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MICROMECHANICAL MODEL FOR PREDICTING THE FRACTURE
TOUGHNESS OF FUNCTIONALLY GRADED FOAMS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
To my parents and to my wife Jin-Sook
All thanks and praises are to Jesus Christ, the Lord of the Universe, for his
blessing, help and guidance.
I would like to express my sincere gratitude to my advisor, Dr. Bhavani Sankar, for
his guidance, his encouragement and his financial support. He is not only my academic
advisor but also a great influence in my life.
My appreciation is also due to Dr. Raphael Haftka, Dr. Peter Ifju, and Dr. Reynaldo
Roque for serving on my supervisory committee and for their valuable comments and
This statement of acknowledgement would be incomplete without expressing my
sincere appreciation and gratitude to both my friends and family. I appreciate the
friendship and encouragement of all the colleagues at the Center of Advanced
Composites (CAC) while working and studying together in the lab. I would like to
extend my appreciation to all my family-church members whose continuous support,
prayers and help were behind me at all times. I would particularly thank my family, my
parents and my brothers in my country, for their continuous support, encouragement and
understanding during my entire school career.
Last, but not least, I would like to thank my lovely wife, adorable son and cute
daughter for their patience and support through the toughest times. They were always
there when I needed them to share my difficulties.
TABLE OF CONTENTS
A C K N O W L E D G M E N T S ................................................................................................. iv
LIST OF TABLES ....................................................... ............ .............. .. vii
L IST O F FIG U R E S .............. ............................ ............. ........... ... ........ viii
A B STR A C T ................................................. ..................................... .. x
1 IN TR OD U CTION ............................................... .. ......................... ..
Reusable Launch Vehicle and Thermal Protection System........................................2
Functionally Graded Foams and Functionally Graded Materials..............................8
Previous Work on Fracture Mechanics of Functionally Graded Materials ................11
Objectives ...................................... ................. ................ ........... 14
S c o p e ........................................................................... 1 4
2 ESTIMATION OF CONTINUUM PROPERTIES..................................................16
Continuum Properties of Homogeneous Foam....................................................16
Continuum Properties of Functionally Graded Foams ................... ...... ............20
Finite Element Verification of Estimated Continuum Properties..............................23
3 FINITE ELEMENT BASED MICROMECHANICAL MODEL ................................26
Overview of M icromechanical M odel .............................. ....................26
M acro -m o d el ..................... .................................. ............ ................ 2 9
Imposing Graded M material Properties ...................................... ............... 30
Methods for Extracting Stress Intensity Factor...............................................32
Convergence Analysis for Macro-model..........................................................36
M ic ro -m o d e l ...............................................................................................................3 7
4 FRACTURE TOUGHNESS OF GRADED FOAMS UNDER MECHANICAL
L O A D IN G ................................................................................................. .... 4 1
Fracture Toughness under Remote Loading............................................................ 41
Study of Local Effect on the Homogeneous Foam under Crack Face Traction.........46
5 FRACTURE TOUGHNESS ESTIMATION UNDER THERMAL LOADING....... 52
Behavior of Foam s under Therm al Loading.................................... .....................52
R results under Therm al Loading........................................... ........................... 57
6 CONCLUDING REMARKS ............................................. ............................. 61
ANALYTICAL SOLUTION FOR BEAM MODEL UNDER THERMAL LOADING..64
L IST O F R E F E R E N C E S ........................................................................ .....................68
B IO G R A PH IC A L SK E TCH ..................................................................... ..................72
LIST OF TABLES
2-1 M material properties of the Zoltex carbon fiber ..................................................... 21
3-1 Comparison between two methods. .............................................. ............... 35
4-1 Fracture toughness of graded and uniform foams. The unit-cell dimensions and
crack length are kept constant, but the strut thickness is varied (c=200yum, crack
length, a =0.03m and a'- 200x10 6). ............................................ .....................43
4-2 Fracture toughness of graded and uniform foams. The unit-cell dimension is
kept constant but the crack length and the strut thickness are varied (c=200 Pm,
ho=40pm and a-=+ 200x 10-6) ................................................................ ............... 43
4-3 Comparison of the fracture toughness for varying unit-cell dimensions with
constant strut thickness (h=20/1m ) ...................... ..... ...................... ........ ....... 44
4-4 Remote loading case uniform displacement on the top edge...............................46
4-5 Remote loading case uniform traction on the top edge ........................................46
4-6 Stress intensity factor, maximum principal stress and fracture toughness for
various crack lengths under crack surface traction. ............................................47
4-7 Fracture toughness estimation from remote loading and crack face traction...........47
4-8 Comparison of fracture toughness under crack surface traction calculated from
the superposition method and the micromechanical model. ...................................51
5-1 Elastic modulus variation of three different models. ...............................................53
5-2 Results of the body under temperature gradient form micromechanical model. .....58
LIST OF FIGURES
1-1 Thermal protection system in Space Shuttle. A) Temperature variation during
re-entry. B) Location of different m materials. ........................................ ..................6
1-2 Schem atic diagram of attaching the tiles......................................... ............... 7
1-3 SEM images of A) low density carbon foam and B) high density carbon foam C)
m metallic foam ................................................. ........................... 9
2-1 Open cell model with rectangular parallelepiped unit cell...................................17
2-2 Micro- and Macro-stresses in open-cell foam............ ........ ...................18
2-3 Flexural deformation of struts under shear stresses. ..............................................18
2-4 Example of variation of elastic modulus and relative density for constant cell
length c=200/m, ho=261m and a=-200x 106.......... ............ ................. ..22
2-5 Boundary conditions for un-cracked plate under uniform extension.....................23
2-6 Comparison of stresses (oyy) obtained using the macro- and micro- models in
graded foam with constant cell size but varying strut cross section. .....................24
2-7 Comparison of stresses (oyy) obtained using the macro- and micro- models in
graded foam with constant strut size but varying cell dimension ..........................25
3-1 Schematic description of both macro- and micro-model. ......................................27
3-2 A typical finite element macro-m odel ..................................... ........ ............... 29
3-3 Example of discrete elastic modulus for macro-model with ten-regions ...............30
3-4 Location of model specimens in the global panel. Each specimen is of the same
size and contains a crack of given length, but the density at the crack tip varies
from specim en to specim en ................................................................................31
3-5 J-Integral for various contours in a macro-model containing 100x50 elements
and the contour numbers increase away from the crack tip. ...................................33
3-6 Stress intensity factor from the stresses normal to the crack plane........................34
3-7 Stress normal to the crack plane ............... ........................................ 35
3-8 Stress distribution of the functionally graded foams..................... ............... 36
3-9 Variation of energy release rate at the crack tip with various size macro-models...37
3-10 Embedded beam element (micro model) in two-dimensional eight-node solid
m odel (M acro m odel) ....................... ...... ............ ................... .. ......38
3-11 Force and moment resultants in struts modeled as beams. .....................................39
3-12 Variation of fracture toughness with the size of micro-models ...........................39
4-1 Edge-cracked model under A) uniform traction or displacement loading and B)
crack surface traction. ......................................... ................. .. ...... 42
4-2 Comparison of fracture toughness of graded and homogeneous foams having
sam e density at the crack tip. .............................................................................. 44
4-3 Comparison of fracture toughness of graded and homogeneous foams. The
graded foams have varying unit-cell dimensions, but constant strut cross section
h = 2 0 u m ...................................... .....................................................4 5
4-4 Fracture toughness estimation from remote loading and crack face traction...........48
4-5 Application of superposition to replace crack face traction with remote traction....49
5-1 Thermal stress distribution output from FEA. ........... ................................ 54
5-2 Thermal stress distribution in homogeneous foam. ............................................55
5-3 Thermal stresses distribution in an FGF; the material properties and the
temperature have opposite type of variation, and this reduces the thermal
stresses ............. ..... .... ........... ............... ...........................55
5-4 Thermal stresses distribution in an FGF; the variation of the material properties
and temperature in a similar manner; and this increases the thermal stresses. ........56
5-5 Maximum and minimum values of normalized thermal stresses...........................56
5-6 Thermal stresses for various aspect ratio models..................................................57
5-7 Thermal stresses for various crack lengths.................................... ............... 58
5-8 Comparison the ratio of maximum principal stress and stress intensity factor........59
5-9 Fracture toughness for various crack lengths. ................... ......................... 60
A-1 A beam of rectangular cross section with no restraint. .............. .............. 64
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MICROMECHANICAL MODEL FOR PREDICTING THE FRACTURE
TOUGHNESS OF FUNCTIONALLY GRADED FOAMS
Chair: Bhavani V. Sankar
Major Department: Mechanical and Aerospace Engineering
A finite element analysis based micromechanical method is developed in order to
understand the fracture behavior of functionally graded foams. The finite element
analysis uses a micromechanical model in conjunction with a macromechanical model in
order to relate the stress intensity factor to the stresses in the struts of the foam. The
continuum material properties for the macromechanical model were derived by using
simple unit cell configuration (cubic unit cell). The stress intensity factor of the
macromechanical model at the crack tip was evaluated. The fracture toughness was
obtained for various crack positions and lengths within the functionally graded foam.
Then the relationship between the fracture toughness of foams and the local density at the
crack tip was studied. In addition, convergence tests for both macromechanical and
micromechanical model analysis were conducted. Furthermore, fracture toughness was
estimated for various loading conditions such as remote loading and local crack surface
loading. Local effect was studied by crack face traction conditions. The principle of
superposition was used to analyze the deviation caused by local loading conditions such
as crack surface traction and temperature gradients. From the thermal protection system
point of view the behavior of graded foams under thermal loading was investigated, and
fracture toughness was estimated. The methods discussed here will help in understanding
the usefulness of functionally graded foam in the thermal protection systems of future
space vehicles. However, further research is needed to focus on more realistic cell
configurations, which capture the complexity of foam and predicts more accurately its
mechanical property changes, such as relative density and modulus in functionally graded
foam, in order to provide more accurate predictions.
Since April 12, 1981, the first launch of the space shuttle-the orbiter Columbia, the
shuttle fleet has played a major role in human space exploration. A large amount of
money has been spent on launching satellites by both the government and the private
sector for the purpose of reconnaissance, communication, global positioning system
(GPS), weather prediction and space exploration (Blosser, 2000). The International
Space Station program also demands the space launch for construction, repair and
service. The private sector has rapidly spread in the last decade. Only a cost effective
launch system can satisfy the increasing demand for lower cost access to space. One of
the major goals of the National Aeronautics and Space Administration (NASA) has been
continued lowering of the cost of access to space to promote the creation and delivery of
new space services and other activities that will improve economic competitiveness.
A thermal protection system (TPS) which protects the whole body of the vehicle is
as crucial as avionics, propulsion, and the structure. A TPS is more limiting than fuel
constraints, structural strength, or engine's maximum thrust. In order to achieve the goals
set by NASA, new TPS concepts have to be introduced, e.g., Integral Structure/TPS
concept. This concept can be achieved because of breakthroughs in the development of
novel materials such as metallic and carbon foam, and functionally graded materials
(FGM). The microstructure of functionally graded metallic and carbon foams can be
tailored to obtain optimum performance for use in integral load-carrying thermal
protection systems due to their low thermal conductivity, increased strength and stiffness.
However, models for strength and fracture toughness of functionally graded foam
materials are in their infancy and it will be the main focus of this research. The details of
the concepts and literature survey will be discussed in subsequent sections.
Reusable Launch Vehicle and Thermal Protection System
Currently, expendable rocket vehicles and the space shuttles are the major launch
systems. An expendable rocket vehicle, such as the US Delta, European Arian, Russian
Proton, and Chinese Long March, is a structure which contains payload, the system-
supporting hardware required to fly and fuel. Expendable rockets can be used only once,
and they are expensive. The space shuttle is only partially reusable because its large
external tank is separated and burns up in the atmosphere during launch. The two smaller
solid rocket boosters land in the ocean and are recovered, but cannot be reused nearly as
many times as the space shuttle itself. Fuel by itself is not comparably expensive, but
tanks to carry it in are, especially if they are only used once such as the space shuttle's
external tank. Furthermore, considerable time for maintenance is required for engines
and thermal protection system (TPS) between flights. The TPS alone is estimated to
require 40,000 hours of maintenance between flights (Morris et al., 1996). The space
shuttle is considered as the first generation reusable launch vehicle due to its partial
In January 1995 NASA announced the development plan for a fully reusable
launch vehicle system and designated the X-33 program. The X-33 program ran for 56
months and was cut by NASA in early 2001 due to the failure of a prototype
Graphite/Epoxy composite fuel tank during the proof test. The failure of the tank
indicates that the material science and composite manufacturing technology was not
After termination of the X-33 program, NASA's Integrated Space Transportation
Plan (ISTP) was formulated in May 2001 to provide safe, affordable, and reliable space
system. As a key component of the ISTP, the Space Launch Initiative (SLI) began with a
goal to achieve the necessary technology development, risk reduction, and system
analysis in order to be used in a second generation reusable launch vehicle (RLV) which
expected to be delivered by 2010. A second generation RLV had these goals:
* Reduce risk of crew loss to no more than 1 in 10,000 missions.
* Reduce payload cost to $1,000 per pound, down from today's $10,000 per pound.
* Be able to fly more often, with less turnaround time and smaller launch crews.
A third generation RLV was planned to start flying around 2025. Its goal was to reduce
cost and improve safety by another order of magnitude.
* Reduce chance of crew loss to 1 in 1,000,000 (equivalent to today's airliners).
* Reduce payload costs to hundreds of dollars per pound.
The RLV was based on single-stage-to-orbit (SSTO) technology. The concept of
SSTO involves a rocket with only one stage carrying crews or cargo to orbit. The RLV
was NASA's true vision for a shuttle successor, but after spending many years NASA
decided to cancel the program because RLV was not attainable using existing technology,
and announced a new strategy that indicated the shuttle would continue flying until at
least 2015. However, in 2003 the space shuttle Columbia was disintegrated during re-
entry to Earth after 16 days in orbit.
After the Columbia tragedy, President Bush announced a new "Vision for Space
Exploration" in January 2004. The President's Vision set NASA in motion to reassess
the space transportation program, and to begin developing a new spacecraft to carry
humans into Earth orbit and beyond. Under the plan, a new spacecraft called the Crew
Exploration Vehicle (CEV) is to be developed and tested by 2008 and the first manned
mission is going to conduct no later than 2014. The first manned lunar landing is
scheduled no later than 2020, and CEV program continues to explore Mars and other
destination in the solar system. NASA hopes to follow this schedule in development of
* 2008 2010 First unmanned flight of CEV in Earth orbit.
* 2011 2014 First manned flight of CEV in Earth orbit.
* 2015 2018 First unmanned flight of Lunar Surface Access Module (LSAM).
* 2016 2018 First manned flight of LSAM.
* 2018 2020 First manned lunar landing with CEV/LSAM system.
* 2020 Start of planning for Mars mission and beyond.
Instead of an airplane-style lifting body used in the space shuttle system, an
Apollo-like capsule design was decided for the CEV because of the fact that the new
CEV design will use the crew and service module design principle. The new CEV design
is virtually identical to the Apollo Command Module except the implementation of the
concept of reusability. The main difference between them is that the new CEV can be
used as many as tem times. Thermal protection system development is the significant
technical obstacles that must be overcome in order to implement reusability and to
improve affordability (vehicle weight reduction) by both new design concept and
material selection such as multifunctional materials that perform structural or other roles.
Harris et al. (2002) surveyed the properties of advanced metallic and non-metallic
material systems. They provided the guidance of emerging materials with application in
order to achieve NASA's long-term goal by addressing materials already under
development that could be available in 5 to 10 years as well as those that are still in the
early research phase and may not be available for another 20 to 30 years.
The main objective of the thermal protection system is protecting the vehicle by
keeping it under acceptable temperature limit and human occupants from heat flow. Heat
sinks and ablative material were used to protect the vehicles before Space shuttle. During
the re-entry process, ablative material is charred and vaporized while the heat sinks
absorb the heat. None of the early vehicles had to be reusable so these materials and
techniques were enough to protect the early vehicles.
In the late 1960s, the space shuttle program was proposed. The program aimed to
produce a vehicle that would be larger than any that had flown in space before.
Conventional aluminum was selected for the main structure and a layer of heat resistant
material for protecting it. The properties of aluminum demand that the maximum
temperature of the vehicle's structure be kept below 175 C in operation. But aero-
thermal heating during the re-entry process creates high surface temperature which is
well above the melting point of aluminum (660 C). Thus, an effective insulator was
needed. A silica-based insulation material was decided for the heat-resistant tiles and
other coverings to protect the Shuttle's airframe. Figure 1-1 shows seven different
materials which cover the external surface of the Space Shuttle according to the
temperature variation during the re-entry.
The materials were chosen by their weight efficiency and stability at high
temperature. The areas of the highest surface temperature in the Shuttle, the forward
nose cap and the leading edge of the wings, are made with Reinforced Carbon-Carbon
(RCC). There are two main types of tiles, referred to as Low-temperature Reusable
Surface Insulation (LRSI) and High-temperature Reusable Surface Insulation (HRSI).
Relatively low temperature of surface where the maximum surface temperature runs
between 370 and 650 C is covered by LRSI. HRSI covers the areas where the maximum
surface temperature runs between 650 and 1,260 C.
955 CR si
5C 42 C SRSI Fnd LRRSI R CC d LRSI
Figure 1-1. Thermal protection system in Space Shuttle. A) Temperature variation
during re-entry. B) Location of different materials. (Courtesy of W. Jordan,
Last accessed February 14th, 2005).
Many of the tiles have been replaced by a material known as Flexible Reusable
Surface Insulation (FRSI), and Advanced Flexible Reusable Surface Insulation (AFRSI)
in the area where the maximum surface temperature does not exceed 400 C. These tiles
are lighter and less expensive than LRSI and HRSI, and using them enabled the Shuttles
to lift heavier payloads. The tiles are brittle and vulnerable to crack under stress. The
tiles could not be mounted directly to the main body structure of the Shuttle due to
expansion and contraction of the aluminum structure by temperature change. Instead of
direct mounting on the structure, the tiles have to mount to a felt pad using a silicone
adhesive, and then the tile and pad combination are bonded to the structure as seen in
Figure 1-2. Tiles are occasionally lost during take off because of the incredible loud
noise as well as aerodynamic forces. Because of this, as well as weight concerns, many
of the fuselage tiles were replaced by FRSI blankets.
Figure 1-2. Schematic diagram of attaching the tiles. (Courtesy of W. Jordan, Source:
http://www2.1atech.edu/j ordan/Nova/ceramics/SpaceShuttle.pdf, Last
accessed February 14th, 2005).
These material developments and techniques enable the partially reusable Space
Shuttle to offer more capability. However, the tiles have their limitations. During both
liftoff and landing, tiles can become damaged and chipped. About 40,000 hours of
maintenance is required between flights (Morris et al., 1996). For fully RLVs, the tiles
would not provide sufficient protection and some other solution would be necessary.
Blosser (1996) emphasized the durability, operability and cost effectiveness as well as
light weight for new TPS to achieve the goal of reducing the cost of delivering payload to
orbit. Most other proposed reusable thermal protection systems have involved some kind
of advanced high-temperature metal.
Metallic TPS is considered as a much-needed alternative to the ceramic-based
brittle tile and thermal-blanket surface insulation currently used on the Space Shuttle.
Metallic TPS offers the significant advantages (Harris et al., 2002).
* Does not require high temperature seals or adhesive development
* Does not require waterproofing or other restorative processing operation between
* Significantly reducing operational cost
* Saving on vehicle weight, when used as part of an integrated aeroshell structural
The TPS forms the external surface of an RLV and is exposed to a wide variety of
environments corresponding to all phases of flight (Dorsey et al, 2004). Thus, the TPS
requirements must apply to any external vehicle airframe surface. Recently, a new
Adaptable, Robust Metallic, Operable, Reusable (ARMOR) metallic TPS concept has
been designed (Blosser et al., 2002) and demonstrated the capability of protecting the
structure from on-orbit-debris and micrometeoroid impact (Poteet and Blosser, 2004).
The concepts of metallic TPS depend primarily on the properties of available materials.
The development of foam and FGM as a core material of TPS panel may offer dramatic
improvements in metallic TPS (Harris et al., 2002). An integrated wall construction is an
approach which the entire structure is designed together to account for thermal and
mechanical loading (Glass et al, 2002). An integrated sandwich TPS with metallic foam
core is studied under steady state and transient heat transfer conditions and compared
with a conventional TPS design (Zhu, 2004).
Functionally Graded Foams and Functionally Graded Materials
Foams are generally made by dispersing gas in a material in liquid phase and then
cooling it to a solid. Solid foams can also be made by dispersing a gas in a solid. These
solid foams are generally called cellular solids, often just called foams. During the last
few decades, many attempts have been made to produce metallic foams, but methods
have suffered from high cost, and only poor quality foam materials were produced. In
the last ten years, improved methods were discovered, and only recently various methods
are available to produce high quality metallic foam. Some start with the molten metal
and others with metal powder. Graded foams can also be manufactured by dispersing
hollow micro-balloons of varying sizes in a matrix medium (Madhusudhana et al., 2004).
The porous structures of carbon and metallic foams are depicted in Figure 1-3.
Foams can be used in many potential engineering applications ranging from light
weight construction to thermal insulation to energy absorption and thermal management.
The mechanical properties of foams are strongly dependent on the density of the foamed
material as well as their cell configuration. For example, the quantities such as elastic
modulus and tensile strength increase with increasing density of foams. Foams can be
used in many potential engineering applications ranging from light weight construction to
thermal insulation to energy absorption and thermal management.
A B C
Figure 1-3. SEM images of A) low density carbon foam and B) high density carbon
foam C) metallic foam.
Foams can be categorized as open-cell and closed-cell foam. In open-cell foams
the cell edges are the only solid portion and adjacent cells are connected through open
faces. If the faces are also solid, so that each cell is sealed off from its neighbor, it is said
to be closed-cell foam (Gibson & Ashby, 2001). In this study, from the thermal
management application point of view only the open-cell foam is considered due to its
large surface area and the ability to transfer heat by working fluid in open porous
structure, if necessary. The combination of open porosity and large specific surfaces
allows a reduction in size of the thermal management system. A reduction in size of the
thermal management system will reduce weight and improve efficiency.
Functionally graded materials (FGMs) are a relatively new class of non-
homogeneous materials in which material properties vary with location in such a way as
to optimize some function of the overall FGM. The FGM concept originated in Japan in
1984 as a thermal barrier material which is capable of withstanding a surface temperature
of 1,725 C and a temperature gradient of 725 C across a cross section less than 10 mm.
Since 1984, FGM thin films have been comprehensively researched and are almost a
The primary advantage of FGM over conventional cladding or bonding is avoiding
weak interfacial planes because material properties are engineered to have relatively
smooth spatial variation unlike a step increase in conventional cladding or bonding.
Thus, FGMs are widely used as coatings and interfacial zones to reduce mechanically and
thermally induced stresses caused by the material properties mismatch and to improve the
bonding strength. Generally, a functionally graded material (FGM) refers to a two-
component composite characterized by a compositional gradient from one component to
the other. In contrast, traditional composites are homogeneous mixtures, and they
therefore involve a compromise between the desirable properties of the component
materials. Since significant proportions of an FGM contain the pure form of each
component, the need for compromise is eliminated. The properties of both components
can be fully utilized. For example, the toughness of a metal can be combined with the
refractoriness of a ceramic, without any compromise in the toughness of the metal side or
the refractoriness of the ceramic side. However, in this study, only the concept of
varying material properties is adopted, and functionally graded foams (FGFs) are
produced by changing the size of unit-cell or the thickness of strut in the foam.
Previous Work on Fracture Mechanics of Functionally Graded Materials
In order to utilize FGMs as reliable engineering materials in structures, among
other properties their fracture mechanics has to be understood. Furthermore, methods to
compute the stress intensity factor (SIF) and energy release rate have to be developed
because the stress intensity factor cannot be measured directly in an experiment, but it
can be found through the relations between SIF and a measurable quantity, such as strain,
compliances or displacement.
Sound fracture mechanics principles have been established for conventional
homogeneous materials so that the strength of a structure in the presence of a crack can
be predicted. However, the fracture mechanics of a functionally graded material which is
macroscopically non-homogeneous is only beginning to be developed. Analytical work
on FGM goes back to the late 1960s when Gibson (1967) modeled soil as a non-
Analytical studies have shown that the asymptotic crack tip stress field in FGMs
possesses the same square root singularity seen in homogeneous materials. Analytical
studies of Atkinson and List (1978) and Gerasoulis and Srivastav (1980) are some of the
earliest work on crack growth in non-homogenous materials in order to evaluate its
integrity. Atkinson and List (1978) studied the crack propagation for non-homogenous
materials subjected to mechanical loads assuming an exponential spatial variation of the
elastic modulus. Gerasoulis and Srivastav (1980) studied a Griffth crack problem for
non-homogeneous materials using integral equation formulations. Delale and Erdogan
(1983), Eischen (1987), Jin and Noda (1994) and Erdogan (1995) showed that the nature
of the inverse-square-root-singularity of crack tip is also preserved for an FGM as long as
the property variation is piecewise differentiable. The work by Delale and Erdogan
(1983) is accredited with having first suggested the standard inverse-square-root stress
singularity for an FGM in which a crack is parallel to the elastic modulus gradient.
Eischen (1987) confirmed their work by using eigenfunction expansion technique in non-
homogeneous infinite plane. Jin and Noda (1994) further confirmed for FGM with
piecewise differentiable property variation. In 1996, Jin and Batra studied crack tip fields
in general non-homogeneous materials and strain energy release rate and stress intensity
factor using the rule of mixture. Based on the early work of Delale and Erdorgan (1983)
that showed the negligibility of the effect of the variation on Poisson's ratio, Erdorgan
and Wu (1997) analyzed an infinite FGM strip under various remote loadings by using an
exponential varying elastic constants and constant Poisson's ratio. Although such
progress has increased the understanding of fracture mechanics of FGM, a suitable stress
intensity factor solution is needed in designing components involving FGM and
improving its fracture toughness.
In engineering context, the closed-form SIF solution is desirable for easier use in
the analysis of fracture of FGM structures for a variety of specimen configurations. The
exact solutions are not available yet and some researchers have attempted to find simple
and approximate closed-form solutions. Yang and Shih (1994) obtained an approximate
solution for a semi-infinite crack in an interlayer between two dissimilar materials using a
known bi-material solution. Gu and Asaro (1997) obtained the complete solution of
semi-infinite crack in a strip of an isotropic FGM under edge loading. The solution was
analytical up to a parameter which is obtained numerically. Then, the solution was
extended to the strip is made of an orthotropic FGM. Ravichandran and Barsoum (2003)
obtained approximate solution and compared the results with the values obtained by finite
element modeling (FEM).
The application of the finite element method to determine crack tip stress fields has
been rapid progress (Broek 1978). A finite element based method for determination of
stress intensity factor in FGM was proposed by Gu et al. (1999). They used standard
domain integral to evaluate the crack-tip field for FGM and studied the effect of non-
homogeneity in numerical computation of the J-integral. They concluded that the
conventional J-integral can provide accurate results as long as the fine mesh near crack
tip is provided. Honein and Hermann (1997) have studied the conservation laws for non-
homogenous materials and proposed a modified path-independent integral. Weichen
(2003) constructed another version of path-independent integrals of FGM by gradually
varying the volume fraction of the constituent materials.
Numerical simulation was carried out by Marur and Tippur (1999) using linear
material property variation in the gradient zone. They studied the influence of material
gradient and the crack position on the fracture parameters such as complex stress
intensity factor and energy release rate. Anlas et al. (2000) calculated and compared the
stress intensity factors obtained for a cracked FGM plate by using several different
techniques- energy release rate, J-integral and a modified path independent integral.
They evaluated the J-integral and a modified J-integral numerically by technique similar
to Gu et al. (1999) and Honein and Hermann (1997) respectively. The results were
compared with the analytical solutions of Erdogan and Wu (1997). Furthermore, the
accuracy of the finite element method and mesh refinement was investigated.
In contrast to above-described analytical studies and numerical investigation, there
are relatively few experimental works on fracture mechanics of FGM. A typical
laboratory technique is the use of photo elasticity. Butch et al. (1999) examined the
surface deformation in the crack tip region by the optical method of Reflection Coherent
Gradient Sensing. They used a graded particulate composition comprised of spherical
glass filler particles in an epoxy matrix as a test specimen. Recently, Rousseau and
Tippur (2002) examined the particulate FGM by mapping crack tip deformation using
optical interferometery. They used a finite element analysis in order to develop fringe
analysis and to provide a direct comparison to the optical measurements.
The objectives of this research are to develop micromechanical models to predict
the fracture toughness of functionally graded foams under various loading conditions -
mechanical and thermal loading as insulation materials for load carrying thermal
protection system, and to develop the understanding of the effect of graded foam solidity
profile on its fracture mechanics. The methods will also be used to understand the effects
of thermal gradients on fracture of homogeneous foams.
Chapter 1 reviewed some background information regarding functionally graded
foams as the thermal protection system of next generation reusable launch vehicles and
some previous works on fracture mechanics of FGM. Chapter 2 discusses the method to
estimate the material properties of functionally graded foams (FGFs). At first,
formulations for homogeneous foam will be established and then the methods will be
extended to the FGFs. Chapter 3 describes the finite element analysis (FEA) of the
micromechanical model. In Chapter 3, macro and micro models for graded cellular
materials are explained with key issues in both models. Chapter 4 discusses the results
under mechanical loading including remote loading uniform traction and uniform
displacement) and local loading (crack face traction). Chapter 5 presents the behavior
and the results under thermal loading on homogeneous foam and FGF. The concluding
remarks are presented in Chapter 6.
ESTIMATION OF CONTINUUM PROPERTIES
The functionally graded foam can be modeled either as a non-homogeneous
continuum, or as a frame consisting of beam elements. The former model will be
referred to as the macro-model and the latter as the micro-model. We require both
models for the simulation of crack propagation in graded foams. The region surrounding
the crack tip is modeled using the micro-model, where as the region away from the crack
tip uses the macro-model. The micromechanical model is treated as an embedded model
around crack tip. The macro-model of the functionally graded foam requires continuum
properties at each point or at least for each element in the finite element model. In this
chapter, the procedures for calculating the continuum properties of a homogeneous
cellular medium (open-cell) is presented, and then the method is extended to functionally
Continuum Properties of Homogeneous Foam
Most of the open-cell foams with periodic microstructure can be considered as
orthotropic materials. Choi and Sankar (2003) derived the elastic constants of
homogeneous foams in terms of the strut material properties and unit-cell dimensions. In
their model they assumed that the strut has a square cross section h x h and the unit-cell is
a cube. In the present approach, the general case is considered wherein the unit-cell is a
rectangular parallelepiped of dimensions c, x c2 x c, as shown in Figure 2-1. The
derivation of formulas for the relative density and elastic modulus are straightforward.
The relative density p*/p, is related to the porosity of the cellular material. A superscript
* denotes the foam properties and a subscript s denotes the solid properties or the strut
properties. The density of the foam can be obtained form the mass and volume of the
unit-cell. Then, the relative density can be expressed as a function of the dimensions of
unit-cell and the strut thickness as shown below:
p*mv (c, + c2 3)h2 2h3
A PA C1C2C3
where m is the mass and Vis the volume of unit-cell.
Figure 2-1. Open cell model with rectangular parallelepiped unit cell.
Elastic modulus can be evaluated by applying a tensile stress a* on unit area of the
unit cell as shown in Figure 2-2. The equivalent force on the strut caused by the stresses
can be written as F = (cq x c3) In micro-scale sense, the force F causes stresses a, in
sectional area h2 (Figure 2-2). Therefore, the stress as in the section h2 and the
corresponding strain E can be expressed as
F cIc3 s C1C3-
F c3u and c c-
U0- = 2 and E=- -2 (2-2)
Sh2 h2 Es h2Es )
where Es is elastic modulus of strut. Therefore, elastic modulus of foam E* can be
derived from Eq. (2-2) as
E, E,, E2 E, E 3 E
C2C3 C13 C12
=... ----,... ,.... 3=
h 0 /~7 .-*" .-"' ~7
Figure 2-2. Micro- and Macro-stresses in open-cell foam.
The derivation of shear modulus is slightly involved and it is described below. We
show the derivation of the shear modulus G1 from the unit-cell dimensions, strut cross
sectional dimensions and the strut elastic modulus. When a shear stress is applied, struts
are deformed as shown in Figure 2-3.
ig IN Curvature=O F2
cC/2 _2 //2
Figure 2-3. Flexural deformation of struts under shear stresses.
Figure 2-3. Flexural deformation of struts under shear stresses.
Bending moment becomes zero at the middle of struts because the curvatures are
zero due to symmetry. The struts are assumed as a beam fixed at the end with a
concentrated force at the middle at a distance c and 2, respectively from the fixed end.
The maximum displacement can be written as,
PL3 F2 PL3 ( 2
S= and 52 = (2-4)
3EI 3EI 3EI 3EJ
where I = (the moment of inertia). The applied shear stress can be written
F F F F
as2 2 Using the relations, =2 the maximum displacements in Eq. (2-
CIC3 C2C3 C1 C2
4) can be rewritten as
1 c1F2 and 82 CC2F2 (2-5)
The shear strain y12 can be derived as
232 251 2c,2 + 2c21 (2-6)
1-J+ -- (2-6)
C2 C1 C1C2
Using Eq. (2-5), the shear strain can be written as,
(CC2 + C)F2
712 12= (2-7)
The shear modulus GI2 can be derived as
G -12 = c2c3 12EJ (2.8)
/712 cC + C1 )F2 C2C3 (1i +2 1
Substituting for the moment of inertia, I
G12 C, (C=j E, (2.9)
The shear modulus in the other two planes can be obtained by cyclic permutation as
G23 CCCt+C ES)
G31 EC2C3 c3)
iG3 C I, --2 + 2 3)
Continuum Properties of Functionally Graded Foams
The properties of a functionally graded foam can be represented by a function of
the coordinate variables x, y and z. The actual functional form depends on the application
and also the type of information sought from the homogenized model of the foam. In this
study, the functions of material properties will be assumed such that the properties
calculated at the center of a cell will correspond to the properties of the homogeneous
foam with that cell as its unit cell. Thus the function is actually defined only at the
centers of the cells of the functionally graded foam. Then, these points will be curve-
fitted to an equation in order to obtain the continuous variation of properties required in
the continuum model. This approach will be verified by solving some problems wherein
the graded foam is subjected to some simple remote loading conditions (uniform
displacement loading) and comparing the resultant stresses from the macro- and micro-
models. The material properties of strut correspond that of Zoltex Panes 30MF High
Purity Hilled carbon fiber studied earlier by Choi & Sankar (2003). The Zoltex Panes
30MF High Purity Hilled carbon fiber is chosen because of the high percentage of carbon
component weight (99.5%). The strut properties are listed in Table 2-1.
Table 2-1. Material properties of the Zoltex carbon fiber.
Density, p, 1750Kg /m
Elastic Modulus, Es 207 GPa
Poisson's ratio, s, 0.17
Ultimate Tensile Strength, o-, 3600 MPa
The relative density of functionally graded foams (FGF) depends on both the
dimensions of the unit-cell and the strut thickness. Therefore, three different cases can be
considered. The first case is the one where the dimensions of the unit-cell remain
constant while the strut thickness varies along the x-axis. In the second case the strut
thickness is kept constant with varying cell length. The last case is varying both of them.
In this paper, the first two cases are studied independently. Furthermore, the material
properties of functionally graded foam can be either increasing or decreasing along the x-
axis. Therefore, the fracture properties of both increasing and decreasing cases are
studied and compared to the homogenous case.
For the case where the strut dimensions vary, the thickness of the square strut is
assumed to vary as
h(x) = h + ax (2-11)
where a is a parameter that determines the degree of gradation of the properties. Then the
properties such as density and elastic constants of the graded foam can be assumed to
vary as given by the equations for homogeneous foams, but changing the constant h by
the function h(x). Figure 2-4 shows the variation of relative density and elastic modulus.
For example, the relative density variation of the functionally graded foam with
varying beam thickness can be written as
P =3 h(x) 2 hx) (2-12)
ps c c
where the unit-cell is assumed as a cube of dimension c (i.e. c = c, = c = c,).
Similar equations can be derived for elastic modulus and shear modulus as
E r =h(x) Es
G 1 h(x)E
-- Elastic Modulus 0 04
0 6 Relative Density
l t 05 003
c \\ 2
e i 002 .
O0 0 000
000 002 0 04 006 0 08 010
Figure 2-4. Example of variation of elastic modulus and relative density for constant cell
length c=200um, ho=261um and a=-200x 106.
For the case where the unit-cell dimensions vary, we can consider the case where h,
c2 and c3 are constants, but cl varies as
cI+l = c1 + p (2-14)
where i denotes the cell number from the left edge (i.e. c represents the size of cell at
left edge) and /is the increment in the cell length in the x direction. Again the properties
of the foam will be calculated at the center of each cell using the equations for
homogeneous foams as given in equations (2-2) through (2-9).
Finite Element Verification of Estimated Continuum Properties
The accuracy of the estimated elastic constants, when a material property
discretization is introduced, is investigated by comparing the stress field from macro-and
micro-models. A simple mechanic problem was solved using both un-cracked macro-
and micro-models. A uniform displacement (70um) was applied along the upper edge of
a rectangular plate using the macro-model, which consists of two dimensional plane
stress elements (eight nodes bi-quadratic, reduced integration element). The elastic
constants of the non-homogeneous material varied as given by Eq. (2-13). In the finite
element model the elastic constants within each element were considered constant. The
boundary conditions are depicted in Figure 2-5.
uO4, uy=0 u=O, uy=0
Figure 2-5. Boundary conditions for un-cracked plate under uniform extension.
The right lower corer was fixed to prevent the rigid body motion. The resulting
displacements along the boundary of a micro-model embedded in the macro-model were
applied as the boundary displacements of the micro-model by using the three-point
interpolation. For the micro-model, each strut was modeled as an Euler-Beroulli beam
element with two nodes and three integration points. In order to verify the validity of
properties used in the macro-model, the stresses in both models are compared. In the
case of macro-model the stresses are obtained as the finite element analysis output. The
outputs in micro-model are the axial force and moment resultant in the beam element. We
convert these forces into equivalent stresses by dividing by the strut cross sectional
area cl x c3 The shaded region on Figure 2-5 represents the micro-model. Both constant
cell length with varying strut thickness and constant strut thickness with varying cell size
cases are considered. In the constant cell length case the cell length is assumed as
200yum. The macro-model consists of 100x50 plane solid elements. The strut cross
section is assumed to vary as a function of x according to the equation h(x) = ho + ax,
where ho =40um and a =-200x 10-6. The region corresponding to the micro-model in the
macro-model consists of 15x5 plane stress elements. The micro-model uses 2,250 beam
4 micro-m odel
Approx 5% difference
0 I I I I I
0 000 0 002 0 004 0 006 0 008 0 010 0 012 0 014 0 016
Figure 2-6. Comparison of stresses (oyy) obtained using the macro- and micro- models in
graded foam with constant cell size but varying strut cross section.
The results for the stress component yy from the macro- and micro-models are
compared in Figure 2-6. The maximum difference in stresses between the macro- and
micro-models is about 5%. In the second case, the strut is assumed to have a square cross
section (h =20pum) and the cell length c, was varied along the x direction with co =2001um
and / =-0.151m. The dimensions of the cell in the 2 and 3 directions, c2 and c3, are kept
constant (100um). The stress component yy from macro-model and micro-model are
compared in Figure 2-7.
0000 0005 0010 0015 0020 0025 0030
Figure 2-7. Comparison of stresses (uyy) obtained using the macro- and micro- models in
graded foam with constant strut size but varying cell dimension.
The above examples illustrate and validate two important concepts that will be used
in this dissertation. We find that modeling graded foams with microstructure as a non-
homogeneous continuum provides good results for micro-stress and displacements.
Some researchers, e.g., Gu et al. (1999), and Santare and Lambros (2000) have used
different properties at the Gauss (integration) points within the element. However, using
the homogenized properties at the center of the continuum element in the FE model
seems to be reasonable and yields accurate results.
FINITE ELEMENT BASED MICROMECHANICAL MODEL
In this chapter, we describe a finite element based micromechanics model for
estimating the fracture toughness of functionally graded foams. The crack is assumed to
be parallel to the material properties gradient direction. At first, we describe a finite
element based micromechanics model for estimating the fracture toughness in order to
understand the key ideas of micromechanical modeling. Detailed macro- and micro-
model descriptions are presented, and the method of extracting stress intensity factor
from the finite element analysis is described in depth. Also, convergence test in both
macro- and micro-models were performed.
Overview of Micromechanical Model
The functionally graded foam, a cellular material is non-homogeneous in the
macro-scale. That is, the microstructure is graded and the foam is treated as a functionally
graded material in macro-scale. The foam can be modeled either as a non-homogeneous
continuum, or as a frame consisting of beam elements to model the struts. The former
model (continuum model) will be referred to as the macro-model and the latter (frame
model) as the micro-model. In the finite element analysis, solid elements are used in the
macro-model and beam elements in the micro-model.
In the finite element analysis model, due to symmetry, only the upper half of the
plate is considered. The lower edge has a zero displacement boundary condition in they-
direction to account for symmetry. As described in the previous chapter, the functional
variation of material properties is estimated by extending the method of calculating the
continuum properties of a homogeneous cellular medium. The eight-node quadrilateral
elements were used to discretize the macro-model and functional variation in material
properties is implemented by having 100 vertical layers, with each layer having a
constant value of material properties. A crack can be created in the functionally graded
foam by removing a set of struts along the intended crack surface in micro-model and by
removing the zero boundary condition along the intended crack surface in macro-model.
A portion of the foam surrounding the crack tip is considered as the micro-model (see
Figure 3-1. Schematic description of both macro- and micro-model. A) Macro-model
consists of plane 8-node solid elements. The region in the middle with grids
indicates the portion used in the micro-model. B) The micro-model consists
of frame elements to model the individual struts. The displacements from the
macro-model are applied as boundary conditions in the micro-model.
The dimensions of the micro-model should be much larger than the cell size (strut
spacing) so that it can be considered as a continuum. For the case of uniform
displacement loading, the upper edge is loaded by uniform displacement in they-
direction. The maximum stresses in the struts in the vicinity of the crack tip are
calculated from the finite element micro-model. From the failure criterion for the strut
material, one can calculate the maximum stress intensity factor that will cause the failure
of the crack tip struts, and thus causing crack propagation in a macro-scale sense. The
key idea in this approach is to be able to calculate the stress intensity factor for a given
boundary displacements or apply a set of boundary conditions that corresponds to a given
stress intensity factor in the macro-scale sense. For this purpose we turn to the macro-
model as shown in Figure 3-1 (A). In the macro-model a much larger size of the foam is
modeled using continuum elements, in the present case, plane solid elements. The micro-
model is basically embedded in the macro-model. The displacements of points along the
boundary of the micro-model are obtained from the finite element analysis of the macro-
model and applied to the boundary of the micro-model by using three points
interpolation. The maximum principal stresses at the crack tip can be calculated from the
force and moment resultants obtained from the micro-model as
otp = (3-1)
Itp A tp
where cp is maximum principal stress at the crack-tip.
Ftp, Mtp are force and moment resultant.
At is cross-sectional area and htp is the thickness of strut.
It, is the moment of inertia.
The strut material is assumed brittle and will fracture when the maximum principal
stress exceeds the ultimate tensile strength. The fracture toughness of the foam is
defined as the stress intensity factor that will cause the crack-tip struts to fail in a micro-
scale sense and cause the crack to propagate in a macro-scale sense. The fracture
mechanism of brittle material is governed by Linear Elastic Fracture Mechanics (LEFM).
Therefore, the fracture toughness can be estimated from the following relation.
-i =- (3-2)
In macro-model, the conventional two dimensional isoparametric plane-stress
elements are used. The problem geometry is shown in Figure 3-1 (A). The material
gradient is in the x-direction. A pre-processor program was coded using MATLAB,
with the parameters such as unit-cell size, strut thickness, crack size, a and /f (defined in
previous chapter) in order to generate a rectangular mesh with eight-node isoparametric
elements with two-degree of freedom at each node and to impose the material gradient in
Figure 3-2. A typical finite element macro-model. A) constant unit-cell size with
varying strut thickness. B) varying(decreasing) unit-cell size with constant
Only half of the model is represented in the finite element analysis by invoking that the
model is symmetric with respect to its midline, x-axis. A zero displacement boundary
condition in y-direction is employed in the lower edge to account for symmetric. Figure
3-2 shows a typical finite element model which consists of 5,000 eight-node
isoparametric elements (50 elements in vertical direction and 100 elements in horizontal
direction) with 1,5300 nodes. It should be noted that the number of elements in a typical
model was decided after convergence test by generating a coarse mesh (smaller number
of elements) and progressively reducing the mesh size (increasing number of elements),
to be discussed in this section later.
Imposing Graded Material Properties
Finite element analysis of functionally graded foam for macro-model requires
imposing the required the variation of material properties in x-direction. The material
properties are graded by either changing the thickness of struts or changing the
dimensions of the unit-cell described as before. Relative density, elastic modulus and
shear modulus vary along the x-axis corresponding to the equations derived in the
0 6 -.. .. Actual Elastic Modulus
S--- Discrete Elastic Modulus
0 00 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 08 0 09 0 10
Figure 3-3. Example of discrete elastic modulus for macro-model with ten-regions.
When functionally graded foam is modeled as a homogeneous solid (macro-model), a
material property discretization is introduced. The discretization is done by grouping
elements in the gradient region into narrow vertical strips and assigning constant values
of estimated material properties at the centroid of the strip of grouped elements. For
example, Figure 3-3 shows the discrete elastic modulus for the ten-region model.
However, the Poisson's ratio is kept constant because the effect of a variation of
Poisson's ratio is negligible (Delale and Erdogan, 1983).
Const crack length
Variation of beam thickness
Figure 3-4. Location of model specimens in the global panel. Each specimen is of the
same size and contains a crack of given length, but the density at the crack tip
varies from specimen to specimen.
The Mode I fracture toughness with various relative densities is conducted in two
different sets for the constant unit-cell lengthcase. The first set is controlling the crack
length while the variation of material properties remains same. The other set is shown in
Figure 3-4. The crack length remains constant while the material properties are
controlled to locate desired relative density at the crack tip. However, the dimensions of
models are fixed (0.1m by 0.5m for macro-model and 0.015m by 0.005m for micro-
model). For the case where the unit-cell dimensions change, the number of elements both
in macro-model (100x50 elements) and micro-model (2,250 elements) are fixed and the
material properties at the crack tip is controlled by c, and /7. Therefore, the dimensions
of models are not fixed.
Methods for Extracting Stress Intensity Factor
Considering only Mode I symmetric loading (mode-mixity=O), the stress intensity
factor at the crack tip is calculated from traditional methods in computational fracture
mechanics i.e. point matching and energy method (Anderson, 2000). The point matching
method is the direct method in which the stress intensity factor can be obtained from the
stress field or from the displacement field, while the energy method is an indirect method
in which the stress intensity factor is determined via its relation with other quantities such
as the compliance, the elastic energy or the J-contour integral (Broek, 1978). The
advantage of the energy method is that the method can be applied as both linear and
nonlinear. However, it is difficult to separate the energy release rate into mixed-mode
stress intensity factor components. In this paper, the crack-tip stress field and J-contour
integral are used to find and verify the stress intensity factor for the point matching and
the energy method respectively.
In energy method, the J-contour integral can be evaluated numerically along a
contour surrounding the crack tip, as long as the deformations are elastic. Generally, J-
contour integral is not path independent for non-homogeneous material. Therefore, J-
contour integral is expected to vary with contour numbers as shown in Figure 3-5. The
contour numbers represent incrementally larger contours around the crack tip. The mesh
refinement governs the size and increments of contours.
200 y = 1.21193E-05x 1.27727E-03x 6.32919E-02x2 2.05304E+00x+ 7.61841E+02
1 6 11 16 21 26 31 36 41 46 51
Figure 3-5. J-Integral for various contours in a macro-model containing 100x50 elements
and the contour numbers increase away from the crack tip.
The first few contours are disregarded due to inaccuracy for most finite element
meshes (Anlas et al., 2000). J-contour integral as r-*0 is obtained by fitting a fourth
order polynomial to the output values of J-contour integral. The limiting value of J-
contour integral can be evaluated numerically as the intercept of the polynomial curve at
y-axis. The value of J-integral for a contour very close to the crack-tip is related to the
local stress intensity factor as in the case of a homogeneous material (Anlas et al., 2000).
Thus, energy release rate, G is identical to the value of J-contour integral as the path of
contour approaches to crack-tip (Gu and Asaro, 1999). Conceptually, energy release rate,
G can be found by the variation of J-contour integrals as shown in Figure 3-5. The stress
intensity factorKI of a functionally graded foam (two-dimensional orthotropic) can be
found from G using the relation (Sih & Liebowbitz, 1968).
2 1M122 )2 22 2 2a12 + t66
G = nK -- + (3-3)
,2 all an 2all
1 1 1
where, al = a22 a33
El E2 E,
a12 = a23 = a3 = 0
1 1 1
a44 = ,a 55 ,a66 -,
G23 G13 G12
Using the point matching method of stress field, the opening mode value of the
stress intensity factor can be calculated from the oyy stress ahead of the crack (Sanford,
K, = Lim [cyy 2r] (0 = 0) (3-4)
The stress intensity factor can be found by plotting the quantity in square brackets against
distance form the crack tip and extrapolating to r = 0. Figure 3-6 shows the one of the
example plot of uV-7r versus distance from the crack tip. A 4th order polynomial
regression is also shown in Figure 3-6. The y-intercept of the curve yields the value of KI.
= -4.70773E+11x4 + 4.70881E+10x3 1.99894E+09x2 + 6.41493E+07x+ 2.42043E+05
2.00E+05 R2 9.99997E-01
0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300
distance from crack tip, r (m )
Figure 3-6. Stress intensity factor from the stresses normal to the crack plane.
The stress intensity factor defines the amplitude of the crack tip singularity and the
conditions near crack-tip (Anderson, 2000). Stress near crack tip increases in portion to
the stress intensity factor. Consider the Mode I singular field ahead the crack tip, the
stress normal to the crack plane, oy can be defined from Eq. (3-5).
yy 2 (3-5)
Figure 3-7 shows the stress normal to the crack plane versus distance form the
crack tip. Where the square-root singularity dominated zone, Eq. (3-5) is valid while
stress far from the crack tip is governed by the remote boundary conditions.
Table 3-1. Comparison between two methods.
Crack Length (m) 0.01 0.02 0.03 0.04 0.05
Relative density 0.094582 0.085536 0.076874 0.068608 0.06075
at the crack-tip
KT From J-Integral
SFrom -Integral 1.2238E+06 1.0699E+06 9.2893E+05 8.4886E+05 7.2691E+05
K From crack-tip 1.2262E+06 1.0715E+06 9.3360E+05 8.0605E+05 6.8363E+05
stress field (Pa-m")
% difference 0.195 0.145 0.503 5.043 5.954
6.00E+06 CPO r,
SK/(2 Pi* r)1/2
Singularity dominated zone
0. 0- 0.005 0.010 0.015 0.020 0.025 0.030
distance from crack tip, r (m)
Figure 3-7. Stress normal to the crack plane.
The stress intensity factor from J-Integral and stress-matching were compared for
various cases in Table 3-1. The maximum difference between the two methods is less
Convergence Analysis for Macro-model
For the convergence test, the model which has constant cell size with varying the
strut thickness is discretized into uniform meshes of 10x5 elements (10 regions), 20x10
(20 regions), 50x25 elements (50 regions), 100x50 (100 regions), 200x100 (200 regions)
and 400x200 (400 regions). Some finite element outputs are shown in Figure 3-8.
Ppm~ma~mrsmn~r I In;as~m l~~u.
.,- :- 1 r n f 1;.i o ,. i :1 .D : :, .11
I*Y4 .... .. ....................
Il r nr lm..... .. .. .. .. .. .. .n
j cr T -a -- -6U /L Ei 3u u .w Fai-i + .
~71n s m:6. l T. Ir~~
R 11llIh.1 F-;uuCitsFu~ +1.42&O
IRinmeii VI."'f BcduirBaLnu 3'uli sFai~;l~ +1.428=402
Figure 3-8. Stress distribution of the functionally graded foams. A) 10x 5 elements
model. B) 20x 10 elements model. C) 100x 50 elements model. D) 200x 100
a~. I MIN~
-1 I I I A
i- -,,I T
'I T.- I W
1 1 F ;
As the number of elements and regions increases, the energy release rate at the
crack tip converges as shown in Figure 3-9. For 100x50 elements model, the variation of
J-contour integral is less than 0.01% compared to the 400x200 element model.
Therefore, 100x50 elements model is used for further analysis in order to maintain
adequate accuracy with reasonable computational time.
Energy release rate Time (sec)
612 --- Energy release rate
611.5 -- Time needed to complete job
6160 -- 400
0 50 100 150 200 250 300 350 400
Number of regions
Figure 3-9. Variation of energy release rate at the crack tip with various size macro-
A portion of macro-mechanical model (ABCD) is taken and used for micro-model
as shown in Figure 3-1 (A). As the 100x50 elements (100 regions) for macro-model and
constant cell length (200 um) for micro-model are used, one macro-model element can be
replaced by 60 beam elements for micro-model as shown in Figure 3-10. The
displacements along the boundaries of micro-model are determined by using three points
interpolation. The corresponding three points can be obtained from the previously
described macro-model analysis. For instance, displacements in the x-direction for each
beam element on micro-model along the three nodal points (a, b and c) can be found as
icro(y) = macro y(y 1) + acro macro y(y + ) (3
a 2/2 b ,2 21c 2/2 (
Solid 2-D clement
b Y ( 0' Beam element |
Sy=- x, u(x,y)
Figue 3-10. Embedded beam element (micro model) in two-dimensional eight-node solid
model (Macro model).
In micro-model, two-node beam elements are used to represent the foam
ligaments/struts. After, the displacements along the boundaries of micro-model, the
maximum principal stress stresses at the crack tip o-tp can be calculated from the results
for force and moment resultants obtained from the micro-model as
Op 2 F+ (3-7)
The fracture toughness of the foam is defined as the stress intensity factor that will
cause the crack-tip struts to fail. We assume that the strut material is brittle and will
fracture when the maximum principal stress exceeds the tensile strength -,.
- 1 I1 I 1 1I L
Figure 3-11. Force and moment resultants in struts modeled as beams.
Since we are dealing with linear elasticity, the fracture toughness can be estimated
from the following relation,
0 2000 4000 6000 8000 10000 12000
Number of Element
Figure 3-12. Variation of fracture toughness with the size of micro-models.
The convergence analysis is conducted to evaluate the variation of fracture
toughness with various sizes of micro-model, 3 x 1 macro-model (170 elements in micro-
model), 6x2 (640), 15x5 (3,850), 21x7 (7,490) and 30x10 (15,200). As model size
increases, fracture toughness converges as shown in Figure 3-12. For 3,850 beam
element model, the error in fracture toughness is less than 0.3 % compared to 15,200-
beam elements model. Therefore, the 3,850-model is chosen for further analysis as a
compromise between the accuracy and computational time. The aforementioned methods
will be extended to graded foams and also to understand the effects of thermal stresses in
FRACTURE TOUGHNESS OF GRADED FOAMS UNDER MECHANICAL
In this chapter, the finite element based micromechanical model discussed in the
previous chapter is used to understand the behavior of functionally graded foam (FGF)
and to estimate their fracture toughness (critical stress intensity factor). We will use the
ABAQUS TM finite element package for performing the simulations. Analysis of FGF
containing a crack under remote loading (uniform displacement) was first carried out.
The results of fracture toughness under uniform displacement are compared with the
homogeneous foam in order to understand the behavior of FGF. Thermal loading can
affect the stress field near crack tip unlike the remote loading case. In order to observe
this local effect, we investigated the case where the pressure applied on the crack surface
for various sizes of crack lengths in homogeneous foam. Then, the results were
compared with the remote traction case. For the remote loading, we considered uniform
displacement and traction on the top edge of model. The principle of superposition was
studied to understand the local effect.
Fracture Toughness under Remote Loading
In this section, fracture toughness of functionally graded foams subjected to
uniform displacements on the top and bottom of the model. The height of the model is
considered same as its width, and is symmetric with respect to its midline, y = 0. The
geometry of the FGF is shown in Figure 4-1 (A) with crack length, a. Only half of the
model is considered in the finite element analysis because the model is symmetric with
respect to its midline, x-axis. The upper edge is loaded by uniform displacement, 70 prm.
A zero displacement boundary condition in the y-direction is applied on the lower edge to
account for symmetry. The material is functionally graded, and the relative density
increases or decreases according to the parameters, a or /, described in Chapter 2. The
parameters determine the degree of gradation of the properties. For the case where the
strut dimensions vary and the cell dimension is constant, parameter a determines the
degree of the gradation (Eq. 2-11). f is used for the case where the cell dimension is
varying in x-direction while the strut dimensions are kept constant (Eq. 2-14).
Y O00o orVo Y
> x >
Figure 4-1. Edge-cracked model under A) uniform traction or displacement loading and
B) crack surface traction.
First, we investigate the case wherein the graded foam has constant unit-cell length
(c=200pm) and the density is varied by changing the strut cross sectional dimensions.
Both cases, increasing and decreasing densities along the x-axis, are considered. Table 4-
1 shows the results form the case which the unit-cell dimensions and crack length are
kept constant, but the strut thickness is varied, such as the FGF model is taken from
imaginary graded global panel from different position. The results from the case which
the model has the constant unit cell but the crack length and strut thickness is varied are
shown in Table 4-2.
Table 4-1. Fracture toughness of graded and uniform foams. The unit-cell dimensions
and crack length are kept constant, but the strut thickness is varied (c=200uPm,
crack length, a =0.03m and a'-200x 106).
Fracture Toughness (Pa-m2)
at the crack-tip Decreasing Increasing Uniform
at the crack-tip
density density density
26 0.028 4.52171E+05 4.56445E+05 4.51326E+05
30 0.039744 6.56122E+05 6.57406E+05 6.56114E+05
50 0.123904 2.24739E+06 2.25108E+06 2.24928E+06
60 0.179334 3.39537E+06 3.39999E+06 3.39819E+06
70 0.241664 4.77575E+06 4.78247E+06 4.77936E+06
Table 4-2. Fracture toughness of graded and uniform foams. The unit-cell dimension is
kept constant but the crack length and the strut thickness are varied (c=200
yim, ho=40pum and a-=200x 106).
Normalized relative Density Fracture Toughness (Pa-m )
crack length the crack-tip Decreasing Increasing Uniform
at the crack-tip
(a/W) density density density
0.1 0.06075 1.10144E+06 1.03627E+06 1.03485E+06
0.2 0.068608 1.25052E+06 1.18201E+06 1.18004E+06
0.3 0.076874 1.33362E+06 1.33619E+06 1.33465E+06
0.4 0.085536 1.49961E+06 1.49980E+06 1.49878E+06
0.5 0.094582 1.67268E+06 1.67266E+06 1.67220E+06
As seen in Table 4-1, Table 4-2 and Figure 4-2, the results from the present
analysis for FGF are very close to those of homogeneous foam. However we see an
interesting trend in Figure 4-10. In both deceasing and increasing density cases, the
fracture toughness deviates from that of uniform density foam for higher densities. When
the density decreases along the crack path, the fracture toughness is slightly higher and
0 0.05 0.1 0.15 0.2
Figure 4-2. Comparison of fracture toughness of graded and homogeneous foams having
same density at the crack tip.
Results for the case of varying unit-cell dimensions are presented in Table 4-3 and
also shown in Figure 4-3. The results again show that the fracture toughness of FGF is
close to that of a homogeneous foam with density same as that at the crack tip of FGF.
Table 4-3. Comparison of the fracture toughness for varying unit-cell dimensions with
constant strut thickness (h=20ym).
Crack length in Relative Fracture Toughness (P,-m )
SetCc Density %
et c (m) c2(m) c3 (m) terms of number Density
of elements at the crack- Graded Foam Homogeneous difference
10 0.0745806 9.62060E+05 9.61172E+05 0.092
20 0.0776305 1.00630E+06 1.00531E+06 0.098
1 -0.15e-6 200e-6 100e-6 100e-6 30 0.0812704 1.05636E+06 1.05533E+06 0.097
40 0.0856898 1.11501E+06 1.11279E+06 0.199
50 0.0911693 1.18440E+06 1.18018E+06 0.356
60 0.0981422 1.26090E+06 1.25952E+06 0.109
70 0.221965 2.05336E+06 2.04649E+06 0.335
60 0.228608 2.15865E+06 2.16343E+06 0.221
50 0.236846 2.30923E+06 2.29987E+06 0.405
2 0.15e-6 50e-6 50e-6 50e-6 40 0.247332 2.47076E+06 2.46204E+06 0.353
30 0.261132 2.66760E+06 2.65985E+06 0.291
20 0.280113 2.92063E+06 2.90924E+06 0.390
10 0.307863 3.24507E+06 3.23901E+06 0.187
Table 4-3. Continued
Crack length in Relative Fracture Toughness (PA,-, )
Set a Density %
Se C0 (m) 2 (m) c3 (m) terms of number Densty
of elements at the crack- Graded Foam Homogeneous difference
50 0.0912307 1.18404E+06 1.18051E+06 0.298
40 0.0982215 1.26352E+06 1.26041E+06 0.246
3 0.15e-6 50e-6 100e-6 100e-6 30 0.107422 1.36134E+06 1.35823E+06 0.228
20 0.120075 1.48715E+06 1.48204E+06 0.344
10 0.138575 1.65340E+06 1.64829E+06 0.310
70 0.0450318 7.45601E+05 7.43076E+05 0.339
4 0.15e-6 200e-6 150e-6 100e-6 50 0.0470359 7.89333E+05 7.85621E+05 0.470
30 0.0495308 8.39136E+05 8.35061E+05 0.486
10 0.0527223 9.11742E+05 8.93837E+05 1.964
70 0.0218062 3.73104E+05 3.72085E+05 0.273
5 0.15e-6 200e-6 200e-6 200e-6 50 0.0230945 3.94813E+05 3.92956E+05 0.470
30 0.0246984 4.19129E+05 4.17251E+05 0.448
10 0.0267500 4.48072E+05 4.46171E+05 0.424
S-3 Set3 (FGF)
--- Set3 (Homogeneous)
S--- Set4 (FGF)
0 0.05 0.10 0.15 0.20 0.25 0.30 0
Figure 4-3. Comparison of fracture toughness of graded and homogeneous foams. The
graded foams have varying unit-cell dimensions, but constant strut cross
section h=20 um.
Study of Local Effect on the Homogeneous Foam under Crack Face Traction
In order to observe and analyze the local effects of the local stresses on the
homogeneous foam, the fracture toughness of the homogeneous foam under crack face
traction (0.5 GPa) is compared with remote loading condition, both uniform traction (0.5
GPa) and displacement (70 /on) on the top edge. For this study, the model in the Figure
4-1 (A) is considered for remote loading condition and Figure 4-1 (B) for crack face
loading condition. The model size is fixed and the crack length is varied in order to
investigate how the local stresses around near crack tip affects the fracture toughness for
various crack lengths. The unit cell size and beam thickness are constant, c=200/nm and
h=20/,u which make the relative density of homogeneous foam equal to 0.028. The
crack length varies from 10% (a/c=50) to 50% (a/c=250) of the plate width.
Table 4-4. Remote loading case uniform displacement on the top edge.
a. 0.1 0.2 0.3 0.4 0.5
(normalized crack length)
c 50 100 150 200 250
Stress Intensity Factor 2.431E+05 2.431E+05 2.431E+05 2.431E+05 2.431E+05
(Maximum Principal 1.939E+09 1.939E+09 1.939E+09 1.940E+09 1.940E+09
mo-x/SIF 7.977E+03 7.974E+03 7.976E+03 7.978E+03 7.980E+03
Fracture Toughness 4.513E+05 4.514E+05 4.513E+05 4.512E+05 4.511E+05
Table 4-5. Remote loading case uniform traction on the top edge.
(normalized crack length) 0.1 0.2 0.3 0.4 0.5
c 50 100 150 200 250
Stress Intensity Factor 1.486E+08 2.988E+08 4.763E+08 6.941E+08 9.776E+08
(Maximum Principal 1.194E+12 2.405E+12 3.832E+12 5.582E+12 7.858E+12
-max /SIF 8.030E+03 8.046E+03 8.044E+03 8.041E+03 8.038E+03
Fracture Toughness 4.483E+05 4.474E+05 4.475E+05 4.477E+05 4.479E+05
As seen on Table 4-4, the stress intensity factor and maximum principal stresses at
the crack tip for homogeneous foam under uniform displacement are almost constant
through various lengths. When the uniform traction is applied on the top edge, the stress
intensity factor and maximum principal stresses at the crack tip increase with increasing
the crack length (Table 4-5). However, the ratio between the maximum principal stress
and the stress intensity factor does not vary much for different crack length. In both
cases, the fracture toughness is independent of crack lengths.
Table 4-6. Stress intensity factor, maximum principal stress and fracture toughness for
various crack lengths under crack surface traction.
a. 0.1 0.2 0.3 0.4 0.5
(normalized crack length)
c 50 100 150 200 250
Stress Intensity Factor 1.486E+08 2.988E+08 4.763E+08 6.941E+08 9.776E+08
(Maximum Principal 1.126E+12 2.337E+12 3.764E+12 5.514E+12 7.791E+12
omax /SIF 7.574E+03 7.820E+03 7.902E+03 7.944E+03 7.969E+03
Fracture Toughness 4.753E+05 4.604E+05 4.556E+05 4.532E+05 4.517E+05
Table 4-7. Fracture toughness estimation from remote loading and crack face traction.
ac Remote loading Crack face
(crack length/unit-cell) (uniform traction on the top edge) traction % difference
50 4.48332E+05 4.75294E+05 6.014
100 4.47406E+05 4.60368E+05 2.897
150 4.47514E+05 4.55565E+05 1.799
200 4.47679E+05 4.53177E+05 1.228
250 4.47851E+05 4.51743E+05 0.869
In the case where a pressure is applied along the crack face crack face traction,
the stress intensity factor and the maximum principal stresses at the crack tip increase
with increasing the crack length as occurred in the case of remote loading with uniform
traction. However, the ratio between the maximum principal stress at the crack tip and
the stress intensity factor also increases with crack length. That means that the fracture
toughness decreases as the crack length increases. The fracture toughness for crack face
traction is higher for shorter cracks, but converges to the value for remote traction
condition for longer cracks. The deviation of the fracture toughness for various crack
length under crack surface traction condition is presented in Table 4-7 and plotted in
2.50E+05 -- Remote loading (unifrom
traction on the top edge)
-A- Crack face traction
0 50 100 150 200 250 300
Figure 4-4. Fracture toughness estimation from remote loading and crack face traction.
The principle of superposition can be applied to the crack face loading condition in
order to explain the differences in fracture toughness presented in Table 4-7. As we have
seen in the previous section, the current method can accurately estimate the fracture
toughness for remote loading conditions. Thus, stresses acting on the crack face (i.e.,
crack face traction condition) can be replaced with tractions that act on the top edge
(remote loading condition) and an uncracked body subjected to tractions, as illustrated in
Figure 4-5. Since the value of stress intensity factor for uncracked body is zero, the two
loading configurations (remote traction and crack face traction) result in same stress
intensity factor in macro scale sense and shown in Table 4-5 and 4-6.
K() = K b) K(c) = K(b) (since Kc) = 0) (4-1)
a (a) (b) a. (c)
Crack Face Traction Remote Traction Uncracked Body
Figure 4-5. Application of superposition to replace crack face traction with remote
However, stresses exist at the crack tip in the uncracked body in micro scale sense as
o(c) = (ro (4-2)
The principle of superposition indicates that the maximum principal stress at the crack tip
under crack surface traction is lower than the maximum principal stress at the crack tip
under remote loading (uniform traction on the top edge) as derived in Eq. (4.3) below:.
o(a) =_(b) Oa (4-3)
max max max (4-)
(a) (b) C C and -(a) <(b) (44)
max max o- and max < Umax
The estimated fracture toughness under surface crack traction is higher than the
fracture toughness under uniform traction on the top edge for short crack by current
micro-mechanical model. By employing the principle of superposition, the analytical
solution for the fracture toughness under crack surface traction denoted by Kjs (= K )i
can be found in terms of the fracture toughness under remote loading K (= K:) The
reciprocal of fracture toughness can be obtained by dividing Eq. (4-4) throughout by the
factor (KI, o,) where o, is the ultimate tensile strength of the strut material and K1 is the
stress intensity factor due to the applied stress o-. Then we obtain
1 1 h2
KS KJ -(4-5)
Kisc KIRC o.K,
The stress intensity factor K, is given by
K, = Yo-, a (4-6)
where oo is the applied stress, a is the crack dimensions, and the form factor Yis a
polynomial in a/Wand depends on geometry and mode of loading (Broek 1978). For
edge cracked model under remote traction (Figure 4-5 (a)) Y is given by
-I aa a 2 4
Y= 1.99-0.41 +18.7 -38.48 +53.85 J (4-7)
Using Eq. (4-6), Eq.(4-5) reduces to
1 1 1
KS Kj h2 (4-8)
Then the fracture toughness under crack surface traction can be derived as
KS Kje a > 50 or a > 0.1 (4-9)
KIC c W
From Eq. (4-9), the fracture toughness under crack surface traction K'S can be
calculated from fracture toughness under remote loading K and compared with the
estimated fracture toughness obtained from the developed micro-mechanical model
(Table 4-8). The local effect caused by local stress was analyzed employing the principle
of superposition. The principal of superposition also indicates that the fracture toughness
under crack surface traction is larger than the fracture toughness under remote loading for
short cracks and tends to converge to the fracture toughness under remote loading for
longer cracks as seen on Figure 4-4.
Table 4-8. Comparison of fracture toughness under crack surface traction calculated
from the superposition method and the micromechanical model.
S Current micro-mechanical
a(m) a/W a/c Analytical solution em ome %diff
0.01 0.1 50 4.76554E+05 4.75294E+05 0.264
0.02 0.2 100 4.64162E+05 4.60368E+05 0.817
0.03 0.3 150 4.58684E+05 4.55565E+05 0.680
0.04 0.4 200 4.55260E+05 4.53177E+05 0.458
0.05 0.5 250 4.52874E+05 4.51743E+05 0.250
FRACTURE TOUGHNESS ESTIMATION UNDER THERMAL LOADING
In this Chapter we investigate the foams under thermal loading using the methods
described in Chapter 3. First we investigate the behavior of functionally graded foams
and homogeneous foams under thermal loading. The second order polynomial variation
of temperature is used in order to investigate the behavior of uncracked foams because
the analytical study of a beam under temperature gradient shows that the second order
polynomial variation is the simplest form to create thermal stresses (Appendix). Sankar
and Tzeng (2002) studied thermal stresses in functionally graded beams by using the
Bernoulli-Euler hypothesis. They assumed that the elastic constants of the beam and
temperature vary exponentially through thickness. They found that the thermal stresses
for a given temperature gradient can be reduced when the variation of the elastic
constants are opposite to that of the temperature gradient. In the present work, we study
the difference between fracture toughness under mechanical and thermal loading. The
purpose here is to understand the effects of thermal stress gradients on fracture toughness.
Behavior of Foams under Thermal Loading
The configuration with constant cell size (200 um) with varying the strut thickness
is used to investigate the behavior of uncracked foams under thermal loading. The
investigation was performed in three different cases homogeneous foam, increasing
density and decreasing density foams. The properties of three different models are shown
in Table 5-1. The subscripts "0" and "f' denote the properties at the x=0 and x=width
(0. Im) respectively. The temperature variation was assumed to be of the form
T=75,000x2+7,500x+30 which makes the temperature to increase along the x-axis from
100C to 1,5300C. The temperatures are nodal values in the finite element analysis. The
temperature according to the given second order polynomial variation is assigned at the
vertical set of nodes. The reference temperature in all the models was 20 C. Therefore,
the ATo and ATf are 100C and 1,5100C for all three models.
Table 5-1. Elastic modulus variation of three different models.
Strut thickness (um) Elastic Modulus (GPa)
ho hf Eo Ef (E/E)
Homogeneous 40 40 8.28 8.28 1
Decreasing material 40 20 8.28 2.07 0.25
Increasing material 33
40 80 8.28 33.12 4
The thermoelastic constant was assumed to be of form f/(x) = foJ(x) with o =
Eo/106, where Eo is the elastic modulus at x=0. Then, the thermal stresses were
normalized with respect to the thermal stress term f/oATo. The axial stress distributions for
the three cases are shown in Figure5-1 and plotted in Figures 5-1through 5-3. When the
variation of the material properties was in opposite sense to the temperature variation, the
thermal stresses were reduced. On the other hand, when the variation of the material
properties was in the same sense as the temperature variation, the thermal stresses
increased compare to that of homogeneous foam. The maximum and minimum stresses
in different cases are compared in Figure 5-5 by bar charts where the values are shown
for modulus ratio, Ef/Eo.
The investigation confirms that the behavior of functionally graded foams
naturally adopt the favorable design for thermal protection system because the nature of
functionally graded foams as load carrying thermal protection system should be such that
the cooler inner layer has high solidity, while the hotter outer layer has low solidity.
I~ ~ ~ ~ ~ ~h TD ~ Iic.QU ABaT/t idid .- Uc E =3 15l51l0. Enn ..... Bl'nai "* 30
.. -- -" r" kn
Figure 5-1. Thermal stress distribution output from FEA. A) Homogeneous foam. B)
FGF with the material properties and the temperature has opposite type of
distribution. C) FGF with the variation of the material properties and
temperature in a similar manner.
Figure 5-2. Thermal stress distribution in homogeneous foam.
Figure 5-3. Thermal stresses distribution in an FGF; the material properties and the
temperature have opposite type of variation, and this reduces the thermal
0.25 0.5 0.75
Figure 5-4. Thermal stresses distribution in an FGF; the variation of the material
properties and temperature in a similar manner; and this increases the thermal
S-10 -- Max. Stress I Min. Stress
0.25 1 4
Modulus Ratio (Ef/E )
Figure 5-5. Maximum and minimum values of normalized thermal stresses.
Results under Thermal Loading
For the analytical solution of model under thermal loading, Euler-Bernoulli beam
theory was used due to its simplification. One of the assumptions for Euler-Bernoulli
beam theory is that the beam should be long and slender (i.e. length >> depth and width).
To compare the finite element results with analytical solution, the beam aspect ratio
(length/width) should be investigated because we have used the model for the aspect ratio
was unity. The width ofuncracked model is 0.05m, and length changed from 0.05 to
0.5m. The temperature gradient is a function of the position. The temperature variation
was assumed to be of the form T(x) = 1- 400 x2 which makes the temperature increases
along the x-axis form 0C to 1 OC. Figure 5-6 shows the thermal stresses developed on
models for different aspect ratios. The model with aspect ratio 10 gives very close
thermal stress distribution as the analytical solution. Thus, we use the model which has
ten times lager length than width in this section. The thermal stress in the models with
various crack length is shown in Figure 5-7.
.)K XAR-10 A
0.0(94:t1 )))2 ()0.0)3 .)4 0. )5
Figure 5-6. Thermal stresses for various aspect ratio models.
SA O c=0.005m
2.50E+03 X c=0.02m
2E O c=0.025m
X O c=0.03m
| 1.50E+03 + c=0.035m
5.00E+02 0-)) A^ X ^O .4)
0.01 A- ^02 "Xx O aO4 5
Figure 5-7. Thermal stresses for various crack lengths.
Table 5-2. Results of the body under temperature gradient form micromechanical model.
a. 0.1 0.2 0.4 0.5 0.6 0.65
(normalized crack length)
25 50 100 125 150 175
Stress Intensity Factor 1.68E+02 1.70E+02 1.08E+02 6.97E+01 3.46E+01 6.45E+00
(Maximum Principal 1.54E+06 1.58E+06 1.03E+06 6.81E+05 3.55E+05 8.82E+04
o-m, /SIF 9185.8 9325.3 9572.6 9782.5 10266.7 13673.4
Fracture Toughness 3.91E+05 3.86E+05 3.76E+05 3.68E+05 3.50E+05 2.63E+05
As described in the previous chapter, the fracture toughness under crack surface
traction converges to the fracture toughness under remote traction as crack size increases.
However, we should notice that the negative stress intensity factor exists and the value
increases as the crack length increases. Therefore, the stress intensity factor decreases
with larger crack size, and consequently the fracture toughness decreases with larger
crack size (Table 5-2). The ratio between the maximum principal stress and the stress
intensity factor is compared with that of remote loading condition shown in Figure 5-8.
The fracture toughness for various crack lengths are shown in Figure 5-9. The result of
estimated fracture toughness under thermal loading is similar to that for crack face
loading, except the sign is reversed. This can be explained by using the principle of
superposition as described in chapter 4.
S4.00E+03 Thermal Loading
0 Remote Loading
0 50 100 150 200
Figure 5-8. Comparison the ratio of maximum principal stress and stress intensity factor.
The results obtained in this chapter indicate that the fracture toughness of a cellular
material depends on the stress gradients produced by thermal stresses. This is similar to
the results obtained in the previous chapter where the fracture toughness was different for
crack surface loading. Thus the nominal fracture toughness obtained from remote
loading tests should be corrected appropriately when stress gradients are presented.
0 50 100 150 200
Figure 5-9. Fracture toughness for various crack lengths.
Finite element based micromechanical methods have been developed to understand
the fracture behavior of functionally graded foams. The finite element analysis used a
micromechanical model in conjunction with a macromechanical model in order to relate
the stress intensity factor to the stresses in the struts of the foam. The stress intensity
factor of the macromechanical model at the crack tip was evaluated by two different
method energy method and point matching method. In the energy method, J-contour
integral was used, and the stress field ahead of crack tip was used to estimate stress
intensity factor in the point matching method. The maximum principal stress at the crack
tip was evaluated from the force and moment resultants obtained from the micro-model.
Then, fracture toughness was estimated by relating the stress intensity factor and the
maximum principal stresses at the crack tip. In addition, convergence tests for both
macromechanical and micromechanical models analyses were conducted.
First we investigated the fracture behavior of functionally graded foam under
uniform displacement remote mechanical loading condition in order to demonstrate the
feature of the current method. Then, the method is extended to another remote
mechanical loading uniform traction. Then, the results of remote loading conditions for
graded foams are compared with the result for homogeneous foam. The fracture
toughness was obtained for various crack positions and lengths within the functionally
graded foam. Then the relationship between the fracture toughness of foams and the local
density at the crack tip was studied. It was found that the fracture toughness of
functionally graded foam is approximately the same as that of homogeneous foam with
the same density at the crack tip.
We also investigated the effects caused by the local stresses. In order to observe
and analyze the local effects of the local stresses on the homogeneous foam, the fracture
toughness of the homogeneous foam under crack face traction is compared with remote
loading conditions, both uniform traction and displacement on top edge. The relationship
between the stress intensity factor and the maximum principal stress was compared. It
was found that the fracture toughness under remote loading condition is independent of
the crack size. However, the ratio between the maximum principal stress at the crack tip
and the stress intensity factor under crack face traction increased with crack length.
Thus, the fracture toughness for crack face traction is higher for shorter cracks, and
converges to the value for remote traction condition as the crack length increases. It is
found that the principle of superposition can be used to adjust the local effect caused by
the differences in maximum principal stresses under crack face loading condition. A
correction factor in terms of crack length is proposed to determine the fracture toughness
of short cracks under crack surface loading.
From the thermal protection system point of view the behavior of functionally
graded foam under thermal loading was investigated. When the variation of the material
properties is in opposite sense to the temperature variation, the thermal stresses is
reduced. On the other hand, when the variation of the material properties is in the same
sense as the temperature variation, the thermal stress increases compared to that in
homogeneous foam. The result of estimated fracture toughness under thermal loading is
similar to that for crack face loading.
The present dissertation demonstrates the use of finite based micromechanical
model to predict the fracture toughness of functionally graded foams by using simple
micro structures. To achieve more accurate prediction, we need to focus on a more
realistic cell configuration, which captures the complexity of foam and predicts more
accurately its mechanical property changes, such as relative density and modulus in
functionally graded foam. The methods discussed here will help in understanding the
usefulness of functionally graded foams in the thermal protection systems of future space
ANALYTICAL SOLUTION FOR BEAM MODEL UNDER THERMAL LOADING
b x, u
Figure A-1. A beam of rectangular cross section with no restraint.
If we assume the both ends are perfectly clamped, the thermal stress, -, is defined
a7 (y) =-aEAT(y) (A-1)
Due to constraints at both ends, the thermal stress prevents extension and bending
of the beam and producing internal force, P and bending moment, M
P = abdy
M = f JTyb dy
If the beam has no restraints against extension, the internal force, P must be
eliminated by a virtual force, pT
PT = P= aEAT(y)bdy (A-4)
If the ends are free to rotate and no external moment applied, the bending moment
M at the ends must be eliminated by a virtual bending moment M' at the ends
M = -M = aEAT(y)ybdy (A-5)
The thermal stress corresponding to the virtual force, pT is
PT 1h h
S= = aEAT(y)b dy aEAT(y) dy (A-6)
'p AEA= dy
A = (2h)* b = 2hb
and the thermal stress corresponding to the virtual bending moment, M' is
T= aT EAT(y)yb dy = aEAT(y)ydy (A-7)
Therefore, the thermal stress 'x in the beam with no constraints at both ends is
o (y)= -aEAT(AT(y) dy + y y aEAT(y)ydy (A-8)
(y) = aE -AT(y)+ AT(y)dy+ 3 ArT(y)ydy (A-9)
Example 1) Constant, AT = CO
fATMdy =[Cyh = 2Ch
fATy dy =Co2 =0
-h 2 h
ax (y) = aE -AT +- Tdy+ ATy dy
= aE -C+ +- (2Coh)
Example 2) Linear variation, AT(y) = CO + Cly
hAT(y)dy= Coy+ Cy2 h = 2Coh
AT(y)ydy= -Coy2 + Cy3 =-Ch3
-h L2 3 -h 3
S 1 + 3y
x(y) = aE -AT(y)+ 2h J AT(y)dy+ 2 J T(y)ydy]
= aE -(Co +Cy) + (2Coh) + (2Ch3)
2h 2h 3
Example 3) Quadratic variation, AT(y) = Co + Cy+C2y2
-h L 2 3 J_
SCoy2 +- Cy3
I AT(y) dy +
(CO +Cly+C2y2) 2Coh2C2h3 3y 2Ch3
2h 3 2h3 3
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Seon-Jae Lee was born in Seoul, Korea, in 1969. He studied Physics in Kyoung-
Won University for three years and entered Department of Aerospace Engineering at
Royal Melbourne Institute of Technology, Melbourne, Australia. He transferred to
Embry Riddle Aeronautical University at Daytona Beach, Florida, and received his
Bachelor of Science in Aerospace Engineering in April 1999. From there he completed
his graduate study of a Master of Science in Aerospace Engineering in May 2002, with
research in the area of vibration analysis of composite box beam. In August 2002, he
joined the Advanced Composites Center at the Department of Mechanical and Aerospace
Engineering, University of Florida, Gainesville, Florida, for his Ph.D. degree. After
completion of his Ph.D. degree, Seon-Jae will begin work at Samsung Techwin in Korea
to contribute his efforts to the development of precise machine.