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- Permanent Link:
- http://ufdc.ufl.edu/UF00086029/00001
## Material Information- Title:
- Inelastic wave propagation under combined stress states
- Creator:
- Myers, Charles Daniel, 1945-
- Publication Date:
- 1973
- Language:
- English
- Physical Description:
- x, 236 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Combined stress ( jstor )
Constitutive equations ( jstor ) Datasets ( jstor ) Inertia ( jstor ) Mathematical variables ( jstor ) Shear stress ( jstor ) Sine function ( jstor ) Stress waves ( jstor ) Trajectories ( jstor ) Velocity ( jstor ) Dissertations, Academic -- Engineering Sciences -- UF Engineering Sciences thesis Ph. D Strains and stresses ( lcsh ) Stress waves ( lcsh ) Wave-motion, Theory of ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis -- University of Florida.
- Bibliography:
- Bibliography: leaves 231-235.
- General Note:
- Typescript.
- General Note:
- Vita.
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000577578 ( ALEPH )
13990362 ( OCLC ) ADA5273 ( NOTIS )
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INELASTIC WAVE PROPAGATION UNDER COMBINED STRESS STATES By CHARLES DANIEL MYERS A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1973 TO PEGGY 2. CKh'NO WLEDC UENT S I would like to thank Professor Martin A. Eisenberg, Chairman of the Supervisory Committee, not only for his untiring efforts during the development and preparation of the material contained in this manuscript, but also for being a counselor, teacher, and friend during both my undergraduate and graduate studies. I am also indebted to Professors L. E. Malvern and E. K. Walsh for their helpful criticism and encouragement during my doctoral studies. In addition, I would like to express my appreciation to the other members of my Supervisory Committee: Professors U. H. Kurzweg, C. A. Ross, and R. C. Fluck. A special word of thanks is extended to Professor N. Cristescu for his many helpful discussions during the development of this dissertation. I thank my wife, Peggy, for her encouragement, moral support, and understanding during the course of my studies. I also thank Peggy for typing and proofreading the drafts of this dissertation. I appreciate the efforts of Mrs. Edna Larrick for the final typing of the manuscript and Mrs. Helen Reed for the final preparation of figures. I acknowledge financial support from the National Defense Education Act, the National Science Foundation, and the University of Florida which made myv studies possible. I also acknowledge the Northeast Florida Regional Computing Center for the use of its IBM 370 Model 165 digital computer without which the scope of this work would have been greatly curtailed. TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . ... ...... iii LIST OF TABLES . . ... ... vi LIST OF FIGURES . .. . vii ABSTRACT . . ... ... ix CHAPTER 1. INTRODUCTION ... . 1 CHAPTER 2. THEORETICAL DEVELOPMENT . .. 12 CHAPTER 3. D]IENSIONLESS EQUATIONS AND NUMERICAL PROCEDURES 29 3.1. Wave Speeds as a Function of the State of Stress .. 29 3.2. Characteristic Solution in Terms of Dimensionless Variables . ... 46 3.3. Numerical Grid for Characteristic Solution .. 51 3.4. Finite Difference Eouations . ... 58 3.5. Solution to the Finite Difference Equations .. 69 3.6. Calculation of the Strains .... . 77 CHAPTER 4. RESULTS AND DISCUSSION . ... 80 4.1. Introduction . ... .. 80 4.2. Grid Size Effects ... . .. 81 4.3. Effects of Radial Inertia . ... .84 4.4. Effects of Strain-Rate Dependence . .. 110 CHAPTER 5. SUIIARY . . ... 118 APPENDIX A. CONSTITUTIVE EQUATIONS . ... 125 A.1. Comments on the Constitutive Equation ... 125 A.2. Rate Independent Incremental Plasticity Theory ... 127 A.3. Rate Dependent Plasticity Theory . .. 135 A.4. Dimensionless Expressions for the Functions 0(s,a) and .(s, . . 137 TABLE OF COIM'ENTS (Continued) Page APPENDIX B. CHARACTERISTICS AND EQUATIONS ALONG THE CHARACTERISTICS . ... 140 B.1. Equations for the Characteristics . .. 140 B.2. Equations along the Characteristics ... 141 B.3. Reducing Equations to Simpler Case . .. 156 B.4. Uncoupled Waves . .... .. 159 B.5. Elastic Waves .... . . 162 APPENDIX C. PROGRAMS FOR DETERMINING THE PLASTIC WAVE SPEEDS . ... ... 164 APPENDIX D. SOLUTION TO THE FINITE DIFFERENCE EQUATIONS IN THE CHARACTERISTIC PLANE . ... 171 D.1. Equations for Fully Coupled Waves . .. 171 D.2. Equations for Uncoupled Waves . ... 176 D.3. Solution at a Regular Grid Point for Fully Coupled Waves . ... 180 D.4. Solution at a Regular Grid Point for Uncoupled Waves ................. 183 D.5. Solution at a Boundary Point (X= 0) for Fully Coupled Waves . ... 185 D.6. Solution at a Boundary Point (X= 0) for Uncoupled Waves . ... 191 APPENDIX E. COMPUTER PROGRAM FOR CHARACTERISTIC PLANE SOLUTION . ... 194 E.I. General Description of the Program . ... 194 E.2. Initial Conditions . . 196 E.3. Calculation of A . .... .. 201 E.4. Input Data . . 202 E.5. Listing of the Program . ... 204 LIST OF REFERENCES . . 231 BIOGRAPHICAL SKETCH . .... 236 LIST OF TABLES Table Page 1 Normalized Longitudinal Stress --) . 36 s 2 Normalized Hoop Stress (-- ............. 38 s 3 Normalized Shear Stress .-) . 40 S LIST OF FIGURES Figure Page 2.1 Coordinate System for the Thin-Walled Tube ... 13 2.2 Stresses on an Element of the Tube . ... 14 3.1 Yield Surface Representation in Spherical Coordinates 31 3.2 Plastic Wave Speeds as Functions of $ and y for Poisson's Ratio of 0.30 . ... 41 3.3 Values of $ at v= 0 for which c =c =c ...... 45 f s 2 3.4 Numerical Grid in the Characteristic Plane ... 53 3.5 Regular Element in Numerical Grid . ... 54 3.6 Boundary Element in Numerical Grid . ... 55 3.7 Location of the Characteristic Lines Passing Through P 57 3.8 Numerical Representation of the Characteristic Lines in a Regular Element . . ... 59 3.9 Representation of the Characteristic Lines in a Boundary Element ... . .. 60 4.1 Grid Size Effects on the Longitudinal Strain at X = 3.75 . . ... 83 4.2 Grid Size Effects on the Longitudinal Velocity at X = 3.75 . . ... 85 4.3 Grid Size Effects on the Stress Trajectories at X = 3,75 . . ... 86 4.4 Longitudinal Strain Versus Time at X = 3.75 for Data Set 1 . . ... 88 4.5 Change in Shear Strain Versus Time at X = 3.75 for Data Set 1 . . .89 LIST OF FIGURES (Continued) Figure 4.6 Transverse Velocity Versus Time for Data Set 1 Without Radial Inertia . . 4.7 Longitudinal Velocity Versus Time for Data Set 1 Without Radial Inertia . . 4.8 Longitudinal Strain Versus Time for Data Set 1 . 4.9 Maximum Radial Velocity Versus X for Data Set 1 With Radial Inertia . . 4.10 Change in Shear Strain Versus Time for Data Set 1 Without Radial Inertia . . 4.11 Longitudinal Strain Versus X for Data Set 1 . 4.12 Stress Trajectories for Data Set 1 Without Radial Inertia . . 4.13 Strain Trajectories for Data Set 1 Without Radial Inertia . . 4.14 Shear Stress Versus Longitudinal Stress for Data S With Radial Inertia . . Page . 93 . 94 96 . 97 . 98 - 99 100 et 1 . 102 4.15 Stress Trajectories for Data Set 1 With Radial Inertia 4.16 Hoop Stress Versus Longitudinal Stress for Data Set 1 With Radial Inertia . . . 4.17 Stress Trajectories for Data Set 2 Without Radial Inertia . . . 4.18 Stress Trajectories for Data Set 2 With Radial Inertia 4.19 Hoop Stress Versus Longitudinal Stress for Data Set 2 With Radial Inertia . . . 4.20 Shear Strain Versus Time for Data Set 2 . 4.21 Change in Longitudinal Strain Versus Time forData Set 2 4.22 Longitudinal Strain Versus Time for Data Set 3 . 4.23 Stress Trajectory at X= 0 for Data Set 3 . 4.24 Stress Trajectory at X= .25 for Data Set 3 . viii 104 105 107 108 109 111 112 114 116 117 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INELASTIC WAVE PROPAGATION UNDER COMBINED STRESS STATES By Charles Daniel Myers August, 1973 Chairman: Dr. M. A. Eisenberg Major Department: Engineering Science, Mechanics and Aerospace Engineering The purpose of this dissertation was to investigate the effects of radial inertia and material strain-rate dependence on the propa- gation of inelastic waves of combined stress along a thin-walled tube. A general quasilinear constitutive equation for multiaxial stress (and strain) states was introduced. The equations of motion and the strain-displacement equations, along with the constitutive equations, were written to form a set of nine simultaneous hyperbolic, quasilinear, partial differential equations. This set of equations was reduced to a set of six equations which was then used to determine the expres- sions for the characteristic lines and the equations along the char- acteristic lines. For combined torsional and longitudinal loading, two distinct wave speeds were found. The values of these two wave speeds were found as functions of the state of stress. Including radial inertia effect in the formulation of the problem was shown to significantly increase the wave speeds for a given stress state. Also certain critical combinations of Poisson's ratio and the "effective tangent mocdulus" caused the two wave speeds to be equal when the shear stress vanished. The equations for the characteristics and the equations along the characteristics were written in terms of dimensionless variables. These equations were then written as first order finite difference equations. A computer code was written in the Fortran IV language, and several problems were solved using an IBM 370 model 165 digital computer. In order to obtain these solutions two particular forms of the constitutive equation were used; one form represented a strain- rate independent material while the other form represented a strain- rate dependent material. The strain at the impact end was considerably larger when radial inertia effects were included than when radial inertia effects were not included in the problem. However, radial inertia effects were found to have little influence on the solution more than two diameters from the impact end. The strain at the impact end was lowered by including strain-rate dependence of the material. For any particular set of initial conditions and boundary conditions, the stress trajec- tories behaved in the same manner, at least qualitatively, whether or not radial inertia effects or strain-rate dependence were included. The details of the stress trajectories were more complicated when radial inertia effects were included since the trajectories were three-dimensional. CHAPTER 1 INTRODUCTION Stress wave propagation is the mechanism by which forces and displacements are transmitted from one part of a structure to another. Stress waves arise when a transient force is applied to a structure, and they propagate through the structure reflecting (at least partially) back into the structure whenever they encounter a boundary. After several reflections the amplitude of the stress waves diminishes and the structure reaches a state of equilibrium. In many engineering problems the time required to reach equilibrium is very short, and for practical purposes the structure can be assumed to reach equilibrium instantly. Problems in which the forces are applied slowly or in which the state of stress is required a long time after the forces are applied are examples of instances when wave propagation effects may be neglected. However, in many cases, the forces are applied rapidly (such as during impact loading or explosive loading), and failure is most likely to occur in the structure almost immediately after the application of these forces. In these cases when it is necessary to determine the state of stress during and immediately following the loading, wave propagation effects may be significant and should be included in the analysis of the problem. In order to understand the development of the theory of stress wave propagation and the application of this theory to modern engineering problems, it is instructive to review briefly the history of wave propagation research. The first serious attempt (at least in this century) to understand nonlinear wave propagation in solids was made by Donnell (1930). In this paper, Donnell used energy principles and impulse-momentum expres- sions to find the particle velocity and the elastic wave speed for longitudinal waves. He also predicted that if a material with a bilin- ear stress-strain curve were impacted at the end by a stress above the yield stress, two stress waves would propagate with distinct velocities. However, after the publication of this paper interest in wave propaga- tion subsided until the early 1940's. A more general theory of longitudinal stress wave propagation was developed independently by Taylor (1940), von Karman (1942) and Rakhmatulin (1945) by assuming that the material exhibited a nonlinear stress-strain curve above the yield point. This stress-strain curve was assumed to be independent of the rate of straining. Using this theory the velocity of propagation of the longitudinal waves was found to be given by 1 da where c is the wave speed, p is the density of the material, C is the da stress, and e is the strain. Thus represents the slope of the stress- de strain curve or the tangent modulus. This theory also considered the stress-strain curve of the material which was obtained for the static case to be valid in the dynamic case. With this assumption, the stress and strain followed a unique functional relationship as long as no unloading occurred. Because of this, the tangent modulus could be written as a function of the stress (or strain) only, so that the velocity of propagation then became a function of the level of stress (or strain). This immediately led to the conclusion that a given level of stress (or strain) propagated at a specific speed, and the stress wave changed shape as it propagated along a prismatic bar for stresses in the nonlinear region of the stress-strain curve. For a bilinear stress-strain curve, the results of Donnell (1930) were again predicted. However, these theories did not account for the lateral inertia effects in the bar or the dependence of the stress-strain curve on the rate of strain, and so more complex theories and constitutive equations were proposed to account for these phenomena. By the late 1940's many investigators including Davis (1938), Manjoine (1940), and Clark and Wood (1950) had experimentally observed the effect of the rate of strain on the stress-strain curve for several materials. In order to incorporate this strain-rate effect into the constitutive equations used to study plastic wave propagation, Sokolovsky (1948a, 1948b) and Malvern (1949, 1951a, 1951b) independ- ently introduced one-dimensional constitutive equations in which the stress was a function of the plastic strain and the plastic strain rate. By selecting a particular form of this constitutive equation, Malvern (1951a, 1951b) was able to obtain a numerical solution which predicted several experimentally observed phenomena. However, his numerical solu- tion did not apparently predict a region of constant strain near the impact end such as had been observed by Duwez and Clark (1947) and others. This new strain-rate dependent constitutive equation also predicted that, if a bar were strained statically above the yield stress and then impacted, the first increment of strain would propagate with the elastic wave velocity and not the velocity given by the tangent modulus.in the strain-rate independent theory. Since this prediction was quite different from that of the strain-rate independent theory several investigators tried to verify one or the other. Bell (1951) published the results of his experiments with aluminum which showed that, for a bar stressed above the yield point, the initial strain pulse propagated with the elastic wave velocity. These results were in accordance with the strain-rate dependent model of Malvern (1951a, 1951b) as were the experimental results of Sternglass and Stuart (1953) which were obtained using copper, Alter and Curtis (1956) which were obtained using lead, Bell and Stein (1962) which were obtained using aluminum, and Bianchi (1964) which were obtained using copper. Encouraged by these experimental results, many investigators continued the development of more general constitutive equations to describe material behavior. Perzyna (1963) generalized the semi-linear constitutive equation of Malvern (1951a, 1951b) to multiaxial states of stress. At about this same time Cristescu (1964) introduced full quasi- linear constitutive relations for a one-dimensional problem. This quasilinear equation was used immediately by Lubliner (1964) to show that the strain-rate independent constitutive equation of Taylor (1940), von Karman (1942), and Rakhmatulin (1945), and the strain-rate depend- ent constitutive equation of Malvern (1951a, 1951b) and Sokolovsky (1948a, 1948b) were both special cases of this more general constitu- tive equation. Later Cristescu (1967a) gave a generalization for multi- dimensional stress states of the quasilinear constitutive equation as well as an extensive summary of the developments in dynamic plasticity until that time. Lindholm (1967) developed a constitutive equation for combined stress states of aluminum which included strain-rate effects and temperature dependence. He also presented extensive data for one-dimensional loading and combined stress loading at several strain rates and temperatures which were used in empirically determin- ing the constants used in his generalized constitutive equation. While these more general constitutive equations were being developed, it was shown by Malvern (1965), by Wood and Phillips (1967), and by Efron and Malvern (1969) that the semi-linear equation of Malvern" (1951a, 1951b) did indeed predict a region of constant or nearly con- stant strain near the impact end if the solution was obtained long enough after impact. Suliciu, Malvern, and Cristescu (1972) have shown that a region of constant strain is not possible for the semi-linear constitutive equation but may be approached asymptotically. They have also shown that a region of constant strain is possible when the quasi- linear constitutive equation is used. However, in the interpretation of experimental results it has been difficult to differentiate between a region of truly constant strain and a region in which the constant strain is approached asymptotically. The experiments of Sternglass and Stuart (1953), Alter and Curtis (1956), and others were believed by many investigators to be proof of the strain-rate dependence of some materials. This led to the exten- sive development of constitutive equations just discussed. However, other investigators sought to explain the experimentally observed phe- nomena by including radial inertia in the formulation of the wave prop- agation problem. Plass and Ripperger (1960) introduced radial inertia effects into the problem of longitudinal impact and used the constitu- tive equation of Malvern (1951a, 1951b). In order to find a character- istic solution, all of the variables were averaged at each cross section 6 and these averaged variables were used. The :results of this work were given by Tapley and Plass (1961) but were somewv.hat inconclusive. More work including radial inertia effects was published by Hunter and Johnson (1964), and a year later DeVault (1965) showed that, at least qualitatively, many observations formerly attributed to a material strain-rate effect could be explained by including radial inertia effects in the formulation of the problem of longitudinal impact of a bar. Shea (1968) obtained good agreement between theory and experiment for the propagation of longitudinal waves in a lead bar. He used the strain- rate dependent constitutive equation of Malvern (1951b) and the "correction" for radial inertia proposed by DeVault (1965). Mok (1972) used the same averaging technique for the variables as Plass and Ripperger (1960) for the problem of longitudinal impact of a bar with radial inertia effects included. He used the strain-rate independent constitutive equations and agreed in essence with DeVault (1965) that radial inertia effects could explain, at least qualitatively, those experimental results usually attributed to strain-rate sensitivity of the material. Since radial inertia is always present in an experi- ment using longitudinal impact it seemed that the only way to conclu- sively determine strain-rate effects in a material would be to perform the experiments using a torsional wave. In an effort to determine the strain-rate dependence of various materials, several investigators have recently conducted theoretical and experimental studies concerning the propagation of torsional waves. Convery and Pugh (1968) gave the results of their experiments in which a tube was stressed statically above the yield stress in torsion and then subjected to a suddenly applied increme-ntal torsional load. The strain caused by this inciremcntal load was lound to propagate with the elastic shear wave velocity. This seemed to be proof that the strain- rate dependent theory was correct, but Convery and Pugh (1968) cau- tioned against that conclusion. For Bell (1960, 1963) and Bell and Stein (1962) had asserted that (based on experimental results with annealed aluminum), while an increment of strain may propagate with the elastic wave velocity, the larger amplitude strains propagate with the wave velocity predicted by the strain-rate independent theory. Nicholas and Garey (1969) tested aluminum samples in torsion at high strain rates and found very little strain-rate dependence. However, Yew and Richardson (1969) were able to measure some strain-rate depen- dence in copper. Another problem which was encountered in wave propagation studies r was that of unloading. The two most common unloading cases were when the applied load was reduced and when waves were reflected from a bound- ary. Unloading was examined for longitudinal plastic wave propagation by Lee (1953) using the strain-rate independent constitutive equation and by Cristescu (1965), Lubliner and Valathur (1969), and Cristescu (1972) using the quasilinear constitutive equation. In all of these investigations, regions of unloading and boundaries between regions of unloading and loading in the characteristic plane were predicted but the results have not been verified experimentally. Many investigators in recent years have become interested in the behavior of materials under combined stress and, more specifically, the propagation of waves of combined stress. One of the first discussions of combined stress wave propagation was given by Ranhmatulin (1958). In this paper he developed the equations which must be solved for elastic-plastic wave propagation under combined stress. Strain-rate independent constitutive equations were used and only the problem for the elastic case was solved. He found that the shear wave did not affect the longitudinal wave in the elastic case. A similar discussion of combined stress wave propagation was presented by Cristescu (1959). Until now nothing has been said about the plasticity theory used. The two plastic strain theories were the total strain theory proposed by IIencky (1924) and the incremental strain theory proposed by Prandtl (1924) and Reuss (1930). These two plasticity theories along with many other developments in plasticity theory were presented in detail by Hill (1950). The different plasticity theories were not presented earlier because in many cases both theories gave the same results. For instance, when a strain-rate independent constitutive equation was used, the two plasticity theories led to identical results when one- dimensional (either longitudinal or torsional) stress wave propagation was studied, when combined stresses were used if the loading was pro- portional, or even when unloading occurred in one-dimensional problems. However, when strain-rate dependent material behavior of nonpropor- tional loading under combined stresses was considered, most investiga- tors used the incremental strain theory. Shammamy and Sidebottom (1967) showed that the incremental strain theory more accurately pre- dicted the experimental results when various metal tubes were subjected to nonproportional static loading in tension (compression) and torsion. Interest in the propagation of waves of combined stress continued and Cli ton (1966) presented the results of his study of combined longi- tudinal and torsional plastic wave propagation in a thin-walled tube. Strain-rate independent material behavior and incremental strain theory were used while radial inertia effects were ignored. The thin- walled tube allowed Clifton to eliminate any dependence on the radial coordinate so that a solution could be obtained in the characteristic plane. (Earlier, Plass and Ripperger (1960) had used a rod and averaged the variables over the cross section in order to eliminate the dependence on the radial coordinate.) The results of this investigation were based on a simple wave solution which resulted from applying a step velocity impact at the end of the tube. Clifton (1966) found that when the tube was stressed into the plastic range, an impact at the end of the tube caused waves with two different speeds to propagate. These waves were called the fast wave and the slow wave, and each wave was found to carry both longitudinal and torsional stresses. Two special cases were examined. The first case involved statically prestressing the tube above the yield stress in torsion and then applying a longi- tudinal velocity at the end. In this case the fast wave caused almost neutral loading, that is, as the fast wave passed a point on the tube, the shear stress decreased and the longitudinal stress increased in such a way that the stress state at that point remained close to the initial loading surface. Then as the slow wave passed the same point, loading occurred so that the stress path was normal to the initial loading sur- face. The second case was for a tube with a static longitudinal plastic prestress impacted by a torsional velocity at the end. In this case the fast wave caused unloading along the longitudinal stress axis followed by an increase in shear stress at a constant value of longitudinal stress and then the slow wave caused loading such lhail the stress path was normal to the initial loading surface. Clifton (1966) also found that for a given initial loading surface, the two wave speeds depended upon the particular stress state on the initial loading surface, and that for one particular initial loading surface the fast and slow wave speeds were equal when the shear stress vanished. This work of Clifton (1966) was a significant step forward in the investigation of waves of combined stress. An extension of this work was presented by Clifton (1968) in which the simple wave solution was used along with unloading at the impact end. In this way certain unload- ing boundaries for combined stress states were determined. Two years later Lipkin and Clifton (1970) published their experimental results from combined stress wave propagation tests and compared these results to the simple wave solution developed earlier. Agreement between the simple wave theory and the experiments was fair. Cristescu (1967b) formulated the problem of combined stress wave propagation in a thin-walled tube using general quasilinear constitu- tive equations but again ignoring radial inertia effects. The equa- tions for the characteristic lines and the equations along these char- axteristic lines were determined. No numerical results were given but the two waves (fast wave and slow wave) were shown to be coupled during loading. Again Cristescu (1971) showed that the coupling of the waves of combined stress depended on the constitutive equations and yield conditions used. This concludes a brief survey of the history of the development of plastic wave propagation theory. No attempt ,vas m3nd1" tc givO a complete historical background. For more information the reader is directed to Hopkins (1961), Kolsky (1963), Olszak, Mroz, and Perzyna (1963), and Cristescu (1967a, 1968). The remainder of this dissertation will be devoted to solving the problem of combined stress wave propagation in a thin-walled tube when radial inertia effects are included. A general quasilinear constitutive equation for multiple states of stress will be presented, and it will be shown to be a generalization of the constitutive equa- tions of both Lipkin and Clifton (1970) and Cristescu (1972). But first the wave propagation problem itself must be developed. CHAPTER 2 THEORETICAL DEVELOPMENT The specific problem to be considered here is that of the propagation of inelastic waves of combined stress along a semi-infinite thin-walled tube, with the effects of radial inertia included. The material consti- tutive equation used is a generalization for multiple states of stress of the quasilinear constitutive equation used by Cristescu (1972) for a single stress component, and is a special case of the very general quasilinear constitutive equation given by Cristescu (1967a). The coor- dinate system used is shown in Figure 2.1, and the stresses on an ele- ment of the tube are shown in Figure 2.2, where r is the mean radius of the tube. The problem is assumed to be axisymmetric so that there is no dependence on e. Since the tube considered is thin-walled, the stresses Sr, Tr, and Trx are assumed to be negligibly small as are the strains e r and e The strain e is not included in the problem. Stability re rx r of the tube wall and thermal effects are not included in the formulation of the problem, and only small strains are used. The strain rate is assumed separable into elastic, plastic, and visco-plastic parts. The radial displacement is very small compared to the tube radius, and plane sections of the tube remain plane. The material is assumed to be isotropic and homogeneous, to obey the von Mises yield condition, and to be isotropically work-hardening. All unloading is assumed to be elastic. Figure 2.1 Coordinate System for the Thin-Walled Tube ax rx Figure 2.2 Stresses on an Element of the Tube 15 The equations of motion in the cylindrical coordinates shown in Figure 2.1 are given by 1 1 C + T + + -T = x,x rx,r r ex,e r rx x,tt 1 1 T + C + -+ ) = p Ur rx,x r,r r r@e, r r r,tt 1 2 T + T + -- T = 0 U 6x,x re,r r e,e r re9 ,tt which, under the assumptions given above, become (xx =p uxtt (2.1) X,x P xtt - -- = u (2.2) r r,tt o Texx = P utt (2.3) where the subscripts following the comma represent partial differentia- tion with respect to the variables x (the coordinate along the tube axis) or t (time). The density of the material is p, and ux, u and u are the displacements of any point in the x, r, and 9 direction, respectively. For the cylindrical coordinates of Figure 2.1, the strain- displacement equations are given by e = r r,r C (1 + (u ) r 9,0 r e = U x x,x r 11 r - r 2(r r,6 + Ur - 1 S= (u + u ) = (u -+ u ) Px 2 x r x, and under the above restrictions, these reduce to the following three equations S= u (2.4) x X,x 1 e = u (2.5) o 1 S= u (2.6) ex 2 e,x Defining the velocities v v, and va as u t, ur,t and ue , respectively, equations (2.1) to (2.6) become xx = x,t (2.7) -r e = r,t v(2.8) o Tx,x= p V,t (2.9) S = v (2.10) x,t X,X 1 = -- v (2.11) xt = vx (2.12) ex,t 2 e,x Under the assumptions used here, the variables no longer depend on r, so that the problem becomes two-dimensional (the independent variables are x and t) and can be solved by the method of characteristics. The equations necessary for completion of the set of simultaneous partial differential equations describing the behavior of the body are the constitutive equations. Ci-rstescr. (1972) uses a full quasilinear constitutive equation for ,a -in:.;le sot'itu diinal stress as Ot- =E t+ (, ~ + (~ ) (2.13) As a generalization of this equation to a constitutive equation governing multiaxial states of stress and strain, the following equation is used + 3+ 3 ij S l+v + (s,A)s +- (s,) (2.14) ij E ij E ijkk A)s- (2.14) s where the dot represents partial differentiation with respect to time, s.. is the deviatoric stress, 6.. is the Kronecker delta, v is Poisson's ij ij ratio, E is Young's modulus, 0(s,iA) and '(s,A) are material response functions as yet unspecified, and s and A are defined as s / s..s.. (2.15) S 2 ij ij /2 .P "P s A ij ij dt + (2.16) v 3 / 13ij E *P and e. is the inelastic portion of the strain rate which, using ij equation (2.14) can be written as P 3 "3 sii P. 3 0(s,A)s + (s,A) (2.17) 3ij 2L I s when the elastic, plastic, and visco-plastic portions of the strain rate are assumed to be separable. The constitutive equation (2.14) is a special case of the equation k1 e. = fk. 0 + gi ij ij kl ij given by Cristescu (1967a). The form of equation (2.14) was chosen as Lhe general constitutive equation because it contains terms which may be considered separately as elastic, plastic, and visco-plastic strain- rate terms, because the inelastic strain-rate tensor is proportional to the corresponding deviatoric stress tensor, and because it reduces to the form of equation (2.13) when the only stress present is the longi- tudinal stress. This simplification to the form of equation (2.13) is shown in Appendix A. The functions 0(s,A) and *(s,A) are functions which depend on the particular material being studied. The function 0(s,A) is a measure of the rate insensitive inelastic work-hardening, and the function r(s,A) is a measure of the visco-plastic strain rate due to the strain-rate sensitivity of the material. In the classical rate independent plastic-- ity theory, t(s,) vanishes. When s < 0 or when s < a (the current "yield stress"), 0(s,4) is set equal to zero. The unloading conditions when ((s,6) = 0 are stated in equation (A.3.1). Two separate materials are modeled in the numerical work done. One is a 3003-H 14 aluminum alloy used in the experimental work of Lipkin and Clifton (1970). This material is assumed to be insensitive to strain rate and the functions 0(s,A) and r(s,A) are obtained using the classical Prandtl-Reuss incremental plasticity theory with iso- tropic work-hardening and the stress-strain curve for uniaxial tension. (See Appendix A.) The other material used is a commercially pure aluminum dead annealed at 1100F. This material is assumed to be strain-rate sensitive, and the functions 0(s,A) and '(s,A) are obtained from the data given by Cristescu (1972). (See Appendix A.) Since the stresses a, T and T are assumed to vanish, r r rx equation (2.141) as applied to the present problem r reduces to 1 v 1 (s,)s+(sA) 2s v 1 1 xt E x,t E E ,t 2 2s e-+-1 ( 0 -( )s+ (s,A)s + (s.A) (2.18) + S -xt E +x,t (sA)s (s, ) 2s where the deviatoric stresses are 1 s = s (2a () x 11 3 x 1 s = s (+ a ) r =33 3 x + ) s = S = s1 = 2 Tx sx 12 21 ex sr = s = . re rx Using these deviatoric stresses, the expression for s becomes 1 1 ds = t L- sijij 2 i sklkl s L 'ijsi 1 3 1 3 a F- s L2 sijsi=-- LssIl + s22s22+ s33s33+ sl2sl2+ s21s21 S 3 o 2 2 2 2 s (Cx + -~ ) + 2T 4s i 1 r -1 s 1 (2 ) (20 )a + 6T7 (2.19) x x,t x ,t x x 2s and equations (2.18) become (2c % ,7"2 .(s,Z) (2- -a7,) (a -2cr) - L ix ,t l-J i-' j et -6(20 -a )T (2c -c ) 42 ex ex (S A)] + t (S,) (2.20) + -2 (s,A) Text +- ((s,A) (2.20) 4s 2s (20 -a,)(a -2 j F- (a -2c ,t E -2 (s A) x,t L+ 2 ( c ,t 4s 4s (6(2crQ )IxTx j 2crQ cr -6 ( 2 ,7 C Y x ) T e x 2 a 9 C 7x + ((s,A) Txt + (s,A) (2.21) L 4s 2s r3T e(2x ) 3Te] 3r e (2e 1) ex, t 2= L0( xt 2(A ,t 4s 4s 18T 3T P1_+ Ox ex + -- 0(sA) T + --- (s,6) (2.22) 4s 2s The equations (2.7) to (2.12) and equations (2.20) to (2.22) form a set of nine simultaneous hyperbolic quasilinear partial differential equations for the nine unknowns ax' cre T6 x' V Vx', V, e', e and e6x. A special case of this system of equations is the set of equations obtained by neglecting radial inertia effects. When radial inertia effects are ignored, the variables a', E and vr are not included directly in the problem formulation. This case can be incorporated into the more general formulation by multiplying a~, e vr, and their derivatives by the dummy variable "a," where "a" has the value of 1 when radial inertia effects are included and the value of 0 when radial inertia effects are neglected. Also the equation of motion, the kine- matic equation and the constitutive equation for motion in the radial direction must be multiplied by "a." Doing this, and defining the quantities 2 1 (2 a% ) A = + -(s,6) 1 E -2( 4s 2 L (2j ac,)(c x x -2 4s 2a~ ) 6Te (2y aOe ) A3 -2 (s,A) 4s A= 4 E (7 2ac )2 + s(-2) 4s 6 T Gex(ax 2aca ) - A = 02(, A) 4s 2 4s (2.23) the nine simultaneous equations (2.7) to (2.12) and equations (2.20) to (2.22) can be written as ,x=vx,t (2.24) a r a = apvt (2.25) ex,x ,t (2.26) x,x a ae = -r v e,t r r o 1 ex,t = Ve,x ex, = Ax, + +aA + A + x,t 1 x,t 2 e,t 3 ex,t as ,t= aA 2 + aAa ,t 9,t 2 x,t 4 e,t 2 aaa x , (s,6) 2s 2aa a '2s 3T 1 a 1 ex , = A3x + A + T + -- (sA) Ex,t 2 3 ,t 3 A5 ,t 2 56 6xt 2S 2s (2.27) (2.28) (2.29) (2.30) (2.31) (2.32) x,t 1 3(5.1_. )_1 Eliminating the strain rates from the last six of these equations, and defining 2a au, x e x- (,) 2s 2aa -a = x (sA) 2s (2.33) the system of equations for 3T Ox 4(s,/) J 2s nine equations reduces to the following system of six the unknown variables ex, 1O, Tx' Vx, v and v a x= xt X,X X,t a - a7 r o ap vr,t r,t T6x,x = Pv,t v = Aa + aA + A T + X x,x 1 x,t 2 9,t 3 ,t x a - -- v = aA +aAo + aAT + a r r aA2 x,t 4 9,t 5 9x,t a o O,x= A3 x,t + aAO 9,t + A5 x,t + 2ex 2- 2 2 2 1 s = (c ax 0 + a + 3T Ox (2.34) (2.35) (2.36) (2.37) Since the equations (2.24) to (2.26) and (2.34) to (2.36) form a system of hyperbolic equations, they can be solved by the method of characteristics. To do this, first the equations for the characteristic lines must be determined, and then the equations along these characteristic where A w, B b (2.38) where p 0 0 0 0 0 0 ap 0 0 0 0 0 0 p 0 0 0 A = (2.39) 0 0 0 A! aA A3 0 0 0 aA2 aA aA5 0 0 0 A3 aA5 A6 0 0 0 -1 0 0 - 0 0 0 0 0 0 0 0 0 0 0 -1 B = (2.40) -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 V x v r w = (2.41) CT x LT:xJ a,. and 0 r o 0 x x av o -2a r @Xe - 2te X and if the slope of the characteristic line (or the wave speed ) is denoted by c where dx dc t then the equation for the characteristic lines is given by cA -B =0 and from the calculations shown in Appendix B, equation (2. [ 2 2 Cr -2,2 c a pc J L(pc ) [a} (2.43) (2.44) 44) yields -2 ( - (pc ) {b + A4j = 0 (2.45) where S-2 -2- -2- a = A 4 A6 + 2aA2A 3 5 aA 1 A5 aA26 A3A (2.46) -2 -2 = A1 -aA2 + A A a. 5 1 4 2 4 6 5 Setting the first factor in (2.45) equal to zero, dx ac = a = 0 (twice) dt and, setting the second factor in (2.45) equal to zero, r= I 1 -2 1il 2 c = L-- (b -4aA4) 4 2ap (2.42) (2.47) (2.48) If the wave speeds in equation (2.48) are denoted by c = i -- -b (b (2. 49) f -L9_ fj 2ap and c = ( 4aA ) (2.50) 2ap where cf is the fast wave speed and cs is the slow wave speed, then the slopes of the characteristic lines are given by c = 0 (twice) c = f c = c s Equations (2.47) and (2.48) are the six equations for the character- istic lines for the set of six simultaneous, hyperbolic, quasilinear partial differential equations of (2.38). When radial inertia effects are not included (that is, when a= 0), the equations (2.47) vanish identically and the remaining four simultaneous equations of (2.38) have the characteristics given by equations (2.48). For this case (a=0) and when 0(s,A) is obtained from incremental, rate independent plasticity theory with isotropic work-hardening, equations (2.48) reduce to those given by Clifton (1966) as shown in Appendix B. The equations along the characteristics can be obtained in two different ways, both of which are discussed in Appendix B. The result- ing equations along the characteristic lines of equation (2.48) are 0 = 5 -A A ]dTex 5 3 4iJ x -2 {r + pc2)(A2 6 -35 2 1L o 1 c c + 2(pc )aA2A5-A3A4 ( dt 2s ( 2aaG) 2s (s,A) dt [(P c) (iA6-A5) -A d-v -- (pc 2) (AiA5-A2A3) -A dvg c S -2 6 - S (P c ) (A115-A 23) 5 dTgx (p 2 ) (A2A6-A3A5 dex pc pc [ 3 ] [ (2Jxa a) )]dt 2 x (s, + (PC )(2 6-AA 5 -A- (s,) 2dt 2s 1 + -. -2 PC 2 -2- -2 (-2 AA)] (pc ) (A16-A3) (pc )(A1 )+1 1 o (a -2aa ) x 6 +2 2s 2 3ex3T + 2 (PC)(A 1 -A 2 -A35 (s,6)]dt 2s 2-3 ( C 2 A jidA -aA 2 d O =p c aA2 -A34dvx- oc c)( 1 4-aA2)4 dv + F(pc2) L (A A4-aA2)-Aj dTx 1 4 2 4~ + ( c2a2--3 + (pc )LaA2A5-3 A4] dx -2 2 ( x c + (pc2)2 aA2AA-A 3J -x (sA) dt 2s 1 2 2 [(p-2) (A"- -2 (A 4 dv, ~ 6 La 2 + --2 [(PC (A4A6-aA)-AX d [(pc ) (A4 - pc [(2a ac7 ) (s,A) dt 2s (2.51) (2.52) (s,A) dt S-2r 9 _2 r'o7 (, 2ao ) ] + a(pc ) (PC ) 1(A.-A2A)-i I) dt 0 + 2(pc(p pc) (A1A 4-aA2)- 4I L- ,) dt (2.53) 2s These three equations each represent four equations, one equation in differential form along each of the four charaxeristic lines of equation (2.48). When the waves are coupled, equations (2.51), (2.52), and (2.53) are identical. That is, by multiplying equation (2.52) by the quantity (pC ) (aA25 3A4) 24 (pc )(A1A A2A3) A5 and using equation (2.48), equation (2.51) is obtained; or by multiply- ing equation (2.52) by the quantity -2 -2 pc (pc 2)(A a2) A -2 (pc )(A A A2A3) A5 and using equation (2.48), equation (2.53) is found. When the numerator and denominator of these multiplying quantities do not vanish, equations (2.51), (2.52), and (2.53) are identical. However, when the waves become uncoupled, a phenomenon discussed in Appendix B, A3 and A5 vanish. In this case the multiplying factors used above become undefined and the equations (2.51), (2.52), and (2.53) are not the same. When A =A =0 the equations (2.51) and (2.53) reduce to equations (B.4.4) and (B.4.6), respectively. Under these conditions, equation (2.52) also reduces to the form of equation (B.4.4). The equations along the characteristics of equation (2.47) may be obtained (since :nlonig Lhese characteristics there is no variation in x) directly from equations (2.25) and (2.35) by multiplying each of these equations by the increment of time, dt. These equations then yield aco S- dt = ap dv (2.54) r r 0 av r dt = aA 2dxa + aA4do + aA 5dTrx + a4 dt (2.55) The equations for the six characteristics and the equations along these characteristics, along with the appropriate initial conditions and boundary conditions, represent a complete mathematical formulation of the problem, and the solution to these equations is the solution to the problem posed here. The solution to these equations will be obtained by using a finite difference numerical technique which will be discussed in the next chapter. CHAPTER 3 DIMENSIONLESS EQUATIONS AND NUMERICAL PROCEDURES 3.1 Wave Speeds as a Function of the State of Stress In this chapter the numerical schemes used to find the solution to the wave propagation problem of Chapter 2 will be presented. In this first section the dependence of the wave speeds on the stress state will be shown. The stresses a T, and have already been assumed r r' rx negligibly small so that the scalar representation of the stress state is given by equation (A.1.2) as (2 9 9x9 2 2 - s = au T + a + 3T 2 (3.1.1) x x 6 9 6 Next, the new variables a ae and Tx will be defined so that the x 9 ex surface s = constant can be represented in terms of these variables as a sphere, and the stress state on this surface in terms of these new variables can be described in terms of spherical coordinates. Now defining, / 1 = 2( + aca) x 2x G a (ae ) (3.1.2) *'Bx- = /3 T x ex equation (3.1.1) can be written as 12 + 2 2T] (3.1.3) s = / + a/ + T (3.1.3) x 9 9xJ and defining the angles y and S6 as shown in Figure 3.1 these new variables defined in equation (3.1.2) Co-n Ibe written as x = s cos y cos 6 e, = s cos y sin 6 (3.1.4) x = s sin y The angle y is the complementary angle to the one normally used in spherical coordinates. It is used here to facilitate comparison of results obtained later on to already published results. From equations (3.1.2) and (3.1.4), 9 s cos v sin 6 / - S s cos v cos 6 x so that the cx and ae x 9 a -axis : 0 = ( + a(') 7 + ac tan = S(aa -a) a x_ 2 9 x axes are located by and tan 6 =- and 6 = -60 and tan 6 and 6 = +60. In order for the equations (3.1.4) to reduce automatically to the simpler case when radial inertia is not considered, the angle 6 is defined as 6 = a6' + (a-1)600 (3.1.7) so that when radial inertia effects are included, a=l, and 6= 6', and when radial inertia effects are not included, a= 0, and 6 = -60, which from equation (3.1.6) automatically causes (a to vanish as it should. e (3.1.5) (3.1.6) x ex Figure 3.1 Yield Surface Representation in Spherical Coordinates Using the uniaxial s;rs --strain curve in the form of equation (A.2.13), the universal str'ess-:strain curve can Ie r ..itten as (3.1.8) S- n S= B ) y and letting Et(s) be the tangent modulus of this curve, yields 1 dA 1 n-l S -+ Bn (s - Et(s) ds y (3.1.9) and from equation (A.2.18) this becomes 1 1 + O(s) Et(s) E () 1 1 E (s) (3.1.10) Now g = n(s) is defined so that where Et(s) = 1(s) E 0 I $(s) < 1 and when 5=1, the material is elastic, and when P=0, the material is perfectly plastic. Using equation (3.1.11) in equation (3.1.10), 0(s)can be written as Inverting equations (3 1 1 (s) = (- -- 1) . .1.2), the stresses a x x 3 / 1 , ao = 0 + -- Ca ao8 O Fx (3.1.12) re given by (3.1.13) 1 1 78x : (3.1.11) and using equation (3.1.4) th-se become i 1 -S CO '- jlC,; S c . 1 1 aoe = s cos y|cos 6 + -- sin T -- s sin y Now substituting equations (3.1.12) and (3.1.14) S(3.1.14) into equation (2.23), F 1 ) 2 2 1 21 I 1+ -1) (1)(cos Y) (2 cos 6 --- sin 6 -cos 6 --- sin 6) 1r 1 1 2 2 1 E -[ (- 1) )(cos y(2 cos --- sin 6 -cos 6 --- sin 6) (cos 6 -- sin 6-2 cos 6 -2 sin 6) 1 1 6 1 1 E -- sin v) (cos v)(2 cos 6 --- sin 6 -cos 6 --- sin 6) S3 I 1 2 2 2N = 1 + 14 1) (cos 6 -- sin 6 -2 cos --- sin 6) cos E1 i (1 1 1 2 ] E- (-1( sin y)(cos y)(cos 6 --- sin 6 -2 cos 6 --- sin 6) 1F 36 1 1 2 [2(+1 +-- (--1) ( sin Y) E 4 B3 1 14 11 2 2 2 S[1 + ( 1) (cos v)(cs6 2 sin 6 cos + 3 sin 6) 1 I 1 1 2 2 2 1- + ( -1)(cos 2y) (-cos2 6+3 sin 6) ) (--) (I -1) (sin y cos y) (cos 6 -,3 sin 6) E 2 J (3.1.15) 1 1 1 2 2 2 1 +(-+ 1) cos y(cos 6 + 2A/3cos 6 sin 6+ 3 sin 26) E v( 1) (sin y cos y)(-cos 6 -/3 sin 6) S2(1+v) + 3(1-1) sin y A3 A4 A5 A 6 or A1 2 A3 A4 A A5 A6 The elastic wave speeds arc defined from equations (B.5.4) as c /- o / c c1 = (3.1.16) (1- ) c2 ~ p and the wave speeds from equation (2.48) can be written in dimension- less form as c = -L b Vb 4aj a (G/p) c2 k2 ap L 4 C2 c b 4A4a 1 Lv j 24Aa] Ea or 12 lv F 2 3- C 2 =- E E2 b (E2b )2 4(EA )(E3 a) (3.1.17) 3- 4 E a By defining the dimensionless functions from equation (3.1.15) as Ai = E A i = 1,2,...,6 (3.1.18) 1 and 3- 2 2 2 a = Ea = A1A4A + 2aA2A3A5 -A3A4- aA1A aA2A6 S(3.1.19) b' = E = A1A aA2 + A4A- aA , 1 4 2 46 5 ./ then the wave speeds in dimensionless form become /2 l+v r; / i ~2 ~) / c \2 c -= Vb b 4A4a = (3.1.20) a 2 A computer program was written to solve this equation for the two positive wave speeds as functions of the angles y and 6 for specified values of v and 8. This pDrol- i- i lin ised in Appendix C. This pro- gram also calculated the viuecs of the normalizz:ed stresses a /s, 0 /s, and ex/s as functions of v and 6. Those results are given in Tables 1, 2, and 3. The wave speeds are shown in Figure 3.2 for the case when 6 =-60o, which corresponds to j = 0. Also plotted in Figure 3.2 are the results given by Clifton (1966). It is obvious that the results are not the same and that including radial inertia effects in the formulation of the problem can have significant effects on the wave speeds and that, for any given state of stress, the waves are always faster when radial inertia effects are included. The results plotted in Figure 3.2 do not correspond to the case when a= That is, although a = 0 when 6 =-60 a does not necessarily vanish for this case. Ahen a= 0, the results obtained were identical to those of Clifton (1966) as they should be, since a= 0 corresponds to the absence of radial inertia effects. An interesting phenomenon can be observed by remembering that the physical presence of radial inertia is due to the Poisson effect. That is, the longitudinal (fast) wave speed would be expected to be the same when a=0 (no radial inertia effects) as when v=0 (the cause of the radial inertia effects vanishes). However, in the fomulation of this problem it is tacitly assumed that Poisson's ratio for the 1 inelastic portion of the material behavior is -, or that the material behavior in the inelastic range is incompressible. When the material is elastic ( = 1), this Poisson effect can be studied directly. Comparing equations (3.1.15) with (B.5.1) when 1 = 1, the wave speeds are given by equation (B.5.4) as \ TABLE 1 NORMALtZLD ,LONGI'L:.UD.NAL STRESS (--) s Gamma Delta 00 100 200 300 400 -900 0.57735 0.56858 0.54253 0.50000 0.44228 -80 0.74223 0.73095 0.69747 0.64279 0.56858 -70 0.88455 0.87111 0.83121 0.76605 0.67761 -600 1.00000 0.98481 0.93969 0.86603 0.76604 -500 1.08506 1.06858 1.01963 0.93969 0.83121 -40 1.13716 1.11988 1.06858 0.98481 0.87111 -30 1.15470 1.13716 1.08506 1.00000 0.88455 -200 1.13716 1.11988 1.06858 0.98481 0.87111 -100 1.08506 1.06858 1.01963 0.93969 0.83121 00 1.00000 0.98481 0.93969 0.86602 0.76604 100 0.88455 0.87111 0.83121 0.76604 0.67761 200 0.74223 0.73095 0.69746 0.64279 0.56858 300 0.57735 0.56858 0.54253 0.50000 0.44228 400 0.39493 0.38893 0.37111 0.34202 0.30253 500 0.20051 0.19746 0.18842 0.17365 0.15360 600 0.00000 0.00000 0.00000 0.00000 0.00000 700 -0.20051 -0.19747 -0.18842 -0.17365 -0.15360 800 -0.39493 -0.38893 -0.37111 -0.34202 -0.30254 -0.56858 -0.54253 900 -0.57735 -0.50000 -0.44228 TABLE 1 (Continued) Gamma Delta 500 600 700 800 900 -900 -800 -70 -600 -500 -400 -300 -200 -100 00 100 200 300 400 500 600 70 800 0.37111 0.47710 0.56858 0.64279 0.69747 0.73095 0.74223 0.73095 0.69747 0.64279 0.56858 0.47709 0.37111 0.25386 0.12889 0.00000 -0.12889 -0.25386 0.28868 0.37111 0.44228 0.50000 0.54253 0.56858 0.57735 0.56858 0.54253 0.50000 0.44228 0.37111 0.28867 0.19747 0.10026 0.00000 -0.10026 -0.19747 0.19747 0.25386 0.30253 0.34202 0.37111 0.38893 0.39493 0.38893 0.37111 0.34202 0.30253 0.25386 0.19747 0.13507 0.06858 0.00000 -0.06858 -0.13507 0.10026 0.12889 0.15360 0.17365 0.18842 0.19747 0.20051 0.19747 0.18842 0.17365 0.15360 0.12889 0.10026 0.06858 0.03482 0.00000 -0.03482 -0.06858 0.00000 0:00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -0.28868 -0.19747 900 -0.37111 -0.10026 0.00000 sa, TABLE, 2 NOWRALIZFD ITOOIP S TiESS S Gamma Delta 00 100 200 300 400 -900 -800 -700 -600 -50c -400 -300 -200 -100 00 100 200 300 40 500 60" 700 800 900 -0.57735 -0.39493 -0.20051 0.00000 0.20051 0.39493 0.57735 0.74223 0.88455 1.00000 1.08506 1.13716 1.15470 1.13716 1.08506 1.00000 0.88455 0.74223 0.57735 -0.56858 -0.38893 -0.19746 0.00000 0.19747 0.38893 0.56858 0.73095 0.87111 0.98481 1.06858 1.11988 1.13716 1.11988 1.06858 0.98481 0.87111 0.73095 0.56858 -0.54253 -0.37111 -0.18842 0.00000 0.18842 0.37111 0.54253 0.69747 0.83121 0.93969 1.01963 1.06858 1.08506 1.06858 1.01963 0.93969 0.83121 0.69746 0.54253 -0.50000 -0.34202 -0.17365 0.00000 0.17365 0.34202 0.50000 0.64279 0.76605 0.86603 0.93969 0.98481 1.00000 0.98481 0.93969 0.86602 0.76604 0.64279 0.50000 -0.44227 -0.30253 -0.15360 0.00000 0.15360 0.30254 0.44228 0.56858 0.67761 0.76604 0.83121 0.87111 0.88455 0.87111 0.83121 0.76604 0.67761 0.56858 0.44228 TABLE 2 (Continued) Gamma Delta 500 600 700 800 900 -900 -800 -700 -600 -500 -400 -300 -200 -100 00 100 200 300 400 500 600 700 800 900 -0.37111 -0.25386 -0.12889 0.00000 0.12889 0.25386 0.37111 0.47710 0.56858 0.64279 0.69747 0.73095 0.74223 0.73095 0.69747 0.64279 0.56858 0.47709 0.37111 -0.28867 -0.19746 -0.10026 0.00000 0.10026 0.19747 0.28868 0.37111 0.44228 0.50000 0.54253 0.56858 0.57735 0.56858 0.54253 0.50000 0.44228 0.37111. 0.28867 -0.19747 -0.13507 -0.06858 0.00000 0.06858 0.13507 0.19747 0.25386 0.30253 0.34202 0.37111 0.38893 0.39493 0.38893 0.37111 0.34202 0.30253 0.25386 0.19747 -0.10026 -0.06858 -0.03482 0.00000 0.03482 0.06858 0.10026 0.12889 0.15360 0.17565 0.18842 0.19747 0.20051 0.19747 0.18842 0.17365 0.15360 0.12889 0.10026 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 /T TABLE 3 NORLALIZED SHEAR STRESS \ T x Gamma Value of x- Gamma Value of Tx s s 00 0.0 500 0.44228 100 0.10026 600 0.50000 200 0.19747 700 0.54253 300 0.28868 800 0.56858 400 0.37111 900 0.57735 _- E c = f2 p (1 av2) S= E - Cs p 2p (1 +) -2 It is now obvious that the fast wave speed is the same when a=0 and v 0 as when a= 1 and v= O. The slow wave speed (and consequently c2) is the same when a=1 as when a=0, although it does depend on v. Because of this dependence of c2 on v, the dimensionless fast wave speed of equation (3.1.20) will have values when a=0 and v O differ- ent from those when a=1 and v =0. Also in Figure 3.2 it can be seen that when a=0 and y=0 the fast and slow wave speeds are the same for = .385 and = .30. There is usually some value of $ for which the two wave speeds are equal at y=0 for each combination of values of v and 6. The condition for which this is true can be obtained from equation (3.1.20) and is ---Without Radial Inertia Clifton (1I66) -- With Radial Inertia and ca = (6 = -600); 6 0 S= .01 y, Degrees Figure 3.2 Plastic Wave Speeds as Functions of $ and y for Poisson's Ratio of 0.30 0, 8 ti'hen =- 0 A1 2 A A3 A4 A = 5 A6 6 b/2 4a A = 0 . equations (3.1.18) and (3.1.19) are 1 + -1(--1)(cos 6 -2v sin 6 cos 6 + 3 sin26) 43 -v + (--1)(cos 6-3 sin25) 0 1 1 2 2 1 (- -1)(cos 6 +23 sin 6 cos 6+ 3 sin 6) 1+45- (3.1.21) (3.1.22) 0 2( :1 + ) 2 2 a = AIAA aAA6 b/ = A1A + AA aA2 and using the same manipulations as in Appendix B, Section 4, equation (3.1.21) becomes 0 = A(A -A ) + aA2 where Al, A2, A4, and A6 are given by equation (3.1.22). Now fl = () = f cos26 2 3 sin 6 cos 6 + 3 sin2 ] f2 =f2(6) = cos26 + 2v sin 6 cos 6 + 3 sin26 f= f(6) = 1 cos26 -3 sin26] 1 z = z( ) 1 c c c (3.1.23) (3.1.24) defining (3.1.25) the critical value of $ is found from 0= L L c f L 2(1-,.,)- (I + zcl) I+ a -(v cf3 0,= z -fl 2+af3 + z ([2(1+l ) -13 f- + a2vf3 + [2(1 +v) -1+ av (3.1.26) The expressions for the critical value of 3 will now be found for the two separate cases of a= and a=1. First, when a= 0, equation (3.1.24) becomes A6 = A 2(1+v) 1+ z f c 1 1 + 2v (3.1.27) z -- c f 1 c z +1 C j and in this case 6= -60" from equation (3.1.7) and fl = cos2(-60) -273 sin (-600) cos (-600) +3 sin2(-600) f -24( )( ) + 3( ) 1 4 4 2 2 2 f =1 and 1 c (1+ 2v) +1 1 S2 ) (3.1.28) c 2(1+ v) When a = 1, 2 1 9 fl 2 af3 6 (cos" -2 ,':3sin cos c + 3 sin- ) 22 2 2 21] (cos52 + 2/3 sin cos 6+3 sin6) -(3 sin6 -cos2)2 ff2 af = 0 1 2 3 and equation (3.1.26) becomes -[2(l +v) -1+ V c [(2)(1 +) -1] f, -f +2vf (1+v) z (3.1.29) S 3 (1 +v) sin 6 cos + 3v sin26 A short computer program was written to calculate the critical values of (using equation (3.1.28) when a=0 and equations (3.1.29) and (3.1.27) when a= 1) for various values of v and 6 when y= 0. This program is shown in Appendix C and the results are plotted in Figure 3.3. The only values of 3 which are physically possible are between 0 and 1 and therefore only values of $ in this range are plotted in Figure 3.3. For all other values of 6, there is no phys- ically possible critical value of S; that is, there is no value of such that the fast and slow wave speeds are equal at y= 0. For the case when a = 0 (6 =-60), for any value of v the critical value of $ is smaller when radial inertia effects are included. v = 0, a 0 v = .25, =.30, a Figure 3.3 Values of 3 at y=00 for which cf cs = c s 2 .6 .4 _ 0 -30 -60 -90 6, Degrees 3.2 Characterirsi.c Solution in Terms ol' uL'i ; loni:iolon ?le s C:: vria bles In order to make the numerical solution in the characteristic plane more general, the equations for the characteristics and the equations along the characteristics given in Chapter 2 will be written in terms of dimensionless variables. The dimensionless variables used are c x 1 s X T = t s = - 2r 2r E 0 0 x ex S S T x E E E V VV V v v v x e r V v =- V =- x 9 r - Sc c c -f s c c c (3.2.1) s c c c r C1 ,(Sf) 2o 0 (S Y(s,A) = -0---- s where c1 given by equation (3.1.16) is the elastic longitudinal wave speed when radial inertia effects are included. In terms of these variables, the functions defined by equations (3.1.18) can be written from equations (2.23) as (2S aS )2 x 9 A1 EAI = 1 9(st) A1 = E2 1 2 O(s,6) 4s (2S aS(S 2aS) A2 = EA2 L + 4s (s, 4s 6T(2S aS ) A3 = E 3 2 2(s) 4s (S 2aS )2 A4 = EA4 2 -(s,) + 4s A = EA = 5 5. A6 = EA6 = 6 6 6T(S 2aS ) 4x 2 9T2 2(1+ v) + 2 @(s,A). s Using these and equations (3.1.19), the fast and slow wave speeds can be written from equations (2.49) and (2.50) as cl c s s ( c1 S {b + (b2 4aA4 (1 -2 2a p = [E { b (b'2 4a'A4 (1 2a p /2s f = -- b' + (b'2 4aAA4)2 2a es b' (b2 4a'A4 L 2a' (3.2.3) and the wave speeds in dimensionless form are given by equations (3.2.3) and (2.47) and (2.48) as (3.2.2) 48 c = cf c = c (3.2.4) c = 0 (twice) where the wave speeds were written in dimensionless form by dividing the wave speeds by cl. This was done because cl is the fastest wave speed possible in the problem considered here, and thus all the dimensionless wae speeds have values in the range -1 c < 1 . ,When radial inertia effects are not included, the maximum value of the fast wave speed is C - Cf p= max so that, for a=0, the maximum value of the dimensionless wave speed is Cf I cfI < max E/ 1 E p (1 -v2) Next, the equations along the characteristics for fully coupled waves will be written in dimensionless form. Along the vertical char- acteristics (c=0), the equations can be written from equations (2.33), (2.54) and (2.55) as -aS dT = a L 1 dV 2(1 -o2) or a [2(1 -v2)S] dT = adV (3.2.5) .I2aS *S a a(2V dT) = aA2dS -A!dS +AArC + -- (s,. j r L 2 x 4 + 5 \ s 0 o (3.2.6) The equations along the nonvertical characteristics (c= cf, cs) can be written in three different forms from equations (2.51), (2.52) and (2.53). In dimensionless form these become, respectively, O = I ---(A4A6 aA5) -j dx A2A5 -A3A dV 1 -v -v 2 2 1- c 2 S[aA2A5 -A3A4] dT + 2 ( 2(A4A6 5 -A4dS c 1 -v [(2 ) r92S aS] + C -2-(A4A6 _A2) -< s "0 o0(s,)dT K 2 \ n r -S 2aS + a L( 22(A A(3 -A A ) A 2V+ 2+ (A2 A6-A -A3A5) 2J L2Vr s( )dT 22 ] L+ 6 V-2 2 L _A o(s')Jd (3.2.7) 2 S= c 2 (A A -AA5 )-A dV - c 1 \ + -( 2,) c <2A c 1 -v + (--. (A2A A3A5) L \1 -v 2A r- 2 L c (A5-A2A )-A dV c L5 2-v 5 - A5 dT - A3A5) A dSx 2Sx aS)dT -A2] Lx o(s dT -i 2 "0 (1- 2 r( e2 \2 3 2 + ( ) [ 2) (A1A6 --) (A1 +A 1 c 1 -v 1 -V and F S 2aS - 2V + ( (sA) dT r s 0o + 6 c 2 (A1A5 A2A3) (s,A)]dT (3.2.8) and 3 2 0 c 2 aA2A A3A4 d c 2) (A1A4-aA2) -A dV (1 -v 2) 1 -v -v + c2 (A1A4 -aA 2)-A4 dT+ L_ 2A5-A3A 4 dSx 1 -v 1 -V 2 2 r -2S a + ) LaA2A-A3A4 L s o(s, l dT 1 -i- + a )L-- (AlA5-A 2A3)-A 2Vr x s(s'A)j dT + 6 2) (A- aA ] )-AA tsi AJdT (3.2.9) 1-v 1 -v WVhen the equations are uncoupled (A3 = A 0), the equations along the characteristics are given by equations (B.4.7) and (B.4.8). In dimen- sionless form, the equations (B.4.8) along the vertical characteristics (c=0) become a[2(a -v )S ] dT = adV (3.2.10) er and 2aS S a[2V dT] = a[AdS + A4dS + -- (s,)dT] (3.2.11) The equations (B.4.7) along the nonvertical characteristics (c= c ) for the fast waves are given by 1 1 -2 2S aS _0 1 dVd + 2- d + ---- dTA4 f c r S 2aS + 2 L2Vr + (- ) o dT (3.2.12) and the equations (B.4.7) along the nonvertical characteristics (c= c ) for the slow waves can be written as s 2 c 6c2 0 = s dV + dT + () o(s,T)d (3.2.13) L(1-v2) u (l-v2) The equations (3.2.5), (3.2.6), and (3.2.7), or (3.2.8), or (3.2.9) are the equations along the characteristics for the fully coupled waves written in differential form in terms of the dimension- less variables. The solution to this set of equations will be obtained numerically by writing them in finite difference form, and then solving the resulting set of algebraic equations simultaneously. When the waves are uncoupled, the equations along the characteristics are given by equations (3.2.10), (3.2.11), (3.2.12), and (3.2.13). These equa- tions will also be written in finite difference form and solved (when applicable) in the same manner as described for fully coupled waves. The procedure for obtaining these finite difference solutions is out- lined in the next sections. 3.3 Numerical Grid for Characteristic Solution Since the slopes of the characteristic lines at any point in the characteristic plane depend upon the state of stress at that point and upon the history of the deformation at the corresponding location along the axis of the tube, the equations for the characteristic lines cannot be determined before the solution (in terms of stresses) is known. Because of this, the slope of the characteristic lines and the solution to the problem must be determined at each point simultaneously. This is done by using the iterative numerical technique described below. The numerical grid shown in Figure 3.4 will be used. There are two types of elements in this grid: boundary elements and regular elements. All of the regular elements are alike, and all the boundary elements are like the right-hand side of a regular element. A detailed picture of a single regular element is shown in Figure 3.5, and a single boundary element is shown in Figure 3.6. The grid is defined in terms of the dimensionless variables given in equations (3.2.1) and (3.2.3). It is diamond shaped with the straight outer lines corre- sponding to the characteristic lines for elastic longitudinal waves with radial inertia effects included. These outer characteristic lines have slopes of either c= +1 or c= -1, which can be seen from equations (3.2.1) and (B.5.4). The vertical straight line corresponds to the two vertical characteristic lines, and the straight inner nonvertical lines correspond to the characteristic lines (through the point P) for the elastic shear waves. For both types of elements, the problem reduces to that of determining the values of the stresses and velocities at the point P, when their values at the points B, R, and L are known. This grid with all the elements constant in size simplifies the writing of the finite difference equations. The diamond shape allows the vertical characteristic lines to automatically connect point P (at which the solution is desired) with point B (at which the solution is known) and makes the finite difference equations along the vertical 53 T AFigX -3.4 Nu l Gd in te Chc Figure 3.4 Numerical Grid in the Characteristic Plane 2AT 2AT 77 Ic = -c = -- -- AI- 1c B -a Figure 3.5 Regular Element in Numerical Grid op 55 T P c = -c = - C 1 I c- c= c =1 B ---- AX ----- k LX Figure 3.6 Boundary Element in Numerical Grid characteristic lines very easy to obtain. The boundary lines for each element are c-= This is the smallest value of c which insures that all characteristic lines passing through tne point P will intersect the line L-B between the points L and D if the lines have a positive slope at P or will intersect the line R-B between the points R and B if they have a negative slope at P. This is true since all of the waves considered here will propagate with a speed less than or equal to the speed of an elastic longitudinal wave with radial inertia effects included. A larger value of c could be used, but the element size would increase (for a given distance along the T axis), and the solution would be inherently less accurate. The straight lines representing the elastic shear wave character- jstic lines are added to the grid elements as a convenience. The results of Section 3.1 show that the fast wave speed always occurs in the range c2 : cf 1 c1 and the slow wave speed always occurs in the range 0 O c s c2 s 2 Therefore, these characteristic lines c=c2 divide each element so that a characteristic line through P lies in one of the upper triangles (P -L -LB or P -R-RB) if it is for a fast wave and in one of the lower triangles (P -B-LB or P -B-RB) if it is for a slow wave. This is shown for the regular grid elements in Figure 3.7. These characteristic lines for the fast and slow waves will not, in general, be straight. P e ---/I \ \=\- c = c2 / \ -0C 2 / c c = -cC c = 0 B -------------------------------------------------------------------------------------------------- ------- ^- x Figure 3.7 Location of the Characteristic Lines Passing Through P 3.4 Finite Difference Equations General Discussion 1hile the actual characteristic lines for the fast and slow waves are seldom straight, they can be represented as straight lines within each grid element without introducing significant errors if the grid elements are small. From the discussion in Section 3.3, it is known that the slope of the characteristic lines at any point cannot be determined before the solution at that point is known. Because of this the solution at the point P (Figures 3.5, 3.6, and 3.7) must be obtained by an iterative technique. Within any grid element, the slope of each characteristic line will be constant during each iteration although the slope of each characteristic line will change from one iteration to the next as the solution at P is approached. These straight lines are used to represent the characteristic lines for th c= c and c = c during each iteration and are shown for the i iter- f s ation as c=cfi and c= cs in Figure 3.8 for a regular grid element and in Figure 3.9 for a boundary grid element. The points LLB, LBB, RBB, and RRB are the intersections of the lines shown in Figure 3.8. Each element has its own coordinate system X -T which is also shown in Figures 3.8 and 3.9, and the finite difference equations are written in terms of this local coordinate system so that the finite difference equations for each element are the same. First order finite difference equations will be written along each characteristic line. The coefficients of the dependent variables in these equations will in general be functions of the stresses and (s,A). Thus, in order to linearize the equations, the coefficients for each T 2 bx Figure 3.8 Numerical Representation of the Characteristic Lines in a Regular Grid Element 60 I ST P c = -C1 C- i= C = -cf R RRBI c=-c )\ s. \ 1 AT RBB c=0 RBB B ^- ---- ^X ---- = X Figure 3.9 Numerical Representation of the Characteristic Lines in a Boundary Grid Element iteration will be calculated using the solution obtained in the previous iteration. In this way the coefficients are :llays known quantities. One other scheme will be used with the coefficients in the finite difference equations in order to reduce the time required for compu- tation. Normally each coefficient used is the average value of that coefficient at the end points of the interval over which the finite difference equations are written. As an example, consider the charac- teristic line from point LLB to point P, and let one term in the finite difference equation along this characteristic line be U(S S ). xP xLLB As a rule the value of the coefficient is calculated as 1 U = (Up + ULLB) If this method is used, the coefficients of each variable in the equa- tions along the characteristics of positive slope will be different from the coefficients of the corresponding variables in the equations along the characteristics of negative slope. For instance, one term in the equation along c=+cf. can be represented as S(SxP xLLB and the corresponding term in the equation along c=:-cfi as UR(Sx SxRRB where 1 U = (U + ULLB) S= + R 2 p RRB and U is the value of the coefficient at point P calculated from the solution from the previous iteration. When the coefficients are calculated in this manner, the number of equations which must be solved simultaneously cannot be conveniently reduced below five. However, if the coefficients are calculated in such a way that the coefficient of any variable in the equation along c=+cfi is equal to the coefficient of that same variable in the equation along c=-cfi (so that UL = UR, etc.), then by adding these two equations and subtracting one from the other, two different equa- tions can be obtained, each with fewer variables than the two original equations. If this procedure is applied to the equations along c=Csi, then the set of five simultaneous equations can be reduced to at most a set of three simultaneous equations and a set of two simultaneous equations. This is shown in Appendix D. Since this set (or sets) of equations must be solved during each iteration, the savings in compu- tation time is significant. One way to make the coefficients of similar terms equal is to calculate the coefficients from values of the variables obtained at point P during the previous iteration such as UL = R = U * The coefficients will be calculated in a somewhat more accurate manner by using a weighted average of the value of each coefficient between point P and point B, that is UL = UR = alUP + (1 )UB This gives the value of the coefficients at a point nearer the center of each grid element. For this work, the value of ai is chosen arbi- trarily as .625, so that the point at which the coefficients are calcu- lated is at approximately the same location along the T'-axis (Figures 3.8 and 3.9) as the centers of the four characteristic lines c= cfi, Csi. The values of all quantities at the points LB, LLB, and LBB will be obtained by linear interpolation between the points L and B. Similarly, the values of all quantities at the points RB, RRB, and RBB will be obtained by linear interpolation between the points R and B. From Figure 3.8, the times T1, T2, and T3 can be written as 2c 2c 2c s 2 f T, = AT T c AT T +c- AT 1 1 +c 2 1 + c 3 1 +c s 2 f and the interpolation constants for the points LB and RB are 2 2c2 CLRB - AT 1+ c2 T2 1 2 CLRBI = 1 AT 1 + c2 Using subscripts to denote the grid point, the values of any quantity F at the points LB and RB are FLB = CLRB*FL + CLRBI*FB LB L B FRB = CLRB*FR + CLRBI*FB RB R B The interpolation constants for the points LLB, LBB, RBB, and RRB are T -T 2(c c) CON1 - AT T2 (1 + cf)( c2) T1 Cs ( + c 2) CON2 = -- s T c2(1 + c ) 2 2 s CON3 = 1 CON1 CON4 = 1 CON2 so that the value of F at each of these points is FLLB = CON1*FL + CON3*FLB FBB = CON2FLB + CON4*FB FRRB = CON1*R + CON3*FRB FRBB = CON2*F + CON4*FB RBB RB B For Fully Coupled Waves When the equations are fully coupled, the equations along the nonvertical characteristics (c=c cs) are given by either equation (3.2.7) (3.2.8), or (3.2.9). Equation (3.2.7) will be used, and the values of AV, A2, A3, A4, A5, and A6 of equation (3.2.2) will th be calculated for the i iteration as described for U earlier in this section and defined as Ali, A2i, A3i A4i A5i, and A6i, respectively. th The coefficients will then be defined for the i iteration as 2 IR c f (A .A If \ 2/ 4i 6i 1 v 2 C R2f = 2 (AA 6i 1 v 2 c / s R is 2 (A 4iA6i 1 v 2 R2s 2) (-2i 6i 1 v 2 - aA.) - 5i - A3iAi) A2i 3i 5i 2i 2 - aA ) - 5i - Ai ) A2i *3i 5i 2i Rfs= aA iA i- Ai A fs 2i 5i 3i 4i Thus, the finite difference equations when the waves are fully coupled can be written directly from equations (3.2.5), (3.2.6) and (3.2.7), and using the last subscript to represent the point in the numerical grids of Figures 3.8 and 3.9 the equation along c = 0 is 2 [SP SB AT = a(V -2a(l v 2AT = a(V2 L 2 rP rB- ) (3.4.2) along c = 0 is rP rB A + 2B\ 2a VB (2LT) = a 2P )(S S ) 2 1 2 ) xP xB A4P + A4B (S + ( 2 (SeP S()A5P + A5B)( B - S ) + (A( T ) OB \ 2 P B 1 P- xp + B SxB + 2 -- + 2aS (2LT)] SoP/ s P (3.4.3) (3.4.1) along c = + c is -Rf 0 f V Cf xP + R f(T p fs P c Rfs xLLB 2 VP 1 -v - V )LLB eLLB (1 2)Rf ) + (S S ) LLB 2 xP xLLB Cf + R [lf1 (2Sxp - aS + 2SLL GP xLLB - aS ) 2AT 1 6LLB 1 + cf_ + FV +V + '1 + (Sxp 2aS +S 2aS 2irP rLLB 1 2 xP xLLB GLLB J [2 T f * T L1 + c j 6c Rfs1 l1 2 2 (Tp + P LLB 1 v 2[rAT /+f (3.4.4) along c = cf is Rlf 0 = (V C xP f c R f fs - V ) + (V V ) xRRB 2 GP RRB 1 v + R (TP RR) + fs P RRB + R f 1i (1 v2)Rf 2 R if(S S ) 2 xP xRRB f (2Sxp aS p + 2SRRB xP 6P xRRB - aS )L cfJ 6RRB 1 + c + aR2Vr +VrRRB + (SP -2aSgp +SxRB aSRR) 2 S2T 1 R fs 1 R 2T- (3.4.5) 26T* L (T (3.4.5) 1+ c 2 P RRB c L1 -C 67 along c = + c is s 1s s -fs c xP xLBB 2 eP 6LBB s l-v (1 V2)R s + Rs 1(2Sxp aS + BB (aS BB+ + aR V + V + -(S -2aS +S -2aSL) s rP rLBB 2 P P xLBB LBB 6c 2R 2 2T 6csRf 1 T ) j (3.4.6) + c 2 L2 TP + LBB Ll+ (3 s 1 s along c = c is s R cR Is s fs 0 =- (V -V ) + (V -V ) c xP xRBB 2 GP eRBB s 1-v (1 v2)R + (T TRBB) + 2 (S SRBB) fs P RBB 2 xP xRBB C s + R11 [+(2Sx -aS +2S -aS )] [2 1S 1 xP GP xRBB GRBB 1 +c s + aR2s [Vrp VrRBB + 1{(Sx -2aS p+ SxRBB -2aSRBB)} c 2 R +C + (T + T ) 1 (3.4.7) L +c2 2 [ (TP + RBB 1 +c -RB J S 1 v s where the values of *P / and s / obtained at point P from the previous iteration are used and oP roB S= al --+ (1 -a) -s (3.4.8) sp/ B with al defined earlier in this section. For Uncoupled Waves Then the waves become uncoupled as described ir Section B. l, the equations along the characteristic lines have a simpler form given by equations (3.2.10) to (3.2.13). Using the averaging technique already described in this section for the coefficients, the equations along the characteristic lines for uncoupled waves simplify. The equation along c = 0 is Sp + SB 2 QP GB -2a(1 -v )( )(2AT) = a(Vp V ) (3.4.9) 2 rP rB along c = 0 is [(A +A A +A a(V + V )(2T) = a (2 2B)(S S) + 4( )(S -s rP rB 2 xP xB 2 OP GB S(2aS -S )* o (2aS -S )P oB + a op + --9B- B (2LT) (3.4.10) 2L Sp, s B along c = + c is 2 S(V -V ) + (S -S S= x LLB 2 xP xLLB f c 1 2,T + 1 {(2S aS + 2SLLB aS ALLB ] i frP r P xLLB GLLB 1 C B 4i + aA V+V + l(S -2aS +iS -2aS ) 2/T + VrPVrLLB+ P G xlLB LLB I (3.4.11) along c = c is 0 = A (vP-x ) + 12 (Sx -S 4 c xP xRRB 2 xP xRRB f Cf + 1{ (2SxP -aS e+2SxRRB-2aS eRB)} -i + aA v +V +'{(S 2aS SX 2aS, 1 26T. 2iL rP YrRRB 1 (SxP-2aS PxRRB- SRRB L (3.4.12) along c = + c is s c C 0= (V p V LBB) + (T -T LBB 2 P LLBB P LBB S- .(Tp + TLBB) 2 A (3.4.13) 2 L2P l-v s along c = c is s c s 0 (V e- V ) + (T T ) -2 eP eRBB P RBB 2 + 22 (P + RBB) (3.4+ 1-v s 3.5 Solution to the Finite Difference Equations The solution to the finite difference equations of Section 3.4 are given here for any iteration. The solutions consist of expressions for VxP V p, VrP, Sx, S and Tp in terms of known quantities, including quantities calculated during a previous iteration. The solutions given in this section are obtained using Cramer's rule as shown in Appendix D, and the definitions of the variables used in Appendix D will not be repeated here. At a Regular Grid Point for Fully Coupled Waves The solution to the finite difference equations along the charac- teristic lines at a regular grid point in the case of fully coupled waves is given here. The longitudinal and transverse velocities from equations (D.3.5) and (D.3.6) are 1 V = (Ds RHSBA D2 RHSDC) (3.5.1) 1 V --(Df RHSDC Ds RHSBA) (3.5.2) When radial inertia effects are included, the stresses at point P are given by equations (D.3.13), (D.3.14), and (D.3.15). These stresses are T = RHSF(D2 D1D7s) + RHSG(D1D D2D4f) + RHSH(D4fD7s D4sD7f)] (3.5.3) 1 r S RHSF(A D DD ) + RHSG(D2D AD ) xP 2L 5Q 7s 23s 2 3f 5Q 7f + RHSH(D3sDf D3fD7s) (3.5.4) S HSF(D1D A D4s) + RHSG(A5QD D1D) P =2 53s 5Q 4s Q 4f 3f + RHSH(D3fD4s D4fD3s) (3.5.5) and the radial velocity of equation (D.1.3) is aVrp = a(D3 Q3S p). (3.5.6) When radial inertia effects are not included, the hoop stress, S p, and the radial velocity, VrP, automatically vanish, and the shear veocty V, 71 stress and the longitudinal stress are given by equations (D.3.18) and (D.3.19), respectively, as S= (D RHSF D4f RHSG) (3.5.7) P 4s 4f 1 S = (D RHSG D RHSF). (3.5.8) xP 3 3f 3s 3 At a Regular Grid Point for Uncoupled Waves When the waves are uncoupled, the solution to the finite differ- ence equations has a much simpler form. In this case, the shear stress, the transverse velocity, and the longitudinal velocity of equations (D.4.1), (D.4.2), and (D.4.3), respectively, are T = (RHSCE + RHSDE) (3.5.9) P 2F2 2s 1 V = f (RHSBEM RHSAEM) .(3.5.11) xP 2F When radial inertia effects are included, the longitudinal and hoop stresses from equations (D.4.8) and (D .4.9) are 1 SP = (D2 RHS3 F5f RHSEEM) (3.5.12) 5 1 S = (F RHSEEM D RHS3) (3.5.13) OP 6 2f2 1 5 and the radial velocity is again given by equation (3.5.6). When radial inertia effects are not considered, both the hoop stress and radial velocity vanish, and the longitudinal stress of equation (D.4.10) is RHS3 S S (3.5.14) xP F22 2f2 At a Boundary Point (X=0) for Fully Coupled Waves In a boundary element, there are only four characteristic lines (cO, c= O, c =-c c= -c ) and consequently only four equations along these characteristic lines. Since the equations along the character- istic lines are written in terms of six unknown variables at point P, the solution at a boundary point can be obtained only if two of these variables are prescribed at each boundary point. The hoop stress and the radial velocity do not enter the formulation of the problem when radial inertia effects are omitted, and therefore these variables are not specified at the boundary. Thus, the four remaining variables, two of which may be specified at any boundary point, are the longitudinal stress, the longitudinal velocity, the shear stress, and the transverse velocity. From a purely physical standpoint, it is also reasonable to specify the longitudinal and transverse variables at the boundary since these are the quantities which are normally associated with the impact at the end of the tube and which can be measured more readily than radial velocity and hoop stress. Only two of the four variables Sx, Vx, T and V can be specified at any one boundary point. Furthermore, at a given boundary point V and S cannot both be specified since they x x are not independent. Also, both T and V6 cannot be given at the same boundary point. Therefore, four combinations of variables to be speci- fied on the boundary will be considered: for Case I, S and T will be x given at the boundary, for Case II, Vx and V will be given, for Case III, S and Ve will be given, and for Case IV, Vx and T will be given. The solution to the finite difference equations at a boundary point for each of these four cases when the waves are fully coupled is given below. Case I: Traction boundary conditions ihen Sp amd are known, then from equations (D.5.1), (D.5.5) and (D.5.6) the solution to the finite difference equations at P is a Sp D (RHSH DS A5QT p) (3.5.15) Vp = (B2s RHS1 Bf RHS2) (3.5.16) 4 V = 1 (B RHS2 B RHS1) (3.5.17) OP A if is and Vrp is given by (3.5.6). Case II: Kinematic boundary conditions When Vxp and Vp are given, the solution at P is given by equations (D.5.11), (D.5.12), and (D.5.13) when radial inertia effects are included as T RHS4(D42 D2 + B7s D1 RHS5(D D2+BfD + RHSH(D4s2B7f D4f2B7)] (3.5.18) Sx [RHS4(D3s D + AD) + RS5(D D + B A) xP L6 3s2 2 5Q 7s 3f2 2 7f5Q -RHSH(D3s2Bf D3f2B7)] (3.5.19) S RHS4(D D A D ) RHS5(D D A D P = RS 3s2 1 A5Q4s2 3f2D1 -5Q4f2 + RHSH(D3f2D4 D4f2D3s2)] (3.5.20) and Vrp is given by equation (3.5.6). When radial inertia effects are not included, Vrp and Sp are zero and the solution given by equations (D.5.16) and (D.5.17) is T = A- (D sRHS4 D RHS5) (3.5.21) P 4 4s2 4f2 7 1 Sxp = (D3 RHS5 D s2RHS) (3.5.22) xP A 3f2 3s2 Case III: Mixed boundary conditions When Sxp and Vp are known, the solution when radial inertia effects are included is given by equations (D.5.22), (D.5.23), and (D.5.24) as S RHS6(D D + AB) RS7(D D + A5Q B 7f) xP L 3s2 2 5Q7s 3f22 5Q7f 8 + RHS8(D3s2B7f D3f2B7s) (3.5.23) Tp F- (D2RHS7 + B7sRHS8) B s(D2RHS6 + B RHS8) (3.5.24) P A Li~f 8 2 2 S 1 Bf(D3s2RHS8 -A5QRHS7) -B (D3f2RHS8-A5RHS6)] (3.5.25) and Vrp again is found from equation (3.5.6). When radial inertia effects are not included, VrP= S= 0 and from equations (D.5.27) and (D.5.28), the solution at P becomes V = (D 3s2RHS6 D 3f2RHS7) (3.5.26) xP A 3s2 3f2 p = 1 (B RHS7 B RHS6) (3.5.27) 9 if Is 9 Case IV: Mixed boundary conditions When Vp and T are known at the boundary, the solution at P is found from equations (D.5.33), (D.5.34), and (D.5.35) when radial inertia effects are included to be i F LP 10 RHS9(Ds2D + D1B7s) RHS10(D4f2D2 + D1B7f) -7 + RHSll(D4s2B7s D4f2B7s)J (3.5.28) SxP 0 B2f(D2RHS10 + B7sRHS11) B2s(D2RHS9 + BsRHS11) (3.5.29) Sp = LB2f(D4s2RHS11 -D1RHS10) B2s(D4f2RHS11 -D1RHS9) (3.5.30) where again Vrp is given by equation (3.5.6). When radial inertia effects are not included, Vr and S vanish, and from equations (D.5.38) and (D.5.39), the solution at P is found to be 11 SP = (B2fRHS10 B2sRHS9) (3.5.32) 11 At a Boundary Point (X= 0) for Uncoupled Waves When the waves are uncoupled, the solutionsto the finite differ- ence equations are obtained at the boundary points for the same four cases outlined above. When radial inertia terms are included in the formulation of the problem, the expression for VrP is given by equation (3.5.6), and in all cases when radial inertia terms are not included both Vrp and S vanish. In all four cases the solutions can be found in Appendix D. Case I: Traction boundary conditions When T and Sxp are known at a boundary point, then from equations (D.6.1), (D.6.2), and (D.6.3) at that point V = (RISDE F Tp) (3.5.33) GP z9 2s P a S = (RHSEEM D S ) (3.5.34) 2 1 VP (RHSBEM aFfSp F2f Sx) (3.5.35) XP F1 5f P 2f xP Case II: Kinematic boundary conditions When Vxp and Vp are prescribed at a boundary point, then the solution at that boundary point is given by equations (D.6.4), (D.6.9), (D.6.10), and (D.6.11). When radial inertia is included the solution is S 1- (D RHS12 Ff RHSEEM) (3.5.36) 12 S P 1 (F RHSEEM D RHS12) (3.5.37) P 2f 1 12 and when no radial inertia effects are included the solution becomes RHS12 S (3.5.38) xP F2f The shear stress in both cases is 1 T (RHSDE Z V ) (3.5.39) P F2 2 OP Case III: Mixed boundary conditions When Sp and Vp are known at a boundary point, then from equations (D.6.12), (D.6.13), and (D.6.14), the solution at that point is Tp (RHSDE Z2 V ) (3.5.40) F2s a S = D2 (RHSEEM DSpS ) (3.5.41) 1 Vxp (RHSBEM F S aF Sep) ( P 2f xP f P). (3.5.42) if Case IV: Mixed boundary conditions Vihen VXP and are given at a boundary point, then the solution at that point is given by equations (D.6.9), (D.6.10), (D.6.11) and (D.6.15) i.e., 1 p = (RHSDE F T ) (3.5.43) oP 2 2s P and Sxp and Sp are given by equations (3.5.36), (3.5.37), and (3.5.38). 3.6 Calculation of the Strains At any grid point P, the solution is obtained by an iterative technique. Once this is done, the values of Sx, Se, T Vx, Vr, and V are known at P as well as at points L, B, and R (see Figures 3.5 and 3.6). The strains at point P can be computed very easily from equations (2.27), (2.28), and (2.29). These equations can be written in dimensionless form using equation (3.2.1) as x x -x x (3.6.1) T 2 (3.6.2) = 2Vr (3.6.3) For a regular grid element, these equations can be written in finite difference form as e V V V xP xB xR xL 2ZT 2t X C. C V -V exP OxB 1 (R L 2AT 2 2AX OP B Vrp VrB 9P eB rP rB = 2 ( -) 2AT 2 where the final subscript on each variable denotes the point in the grid element where that variable is evaluated. Now, since the outer dX grid lines defined in Section 3.3 have slopes of c = 1, AX and AT are equal so that the expressions for the strains at point P are C = x+ V V (3.6.4) xP xB xR XL 1 CxP = exB + (V V ) (3.6.5) e = CeB + 2(Vrp + VrB) AT (3.6.6) For a boundary grid element, equations (3.6.1), (3.6.2), and (3.6.3) can be written in finite difference form as 1 e C V -1 (V + V ) xP xB xR 2 xP xB 2LT 4x GxP OxB 1 R 2 (Vp + B) 2AT L AX eP V + V eP BA2 [frP 2 rB 2 T 2 and again since AX = AT, the strains at the boundary point P are given by 79 S= xB + 2V V VxB (3.6.7) xP xB xR xP xB 1 xP = eB + VR 2 (Ve + ) (3.6.8) e" = 6eB + 2(VrP + VrB) AT (3.6.9) where equations (3.6.6) and (3.6.9) are the same expression. CHAPTER 4 RESULTS AND DISCUSSION 4.1 Introduction In Chapter 2 the problem of inelastic wave propagation was formulated and the equations for this problem were found. In Chapter 3 these equations were written in finite difference form and from them expressions for the stresses and the velocities at the points in the numerical grid (Figure 3.4) were determined. Next a computer code (shown in Appendix E) was written to facilitate the calculation of the stresses, velocities, and strains at the grid points in the charac- teristic plane. Now, in this chapter the results obtained by using this computer code will be discussed for several different combinations of initial conditions and boundary conditions. The computer code is written so that the boundary conditions are specified by reading in values of two variables at each grid point along the boundary (X=0). By specifying the boundary conditions in this manner, any variable given as one of the boundary conditions can have any functional shape. All of the data presented in this chapter were obtained using the kinematic boundary conditions (Case II), that is, by assigning values to the longitudinal velocity (V ) and the trans- verse velocity (V ) at the impact end of the tube. Furthermore, the same functional form was chosen for the two velocities in each case. This form consists of assuming that each of the velocities at the boundary increases linearly up to its final value (denoted by Vxf or VG ) during a period of time called the rise time (T ) and then remains constant. That is T TV Vqf if 0 T < T T R R V@ (X = 0) = Vef if T > TR T Vxf if 0 < T < T Tr R R R V (X = ) = Vxf if T > TR Now that the computer code is set up, it would be advantageous to compare the results from it to data which have already been published. This is done in the following section by using the data of Lipkin and Clifton (1970), and some interesting effects of the size of the numer- ical grid are noted. Then, finally, the effects of radial inertia and strain-rate dependence on the propagation of inelastic stress waves are discussed. 4.2 Effects of Numerical Grid Size Lipkin and Clifton (1970) published the results of three different experiments where a thin-walled tube was given an initial static shear stress and then impacted longitudinally. In this section the initial conditions and boundary conditions from one of these experiments will be used and the results obtained from the computer code will be compared with the experimental and theoretical results of Lipkin and Clifton (1970). The data vhich will be used are 0 T O 0 x = initial static shear stress = 3480 psi = initial static longitudinal stress = 0 Xf = final longitudinal boundary velocity = 500 ips vf = final transverse boundary velocity = 23 ips = rise time = 9.6 p sec which can be written in terms of the dimensionless quantities for input to the computer code as 0 T -o 6x E = .0003480 0 o0 S x 0 S O 0 x E Cl T R 2r No radial inertia effects or rate this section. v xf Xf V .002404 x f c1 ef V .0001106 f c1 t = 4.00 dependence will be considered in The results from three different computer runs will now be made. Each computer run used these initial conditions and boundary conditions but had different grid sizes. The three grid sizes used were AX=AT = .25, AX=AT= .125, and AX=AT= .05. The longitudinal strain versus time obtained by using the computer code in Appendix E is shown in Figure 4.1 along with the experimental results and the simple wave solution of Lipkin and Clifton (1970). From this it can be seen that Simple Wave Solution, Lipkin and Clifton (1970) Experimental Results, Lipkin and Clifton (1970) ----- X .05 ---- X .125 ---- X= .25 Figure 4.1 u G60 Time, T Grid Size Effects on the Longitudinal Strain at X = 3.75 -.012 -.008 -.00,4 for the small grid size the strain follows closely the strain obtained by Lipkin and Clifton (1970) for a simple ;'.ave with ;n instantaneously applied velocity at the boundary. For the larger grid sizes the strain- versus-time curve is smoother and follows more closely the experimental results of Lipkin and Clifton (1970). Apparently, the larger grid sizes tend to smooth out the data and eliminate the distinction between the fast and slow wave speeds. For instance, in Figure 4.1, the' simple wave solution of Lipkin and Clifton (1970) exhibits a region where the longitudinal strain has the constant value of 0.00085. The strain remains at this constant value from just after the fast wave passes until the arrival of the slow wave. From these computer runs other quantities of interest can also be plotted and the same grid size effect can be observed. This is shown in Figure 4.2 for the longitudinal velocity versus time. The grid size has a much smaller effect on the stress trajectory than on the time history curves. The stress trajectory is shown in Figure 4.3. Because the details of the solution depend on the size of the numerical grid, all subsequent computer runs will be made using a small grid. This small grid size necessitates a large amount of computer time to obtain a solution more than 1.0 diameter from the impact end, and most of the results given below are obtained near the end of the tube. 4.3 Effects of Radial Inertia In order to determine the effects of radial inertia, four separate computer runs were made using the computer code in Section E.5. The generalization of the uniaxial stress-strain curve of Lipkin and Clifton 4/' // -X= .05 ,/I -*- X = .125 / ----X = .25 -J/ 20 10 0 Time, T Figure 4.2 Grid Size Effects on the Longitudinal Velocity at X = 3.75 C V x r4 o X o >-, -4( o '0 0 1-1 C3 C, *H 4 -- -.--- X =.05 *-- X = .125 ----X = .25 ..... 5 |.rLr' Yield Surface after Static Preload -6 -9 Longitudinal Stress, S x '04 A Figure 4.3 Grid Size Effects on the Stress Trajectories at X = 3.75 4 2 0 (1970) was used. This constitutive equation (shown in Appendix A) was for strain-rate independent material behavior. The first two computer runs (one including and one not including radial inertia effects) were made using the initial conditions and the boundary conditions which Lipkin and Clifton (1970) used in one of their experiments. These input data used were T = 4.00 R AX = AT = .050 S= 0 x Data Set 1 T = .0003480 V = .002404 x -f V = .0001106 1 f These data represent a tube with an applied static pretorque (above the yield stress) impacted longitudinally at one end. The time history curves of the longitudinal strain and the change in shear strain are shown in Figures 4.4 and 4.5, respectively, for the section of the tube 3.75 diameters from the impact end. The simple wave solu- tion and the experimental results of Lipkin and Clifton (1970) are also shown in these figures. It can be seen in Figure 4.4 that the longitudinal strain obtained in this work follows the experimental results more closely than does the simple wave solution. Most of the improvement over the simple wave solution is the result of using a finite rise time (T = 4.0) for the impact velocity. The fast wave has passed the point X= 3.75 at the time when the longitudinal strain I- - i/ / / / - -=s-c. Solution Without Radial Inertia ---- Solution With Radial Incrtia ---Simple Wave Solution Lipkin and Cliflton (i1 70) --- Experimental Results Lipkin and Clifton (1.'70) Figure 4.4 Longitudinal Strain Versus Time at X = 3.75 for Data Set 1 -. 012 - -.008 -.004_ .7 - I I I 0 20 40 60 Time, T ~------ .003 Solution Without Radial Inertia -----Solution With Radial Inertia -- Simple Wave Solution Lipkin and Clifton (1970) - -Experimental Results Lipkin and Clifton (1970) I - I 4- / Time, T -. I 1 -. 001 Figure 4.5 Change in Shear Strain Versus Time at X = 3.75 for Data Set 1 has reached the value of 0.00085. For the simple wave solution this time is approximately T 6 Where for the finite rise time (T 41) this time is approximately T=10. The difference in time when the fast wave has passed can thus be accounted forby the finite rise time. As the slow wave passes a point on the tube, the longitudinal com- pressive strain begins to increase to values larger than 0.00085. The higher levels of strain (e x -.008) occur later (in the results given x here) than in the simple wave solution. Again this can be accounted for by the finite rise time. The inclusion of a finite rise time in the theoretical solution gives results which resemble the experimental data more closely than the simple wave solution. It can also be seen that including radial inertia effects in the formulation of the problem gives longitudinal strains which are somewhat closer to the experimental data than the corresponding strains when radial inertia effects are ignored. The change in shear strain versus time curve in Figure 4.5 exhibits the same rise time effect as the longitudinal strain. The results obtained here are much closer to the experimental data than the results for the simple wave solution. The final value of the shear strain appears to be low. Since this shear strain is calculated from the values of the transverse velocity, it may be that the final value of the transverse velocity should be larger. This can be seen more easily by examining the transverse velocity at several distance from the impact end as shown in Figure 4.6. A transverse velocity is induced when the tube is impacted with a longitudinal velocity, if the tube is statically preloaded in torsion. From Figure 4.6 it can be seen that the transverse velocity induced |