29, T* = 0.89 PAGE 127 127 The thermoviscoelastic creep and relaxation models agree well with the long term behavior of the collagen implants as shown in Figure 5 12 and Figure 5 13 It also has the advantage that the empirical data k and k are time independent. The model succeeds in reproducing the creep and relaxation responses of the material with one set of material constants derived from a different test. Recall that parameters G( ), strain data and here the same values are used to predict creep and relaxation responses The thermoviscoelastic model, however, is not very accurate in predicting the immediate or short term responses hence a more robust model may be required. Introduction The Schapery single integral theory (Lou and Sch apery 1971; Schapery 1964, 1966) is an outgrowth of irreversible thermodynamics developed by Biot and others (Truesdell 1984) and is the most widely used technique to represent the nonlinear time behavior of polymers (Lou 1971; Provenzano et al. 2002; Scha pery 1974). An excellent description of the original thermodynamic derivation is given by Hiel et al. (1984). According to Schapery (1969), the stress in a viscoelastic material subjected to a variable strain is represented by the following single integral expression: ( 5 26 ) where h h 1 h 2 a are strain dependent material parameters that have thermodynamic significance; in the original formulation the first three terms arise from third and higher order dependence of strain on the Helmholtz free energy (Schapery 1966, 1969) and t he parameter a is a shift factor which modulates the time scale much PAGE 128 128 in the same way that the temperature dependent shift factor a T modulates the time scale for temperature effects. E is the transient modu lus. Since the final refers to the behavior of infinite time, which is idealistic, the quantity E for practical purposes is the modulus value of the last data point in the experiment time frame. Ideally, the final value occurs after some steady state beha vior has been approached. The shifted strain dependent time scale, is defined as (Provenzano et al. 2002), ( 5 27 ) The variable of integration, ( 5 28 ) When h e = h 1 = h 2 = a = 1, Eq. ( 5 26 ) reduces to the Boltzmann superposition integral for linear viscoelasticity. However, when implementing the above theory to nonlinear behavior, one or more of the strain dependent functions (h e = h 1 = h 2 = a e ), but not all, can be assumed to equal unity (D illard et.al 1978; Lou and Schapery 1971; Touti and Cederbaum 1997). In polymers and fibrous composite materials it is common to set h 1 and a e to unity (Dillard et.al 1978; Lou and Schapery 1971; Schapery 1969; Touti and Cederbaum 1997), this case will be applied in this study. Following the modeled as a power law: ( 5 29 ) where C and n are strain independent materials parameters for a constant temperature (isothermal) loading process. (Dillared et al. 1987). When Eq. ( 5 29 ) is substituted into Eq. ( 5 26 ) the stress becomes: PAGE 129 129 ( 5 30 ) Substituting the Heaviside function into Eq. ( 5 30 ) 0 and setting h 1 = a e = 1 results in: ( 5 31 ) Noting the E n of Eq. ( 5 31 ) take the role of final and transient stresses, respectively, that E constants determined by curve fitting, it can be seen that Eq. ( 5 31 ) predicts the strain dependent elastic and rate behavior with two parameters that are functions of strain, h e and h 2 Schapery (1969) presented a complementary relation for creep in which nonlinear stress dependent ( 5 32 ) and, ( 5 33 ) where g 0 g 1 g 2 a are material parameters which depend on stress. The first three parameters arise from third and higher order dependence of stress on the Gibbs free energy where as the embedded function a in the reduced time expressions is used to shift the viscoelastic time scale much in the same way that the temperature dependent shift factor a T modulates the time scale for temperature effects. J 0 the instantaneous and transient creep compliances, J 0 J 0 PAGE 130 130 Application of Schapery's Theory T he nonlinear theory of Schapery (1969) was applied to experimental stress relaxation data of collagen grafts and is shown in Figure 5 14 The fit is good and the mode l predicts not only the strain dependent stress but also the strain dependent relaxation rate of the materials. Such agreement was not achieved with the quasilinear theory. Figure 5 14 Nonlinear theory of Schapery applied to describe the relaxation behavior of dermal collagen grafts PAGE 131 131 Table 5 4. Experimental material constants used to nonlinear viscoelastic model, coefficients with 95% confidence bound, R 2 = 0.92 Coefficient KV3 784 KV3 296 C 2.677 2.69 n 0.0586 0.07938 E 0.4643 0.4643 h e 0.01836 0.0174 h 2 0.9827 0.984 Concluding Remarks The aim of this chapter was to develop a constitutive law for dermal collagen grafts that can be extended to include thermodynamic variables (in particular temperature and humidity). We have shown that the quasilinear viscoelasticity model, the most common ly applied theory in biomechanics is capable of describing the strain dependent relaxation behavior of collagen grafts fairly well. However, the relaxation rate predicted by the QLV theory is significantly slower than what the grafts actually exhibited dur ing the unaxial relaxation experiment. The thermodynamic model of Haslach (2004) which is borne out of non equilibrium thermodynamics does predict the relaxation and creep behaviors from the same set of experimental data. The resulting model predictions ag ree reasonably well with the long term response than they do with the short term behavior. It's believed that such disparity in accuracy of prediction into part of the stress time curve can be resolved by making the constants to vary with time rather than keeping them constant (Haslach believed they are constants). Schapery's theory captures the nonlinearity in much superior way than both Haslach (2004) and the QLV (1972) theories. The predictions are accurate for both stress and relaxation rate. PAGE 132 132 QLV has b een, and remains, a valuable tool in the field of biomechanics. However, was experimentally observed. Therefore, in case if, accurate predictive model is of any im portance, we recommend the nonlinear theory of R.A Schapery over QLV. PAGE 133 133 CHAPTER 6 EQUIBIAXIAL CHARACTE RIZATION USING BUBBL E INFLATION TEST Introduction The uniaxial tension test is widely used to provide basic information about the mechanical prope rty of most industrial materials. In this test, a cylindrical specimen is exposed to a gradually increasing unidirectional tensile force while simultaneously measuring its load extension behavior. The latter is then used to construct stress strain curves o f the material from which such useful information as, the elastic limit, yield strength, and Young's modulus are derived. The unaxial test, however, has certain limitations when it comes to testing the mechanical behavior of dermal collagen grafts. Figure 6 1 Light image of saturated gel showing fibrillar network embedded in aqueous matrix (Knapp 1997). Generally, dermal grafts, Figure 6 1 are a fluid saturated fibrous collagen network imbedded in a glycosaminoglycan (GAG) matrix and when exposed to a single axis stretch the network responds by; (1) reorienting the fibers in the direction of the st retch axis followed, (2) compaction of the entire matrix and finally (3) extension of the short fibrillar followed by the long ones. A similar phenomenon has been o bserved by other PAGE 134 134 experimenters. Viidik (1990) observed that at zero stress the collagen fibe rs are coiled, bent or buckled and when a tensile load is applied the number of collagen fibers that are pulled straight and brought to action increased. Therefore, it is difficult to know with certainty whether the uniaxial stretch is due to an actual fib ril extension or just a mere reorientation of the network as a whole. Figure 6 2 Schematic drawing of circumferentially sutured graft with the microstructure section view showing randomly arranged fibers inside aqueous matrix. The limitations of the uniaxial tensi on test can be explained by referring to the way the grafts are loaded during service. During a typical repair surgery, dermal grafts are stitched on the host body using several circumferential suture lines, Figure The tensile load that the circumferenti al sutures apply is intrinsically multidirectional, and as such there is no one dominant stretch axis that would cause progressive reorientation or compaction of fibers as was observed during uniaxial stretch tests. Hence a material property data obtained from a tensile test experiment alone will not be able to capture certain mechanical phenomena that exist during the actual service condition of the grafts. Therefore, the validity of tensile test data and constitutive model built based on it has to be veri fied with more material tests. Dermal graft Wound Boundary Microscopic view Circumferential Suture PAGE 135 135 Confined compression test has been used to characterize the mechanical behavior of collagen grafts (Chandran et al. 2004; Girton et al. 2002). In one such test, a porous piston is pressed against collagen gel while measuring the displacement of the piston vs. the load applied. The stress strain curve obtained is then used to build a constitutive model. In compression test s the network stiffness is considerably reduced and the problems associated with febrile reorientation do n ot exist. However, the resistance to interstitial flow during compression test may introduce a new mode of localized fibril deformation, as such it may subject an otherwise buckling fibril network into biaxial tension (Barocas et al. 1995; Elsdale et al. 1 972). Compression test data depends not only on the solid matrix deformation but also on the movement of fluid in and out the pores during the deformation. Since fluid plays a role in load transfer the stress strain behavior becomes time dependent much li ke viscoelasticity. Therefore, the stress relaxation response to a compressive loading may arise either from the viscosity intrinsic to the graft material or from the resistance to interstitial fluid flow. The stress relaxation due to interstitial flow is a totally different exhibited by any fluid saturated porous elastic medium. Stress re laxation that arises from the viscosity of the matrix, called viscoelasticity, is an intrinsic mechanical property of the material that we are dealing with. In a compression test of grafts the viscoelastic response is coupled with the poroelastic (Biphasic) response. Decoupling of the two responses is not a trivial task and involves formulation of Darcy's constitutive equation for the fluid flow in the porous matrix and solving the corresponding Navier Stokes equations (Mow et al. 1997; PAGE 136 136 Velegol 2001). If one chooses to characterize the materials by exhaustively solving these equations then the experimental investigation problem becomes a theoretical problem and may not yield the desired result. Equal Biaxial (Equi b iaxial) test involves pulling a specimen at regularly spaced points thereby effectively removing the presence of a single dominant axis of stretch. Unlike the uniaxial tension no significant network r estructuring or compaction is occurred. In contrast to compression test interstitial flow is not constrained and the fluid is free to escape through the pore therefore the issue associated with poroelasticity is negligible. Moreover, the pure state of stra in that could be achieved with equibiaxial test will result in a more accurate material model. Literature Survey on Equibiaxial Testing Equibiaxial tests are more complex than simple uniaxial tension or compression tests and hence there are only a few pub lished works available, which are mostly on non biological materials such as polymers, textile and plastics. The inflation of circular membrane has been of interest to experimenters for many years. Bubble inflation with compressed air was used for the firs t time by Treolar in 1944, to study the behavior of natural rubber and its burst mechanism. In 1951, Rivlin used a bulge test to identify Neo Hookean and Mooney Rivlin material parameters of a hyperelastic membrane, and showed that the equibiaxial state of strain at the bubble pole is equivalent to pure compression. In 1973 Paisant conduct an inflation test to study the pressure vs. strain characteristics of natural rubber membrane. Vernon et al. (1972) developed a constitutive model from first principles to describe the behavior of inflated elastomeric sheets and adopted a bubble inflation test to calibrate his mathematical model. PAGE 137 137 Numerical procedure using bubble height to compute the rheological parameters for a given constitutive equation of an inflated membrane. Almost all bubble inflation studies were focused on polymers and hyperelastic materials. With little adaptation, howev er, some of the techniques used for hyperelastic materials can be used to characterize the equibiaxial property of collagen dermal grafts as presented in this chapter. Bubble I nflation T est of Dermal Grafts Equibiaxial loading of membranous grafts can be p erformed by using a mechanical frame having a mechanism for stretching the membrane in a number of equally spaced directions. Figure 6 3 shows one possible way of tes ting a circular specimen of uniform thickness in equibiaxial loading. Figure 6 3 A mechanical test frame used for equibiaxial extension of elastomer (Axel TM physica l testing services Inc.) As shown in the figure, the specimen is hooked in 16 places along its boundary by means of small grip p ing hooks. Each hook is connected by means of a wire to a screw that applies equal displacement to each wire. A pulley system al lows the wire to be pulled at a constant stretch rate. The displacement of each wire is coordinated by the PAGE 138 138 arrangement of the pulley and the drive motor. Full field strain from the surface of the specimen can be measured using optical system such as the DI C. The mechanical frame has the advantage of direct force measurement and is readily adaptable for probing porous materials such as collagen grafts. Moreover, the arrangement allows relaxation and creep tests to be carried out without any major challenge. However, it is very complex and requires precise alignment of its components particularly the pulling wires must spaced precisely around the perimeter of the test specimen. In addition, there is always a variation of stress distribution towards the center of the specimen which may result in a significant shear stress. Stress concentrations in the test specimen are relatively greater than those occurring in bubble inflation tests and the effect of inertia and friction inherent in the frame may add to offset the measurement error beyond the acceptable limit. In a bubble inflation test, a uniform membrane is clamped at the rim of a circular cylindrical. A compressed air is then allowed to gradually flow into the cylinder while simultaneously measuring the radi us of curvature, full field strain and the applied inflation pressure. The inflation of a circular membrane into a spherical bubble (bulge) results in a truly equibiaxial deformation at the pole of the bubble. Except for a slight bending near the rim, whi ch is of less significance for hydrated collagen grafts, the method has none of the drawbacks that the mechanical stretch frame has. Moreover, the equipment and its parts can be manufactured economically. PAGE 139 139 Figure 6 4 Pictorial view of bubble inflation apparatus use for testing the behavior of collagen dermal grafts Materials and Methods The bubble inflation test was performed on a custom made material tester shown in Figure 6 4 In the process, a circular scaffold is mounted between two metal disks c ontaining a circular aperture (85 mm diameter) and clamped on a support. The specimen is 88 mm in diameter and 2 mm thickness. The air pressure is regulated by a control valve attached to the cylinder. The pressure relief valve maintains a constant hold pr essure during a bubble inflation creep test. An absolute pressure sensor with a digital display mounted on the cylinder measures the inflation pressure in the chamber and sends readout to a LabView program that records pressure versus time. The bubble prof ile and full field strain data are acquired using a digital image correlation (DIC) technique. The DIC system employed here has the ability to measure not only the deformation and strain fields but also the whole field radius of curvature and the inflatio n PAGE 140 140 rate of the bubble. The la t ter was indirectly calculated from the rate of change of the bubble profile (or the w field) at the pole. The hoop stress at the pole is calculated as: ( 6 1 ) where p is the inflation pressure, t is the thickness and r is the radius curvature of the spherical bubble at its pole. DIC algorithm is not suitable for use (at least in this case with VIC 3d version 2006) due to the high amount of noise present in the strain and deformation data. The unsu itability of a DIC based curvature measurement from noisy deformation data can be traced to the correlation algorithm. In a DIC technique, the subpixel deformations (u, v, w fields) are the quantities that are directly measured by the visual correlation sy stem ; on the other hand the radius of curvature is calculated numerically. The strain field is calculated by differentiating the deformation tensor once i.e. whereas the radius of curvature is obtained by differentiating the defor mation vector twice i.e. Analytical differentiation is known to amplify noise and the radius of curv ature, being the second derivative is prone to be too noisy For example, i f a low frequency noise of, say, u = 0.001* cos ( 10 6 x ) is present in the displacement measurement, then the first derivative becomes d u /d x = 0.001*10 6 sin ( 10 6 x ) and the second derivative becomes d u 2 /d x 2 = 0.001*10 12 cos ( 10 6 x ). Hence a noise with much small er amplitude will be magnified in the final calculation. The displacement measurement of dermal grafts, regardless of the test setup, contains a significant noise PAGE 141 141 due to the inherent reflection of light from the wet and highly moist surface. To overcome this issue, we adopted geometrical methods to calculate the radius of curvature as will be discussed in subsequent section. PAGE 142 142 Figure 6 5 Schematic view of the experimental setup showing section view of the bubble inflation test equipment and the arrangement of digital image correlation system DIC System Hyperelastic Latex Pressure Control Valve Pressure Measurement Specimen CCD C ameras Compressed Air Inlet PAGE 143 143 Sample Prep aration Sample handling and preparation for the bubble inflation test is similar to that for the uniaxial test described in detail in Chapter 3. The specimens were cut to a circular shape and speckled using a Sharpi marker and then hydrated in 0.9% salin e solution for 30 minutes. During the hydration process a precaution has been taken not to pour the saline solution directly over the top of the graft s as this will cause the fibrill ar to deform and the graft as a whole to bow. After the 30 min hydration p eriod the specimens were superimposed with extra thin Thera Band hyperelastic latex and clamped on the rim of the test cylinder. The latex is placed on the coarse side of the scaffold that faces the pressure cylinder. Figure 6 6 shows a snap shot of the specimen preparation and accessories used during the test. Figure 6 6 Sample preparation for the equibiaxial test The latex serves as backing material and is required to prevent the compressed air from leaking out through the pores of the test samples. Specimen Latex backing Back rim of test rig PAGE 144 144 In general, the bubble inflation test consists of the following three main steps: 1. Clamp the specimen latex assembly firmly on the rim of the test cylinder so it won't slip out during inflation. The rim of the inflation test device has a wave type jaw that prohibits slippage of the scaffold membrane hence there is no risk of a test being aborted, 2. Apply air pressure and record bubble height, strain, and inflation pressure with time. The digital image correlation system can be programmed to take digital image s at preset intervals. The digital pressure sensor sends pressure read out to a LabView virtual instrument that records pressure vs. time, 3. Once the maximum set pressure is reached, it is held constant by a relief valve which serves as a control device by r eleasing excess pressure out of the cylinder, As said earlier, the main issue that was encountered during the bubble inflation process was the problem associated with maintaining enough pressure in the cylinder to inflate the porous membrane. Dealing with Membrane Porosity The grafts have a pore volume reaching 99% of the total volume of the matrix. To be able to carry out the bubble inflation test, one has to find a way to create a pressure inside the test cylinder by preventing the outflow of compressed air through the pores. One way of doing so is to use a very thin hyperelastic membrane as a backing material to seal the pressure side of the specimen. In this test 0.25mm thick Thera Band latex was overlaid on the specimen and clamped on the rim of the p ressure cylinder. On a separate inflation test it was found that the latex yielded at about 0.25psi well below the 100psi ranges that the specimens were tested implying that the latex imparts negligible reinforcement to the test samples, Figure 6 7 As shown in Fi gure 6 8 the maximum pressure that was achieved without the latex ba cking was about 40psi, moreover, the air flow though the pores had pushed the water to the surface resulting in a poor and noisy DIC data. PAGE 145 145 Figure 6 7 Comparison of pressure vs. strain curves of an Extra thin Thera Band Latex with a thin brown latex. The Former was used as a backing material during inflation testing. The reinforcement of the extra thin band on brown thin shown as 'combined' is insignificant Fi gure 6 8 Inflated porous matrix with and without latex backing. Without a backing material the maximum pressure achieved was 39.6 psi whereas with latex the pressure buildup was unlimited. The samples were speckled using Sharpie m arker. PAGE 146 146 Curvature M odeling Geometrical Formulation Consistent with experimental observation, the inflation of hydrated collagen grafts can be assumed to comply with the membrane theory. For an isotropic material, an inflated bubble contour exhibits a wel l defined axisymmetry and therefore the deformation at the pole, where the cylinder axis intersects the bubble, is essentially in a state of equibiaxial deformation. Figure 6 9 A state of equibiaxial deformation exists at the pole of an inflated bubble The geometric shape of an inflated bubble changes according to the depth of inflation. A test carried out on a nonlinear elastomeric sheet (Reuge et al. 2001) has shown that the shape of an inflated bubble remains to be hemispherical for a bubble height to rim diameter ratio h/d of up to 0.5. However, once the bulge height exceeds h/d > 0.5, the contour become s an ellipsoid and its meridian follows the shape of an ellipse till it burst s Figure 6 10 The scaffolds tested here are inflated within the limit h/d < 0; hence, it is assumed that the bubble remains hemispherical during inflation. With this assumption the radius of curvature R is calculated from: ( 6 2 ) rr PAGE 147 147 where h is the bubble height at the pole which is continuously measured throughout the test and d is the diameter of the rim. Figure 6 10 Measured and approximated contours of a hyperelastic membrane, for h/d < 0.5 the bubble shape is spherical and for h/d > 0.5 it becomes ellipsoid (Reuge et al. 2001) Biaxial Stress Strain Relations The stresses induced in a thin wall bubble depend on the bubble contour and thickness, and the inflation press ure. With t 0 as the initial thickness of the membrane, the hoop stress at the pole can be obtained by (see also Figure 6 11 ): ( 6 3 ) ii are the stresses defined in the sense of Cauchy and Euler, T ii are stresses define d in the sense of Lagrange and Piola, and S ii are stresses defined in the sense of Kirchhoff. For compressible materials, the Kirchhoff stress can be modified to take into 0 to the deformed state (Malvern 1969): Bubble width d Bubble width d (a) Spherical Bubble height h Bubble height h (b) Elliptic PAGE 148 148 ( 6 4 ) Figure 6 11 Dimension of inflated specimen showing the change in thickness during test In structures undergoing large deformation (aka finite strain problem), it is more accurate to use the Cauchy Euler stress than the Lagrange Piola stress definition. The Cauchy Euler stress, however, requires that the instantaneous thickness as a function of internal pressure be determined. In an ideal and more sophisticated test setting, the instantaneous membrane thickness is measured using devices such as laser scan micrometers (such as Keyence LS5000) or magnetic probe integrated with the test setup. Th e instantaneous bubble thickness can also be estimated numerically with the help of DIC measured surface strain and the assumption of material incompressibility. The assumption of incompressibility requires that the product of the three principal stretch r atios be unity i.e. r = 1. If the initial thickness is t o then the instantaneous thickness becomes t = r t o T he stretch ratio s = = are directly obtainable from the digital image correlation system or one may calculated by: ( 6 5 ) where E and E are the Green Lagrangian strain values measured by the DIC system. t t 0 d PAGE 149 149 Planar verses Bubble Inflation Equivalence Inflation of flat elastic membrane into a hemispherical geometry decreases the radius of curvature from R 0 to R (initially R 0 = ) and the thickness from t 0 to t. On the contrary a uniaxial test and a planar Equibiaxial test, such as the type shown in Figure 6 3 an increase i n initial sample length l 0 to l and shrinks the thickens t 0 to t. Hence, direct comparison of test data obtained from unaxial and bubble inflation tests might lead to an erroneous conclusion unless the following relations are considered: ( 6 6 ) where is the biaxial Cauchy stress, pe is Lagrangian planar equibiaxial stress, bf is Lagrangian bubble inflation stress, and is stretch. The nominal Lagrangian hoop stresses calculated by ( 6 4 ) needs to be scaled before comparison w it h uniaxial test data is made. ( 6 7 ) Table 6 1 Stress Strain equivalence between uniaxial, bubble inflation and planar tests (Reuge et al.2001) Uniaxial Test Planar Equibiaxial Test Bubble Inflation Test (spherical) Incompressibility, I 3 = 1 Nominal True True Nominal 2 Comparison between biaxial stresses obtained from the bubble inflation test to a uniaxial test should always be made using equations listed in Table 6 1 Figure 6 12 shows the equibiaxial stress strain curves of four samples tested using the bubble PAGE 150 150 inflat ion test method described in this section, a uniaxial stress strain response of a representative sample is also shown for comparison. Figure 6 12 Comparison of equibiaxial and Uniaxial stress strain behavior of collagen grafts PAGE 151 151 Figure 6 13 Average equibiaxial stress strain response of collagen grafts (error bars at one standard deviation) DIC Based Detection of Anisotropy The bubble inflation test can also be used to study the directional property of c ollagen grafts. The digital image correlation pattern encodes displacement information enough to detected anisotropy. In case of planar biaxial or uniaxial tests, inclination and frequency of a DIC pattern with respect to the principal stretch directions w ill indicate the existence of a preferred material orientation distinct from the loading direction. In case of bubble inflation test, however, the existence and extent of anisotropy is indicated by the very formation of unsymmetrical and random fringe patt erns on the inflated bubble surface. To help understand DIC based anisotropy we included the pattern of the hyperelastic latex used as backing material during the inflation test. According to the way the latex is often made, the Thera Band latex may have at most two directional properties, one in the rolling direction and a second in the transverse direction and we should anticipate a bidirectional strain field. As it turns out, the DIC strain patterns of the inflated latex have an elliptic shape where the latex is somehow stiffer in the rolling direction as indicted in Figure 6 14 The w field and principal strains are perfectly circular as shown in Figure 6 15 The latex is specified as isotropic by its manufacturer and yet the DIC patt ern resolves the very small anisotropy present in the material. PAGE 152 152 Figure 6 14 DIC strain field of inflated Thera Band latex, elliptic pattern indicates bidirectionality of the membrane. Figure 6 15 DIC pattern showing the w field and principal strain on the surface of an inflated Thera Band latex. Full field strain images of the graft are shown in F igure 6 16 and Figure 6 17 Contrary to the isotropic latex material, the DIC pattern doesn't have a well defined directional preference. This observation is consistent with the fa ct that during in vivo reconstitution, collagen fibers are randomly arranged via entropy driven process. Recent studies have shown that collagen fibers are normally distributed about a mean preferred direction, however this assertion can't be proved. PAGE 153 153 F igure 6 16 DIC strain field of inflated collagen graft, random patterns indicates anisotropy of the material Figure 6 17 DIC w field and principal strain of inflated collagen scaffold Concluding Remar k s The ubiquitous uniaxial loading experiment does not describe the full mechanical behavior of collagen dermal grafts. The main reason was that, in a single axis stretch test the fibrillar network of inside the aqueous matrix aligns itself with the principal axis of loading and as a result a considerable compaction of the specimen happens. Such PAGE 154 154 compaction, however, does not normally happen in service and hence the mechanical response that is obtained through uniaxial test may not be fully representative. An equibiaxial extension mimics the service loading condition of collagen grafts very well, since the grafts are often attached to the host body using circumferential stitches. A simpler way of probing the equibiaxial behavior is to use a bubble inflation (or a bulge) experiment. In a bubble inflation test, a circular graft was cut and clamped around the periphery of custom made equipment, Figure 6 5 C ompressed air press ure is applied inside the test cylinder to inflate the specimen into a spherical shape to induce a state of pure equibiaxial strain at the pole of the bubble. Strain is measured using a DIC system and pressure was measured using a digital pressure transdu cer. With the assumption of spherical symmetry at the bubble pole the stress was determined and the equibiaxial stress strain curve was drawn. In general, the equibiaxial stress is larger than a similar uniaxial test. Moreover, the range of the hi ghly complaint region is smaller in the equibiaxial stress strain curve implying that the network has not been restructured to the extent that it does in a uniaxial test. Some reorientation, however, was still observed. Interestingly, the stiffness of, whi ch is the slop e of the second part of the stress strain curve, is almost the same in both the uniaxial and biaxial tests. Perhaps, this might allow the use of a constant multiplication factor to relate the stress from uniaxial tests to that of equibiaxial test as in: Equibiaxial_Stress = Constant Uniaxial stress. Determination of the constant has been left as a future work. PAGE 155 155 CHAPTER 7 THREE DIMENSIONAL CO NSTITUTIVE MODELS BASED ON BUBBLE INFL ATION TEST Introduction Three dimensional problems are often solv ed using numerical techniques such as finite element or finite difference methods. The finite element method requires implementation of a multiaxial constitutive material model for accurate simulation. In general, f ormulation of a closed form three dimensi onal constitutive material model is not a trivial task as it involves working with a large number of t e nsorial quantities in a non commutative ring. The one dimensional nonlinear viscoelastic constitutive models such as the Colman Noll, Pipkin Roger, Fung' and Schapery's nonlinear theory can be extended into a 3D model by replacing the scalar creep and relaxation functions with their tensorial equivalent. This chapter presents 3D viscoelastic models based on the bubble inflation test data of Chapter 6. The models can be readily implemented into a user defined material subroutine in any finite element software. Three Dimensional Thermoviscoelastic Model The three dimensional thermoviscoelastic model presented here follows the evolution equation formulation of Chapter 6 Recall that the evolution equat ion is a system of differential equations describing the long term behavior of viscoelastic material as represented by an energy function of the type used in hyperelasticity. The evolution equations which represent non equilibrium transient responses such as creep and stress relaxation are derived from the maximum energy dissipation principle, which supplements the second law of thermodynamics. PAGE 156 156 The develop ment is based on the principal stretch (at the bubble pole) as representative of the equibiaxial defor mation process, and whose conjugate pairs are the Green Lagrange strain E and the second Piola Kirchhoff stress, S. In the bubble inflation creep test, the change in the radius of curvature is very small compared to the surface strain hence the non equilib rium process is considered to be stress controlled rather than strain controlled (since it is pressure that we are controlling). Consequently, the evolution equation for the stress controlled viscoelastic creep is : ( 7 1 ) here is the Green strain rate or creep rate, is Helmholtz energy function, S(t) is the Piola Kirchhoff stress and k is time independent material constant. Expanding Eq. ( 7 1 ) into a matrix form: ( 7 2 ) where E 2 and E 3 are the orthogonal Green strain components at the pole of the inflated spherical bubble as shown in Figure 7 1 Figure 7 1 Formulation of thermoviscoelastic model using Green strain components E 2 and E 3 E 2 E 3 PAGE 157 157 The Green strain rate is directly obtained from digital image correlation test data, it can also be calculated as: ( 7 3 ) Holzapfel et al. (2000) formulated the Helmholtz free energy function with the assumption of material incompressibility in the long term where the isochoric deformation satisfies 1 2 3 = 1. The later requires the deformation gradient tensor t to be, where then by definition C onsequently, the long term anisotropic elastic function for the viscoelastic grafts become s : ( 7 4 ) where and k 1 k 2 and c are material constants. The first term of Eq. ( 7 4 ) is a neo H ooke a n model which accounts for the isotropic behavior of the glycosaminoglycan (GAG) matrix and the exponential terms in the series account for the anisotropic collagen network. If the deformation gradient tensor F is diagonal, the Holzapfel et al ( 2002) model given in Equation ( 7 4 ) becomes a three dimensional version of the Fung (1972 ) model except that the material constants are given a more elaborate thermodynamic explanation. Equation ( 7 4 ) is further simplified to have the form: ( 7 5 ) The first and second derivatives of the Holzapfel hyperelastic model Eq. ( 7 5 ) are required for use in the thermoviscoelastic evolution model of Eq. ( 7 2 ) Therefore differentiating Eq. ( 7 5 ) : PAGE 158 158 ( 7 6 ) Substituting Eq. ( 7 6 ) into the b iaxial evolution Eq. ( 7 2 ) and solving the resulting equation would result in a thermodynamically consistent creep model. ( 7 7 ) Application to Inflated Membrane Recall that in the bubble inflation test the state of strai n at the bubble pole is equibiaxial which means the sphere deforms uniformly in all directions. It is, therefore, possible to use a single state variable to describe the creep behavior of an inflated graft. We describe creep of the bubble in terms of the c hange in radius of a small circular area circumscribing the pole of the bubble as shown in Figure 7 2 PAGE 159 159 Figure 7 2 Parameters describing the creep behavior of inflated bubble, A is the initial radius, and a(t) radius at time t of a circular region enclosing the pole The in plane stretch ratio of the circular region enclosing the pole can be defined using the change in the perimeter of the circle as: ( 7 8 ) where a(t) is the current or deformed radius and A is the original undeformed radius of the circle as shown in Figure 7 2 The incompressibility constraint on the graft material requires that the volume of the control region be preserved, that is 2 With this formulation, the entire bubble inflation problem can be treated as a two dimensional problem with the gradient matrix F having Kirchhoff stress S, can be related to the inflation pressure and the stretc h acceleration by the equation of motion in the radius direction as: ( 7 9 ) where T is the Cauchy stress. The second Piola Kirchhoff stress is related to Cauchy stress by S=JF 1 TF t 2 so that T = S. Therefore the stress is: a(t) Bubble Pole A PAGE 160 160 ( 7 10 ) Figure 7 3 Thin walled isotropic spherical bubble for which Eq. ( 7 10 ) is formulated Using the non equilibrium thermodynamic formulation ( Haslach 2004 ; Holzapfel et al. 2000 ), the energy function representing the long ter m behavior is: ( 7 11 ) At the experimental point near the pole we have Hence: ( 7 12 ) where c t = c 1 + c 2 + c 3 Differentiating the energy function ( 7 12 ) yields: ( 7 13 ) PAGE 161 161 Figure 7 4 Typical strain field on the surface of an inflated dermal graft The general evolution equation governing the creep behavior is now written as : ( 7 14 ) Differentiating the energy function Eq. ( 7 14 ) and using we will arrive at the expression for the stretch acceleration of the control area: ( 7 15 ) Combining Equations ( 7 13 ) and ( 7 15 ) and rearranging terms would yield the governing constitutive equation for the viscoela stic creep acceleration of collagen dermal grafts as: PAGE 162 162 ( 7 16 ) The constants k, c and c t in Eq. ( 7 16 ) are calibrated using the bubble inflation test data obtained in Chapter 6. Equation ( 7 16 ) is numerically solved using MATLAB's stiff inte grator ode15s solver t hat implements an explicit Runge Kutta pair. The material parameters are c t = 13, c = 0.88, and k = 12 average for the three test specimens. Figure 7 5 shows the equibiaxial creep strain of collagen dermal grafts at three different pr essure levels. As shown in the f igure, better agreement between the analytical prediction and the experimental result is obtained at higher input pressure (100psi). Moreover, t he predicted creep rates are slightly higher in smaller strain amplitude than they shou ld be. PAGE 163 163 Figure 7 5 The equibiaxial viscoelastic creep response of collagen grafts to multiple pressure inputs Thin inflated membrane theory is used with Haslach's (2004) evolution equation to formulate the creep law. MATLAB's stiff integrator ode 15 s was used to solve the problem. Three Dimensional QLV Theory Within the framework of QLV theory, it is possible to utilize hyperelastic strain energy function to describe the instantaneous nonlinear viscoelastic anisotropic 3D defor mation of collagen grafts. A similar hyperelastic model has been used by Tonuk et PAGE 164 164 al. (2004), Li et al. (2006) and Bischoff (2006) to characterize ligament viscoelasticity The multiaxial stress history is given by: ( 7 17 ) where is a hyperelastic strain energy function and G(t) is the reduced relaxation modulus. Fung's quasilinear model requires the relaxation function and the derivative of stress with strain be defined. For an incompressible material subjected to a hydrostatic pressure p, the stress is given by: ( 7 18 ) Differentiating the stress S ij with respect to strain E kl gives: ( 7 19 ) Consequently, the relaxation stress becomes: ( 7 20 ) The hyperelastic strain energy function can take any of the known finite deformation models such as Aruda Boyce, Ogden or polynomial forms. Many authors (Lai 1968; Lubarda 2004) have proposed a three term Mooney Rivlin strain energy function to be used with most biological solids: ( 7 21 ) PAGE 165 165 where c 10 c 01 c 11 are experimental materials constants and I 1 I 2 are the first and secon d stress invariants. The later is written in terms of the deformation tensor C and expanding to yield: ( 7 22 ) With the assumption of material incompressibility i.e. 1 2 3 = 1, and zero shear stress, that implies working with principal stresses, we obtain the deformation gradient and right Cauchy tensors as: ( 7 23 ) The above diagonal matrices are obtained by imposing the incompre ssibility assumption that: ( 7 24 ) This formulation particularly works well with the bubble inflation test scheme as stress and deformation are all measured near the pole. The Mooney Rivl i n strain energy function Eq. ( 7 21 ) is further simplified using the expanded form of Eq. ( 7 22 ) to yield: ( 7 25 ) PAGE 166 166 The relaxation stress, the second Piola Kirchhoff stress in direction 11 can be calculated as: ( 7 26 ) Some authors choose the stress relaxation function to have the form of a prony series as ( Tong 1978; Fung 1969): ( 7 27 ) where d 1 and d 2 represent short and long term relaxation magnitudes, and 1 and 2 represent short and long term relaxation time constants. For the case of a ramp strain input, the integral equation ( 7 26 ) would be split into the time it takes to ramp up the strain, and then the time past this ramp time. The relaxation equation presented here can only be used with strain control test setup whereas the bubble inflation test uses a stress control (pressure control) hence these equation cannot be directly applied. Three dimensional Generalization of Schapery's Theory Using the principles of irreversible thermodynamic descriptions of the state of a vis coelastic material subjected to external loads, Schapery (1966, 1969) developed the following single integral representation for creep strain resulting from the application of a single axis stress. ( 7 28 ) and, PAGE 167 167 ( 7 29 ) Here g 0 g 1 g 2 are material parameters which are dependent on stress, is time scale is variable of integration, D 0 is the initial or elastic compliance and is the transient creep compliance function. The parameter a is a shift factor which modulates the time scale much in the same way that the temperature dependent shift facto r a T modulates the time scale for temperature effects. Similarly, the single integral Schapery stress relaxation for a viscoelastic material subjected to a single axis stretch is given by: ( 7 30 ) and ( 7 31 ) where h h 1 h 2 are material parameters which are dependent on strain, E is the equilibrium or final value of the elastic modulus is the transient modulus, a is the shift factor. The above equations are applicable for the case of a single axis (or uniaxial ) loading and have to be generalized in to a three dimensions. The generalization of the one dimension al single integral Schapery type constitutive equations into 3D can be made through mathematical manipulation by replacing the scalar elastic and linear viscoelastic properties with their tensorial equivalents Hence for a three dimensional problem, the no nlinear viscoelastic creep model shown in Eq. ( 7 28 ) will have the form: PAGE 168 168 ( 7 32 ) where the transient compliance matrix has the form: ( 7 33 ) Similarly, the three dimensional stress relaxation function is derived from Eq. ( 7 30 ) by representing the scalar material data by their equivalent tensorial form as: ( 7 34 ) where the transient elastic matrix has the form: ( 7 35 ) The three dimensional Schapery type models presented above are derived from the single integral formulation using pure mathematical manipulation. This was required because the original three dimensional Schapery models (which were primary formulated for po lymer materials) are too general with additional terms that do not add much information to the isotropic formulation that we are developing here. The original 3D Schapery models for strain and stress are: ( 7 36 ) PAGE 169 169 where and are the strain or stress energy functions associated with nonlinear elastic materials, ij Eq. ( 7 36 ) are general in the sense that they can account for various levels of material symmetry or nonlinear dependency. However, they are very complicated and will not add much to the accuracy of the dermal grafts considered here. In fact, we'll show that with the bubble inflation test data the dimensionality of the problem is further reduced into a two dimensi onal case. Application of 3D Schapery's Theory to Bubble I nflation Test Data Determination of the material parameters necessary for the application of Schapery's equation is best done as follows. The pressure applied during experiment is a single step pres sure which can be written as: ( 7 37 ) where U(t) is the unit step function and P 0 is the constant inflation pressure maintained by the pneumatic system ( Figure 6 3 ) also illustrated in graphical form in Figure 7 6 Substituting Eq. ( 7 37 ) into ( 7 10 ) yields: ( 7 38 ) Figure 7 6 Constant inflation pressure is maintained during inflation creep test of dermal grafts P 0 T ime Pressure t 1 PAGE 170 170 We intend to use Eq. ( 7 38 ) with ( 7 28 ) hence we need to obtain the derivatives of E quation ( 7 38 ) ( 7 39 ) The derivative of the stretch acceleration (or the third derivative of stretch) doesn't have physical meaning as jerk is not an anticipated phenomenon during creep as the later is a relatively slow p rocess Hence Eq. ( 7 39 ) is simplified by dropping the term into: ( 7 40 ) Substituting Eq. ( 7 40 ) and ( 7 38 ) into Eq. ( 7 28 ) yields: ( 7 41 ) In terms of stretch, ( 7 42 ) PAGE 171 171 Here the integrand must be evaluated at = 0 (step pressure loading) As a result, the effective times may be calculated as: ( 7 43 ) and ( 7 44 ) It is necessary to specify in Eq. ( 7 42 ) to be able to solve the integral equation and for that we use a common power law used for most polymers (Wineman et al. 2000; Huang 1987; Ferry 1961) : ( 7 45 ) where D 0 D 1 and n are material constants and is given by: ( 7 46 ) Substituting Eq. ( 7 46 ) into Eq ( 7 42 ) and rearranging terms yield the following constitutive law: ( 7 47 ) Equation ( 7 47 ) describes the viscoelastic creep behavior of an inflated dermal graft tested in this research work. It is a genuine adaptation of the irreversible thermodynamic theory of Schapery (1969) with thi n inflated membrane theory for use with the bubble inflation test data. This approach has not been reported in the literature. PAGE 172 172 Finding Material Parameters The constants D 0 D 1 and n as well as the stress dependent parameters g 0 g 1 g 2 and a in Eq. ( 7 47 ) or in other words five material properties are needed to fully describe the nonlinear creep behavior of dermal grafts. These parameters are determine d from the experimental data shown in Figure 7 7 Figure 7 7 Creep responses at multiple inflation pressure (linear and log log scales ) D 0 and D 1 are linear material parameters and they need to be determined from the experimental data in the linear stress regions before the nonlinear parameters g 0 g 1 and g 2 can be properly determined. For linear viscoelastic response, when the transient strain is large compared to the initial step strain input, the strain vs. time on a log log plot is a straight line at long times and the slope of the line is the power law e xponent n. To determine the other material parameters equation ( 7 47 ) is numerically solved using MATLAB's integrator ode23s that implements Runge Kutta pair It is a low accuracy integrator but has the advantage of solving stiff systems and has been found to work very well with most of the numerical models presented throughout this work. PAGE 173 173 The integrator needs to be stiff to solve the relaxation and creep probl ems particularly at the transient regime where there is a rapid change in the dependent variable (stress or strain) due to the step loading Figure 7 8 shows the mod el predictions and the experimental data in linear and log log plots. The model predicts the observed experimental data well with overestimation error in the order of 2 5%. One possible explanation for this error could be the fact we used the transient cre ep data in the linear range to calibrate the nonlinear material parameter. Schapery and his co workers used the recovery data instead of the transient creep data to estimate the nonlinear material parameters g 0 g 1 g 2 In t his work, however, due to test r elated limitations we are not able to reproduce creep recovery data of the dermal grafts hence all material data is calibrated based on transient creep response. Table 7 1 shows the viscoelastic creep material data obtained from inflation tests conducted a t three different inflation pressures. Table 7 1. Viscoelastic material parameters used with Eq. ( 7 47 ) and Schapery type thermodynamic model adapted for bub ble inflation experiments Coefficient Value D 0 3.3 D 1 1.25 g 0 0.9765 g 1 g 2 /a 1.0848 n 0.2289 It's important to note that Schapery type nonlinear characterizations are best used for materials that do not have residual or permanent deformation when the stress is removed. However, we have observed that dermal grafts exhibited considerable PAGE 174 174 permanent d eformation that needs to be taken into account in modeling the transient creep response. Figure 7 9 shows the dermal grafts several hours after unlading where it is sh own that the graft remained permanently stretched after the 100 psi pressure load is completely removed. Figure 7 8 Creep predictions and experimental data at multiple stress levels PAGE 175 175 Figure 7 9 Residual creep deformation of collagen dermal grafts several hours after unloading 100psi pressure Concluding Remarks The purpose of this chapter was to develop predictive models for describing the viscoelastic creep response of dermal grafts using the ex perimental data obtained from the bubble inflation test. Th e model development was required because almost all multiaxial constitutive models are built from experimental setup that involve s only a mechanical stretch frame where a planar sample is pulled in equally spaced direction. Such models, however, cannot be directly applied to the bubble inflation experiment because the latter h as a different loading mechanic ( a uniform transverse pressure on a spherical specimen vs. a direct tensile force on a planar specimen ) We have shown that the creep behavior of the inflated graft can be described using the motion of small circular region near the pole of the bubble. The stretch ratio was defined as the change in length of the perimeter of the circle overtime a nd then a mathematical expression for the stretch acceleration is given. The C hapter has considered thermodynamically consistent viscoelastic models to describe the observed creep behavior. Both the thermovis c oelastic creep theory and Schapery's nonlinear model have shown a good agreement with the experimental data PAGE 176 176 of the bubble inflation test. The apparent small discrepancy in the predictions made by Schapery's model can be improved by further incorporating the creep recovery data in to the analytical mode l. I would like to emphasize here that none of the above models have been used to describe the behavior of dermal grafts before Schapery's theory is mostly known to authors in polymer science and has never been explored for use in modeling biological materials such as the dermal grafts. The same is true to with thermoviscoelastic constitutive theory In addition, I have presented a new approach that enables use of bubble inflation data with nonlinear viscoelasticity theories. In sum, b ubble inflation test mimics the in service loading condition of dermal grafts better than mechanical stretch fram es and a thermodynamically admissible viscoelastic material model that is borne out of such a test will certainly make s more sense than spring and dashpot arrangement that many experimenters are fixated with. PAGE 177 177 CHAPTER 8 CONCLUSIONS AND FUTU RE WORK The results given in this work detail a broad research effort to understand the mechanical behavior of collagen dermal grafts. Collagen dermal grafts are viscoelastic rather than elastic and a full description of the mechanical behavior require s a nonlinear thermodynamically consistent viscoelastic material model born out of experiments. Experimentally, the mechanical pr operty of the dermal graft has been studied using uniaxial and equibiaxial material testing protocols. In the uniaxial test, the instantaneous elastic response of the graft to a step stretch has been investigated by running the crosshead of the uniaxial te st machine at its highest speed (500 mm/min). The high stretch rate prevents simultaneous relaxation of the specimen during deformation and thereby enables separation of the elastic and viscous responses of the viscoelastic graft material. Typically the s tress strain curve is nonlinear, with a highly compliant region followed by a stiff linear region. In the highly compliant region the stretch results from gradual and progressive reorientation of the collag en fibers inside the highly porous fluid saturated glycosaminoglycan (GAG) matrix. With increasing stretch, the specimen compaction led to considerable resistance to further fibril reorientation and led to a stiff region. In the stiff linear region, the second part of the stress strain curve, the collagen fibril extension is the principal mode of deformation. The response to ramp load depends on both strain and stretch rate, and as such for the same strain level a higher stretch rate resulted in a higher peak stress and an apparently higher elastic modulu s. Isochronous stress strain curves are better suited to describe such behavior than the conventional stress strain curves particularly when only PAGE 178 178 ramp data is available. In general, the instantaneous elastic behavior is represented by Neo Hookean or Moony Rivlin hyperelastic strain energy function. The dermal graft s exhibit considerable viscoelastic stress relaxation. The stress relaxation behavior was examined by applying a step stretch and holding the specimen at a constant strain while simultaneously mea suring the change in stress. It is observed that rapid stress relaxation occurs within 10 to 12 seconds after the strain was held and a steady state stress has reached in less than 3 minutes, however, the relaxation continued at a much slower relaxation ra te. The relaxation modulus and relaxation rates of collagen grafts depend on the strain level implying the presence of considerable material nonlinearity that cannot be described by linear constitutive laws. Nonlinear thermodynamically consistent viscoela stic constitutive laws are divided into: rational (or functional) thermodynamic approach (Truesdell 1984; Coleman and Noll 1962; Haslach 2004) and the irreversible (non equilibrium) thermodynamic approach (S c hapery 1966, 1969, Lou and S c hapery 1971). In th e rational thermodynamics, the Helmholtz free energy is represented by a pair of conjugate thermodynamic variables such as strain and stress or temperature and entropy, etc, and then the constitutive equation is formed by taking the derivatives of the free energy with respect to the appropriate conjugate variable (strain for relaxation, and stress for creep). The irreversible thermodynamic formulation, on the other hand, uses certain internal variables in order to describe the internal state of a material. Analytical equations that describe the evolution of the internal state variables then become the governing constitutive equations. PAGE 179 179 Chapter 4 has proposed an elastic image registration technique to undo the refraction induced distortion through control poi nt matching and spatial transformation. The elastic image registration technique uses a local weighted mean (LWM) transform ( that relies on different mathematical functions at different regions of an image ) to have a more accurate local alignment of curved regions of a refracted image with its non refracted counterpart. To obtain the transformation functions, refracted and non refracted images of a template panel were compared and brought into alignment and the mapping functions that achieved the registrati on with the smallest error are retrieved. The same transformation is then applied to underwater images of a test specimen that are acquired in similar experimental conditions as in the template Strain and deformation measurements of a case study have show n a good agreement between the reconstructed and the reference DIC patterns. In Chapter 5 we have shown that the quasilinear viscoelastic (QLV) constitutive model, the most widely used theory in biomechanics is capable of describing the strain dependent r elaxation stress of collagen grafts fairly well. However, the relaxation rate predicted by the QLV theory is significantly slower than what was actually observed in the unaxial relaxation tests. The thermodynamic model of Haslach (2004) which is based on of non equilibrium thermodynamics predicted the rela xation and creep behavior from the same set of experimental data. The resulting model predictions agree reasonably well with the long term relaxation and creep responses than they do with the short term b ehavior. We have pointed out that such disparity in short term prediction can be circumvented by allowing the constants k and k to vary with time rather than keeping them constant. PAGE 180 180 Schapery's nonlinear theory, founded on irreversible thermodynamics, was able to describe the viscoelastic nonlinearity of dermal grafts in a much superior way than the t hermoviscoelastic and quasilinear theories. The predictions are accurate in both stress and stress relaxation rate. In Chapter 6 we introduced an integrated b ubble inflation and digital image correlation technique to investigate the equibiaxial behavior of the material. This was required because the uniaxial loading experiment does not describe the full mechanical behavior of grafts since it probes the material property in its very unnatural form. The compaction of collagen fibers inside the aqueous matrix does not normally happen in service and hence the mechanical response that is obtained through a uniaxial test may not be fully valid. On the contrary, an equ ibiaxial extension test mimics the in service loading condition of dermal grafts very well, since the grafts are often attached to the host body with circumferential stitches. A simpler way of probing the equibiaxial behavior is to use a bubble inflation (or a bulge) experiment. In a bubble inflation test, a circular graft was cut and clamped around the periphery of custom made equipment; a compressed air pressure is applied to inflate the specimen into a spherical bubble shape. In the process, full filed strain data was acquired using an optical system while pressure was measured using a digital pressure gage. With the assumption of spherical symmetry at the bubble pole the stress was determined and the equibiaxial stress strain curves were drawn. In gene ral, the equibiaxial stress is greater than a comparable uniaxial test at the same strain. In addition, the range of the highly complaint region of the stress strain curve is less than t he equibiaxial test implying that the network did not restructure to t he PAGE 181 181 extent that it did in a uniaxial test. Some reorientation, however, had been observed in the equibiaxial test. Interestingly, the stiffness of the dermal grafts in equibiaxial loading, as defined by the slope of the stress strain curve of the 'stiff' r egion, is almost the same as the stiffness obtained from a comparable uniaxial test. This important observation allows the use of a constant multiplicative factor to relate the stress strain data from uniaxial test s to that of equibiaxial test. Chapter 7 d eveloped predictive models for describing the viscoelastic creep response of dermal grafts using experimental data obtained from the bubble inflation test. The model development was required because most multiaxial constitutive theories are built from expe rimental setups that cannot be directly applied to the bubble inflation experiment. It is shown that the creep behavior of an inflated graft can be described by observing the motion of small circular reg ion near the pole of the bubble where the stretch rat io is expressed as the change in length of the perimeter of the circle over time. There are several factors that are not considered in the current research and may be a future study of interest One such factor that needs to be addressed is the porosity o f the material. Dermal grafts are composed of porous solid matrix with fluid filling the pores. The overall mechanical response to load and strain depends not only on the deformation of the fibers and GAG matrix but also on the movement of the fluid in and out of the pores during deformation. Since fluid plays a role in load transfer, it is anticipated that the stress strain, stress relaxation and creep behavior may all contain a PAGE 182 182 sizable contribution from the fluid movement. Such fluid solid interaction nee ds to be modeled separately using constitutive laws as poro elasticity and biphasic theories. The experimental and computational frameworks developed in this thesis have several implications for future research. The data on the viscoelastic property of derm al grafts can be useful to qualitatively study how cells contact and apply forces to the graft since the time dependent property of the dermal grafts is now known Future researchers can focus on the effect of a particular viscoelastic property on cell behavior such as cell migration, attachment and contraction. Hopefully with the mathematical models presented here, a better understanding of tissue regeneration thr ough implantable grafts can be achieved thereby realizing a successful and complete regeneration of skin and even other tissues in our mortal body. PAGE 183 183 APPENDIX A THERMODYNAMICALLY CO NSISTENT CONSTITUTIV E FORMULATION This appendix reviews the fundamentals of thermodynamic based constitutive formulat ions. An irreversible process of a continuum system is usually characterized by a change in the microstructure as a result of thermomechanical loading. The entropy always increases in an irreversible process (Clausius, 1850) and the energy dissipation is m anifested by a change in temperature or heat rate. In thermodynamics, internal energy is measured by thermodynamic potentials, such as the Helmholtz free energy, enthalpy, and Gibbs free energy. Thermodynamics potentials can be expressed as functions of st ate variables, such as temperature, microstrcuture and damage parameter (Renardy et al. 1987; Schapery 1997) The Helmholtz free energy is a portion of the total internal energy produced by doing work at constant temperature and volume, while enthalpy is a portion of the internal energy that is released with heat or temperature for a volume change, and Gibbs free energy is the net energy in the system at certain temperature and pressure. In order to characterize the state of a system, thermodynamic paramete rs, called state variables, are needed. The choice of a state variable depends on the physical phenomena of the system. Under isothermal conditions, the free energy of a continuum can be expressed as a function of the total strains or stresses and other po ssible internal state variables (ISVs). The Helmholtz free energy, is generally expressed in term of the total strain and ISVs m : ( A 1 ) The rate of entropy production expressed in the Clausius Duhem inequality is given by: ( A 2 ) PAGE 184 184 expressed by: ( A 3 ) Substituting Eq. ( A 3 ) into Eq. ( A 2 ) : ( A 4 ) The local state law (Lemaitre et al. 1990) states that the behavior of a material at a given point is completely defined by the knowledge of its state variables at the current time. Therefore, can be taken as an arbitrarily variable and is not dependent on the rate of entropy production, Thus, the inequality in Eq. ( A 4 ) leads to the following equations: ( A 5 ) The thermodynamic forces, f m are associated with the internal variables, m and are defined as: ( A 6 ) The inverse of Helmholtz free energy is defined as Gibbs free energy (complementary energy) and is expressed in term of the stress tensor and other ISVs, m as: ( A 7 ) The inequality similar to the one in Eq. ( A 4 ) will lead to: PAGE 185 185 ( A 8 ) A material, whose free energy ( or G) is independent of the ISVs, is elastic. For such material, with the energy balance equations, the inequalities in Eqs. ( A 5 ) and ( A 8 ) are trivially satisfied. The evolution of IS Vs is defined by prescribing the rate as a potential function of the stress. This can be generally expressed as ( Corr et al. 2001; Dafermos 1968; Eringen 1962 ) : ( A 9 ) The above equation determines a dissipative irreversible process. In the case of viscoelastic material, t he potential functions g m are usually smooth in terms of their dependent variables (Lubliner 1972). The Schapery nonlinear viscoelastic model for an isotropic material is derived from the above thermodynamically irreversible process ( TIP ) formulation with evolving ISVs (Schaper y 196 6 1969, 1997). The constitutive model assumes small deformations. Isothermal conditions are assumed such that strains are the only observable state variables and temperature effects can be carried through the material properties. Moreover, the materi al is considered to be thermorheologically simple, in which time can be scaled with temperature and humidity functional. Thus, the mechanical behavior can be characterized through this reduced time. It is assumed that ISVs are associated with small energy changes, which allows expressing Gibbs free energy as a second order function in terms of its ISVs (Schapery 1997). This is generally expressed as: PAGE 186 186 ( A 10 ) where G 0 A m and B mn are functions of t he stresses and possibly time t if aging effects are considered. The strain tensor is ob tained by neglecting the second order terms of in Eq. ( A 10 ) : ( A 11 ) In order to describe the changes in the ISVs, evolution equations are needed to define the relationship between the ISVs and t he thermodynamic forces (stability conditions). These are the well known Kuhn Tucker conditions (Renardy et al. 1987) : ( A 12 ) Following Schapery (1997), the strains in Eq. ( A 11 ) may be re presented schematically by a mechanical analogy consisting of springs and dashpots, as seen in Figure A 1 Each m is proportional to the strain of a one cell (a spring with modulus E m and a dash p ot with viscosity m connected in parallel). The elastic strain of the spring with modulus E 0 is the term in Eq. ( A 11 ) The second term represents the transient part. Figure A 1 Generalized Voigt model (Schapery 1997) Next, an associate poten tial formulation is used, in Eq. ( A 9 ) to relate the rate of ISVs to the thermodynamics forces: E 0 E 1 E 2 1 2 m 1 2 m E m PAGE 187 187 ( A 13 ) where C mn is a constant, positive definite and symmetric matrix; a 1 is a positive scalar, 1 =1. The positive definite characteristic of C mn insures that the free energy is minimum at equilibrium. The symmetry of C mn Renardy et al. 1987 ). Time dependence of a 1 is introduced explicitly in the case where physical aging is considered. Next, Eq. ( A 12 ) can be rewritten as: ( A 14 ) The thermodynamic forces in Eq. ( A 12 ) are combined with the inverse of Eq. ( A 13 ) yields: ( A 15 ) Therefore, ( A 16 ) and assuming: ( A 17 ) where D mn is a constant, positive definite, and symmetric matrix; a 2 is a positive scalar, which M, and other ISVs and a 2 =1 at the reference state. Similar to C mn a positive definite characteristic of D mn insures that free energy is m inimum at equilibrium (Schapery 1969) Dividing Eq. ( A 15 ) by a 2 and introducing a reduced time and its rate d yield to: PAGE 188 188 ( A 18 ) Since C mn and D mn are symmetric and positive definite matrices, it is always possible to diagonalize the above relations. Thus, another set of principal ISVs can be used to write uncoupled set of equations in terms of the new principal ISVs: ( A 19 ) The general solution of Eq. ( A 19 ) is: ( A 20 ) where m is a positive retardation time. The term is a function of time t time at which the input function is applied. Next, another simplification is introduced to allow the characterization of master cr eep functions and account for environmental and aging effects through a reduced time. The parameter A m is assumed to be : ( A 21 ) where E m m are constants. Stress dependent M, t, and possibly other variables that relate it to structural changes. In the linear viscoelastic range The term accounts for thermal and moisture expansion effects. Thus, is equal to 0 at the reference state. Substituting Eqs. ( A 21 ) and ( A 20 ) into Eq. ( A 11 ) yields: PAGE 189 189 ( A 22 ) For simplicity, the dependence of on is excluded. If it is necessary to account for stress dependent expansion in Am, this can be done by modifying Thus, the terms and are: ( A 23 ) The terms and are transient components of the mechanical and creep expansion compliances, respectively. The compliances in Eq. ( A 23 ) are dependent on the reduced time and lead the strain response to a unit value of and Therefore, it is appropriate to call them as master creep compliances. The total strains are then obtained by substituting simplification of Eq. ( A 18 ) into Eq. ( A 11 ) and using the definition of reduced t ime as in Eq. ( A 18 ) Consider a strain response due to a mechanical loading and introduce nonlinear parameters that reflect higher orders Eq. ( A 11 ) can be rewritten in order to incorporate nonlinea r effects as: PAGE 190 190 ( A 24 ) where is the instantaneous uniaxial elastic compliance, is the uniaxial transient compliance, g 0 g 1 g 2 and a are the nonlinear parameters, and is the reduced time (effective time) given by: ( A 25 ) The upper right superscript of a given term is used to denote an explicit variable of this term or function. In general, the nonlinear material parameters: g 0 g 1 g 2 and a can be dependent on the stress, temperature, moisture and possible other variables. With fixed environmental conditions the parameters: g 0 g 1 g 2 and a are stress d ependent only. These functions are always positive and equal to one for relatively small values of stress magnitudes (Boltzmann 's linear theory ). The parameter g 0 is the nonlinear instantaneous elastic compliance and measures the reduction or increase in s tiffness. The transient creep parameter g 1 measures the nonlinearity effect in the transient compliance. The parameter g 2 accounts for the load rate effect on the creep response. The parameter a acts as a stress dependent time scaling factor. The function a T is a temperature dependent that is used to define a time scale shift factor for thermorheologically simple material. The parameter a e is a time shift factor due to aging effects. 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J Orthop 8:712 721 Woo SL Y, Simon BR, Kuel SC, Akeson WH (1980) Quasi linear viscoelastic properties of normal articular cartilage. J Biomech Eng 102:85 90 189 J. Biomed Mater. Res., 14 pp. 65 81 Zitova B, Flusser J (2003) Image registration methods: a survey, Image and Vision Computing, 21: 977 1000 PAGE 202 202 BIOGRAPHICAL SKETCH Mulugeta Haile received his Bachelor of Science degree in mechanical engineering and a master's degree in machine design engineering from AAU and IIT Roorkee in 2001. In January 2007, he was admitted to the graduate program of the Department of Mechanical and Aerospace Engineering at the University of F lorida where he just completed his work towards a Doctor of Philosophy degree in the general research area of experimental and computational mechanics. He also holds a master s degree in electrical and computer engineering with concentrations on digital si gnal processing from UF. Haile's research interest s lie mainly on the development of experimental techniques and mathematical models to characterize the behavior of time dependent materials, self healing polymers and composites. He also enjoys working on electro mechanical systems that are used in structural health monitoring of mechanical and aerospace systems. 