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ANALYTICAL AND FINITE ELEMENT BASED MICROMECHANICS FOR FAILURE THEORY OF COMPOSITES
By
SAI THARUN KOTIKALAPUDI
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2017
© 2017 Sai Tharun Kotikalapudi
To Amma and Nanna for the incessant support and always believing in me
4
ACKNOWLEDGMENTS
I would like to express my gratitude to my thesis advisor Dr. Bhavani V. Sankar
for being a very supporting guide and mentor, and for giving me a chance to participate
in his research program critical to composite industry. He has consistently steered me in
the right direction and been a lamppost on this hazy road of exploration. I would also
like to thank Dr. Ashok V. Kumar for willing to be a member of the supervisory
committee and offer constructive criticism wherever needed.
I am grateful for my parents and Srinivas Namagiri for all their support and
encouragement. Without them this day would have never dawned. I am beholden to
Sumit Jagtap for his guidance on Abaqus as well as to Apoorva Walke and Maleeha Babar
for being a constant source of moral support. I would also like to acknowledge all my
teachers from University of Florida and SASTRA University, and my friends and family
for being there for me in the time of need.
5
TABLE OF CONTENTS page
ACKNOWLEDGMENTS .................................................................................................. 4
LIST OF TABLES ............................................................................................................ 6
LIST OF FIGURES .......................................................................................................... 8
LIST OF ABBREVIATIONS ........................................................................................... 11
ABSTRACT ................................................................................................................... 12
CHAPTER
1 INTRODUCTION .................................................................................................... 14
Literature Review .................................................................................................... 14
Research Scope ..................................................................................................... 15
2 ANALYTICAL EQUATIONS .................................................................................... 20
Introduction to the Three-Phase Model ................................................................... 20 Halpin Tsai Formulation for Composite Properties ................................................. 21 Longitudinal and Hydrostatic Stress Equations ....................................................... 23
Longitudinal Shear Stress in the x-y plane.............................................................. 27
Longitudinal Shear Stress in the x-z plane.............................................................. 32 Biaxial tension/compression in y-z plane ................................................................ 37 Transverse Shear Equations .................................................................................. 43
3 FINITE ELEMENT ANALYSIS AND COMPARISON .............................................. 50
Modelling and analysis of Hexagonal RVE ............................................................. 50 Comparison with analytical model .......................................................................... 61
4 ANALYTICAL MODEL RESULTS AND DISCUSSION ........................................... 65
Results for Kevlar/Epoxy ......................................................................................... 65
Carbon/Epoxy plots ................................................................................................ 69 Effects of Interface .................................................................................................. 73 Volume fraction analysis ......................................................................................... 81
Summary ................................................................................................................ 84
5 CONCLUSIONS AND FUTURE WORK ................................................................. 87
LIST OF REFERENCES ............................................................................................... 90
BIOGRAPHICAL SKETCH ............................................................................................ 92
6
LIST OF TABLES
Table page 2-1 Comparison of macro stresses with average micro stresses for longitudinal
shear stress ........................................................................................................ 22
2-2 Comparison of macro stresses with average micro stresses for normal Stress and in plane shear stress ................................................................................... 22
3-1 Properties of Kevlar/Epoxy used in the FEA ....................................................... 52
3-2 Coefficients of stiffness matrix obtained from unit strain analysis ....................... 56
3-3 Transverse strengths at various points for Kevlar/Epoxy (plane strain) .............. 60
3-4 Comparison of various transverse strengths for Kevlar/Epoxy (plane strain) ..... 63
3-5 Comparison of maximum principal stress for Kevlar/Epoxy (plane strain) .......... 63
3-6 Comparison of maximum von Mises stress for Kevlar/Epoxy (plane strain) ....... 63
3-7 Comparison of average of top 10% maximum principal stresses for Kevlar/Epoxy (plane strain) ................................................................................ 63
3-8 Comparison of average of top 10% von Mises stresses for Kevlar/Epoxy (plane strain)....................................................................................................... 63
3-9 Comparison of 10th percentile maximum principal stress for Kevlar/Epoxy (plane strain)....................................................................................................... 64
3-10 Comparison of 10th percentile maximum von Mises stress for Kevlar/Epoxy (plane strain)....................................................................................................... 64
4-1 Properties of Kevlar/Epoxy ................................................................................. 66
4-2 Strengths at various points for Kevlar/Epoxy (MPa) ........................................... 69
4-3 Properties of Carbon-T300/Epoxy-5208 ............................................................. 70
4-4 Predicted strengths of T300/5208/Carbon/Epoxy ............................................... 73
4-5 Comparison of strengths for Kevlar/epoxy including interface failure obtained using ADMM ....................................................................................................... 80
4-6 Comparison of strengths for Carbon/Epoxy including interface failure obtained using ADMM ........................................................................................ 81
7
4-7 Comparison of strengths for several composites with analytical model strengths ............................................................................................................. 85
4-8 %Difference of strengths for several composites relative to reference strengths ............................................................................................................. 85
8
LIST OF FIGURES
Figure page 1-1 Depiction of a RVE for the analytical model ....................................................... 16
1-2 Decomposition of macro stresses applied to an RVE of a fiber composite ......... 16
1-3 Macro stresses applied on the unit cell. (similar to 𝜏12 , 𝜏13 will be acting in the 13 plane and 𝜏23 will be acting in the 2-3 plane) .......................................... 17
1-4 Decomposition of applied state of macro stresses into five cases ...................... 18
2-1 Three-phase model ............................................................................................ 20
3-1 Representative volume element of a hexagonal unit cell .................................... 50
3-2 Coordinate system used in ABAQUS and principal coordinate system .............. 51
3-3 Sectional view and dimensions of the RVE ........................................................ 51
3-4 Meshed RVE, red bounded regions represent fiber and green unbounded region represents matrix ..................................................................................... 52
3-5 Element type used for meshing and analysis ..................................................... 53
3-6 Boundary conditions and loading in unit strain analysis (A) Direction 2 (B) direction 3 ........................................................................................................... 54
3-7 Schematic of the procedure followed to obtain Stiffness matrix.......................... 55
3-8 Initial and deformed hexagonal RVE under unit strain in A) 2nd Direction B) 3rd direction C) 2nd and 3rd direction..................................................................... 56
3-9 Schematic of procedure to plot a failure envelope in 2-3 plane .......................... 59
3-10 Failure envelopes of Kevlar/Epoxy in transverse direction obtained through unit strain analysis .............................................................................................. 60
3-11 Comparison of analytical and finite element model failure envelopes using maximum stress theory in the transverse plane (2-3 plane) ............................... 61
3-12 Comparison of analytical and finite element model failure envelopes using quadratic theory in the transverse plane (2-3 plane) .......................................... 62
4-1 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy on 𝜎1 − 𝜎2 plane ................................................................................................................... 66
9
4-2 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy in the 𝜎2 −𝜎3 plane .............................................................................................................. 67
4-3 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy for longitudinal shear in the 1-2 or 1-3 plane ........................................................... 67
4-4 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy subjected to both longitudinal and transverse shear stresses............................................. 68
4-5 Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy for shear in longitudinal direction and stress in fiber direction ............................................... 68
4-6 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 1-2 plane ................................................................................................................... 70
4-7 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 2-3 plane ................................................................................................................... 71
4-8 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for shear in longitudinal directions ..................................................................................... 71
4-9 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy subjected to both longitudinal and transverse shear stresses............................................. 72
4-10 Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for longitudinal shear and normal stress in fiber direction ........................................ 72
4-11 Interface effects on failure envelopes for Kevlar/Epoxy in 1-2 plane using maximum stress theory ...................................................................................... 74
4-12 Interface effects on failure envelopes for Kevlar/Epoxy in 1-2 plane using quadratic theory .................................................................................................. 75
4-13 Interface effects on failure envelopes for Kevlar/Epoxy in 2-3 plane using maximum stress theory ...................................................................................... 75
4-14 Interface effects on failure envelopes for Kevlar/Epoxy in 2-3 plane using quadratic theory .................................................................................................. 76
4-15 Interface effects on failure envelopes subjected to both longitudinal and transverse shear stresses using maximum stress theory ................................... 76
4-16 Interface effects on failure envelopes subjected to both longitudinal and transverse shear stresses using quadratic theory .............................................. 77
4-17 Interface effects on failure envelopes for Carbon/Epoxy in 1-2 plane using maximum stress theory ...................................................................................... 77
10
4-18 Interface effects on failure envelopes for Carbon/Epoxy in 1-2 plane using quadratic theory .................................................................................................. 78
4-19 Interface effects on failure envelopes for Carbon/Epoxy in 2-3 plane using maximum stress theory ...................................................................................... 78
4-20 Interface effects on failure envelopes for Carbon/Epoxy in 2-3 plane using quadratic theory .................................................................................................. 79
4-21 Interface effects on failure envelopes for Carbon/Epoxy for envelopes subjected to both longitudinal and transverse shear stresses ............................ 79
4-22 Interface effects on failure envelopes for Carbon/Epoxy longitudinal shear and normal stress in fiber direction using maximum stress theory ..................... 80
11
LIST OF ABBREVIATIONS
ADMM Analytical Direct Micromechanics Method
DMM
FEA
PBC
RVE
Direct Micromechanics Method
Finite Element Analysis
Periodic boundary Conditions
Representative Volume Element
12
Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
ANALYTICAL AND FINITE ELEMENT BASED MICROMECHANICS FOR FAILURE
THEORY OF COMPOSITES
By
Sai Tharun Kotikalapudi
December 2017
Chair: Bhavani V. Sankar Major: Mechanical Engineering
An analytical method using elasticity equations to predict the failure of a
unidirectional fiber-reinforced composite under multiaxial stress is presented. This
technique of calculating micro-stresses using elasticity equations and estimating the
strengths of a composite is based on the Direct Micromechanics Method (DMM).
Prediction of failure using phenomenological failure criteria such as Maximum Stress,
Maximum Strain and Tsai-Hill theories have been prevalent in the industry. However,
DMM has not been used in practice due to its prohibitive computational effort such as
the finite element analysis (FEA). The present method replaces the FEA in DMM by
analytical methods, thus drastically reducing the computational effort.
A micromechanical analysis of unidirectional fiber-reinforced composites is
performed using the three-phase model. A given state of macro-stress is applied to the
composite, and the micro-stresses in the fiber and matrix phases and along the fiber-
matrix interface are calculated. The micro-stresses in conjunction with failure theories
for the constituent phases are used to determine the integrity of the composite. The
analytical model is first verified by comparing with results from finite element based
micro-mechanics. Then, it is used to study the failure envelopes of various composites.
13
The effects of fiber-matrix interface on the strength of the composite is studied. The
results are compared with those available in the literature. It is found that the present
analytical Direct Micro-Mechanics (ADMM) predicts the strength of composites
reasonably well.
14
CHAPTER 1 INTRODUCTION
Literature Review
With the growing application of fiber composites, and with the tremendous
progress in low-cost manufacturing of composite structures, e.g., wind turbine blades,
there is a need to develop efficient predictive methodologies for the behavior of
composites. This should include probabilistic methods to aid in nondeterministic
optimization tools used in design. While methods to predict stiffness properties are well
established, methods to predict failure and fracture properties are still evolving.
Computational material science is the new field of study which attempts to use modern
computational analysis tools to perform multiscale analysis beginning from atomic scale
all the way up to structural scale.
While large scale computational methods are being advanced, there is always a
need for simple and efficient analytical methodologies. This is especially true for
strength prediction and failure behavior of composites. Currently available methods for
strength prediction either use numerical simulations such as finite element analysis [1]
or very simple methods such as mechanics of materials models (MoM) [2]. The former
can be expensive and time consuming, and the latter is only an approximate estimate to
be useful in practical design applications. The current study is aimed at developing an
analytical micromechanics method that is better than MoM models, but still not as
complicated as FEA based micromechanics. To this end we use the principles of Direct
Micromechanics Method [3] developed in 1990s in conjunction with the classical three-
phase elasticity model for unidirectional fiber composites [4].
15
Several failure theories for composite materials are available in the literature.
Majority of them use experimental results along with an empirical (phenomenological)
approach to plot the failure envelopes.
Contemporary failure theories, developed for unidirectional composites such as
Maximum Stress Theory, Maximum Strain Theory and Tsai-Hill Theory have been
thoroughly studied and implemented by various researchers and design engineers.
Direct micromechanics method (DMM) has a propitious approach to predict failure
strengths for an orthotropic composite material. First proposed by Sankar, it has been
widely used to analyze various phenomenological failure criteria, e.g., Marrey and
Sankar [5], Zhu et al. [6], Stamblewski et al. [7], karkakainen and Sankar. DMM
encompasses analytical techniques which is an alternative approach for physical testing
and experimental procedures. A micromechanical model is subjected to multiple macro
stresses which produce micro stresses in each element of the finite element model. The
micro stresses are used to devise a failure envelope considering various failure criteria
for fiber and matrix such as maximum principal stress theory and von Mises criterion.
The interface of fiber/matrix played a pivotal role in the failure of envelopes which was
also considered in DMM. Interfacial tensile stress and interfacial shear stress in the
composite are very sensitive properties that depend on various factors.
Research Scope
In this section, the research procedure followed will be discussed in detail. The
RVE for the analytical approach has been modeled as circular fiber surrounded by an
annular region of matrix. The description of analytical model is presented in Chapter 2.
16
Figure 1-1. Depiction of a RVE for the analytical model.
The applied macro stresses on the composite are denoted by the Cauchy stress
tensor [𝜎𝑥 𝜎𝑦 𝜎𝑧 𝜏𝑦𝑧 𝜏𝑥𝑦 𝜏𝑥𝑧], Note that throughout this study the principal material
coordinate system of the fiber composite will be denoted either by the standard 1-2-3
coordinates or x-y-z coordinates used in the commercial finite element software
ABAQUS. The two normal stresses 𝜎𝑦 and 𝜎𝑧 can be decomposed into two cases
hydrostatic stress state such that 𝜎𝑦 = 𝜎𝑧 = 𝜎𝐻 and a biaxial tension/compression
such that 𝜎𝑦 = −𝜎𝑧. The chart in Figure1-2 depicts various analysis of the stress cases
needed to complete the DMM.
Figure 1-2. Decomposition of macro stresses applied to an RVE of a fiber composite
Applied stress
Normal stress
Axail and hydrostaic
Biaxial tension and compresion
Shear stress
Transverse
Shear YZ
Longitudinal Shear XZ
Longitudinal Shear XY
Fiber
Matrix
Composite
17
Application of the stress field on the Representative Volume Element (RVE) of
the composite through individual cases generates macro strains. Every case has
distinct analytical equations for calculating the micro stresses in the fiber and matrix
phases. The load factors are calculated based on the type of failure criterion used either
maximum stress or some form of quadratic failure criterion, e.g. von Mises for isotropic
materials.
Figure 1-3. Macro stresses applied on the unit cell. (similar to 𝜏12 , 𝜏13 will be acting in the 13 plane and 𝜏23 will be acting in the 2-3 plane)
𝜏12
𝜏12
𝜏12
𝜏12
𝜎1
𝜎3
𝜎3
𝜎2 𝜎2
𝜎1
18
Figure 1-4. Decomposition of applied state of macro stresses into five cases A) Hydrostatic and longitudinal stress; B) Biaxial tension and compression; C) Shear in 2-3 plane; D) Shear in 1-2 plane; E) Shear in 1-3 plane
𝜏23
𝜏23 𝜏23
𝜎1
𝜎1
𝜎𝐻 𝜎𝐻
𝜎𝐻
𝜎𝐻 =(𝜎2 + 𝜎3)
2
(A) Case i
𝜎𝑇𝐶
−𝜎𝑇𝐶
−𝜎𝑇𝐶
𝜎𝑇𝐶 =(𝜎2 − 𝜎3)
2
(B) Case ii
(E) Case v
𝜏13
(C)Case iii
𝜏23
(D)Case iv
𝜏12
19
Figure 1-5. Schematic depiction of DMM followed to obtain failure envelopes
Figure 1-3 shows the six stresses acting on the unit cell of the composite which
are divided into five cases as shown in figure 1-4. For each case the micro stresses are
calculated at several locations using stress equations which is explained in detail in
chapter 2. Figure 1-5 portrays a schematic of the process followed in Direct
Micromechanics Method (DMM) to obtain the failure envelopes and strengths. Chapter
2 elaborates the stress equations used for micromechanical analysis and validation of
using energy methods. The analytical equations employed are further validated in
Chapter 3 through unit strain analysis in finite element analysis software ABAQUS.
Chapter 4 consists of the results obtained from the analytical model wherein a thorough
study is performed by considering two different materials i.e. isotropic and transversely
isotropic. A meticulous comparison on different strengths of composites with present
data is included in Chapter 4. A study is performed in Chapter 4 to understand the effect
of fiber volume fraction on the strength properties for few materials.
Macro stressesIsolating
stresses/formulating stress equations
Micro stresses
Eigen values/principal
stressesLoad Factors
Failure envelopes and strengths
20
CHAPTER 2 ANALYTICAL EQUATIONS
Introduction to the Three-Phase Model
In this section, the three-phase concentric cylinder composite assemblage model
is described. The three phase model proposed by Christensen [8] had been
successfully used in the past for predicting the elastic constants of fiber composites,
e.g. Flexible-resin/glass-fiber composite lamina [9]. In the present study we investigate
the use of three phase model to predict the strengths of unidirectional fiber composites.
Figure 2-1. Three-phase model
The model shown in Figure 2-1. consists of a single cylindrical inclusion (fiber)
embedded in a cylindrical annular region of matrix material. The composite cylinder is in
turn embedded in infinite medium properties of which are equal to that of the composite
material studied. The fiber and the composite are assumed transversely isotropic and
the matrix is isotropic. This enables us to use a polar coordinate system for the analysis.
Furthermore, the entire assemblage is in a state of generalized plane strain as the strain
2
1
3
r b
∞
a Fiber
Matrix
Composite
21
휀1 must be uniform and the same in all three phases. Thus, the problem becomes a
plane problem. Since we are using the model to calculate the micro-stresses in the fiber
and matrix for a give macro-stress state, the elastic constants of all three phases must
be available for the stress analysis. As a first step, the Rule of Mixtures and Halpin-Tsai
equations are used to estimate the elastic constants of the composite. In each analysis
energy equivalence verifies the validity of the input composite elastic constants as
explained in subsequent sections.
Halpin Tsai Formulation for Composite Properties
Halpin-Tsai equation is a widely used semi-empirical formulation for transverse
moduli 𝐸2, 𝐺12 of unidirectional fiber composites. The general form of Halpin-Tsai
equations for a property, say 𝑃, is as follows:
𝑃𝑐 = 𝑃𝑚 (1 + 𝜉𝜂𝑉𝑓
1 − 𝜂𝜈𝑓) (2-1)
Where,
𝜂 = ((𝑃𝑓/𝑃𝑚) − 1
(𝑃𝑓/𝑃𝑚) + 𝜉) (2-2)
𝑃𝑐: Property of the composite
𝑃𝑓: Property of the fiber
𝑃𝑚: Property of the matrix
𝜉: Curve fitting parameter
𝑉𝑓: Fiber volume fraction
The above formula was obtained using curve fitting the result for square array of
circular fibers. It is found that for 𝜉 = 2, an excellent fit is obtained for transverse
modulus 𝐸2. Whereas for shear modulus 𝐺12, the value 𝜉 = 1 was in excellent
22
agreement with the Adams and Doner [10] solution. In both cases a fiber volume
fraction of 𝑉𝑓 = 0.55 is used.
Since the analytical model in this paper has a circular fiber in an annular region
of matrix, the curve fitting parameter has been adjusted to formulate more accurate
predictions of moduli - 𝐸2, 𝐺12. The curve fitting parameter 𝜉 was estimated for the
present case by comparing the applied macro stresses in each case to the volume
average of the corresponding micro stresses. The modified values of 𝜉 are: 𝜉 = 1.16 for
transverse Young’s modulus 𝐸2, 𝜉 = 1 for transverse shear modulus 𝐺23 and 𝜉 = 1 for
longitudinal or axial shear modulus 𝐺12.
Table 2-1. Comparison of macro stresses with average micro stresses for longitudinal shear stress
case Fiber Matrix Average stresses
𝜏𝑥𝑧 𝜏𝑥𝑦 𝜏𝑥𝑧 𝜏𝑥𝑦 𝜏𝑥𝑧 𝜏𝑥𝑦
𝜏𝑥𝑧=1 1.235 0 0.647 0 1 0
𝜏𝑥𝑦=1 0 1.235 0 0.647 0 1
Table 2-2. Comparison of macro stresses with average micro stresses for normal stress
and in plane shear stress Case Fiber Matrix Average stresses
𝜎𝑥 𝜎𝑦 𝜎𝑧 𝜏𝑦𝑧 𝜎𝑥 𝜎𝑦 𝜎𝑧 𝜏𝑦𝑧 𝜎𝑥 𝜎𝑦 𝜎𝑧 𝜏𝑦𝑧
𝜎𝑥=1 1.612 0 0 0 0.081 0 0 0 1 0 0 0
𝜎𝑦=1 -0.18 1.166 -0.07 0 0.274 0.752 0.105 0 0 1 0 0
𝜎𝑧=1 -0.18 -0.07 1.166 0 0.274 0.105 0.752 0 0 0 1 0
𝜏𝑦𝑧=1 0 0 0 1.236 0 0 0 0.647 0 0 0 1
23
The adjusted curve fitting parameter 𝜉 for transverse modulus is 1.16,
considering the average stresses in the composite compared satisfactorily with the input
non-zero stress for each case as depicted in tables 2-1 and 2-2.
To summarize, longitudinal modulus 𝐸1 of the composite is calculated from rule
of mixtures:
𝐸1 = 𝐸1𝑓𝑉𝑓 + (1 − 𝑉𝑓)𝐸𝑚 (2-3)
Here, 𝐸1: Longitudinal modulus of the composite
𝐸1𝑓: Longitudinal modulus of the fiber
𝐸𝑚: Young’s modulus of matrix
𝑉𝑓: Fiber volume fraction
Transverse modulus 𝐸2, longitudinal shear modulus 𝐺12 and in-plane shear
modulus 𝐺23 are obtained from Halpin-Tsai equations.
Longitudinal and Hydrostatic Stress Equations
In this section, the displacement equations for the cases of longitudinal and
hydrostatic stresses are derived. Since we are using the composite cylinder model,
cylindrical coordinate system is used. The above two cases are also axis-symmetric.
Since the composite cylinder is under plane strain condition and an axisymmetric
model is assumed, we have only one non-zero displacement, which is the radial
displacement, 𝑢𝑟 . The displacement equation for all the three phases (fiber, matrix and
composite) is given below.
𝑢𝑟𝑖 = 𝐴𝑖𝑟 +𝐵𝑖
𝑟 (2-4)
24
Here, 𝑢𝑟𝑖 radial displacement in phase 𝑖 and 𝐴𝑖 , 𝐵𝑖 are constants to be
determined using various interface and boundary conditions.
The radial displacement equation for the fiber phase is shown below. Here, the
subscript 𝑓 denotes all the variables are pertaining to the fiber phase.
𝑢𝑟𝑓 = 𝐴𝑓𝑟 +𝐵𝑓
𝑟 (2-5)
One can deduce 𝐵𝑓 = 0 as the displacement at the center of the fiber is zero. The
strains are derived as follows:
휀𝑟 =𝜕𝑢𝑟
𝜕𝑟→ 휀𝑟𝑓 = 𝐴𝑓 (2-6)
휀𝜃 =1
𝑟
𝜕𝑢𝜃
𝜕𝜃+
𝑢𝑟
𝑟→ 휀𝜃𝑓 = 𝐴𝑓 (2-7)
The radial displacement equation for the matrix phase is derived below. The
subscript 𝑚 denotes all the variables are pertaining to the matrix phase.
𝑢𝑟𝑚 = 𝐴𝑚𝑟 +𝐵𝑚
𝑟 (2-8)
Following procedures as in fiber phase above, we get
휀𝑟𝑚 = 𝐴𝑚 −𝐵𝑚
𝑟2 (2-9)
휀𝜃𝑚 = 𝐴𝑚 +𝐵𝑚
𝑟2 (2-10)
The radial displacements in the composite phase are shown below. Here, the
subscript 𝑐 denotes all the variables are pertaining to the composite phase.
𝑢𝑟𝑐 = 𝐴𝑐𝑟 +𝐵𝑐
𝑟 (2-11)
휀𝑟𝑚 = 𝐴𝑐 −𝐵𝑐
𝑟2 (2-12)
휀𝜃𝑐 = 𝐴𝑐 +𝐵𝑐
𝑟2 (2-13)
25
Considering a composite unit cell, the strains along the longitudinal direction can
be equated as follows:
휀𝑥𝑓 = 휀𝑥𝑚 = 휀𝑥𝑐 = 휀0 (2-14)
Where, 휀0 is the longitudinal strain in the fiber direction applied to the composite.
Continuity Equations
Continuity of displacement and radial stresses must be ensured along the
fiber/matrix interface and matrix/composite interface. The interface continuity equations,
say between surfaces i and j, are given below:
𝑢𝑟𝑖(𝑟) = 𝑢𝑟𝑗(𝑟) (2-15)
𝜎𝑟𝑖 = 𝜎𝑟𝑗 (2-16)
From the equation of continuity of displacement along the fiber/matrix interface
𝑢𝑟𝑓(𝑎) = 𝑢𝑟𝑚(𝑎) (2-17)
𝐴𝑓𝑎2 − 𝐴𝑚𝑎2 − 𝐴𝑐 = 0 (2-18)
Here,
𝑎: Radius of the Fiber Phase
From the equation of continuity of radial stress along the fiber/matrix interface
𝜎𝑟𝑓(𝑎) = 𝜎𝑟𝑚(𝑎) (2-19)
The constitutive relation for transversely isotropic materials can be written as
{
𝜎𝑥
𝜎𝑟
𝜎𝜃
} = [𝐶11 𝐶12 𝐶12
𝐶12 𝐶22 𝐶23
𝐶12 𝐶23 𝐶22
] {
휀𝑥
휀𝑟
휀𝜃
} (2-20)
Hence, the stress continuity Equation (2-19) takes the form,
[𝐶12 𝐶22 𝐶23]𝑓 {
휀𝑥
휀𝑟
휀𝜃
}
𝑓
= [𝐶12 𝐶22 𝐶23]𝑚 {
휀𝑥
휀𝑟
휀𝜃
}
𝑚
26
𝐶12𝑓휀𝑥𝑓 + 𝐶22𝑚휀𝑟𝑓 + 𝐶23𝑓휀𝜃𝑓 = 𝐶12𝑚휀𝑥𝑚 + 𝐶22휀𝑟𝑓 + 𝐶23휀𝜃𝑚 (2-21)
For simplicity, we use the following notations
𝐶22𝑓 + 𝐶23𝑓 = 𝛼𝑓 (2-22)
𝐶22𝑓 − 𝐶23𝑓 = 𝛽𝑓 (2-23)
𝐶22𝑚 + 𝐶23𝑚 = 𝛼𝑚 (2-24)
𝐶22𝑚 − 𝐶23𝑚 = 𝛽𝑚 (2-25)
Then Equation (2-21) can be simplified as
𝐴𝑓𝛾𝑓 − 𝐴𝑚𝛾𝑚 +𝐵𝑚𝛿𝑚
𝑎2+ 휀𝑥(𝐶12𝑓 − 𝐶12𝑚) = 0 (2-26)
Similarly, considering continuity of displacement along the matrix/composite
interface we get
𝑢𝑚(𝑏) = 𝑢𝑐(𝑏) (2-27)
Equating displacements of matrix and composite phases
𝐴𝑚𝑏2 + 𝐵𝑚 − 𝐴𝐶𝑎2 − 𝐵𝐶 = 0 (2-28)
Here, 𝑏 is the radius of matrix phase
Similarly, continuity of radial stress along the matrix/composite interface yield
𝜎𝑟𝑚(𝑏) = 𝜎𝑟𝑐(𝑏) (2-29)
[𝐶12 𝐶22 𝐶23]𝑚 {
휀𝑥
휀𝑟
휀𝜃
}
𝑚
= [𝐶12 𝐶22 𝐶23]𝑐 {
휀𝑥
휀𝑟
휀𝜃
}
𝑐
Let
(𝐶22𝑐 + 𝐶23𝑐) = 𝛾𝑐 (2-30)
(𝐶22𝑐 − 𝐶23𝑐) = 𝛿𝑐 (2-31)
Then Equation (2-29) can be simplified as follows
27
𝐴𝑚𝛾𝑚 − 𝐴𝐶𝛾𝑐 −𝐴𝑚𝛿𝑚
𝑏2+
𝐵𝐶𝛿𝑐
𝑏2+ 휀𝑥(𝐶11𝑚 − 𝐶11𝑐) = 0 (2-32)
Boundary Conditions
At 𝑟 = ∞ the radial stress at the boundary of the composite 𝜎𝑟𝑐 = 𝜎𝐻
From the constitutive relation {𝜎} = [𝐶]{휀} we get
[𝐶12 𝐶22 𝐶23]𝑐 {
휀𝑥
휀𝑟
휀𝜃
}
𝑐
= 𝜎𝐻 (2-33)
𝜎𝐻 = 𝐴𝑐𝛾𝑐 + 휀𝑥𝐶12𝑐 (2-34)
Here, 𝜎𝐻 is the hydrostatic stress applied at the boundary of the composite
The constants 𝐴𝑓 , 𝐴𝑚, 𝐵𝑚, 𝐴𝑐, 𝐵𝑐 can be solved from Equations (2-18), (2-26), (2-
32) and (2-34). The hydrostatic stress 𝜎𝐻 is the remote stress applied to the composite.
The micro stresses in fiber phase and matrix phase for longitudinal and hydrostatic
stresses are calculated from the following equations:
𝜎𝑥𝑓 = 𝐶11𝑓휀𝑥𝑓 + 𝐴𝑓𝐶12𝑓 + 𝐴𝑓𝐶12𝑓 (2-35)
𝜎𝑟𝑓 = 𝐶12𝑓휀𝑥𝑓 + 𝐴𝑓𝐶22𝑓 + 𝐴𝑓𝐶23𝑓 (2-36)
𝜎𝜃𝑓 = 𝐶12𝑓휀𝑥𝑓 + 𝐴𝑓𝐶23𝑓 + 𝐴𝑓𝐶22𝑓 (2-37)
𝜎𝑥𝑚 = 𝐶11𝑚휀𝑥𝑚 + (𝐴𝑚 −𝐵𝑚
𝑟2) 𝐶12𝑚 + (𝐴𝑚 +
𝐵𝑚
𝑟2) 𝐶12𝑚 (2-38)
𝜎𝑟𝑚 = 𝐶12𝑚휀𝑥𝑚 + (𝐴𝑚 −𝐵𝑚
𝑟2) 𝐶22𝑚 + (𝐴𝑚 +
𝐵𝑚
𝑟2) 𝐶23𝑚 (2-39)
𝜎𝜃𝑚 = 𝐶12𝑚휀𝑥𝑚 + (𝐴𝑚 −𝐵𝑚
𝑟2) 𝐶23𝑚 + (𝐴𝑚 +
𝐵𝑚
𝑟2) 𝐶22𝑚 (2-40)
Longitudinal Shear Stress in the x-y plane
In this section, the equations for the case of longitudinal shear stress are derived.
Since we are using the composite cylinder model, cylindrical coordinate system is used.
28
The displacement equation for all the three Phases (fiber, matrix, and composite) is
given below:
𝑢𝑟𝑖 = 𝐶𝑖𝑥𝑐𝑜𝑠𝜃 (2-41)
𝑢𝜃𝑖 = −𝐶𝑖𝑥𝑠𝑖𝑛𝜃 (2-42)
𝑢𝑥𝑖 = (𝐴𝑖𝑟 +𝐵𝑖
𝑟) 𝑐𝑜𝑠𝜃 (2-43)
Here, 𝑢𝑟𝑖, 𝑢𝜃𝑖, 𝑢𝑥𝑖 are radial, angular, and axial displacements in the phase 𝑖
respectively. 𝐴𝑖, 𝐵𝑖, 𝐶𝑖 are constants varying with stresses in the phase 𝑖.
Strain Derivations
The six strain components [휀𝑥 휀𝑟 휀𝜃 𝛾𝜃𝑥 𝛾𝑟𝑥 𝛾𝑟𝜃] are derived in the shear model.
From the basic strain formulations for cylindrical system we get,
휀𝑟 =𝜕𝑢𝑟
𝜕𝑟→ 휀𝑟 = 0 (2-44)
and
휀𝜃 =𝑢𝑟
𝑟+
1
𝑟
𝜕𝑢𝜃
𝜕𝜃→ 휀𝜃 = 0 (2-45)
and
휀𝑥 =𝜕𝑢𝑥
𝜕𝑥→ 휀𝑥 = 0 (2-46)
It is also observed that the transverse shear strain also vanishes:
𝛾𝑟𝜃 =𝜕𝑢𝜃
𝜕𝑟+
1
𝑟
𝜕𝑢𝜃
𝜕𝜃−
𝑢𝜃
𝑟= 0 (2-47)
Hence, only two shear strains will exist in the body, which are derived as follows:
𝛾𝜃𝑥 =𝜕𝑢𝜃
𝜕𝑥+
1
𝑟
𝜕𝑢𝑥
𝜕𝜃= − [𝐴 +
𝐵
𝑟2+ 𝐶] 𝑠𝑖𝑛𝜃 (2-48)
and
𝛾𝑟𝑥 =𝜕𝑢𝑟
𝜕𝑥+
𝜕𝑢𝑥
𝜕𝑟= [𝐴 −
𝐵
𝑟2+ 𝐶] 𝑐𝑜𝑠𝜃 (2-49)
29
Boundary Conditions
Since the displacement at point 𝑟 = 0 is finite, the constant pertaining to the fiber
phase 𝐵𝑓 must be zero. At 𝑟 = ∞, the longitudinal displacement 𝑢𝑥𝑐 in the composite
phase must be finite, hence 𝐴𝑐 = 0. The longitudinal shear strains in the composite
phase at 𝑟 = ∞ can be derived from Equations (2-48) and (2-49) are shown below:
𝛾𝜃𝑥 = −𝐶𝑐𝑠𝑖𝑛𝜃 (2-50)
𝛾𝑟𝑥 = 𝐶𝑐𝑐𝑜𝑠𝜃 (2-51)
The 3X3 rotation matrix for transformation of cylindrical coordinates to Cartesian
coordinates is
𝑅 = [𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 0
−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 00 0 1
]
Since the transverse shear strain is zero, the rotation matrix can be simplified to
a 2X2 matrix. Transforming the shear stresses from Equations (2-50) and (2-51) into
Cartesian form we get,
{𝛾𝑧𝑥
𝛾𝑥𝑦} = [
𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
] {𝛾𝜃𝑥
𝛾𝑟𝑥} (2-52)
{𝛾𝑧𝑥
𝛾𝑥𝑦} = {
0𝐶𝑐
} (2-53)
It can be observed that the shear strain 𝛾𝑥𝑦 is the only shear strain presiding in
this model and can be equated to 𝐶𝑐 (Constant in composite phase).
From the shear stress-strain relations formula, we deduce the following relation
𝐶𝑐 = 𝛾𝑦𝑥 =𝜏0
𝐺𝑥𝑦𝑐 (2-54)
30
Here, 𝐶𝑐: Constant 𝐶 in the composite phase 𝐺𝑥𝑦
𝑐 : Longitudinal shear modulus of the composite derived from Halpin-Tsai
Equation Continuity Equations
Continuity of displacement and radial stresses must be ensured along the
fiber/matrix interface and matrix/composite interface. The continuity equations are
shown below:
𝑢𝑟𝑖(𝑟𝑖) = 𝑢𝑟𝑗(𝑟𝑖) (2-55)
𝑢𝑥𝑖(𝑟𝑖) = 𝑢𝑥𝑗(𝑟𝑖) (2-56)
𝜏𝑟𝑥𝑖(𝑟𝑖) = 𝜏𝑟𝑥𝑗(𝑟𝑖) (2-57)
Here, 𝑢𝑟𝑖, 𝑢𝑟𝑗 ,𝑢𝑥𝑖, 𝑢𝑥𝑗 are the radial and axial displacements in consecutive
phases 𝑖, 𝑗 and 𝜏𝑟𝑥𝑖, 𝜏𝑟𝑥𝑗 are the radial shear strains in the corresponding phases.
From Equation (2-55) continuity of displacement along the fiber/matrix interface
at 𝑟 = 𝑎
𝑢𝑟𝑓(𝑎) = 𝑢𝑟𝑚(𝑎) → 𝐶𝑓𝑥𝑐𝑜𝑠𝜃 = 𝐶𝑚𝑥𝑐𝑜𝑠𝜃 (2-58)
Similarly, at 𝑟 = 𝑏
𝑢𝑚(𝑏) = 𝑢𝑐(𝑏) → 𝐶𝑚𝑥𝑐𝑜𝑠𝜃 = 𝐶𝑐𝑥𝑐𝑜𝑠𝜃 (2-59)
From Equations (2-58) and (2-59) we can infer that 𝐶𝑓 = 𝐶𝑚 = 𝐶𝑐
Let the constants 𝐶𝑓, 𝐶𝑚 and 𝐶 be equal to 𝐶:
𝐶𝑓 = 𝐶𝑚 = 𝐶𝑐 = 𝐶 (2-60)
Ensuring equal axial displacements at the fiber/matrix interface 𝑟 = 𝑎 we obtain
𝑢𝑥𝑓(𝑎) = 𝑢𝑥𝑚(𝑎) → 𝐴𝑓𝑎 + 𝐴𝑚𝑎 −𝐵𝑚
𝑎= 0 (2-61)
where, 𝐴𝑓 , 𝐴𝑚 and 𝐵𝑚 are constants in fiber and matrix phases
31
From Equation (2-57) continuity of shear stress along the fiber/matrix interface
𝜏𝑟𝑥𝑓(𝑎) = 𝜏𝑟𝑥𝑚(𝑎) → 𝐺𝑟𝑥𝑓
(𝐴𝑓 + 𝐶) − 𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑎2+ 𝐶) = 0 (2-62)
where, 𝐺𝑟𝑥𝑓
, 𝐺𝑟𝑥𝑚 are the longitudinal shear moduli of the fiber phase and matrix
phase respectively
Similarly, considering axial displacements at the matrix/composite interface we
get,
𝑢𝑥𝑚(𝑏) = 𝑢𝑥𝑐(𝑏) →𝐵𝑐
𝑏−𝐴𝑚𝑏 −
𝐵𝑚
𝑏= 0 (2-63)
Here, 𝐴𝑚, 𝐵𝑚𝐵𝑐 are constants in matrix phase and composite phase varying with
applied stresses. Similarly, continuity of shear stress along the matrix/composite
interface yields:
𝜏𝑟𝑥𝑚(𝑏) = 𝜏𝑟𝑥𝑐(𝑏) → 𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑏2+ 𝐶) − 𝐺𝑟𝑥
𝑐 (−𝐵𝑐
𝑏2+ 𝐶) = 0 (2-64)
where, 𝐺𝑟𝑥𝑚, 𝐺𝑟𝑥
𝑐 are the longitudinal shear moduli of the matrix phase and
composite phase respectively
The five constants (𝐴𝑓 , 𝐴𝑚, 𝐵𝑚, 𝐵𝑐, 𝐶) are solved from the following five equations:
𝐴𝑓 + 𝐴𝑚𝑎 −𝐵𝑚
𝑎= 0 (2-65)
𝐺𝑟𝑥𝑓
(𝐴𝑓 + 𝐶) − 𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑎2+ 𝐶) = 0 (2-66)
𝐵𝑐
𝑏−𝐴𝑚𝑏 −
𝐵𝑚
𝑏= 0 (2-67)
𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑏2+ 𝐶) − 𝐺𝑟𝑥
𝑐 (−𝐵𝑐
𝑏2+ 𝐶) = 0 (2-68)
𝐶 =𝜏𝑥𝑦
𝐺𝑥𝑦𝑐 (2-69)
32
The micro stresses in fiber and matrix phases for longitudinal shear XY case are
calculated from the following equations:
𝜏𝜃𝑥𝑓 = −𝐺𝑟𝑥𝑓
(𝐴𝑓 + 𝐶) 𝑠𝑖𝑛𝜃 (2-70)
𝜏𝑟𝑥𝑓 = 𝐺𝑟𝑥𝑓
(𝐴𝑓 + 𝐶) 𝑐𝑜𝑠𝜃 (2-71)
𝜏𝜃𝑥𝑚 = −𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑟2+ 𝐶) 𝑠𝑖𝑛𝜃 (2-72)
𝜏𝑟𝑥𝑚 = 𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑟2+ 𝐶) 𝑠𝑖𝑛𝜃 (2-73)
Longitudinal Shear Stress in the x-z plane
In this section, the displacement equations for the case of longitudinal shear
stress are derived. Since we are using the composite cylinder model, cylindrical
coordinate system is used.
The displacement equation for all the 3 Phases (Fiber, Matrix, and Composite) is
given below:
𝑢𝑟𝑖 = 𝐶𝑖𝑥𝑠𝑖𝑛𝜃 (2-74)
𝑢𝜃𝑖 = 𝐶𝑖𝑥𝑐𝑜𝑠𝜃 (2-75)
𝑢𝑥𝑖 = − (𝐴𝑖𝑟 +𝐵𝑖
𝑟) 𝑠𝑖𝑛𝜃 (2-76)
Here, 𝑢𝑟𝑖, 𝑢𝜃𝑖, 𝑢𝑥𝑖 are radial, angular, and axial displacements in the phase 𝑖
respectively. 𝐴𝑖, 𝐵𝑖, 𝐶𝑖 are constants varying with stresses in the phase 𝑖
Strain Derivations
The 6 fundamental strains [휀𝑥 휀𝑟 휀𝜃 𝛾𝜃𝑥 𝛾𝑟𝑥 𝛾𝑟𝜃] are derived in the shear model.
From the basic strain formulations for cylindrical system we get,
휀𝑟 =𝜕𝑢𝑟
𝜕𝑟→ 휀𝑟 = 0 (2-77)
33
and
휀𝜃 =𝑢𝑟
𝑟+
1
𝑟
𝜕𝑢𝜃
𝜕𝜃→ 휀𝜃 = 0 (2-78)
and
휀𝑥 =𝜕𝑢𝑥
𝜕𝑥→ 휀𝑥 = 0 (2-79)
It can be observed that the longitudinal strain by the application of a discrete
shear stress in this model is zero.
Now the shear strains are calculated, since the shear is applied along the
longitudinal direction it is obvious that transverse shear is zero.
𝛾𝑟𝜃 =𝜕𝑢𝜃
𝜕𝑟+
1
𝑟
𝜕𝑢𝜃
𝜕𝜃−
𝑢𝜃
𝑟= 0 (2-80)
Apparently due to the nature of shear stress only two shear strains will exist in
the body, which are derived as follows:
𝛾𝜃𝑥 =𝜕𝑢𝜃
𝜕𝑥+
1
𝑟
𝜕𝑢𝑥
𝜕𝜃= − [𝐴 +
𝐵
𝑟2− 𝐶] 𝑐𝑜𝑠𝜃 (2-81)
𝛾𝑟𝑥 =𝜕𝑢𝑟
𝜕𝑥+
𝜕𝑢𝑥
𝜕𝑟= − [𝐴 −
𝐵
𝑟2− 𝐶] 𝑠𝑖𝑛𝜃 (2-82)
Boundary Conditions
Since the displacement at point 𝑟 = 0 is finite, the constant pertaining to the fiber
phase 𝐵𝑓 must be zero. At 𝑟 = ∞, the longitudinal displacement 𝑢𝑥𝑐 in the composite
phase must be finite, hence 𝐴𝑐 = 0. The longitudinal shear strains in the composite
phase at 𝑟 = ∞ can be derived from Equations (2-81) and (2-82) are shown below:
𝛾𝜃𝑥 = 𝐶𝑐𝑠𝑖𝑛𝜃 (2-83)
𝛾𝑟𝑥 = 𝐶𝑐𝑐𝑜𝑠𝜃 (2-84)
34
The 3X3 rotation matrix for transformation of cylindrical coordinates to Cartesian
coordinates is
𝑅 = [𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 0
−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 00 0 1
]
Since the transverse shear strain is zero the rotation matrix can be simplified to a
2X2 matrix. Transforming the shear stresses from Equations (2-83) and (2-84) into
Cartesian form we get,
{𝛾𝑧𝑥
𝛾𝑥𝑦} = [
𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
] {𝛾𝜃𝑥
𝛾𝑟𝑥} (2-85)
{𝛾𝑧𝑥
𝛾𝑥𝑦} = {
𝐶𝑐
0} (2-86)
It can be observed that the shear strain 𝛾𝑧𝑥 is the only shear strain presiding in
this model and can be equated to 𝐶𝑐 (Constant in composite phase). From basic shear
stress formula, we can deduce the following relation:
𝐶𝑐 = 𝛾𝑧𝑥 =𝜏𝑧𝑥
𝐺𝑟𝑥𝑐 (2-87)
Here,
𝐶𝑐: Constant C in the composite phase
𝐺𝑧𝑥𝑐 : Longitudinal shear modulus of the composite derived from Halpin-Tsai
equation
Continuity Equations
Continuity of displacement and radial stresses must be ensured along the
fiber/matrix interface and matrix/composite interface. The continuity equations are
shown below:
𝑢𝑟𝑖(𝑟𝑖) = 𝑢𝑟𝑗(𝑟𝑖) (2-88)
35
𝑢𝑥𝑖(𝑟𝑖) = 𝑢𝑥𝑗(𝑟𝑖) (2-89)
𝜏𝑟𝑥𝑖(𝑟𝑖) = 𝜏𝑟𝑥𝑗(𝑟𝑖) (2-90)
Here, 𝑢𝑟𝑖, 𝑢𝑟𝑗 ,𝑢𝑥𝑖, 𝑢𝑥𝑗 are the radial and axial displacements in consecutive
phases 𝑖, 𝑗 and 𝜏𝑟𝑥𝑖, 𝜏𝑟𝑥𝑗 are the radial shear strains in the corresponding phases. From
Equation (2-88) continuity of displacement along the fiber/matrix interface 𝑟 = 𝑎 we can
interpret
𝑢𝑟𝑓(𝑎) = 𝑢𝑟𝑚(𝑎) → 𝐶𝑓𝑥𝑠𝑖𝑛𝜃 = 𝐶𝑚𝑥𝑠𝑖𝑛𝜃 (2-91)
Similarly, at 𝑟 = 𝑏
𝑢𝑚(𝑏) = 𝑢𝑐(𝑏) → 𝐶𝑚𝑥𝑠𝑖𝑛𝜃 = 𝐶𝑐𝑥𝑠𝑖𝑛𝜃 (2-92)
From Equations (2-91) and (2-92) we can infer that 𝐶𝑓 = 𝐶𝑚 = 𝐶𝑐
For simplicity, the following assumption has been made and will be considered
for future derivations and equations
𝐶𝑓 = 𝐶𝑚 = 𝐶𝑐 = 𝐶 (2-93)
Ensuring equal axial displacements about the fiber/matrix interface at 𝑟 = 𝑎 we
get
𝑢𝑥𝑓(𝑎) = 𝑢𝑥𝑚(𝑎) → −𝐴𝑓𝑎 + 𝐴𝑚𝑎 +𝐵𝑚
𝑎= 0 (2-94)
Here, 𝐴𝑓 , 𝐴𝑚 and 𝐵𝑚 are constants in fiber and matrix phases varying with applied
stresses
From Equation (2-90) continuity of radial shear stress along the fiber/matrix
interface
𝜏𝑟𝑥𝑓(𝑎) = 𝜏𝑟𝑥𝑚(𝑎) → −𝐺𝑟𝑥𝑓
(𝐴𝑓 − 𝐶) + 𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑎2− 𝐶) = 0 (2-95)
36
Here, 𝐺𝑟𝑥𝑓
, 𝐺𝑟𝑥𝑚 are the longitudinal shear moduli of the fiber phase and matrix
phase respectively. Similarly, considering axial displacements at the matrix/composite
interface we get
𝑢𝑥𝑚(𝑏) = 𝑢𝑥𝑐(𝑏) →𝐵𝑐
𝑏−𝐴𝑚𝑏 −
𝐵𝑚
𝑏= 0 (2-96)
Here, 𝐴𝑚, 𝐵𝑚𝐵𝑐 are constants in matrix phase and composite phase varying with
applied stresses
Similarly, continuity of radial shear stress along the matrix/composite interface
yields
𝜏𝑟𝑥𝑚(𝑏) = 𝜏𝑟𝑥𝑐(𝑏) → −𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑏2− 𝐶) + 𝐺𝑟𝑥
𝑐 (−𝐵𝑐
𝑏2− 𝐶) = 0 (2-97)
Here,
𝐺𝑟𝑥𝑚: Longitudinal shear modulus of the matrix phase
𝐺𝑟𝑥𝑐 : Longitudinal shear modulus of the composite
The five constants (𝐴𝑓 , 𝐴𝑚, 𝐵𝑚, 𝐵𝑐, 𝐶) are solved from the following five equations:
−𝐴𝑓 + 𝐴𝑚𝑎 +𝐵𝑚
𝑎= 0 (2-98)
−𝐺𝑟𝑥𝑓
(𝐴𝑓 + 𝐶) + 𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑎2+ 𝐶) = 0 (2-99)
𝐵𝑐
𝑏−𝐴𝑚𝑏 −
𝐵𝑚
𝑏= 0 (2-100)
−𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑏2+ 𝐶) + 𝐺𝑟𝑥
𝑐 (−𝐵𝑐
𝑏2+ 𝐶) = 0 (2-101)
𝐶 =𝜏𝑥𝑦
𝐺𝑥𝑦𝑐 (2-102)
The micro stresses in each phase are calculated from the following equations
37
𝜏𝜃𝑥𝑓 = −𝐺𝑟𝑥𝑓
(𝐴𝑓 − 𝐶) 𝑐𝑜𝑠𝜃 (2-103)
𝜏𝑟𝑥𝑓 = −𝐺𝑟𝑥𝑓
(𝐴𝑓 − 𝐶) 𝑠𝑖𝑛𝜃 (2-104)
𝜏𝜃𝑥𝑚 = −𝐺𝑟𝑥𝑚 (𝐴𝑚 +
𝐵𝑚
𝑟2− 𝐶) 𝑐𝑜𝑠𝜃 (2-105)
𝜏𝑟𝑥𝑚 = −𝐺𝑟𝑥𝑚 (𝐴𝑚 −
𝐵𝑚
𝑟2− 𝐶) (2-106)
Biaxial tension/compression in y-z plane
In this section, the equations for the case of pure shear in the transverse plane
are derived. This two-dimensional problem can be dealt by assuming a suitable Airy
stress function ∅ which is shown below:
∅ = (𝐴𝑟2 + 𝐵𝑟4 +𝐶
𝑟2+ 𝐷) 𝑐𝑜𝑠2𝜃 (2-107)
Since plane strain is assumed, the three non-zero stresses pertaining to the 2-D
problem [ 𝜎𝑟 𝜎𝜃 𝜏𝑟𝜃] are derived from the Airy stress function as shown below. The
radial stress at any point in the model can be derived using the elasticity equation [11]
as follows
𝜎𝑟 =1
𝑟
𝜕∅
𝜕𝑟+
1
𝑟2
𝜕2∅
𝜕𝜃2 (2-108)
𝜎𝑟 = (2𝐴 + 4𝐵𝑟2 −2𝐶
𝑟4) 𝑐𝑜𝑠2𝜃 + (−4𝐴 − 4𝐵𝑟2 −
4𝐶
𝑟4−
4𝐷
𝑟2) 𝑐𝑜𝑠2𝜃
𝜎𝑟 = − (2𝐴 +6𝐶
𝑟4+
4𝐷
𝑟2) 𝑐𝑜𝑠2𝜃 (2-109)
The tangential stress derivation is shown below
𝜎𝜃 =𝜕2∅
𝜕𝑟2→ 𝜎𝜃 = (2𝐴 + 12𝐵𝑟2 +
6𝐶
𝑟4) 𝑐𝑜𝑠2𝜃 (2-110)
The transverse shear in the plane is derived as follows
38
𝜏𝑟𝜃 = −𝜕
𝜕𝑟(
1
𝑟
𝜕∅
𝜕𝜃) (2-111)
𝜏𝑟𝜃 = −𝜕
𝜕𝑟(−
1
𝑟(𝐴𝑟2 + 𝐵𝑟4 +
𝐶
𝑟4+ 𝐷) 2𝑠𝑖𝑛2𝜃)
𝜏𝑟𝜃 = (2𝐴 + 6𝐵𝑟2 −6𝐶
𝑟4−
2𝐷
𝑟2) 𝑠𝑖𝑛2𝜃 (2-112)
The radial strain at any point is given by the following equations
휀𝑟 =1
𝐸′(𝜎𝑟 − 𝜈′𝜎𝜃) 𝑜𝑟 휀𝑟 =
𝜕𝑢𝑟
𝜕𝑟 (2-113)
Here, 𝐸′ is the plane strain modulus and 𝜈′ is the plane strain Poisson’s ratio
which is given by
𝜈′ = (𝜈
1 + 𝜈)
The radial displacement is calculated from the Equation (2-113) as follows
𝑢𝑟 = ∫(𝜎𝑟 − 𝜈′𝜎𝜃)𝑑𝑟 (2-114)
Substituting 𝜎𝜃 from Equation (2-110) in the above equation we get
𝑢𝑟 =1
𝐸′∫ ((−2𝐴 −
6𝐶
𝑟4−
4𝐷
𝑟2) 𝐶𝑜𝑠2𝜃 − 𝜈′ (2𝐴 + 12𝐵𝑟2 +
6𝐶
𝑟4) 𝐶𝑜𝑠2𝜃) 𝑑𝑟
𝑢𝑟 =𝐶𝑜𝑠2𝜃
𝐸′∫ ((−2𝐴 −
6𝐶
𝑟4−
4𝐷
𝑟2) − 𝜈′ (2𝐴 + 12𝐵𝑟2 +
6𝐶
𝑟4)) 𝑑𝑟 (2-115)
𝑢𝑟 =𝐶𝑜𝑠2𝜃
𝐸′((−2𝐴𝑟 +
2𝐶
𝑟3+
4𝐷
𝑟) − 𝜈′ (2𝐴𝑟 + 4𝐵𝑟3 −
2𝐶
𝑟3))
The radial displacement at any point on the surface can be obtained by the
following equation:
𝑢𝑟 =𝐶𝑜𝑠2𝜃
𝐸′((−2𝐴𝑟(1 + 𝜈′) − 4𝜈′𝐵𝑟3 +
2𝐶
𝑟3(1 + 𝜈′) +
4𝐷
𝑟)) (2-116)
39
The tangential strain at any point is given by the following equations:
휀𝜃 =1
𝐸′(𝜎𝜃 − 𝜈′𝜎𝑟) → 휀𝜃 =
𝑢𝑟
𝑟+
1
𝑟
𝜕𝑢𝜃
𝜕𝜃 (2-117)
Equation (2-117) can be modified to obtain the angular displacement gradient
with respect to 𝜃 as follows:
𝜕𝑢𝜃
𝜕𝜃= 𝑟 (
1
𝐸′(𝜎𝜃 − 𝜈′𝜎𝑟) −
𝑢𝑟
𝑟) (2-118)
Substituting 𝜎𝜃 and 𝜎𝑟 from Equations (2-110) and (2-109) we get the following
relations:
𝜕𝑢𝜃
𝜕𝜃= 𝑟 (
1
𝐸′((2𝐴 + 12𝐵𝑟2 +
6𝐶
𝑟4) 𝐶𝑜𝑠2𝜃 + 𝜈′ (2𝐴 +
6𝐶
𝑟4+
4𝐷
𝑟2) 𝐶𝑜𝑠2𝜃) −
𝑢𝑟
𝑟)
𝜕𝑢𝜃
𝜕𝜃= (
𝐶𝑜𝑠2𝜃
𝐸′(2𝐴𝑟(1 + 𝜈′) + 12𝐵𝑟3 +
6𝐶
𝑟3(1 + 𝜈′) +
4𝐷𝜈′
𝑟)
−𝐶𝑜𝑠2𝜃
𝐸′((−2𝐴𝑟(1 + 𝜈′) − 4𝜈′𝐵𝑟3 +
2𝐶
𝑟3(1 + 𝜈′) +
4𝐷
𝑟)))
The obtained angular displacement gradient is as follows:
𝜕𝑢𝜃
𝜕𝜃= (
4𝐶𝑜𝑠2𝜃
𝐸′(𝐴𝑟(1 + 𝜈′) + 𝐵𝑟3(3 + 𝜈′) +
𝐶
𝑟3(1 + 𝜈′) +
𝐷
𝑟(𝜈′ − 1))) (2-119)
In the above equations 𝐴, 𝐵, 𝐶, 𝐷 are constants specific to each phase and vary
with applied stresses.
Since the stresses in fiber at 𝑟 = 0 are finite 𝐶𝑓 𝑎𝑛𝑑 𝐷𝑓 are equal to zero.
Continuity of 𝜎𝑟 , 𝜏𝑟𝜃, 𝑢𝑟 , 𝜕𝑢𝜃/𝜕𝜃 must be satisfied at 𝑟 = 𝑎 (fiber/matrix interface) and
𝑟 = 𝑏 (matrix/composite interface)
Continuity of radial stress along fiber/matrix interface yields the following
equation:
40
𝜎𝑟𝑓(𝑎) = 𝜎𝑟𝑚(𝑎) (2-120)
𝐴𝑓 − 𝐴𝑚 −3𝐶𝑚
𝑎4−
2𝐷𝑚
𝑎2= 0 (2-121)
Similarly, incorporating continuity of radial stress along the matrix/composite
interface we get,
𝜎𝑟𝑚(𝑏) = 𝜎𝑟𝑐(𝑏) (2-122)
−𝐴𝑚 −3𝐶𝑚
𝑏4−
2𝐷𝑚
𝑏2+ 𝐴𝑐 +
3𝐶𝑐
𝑏4+
2𝐷𝑐
𝑏2= 0 (2-123)
Continuity of radial displacement must be ensured along the fiber/matrix interface
𝑢𝑟𝑓(𝑎) = 𝑢𝑟𝑚(𝑎) (2-124)
1
𝐸𝑓′ (−2𝐴𝑓𝑎(1 + 𝜈𝑓
′ ) − 4𝜈𝑓′ 𝐵𝑓𝑎3)
−1
𝐸𝑚′
((−2𝐴𝑚𝑎(1 + 𝜈𝑚′ ) − 4𝜈𝑚
′ 𝐵𝑚𝑎3 +2𝐶𝑚
𝑎3(1 + 𝜗𝑚
′ ) +4𝐷𝑚
𝑎))
= 0 (2-125)
Similarly, considering the continuity of radial displacement along
matrix/composite interface we get the following equation:
1
𝐸𝑚′ ((−2𝐴𝑚𝑏(1 + 𝜈𝑚
′ ) − 4𝜈𝑚′ 𝐵𝑚𝑏3 +
2𝐶𝑚
𝑏3 (1 + 𝜈𝑚′ ) +
4𝐷𝑚
𝑏)) −
1
𝐸𝑐′ ((−2𝐴𝑐𝑏(1 + 𝜈𝑐) +
2𝐶𝑐
𝑏3(1 + 𝜈𝑐) +
4𝐷𝑐
𝑏)) = 0 (2-126)
Continuity of shear stress must be ensured along the fiber/matrix interface
𝜏𝑟𝜃𝑓(𝑎) = 𝜏𝑟𝜃𝑚(𝑎) (2-127)
(2𝐴𝑓 + 6𝐵𝑓𝑎2) − (2𝐴𝑚 + 6𝐵𝑚𝑟2 −6𝐶𝑚
𝑎4−
2𝐷𝑚
𝑎2) = 0 (2-128)
Similarly, considering the continuity of shear stress along matrix/composite
interface we get the following equation:
41
(2𝐴𝑚 + 6𝐵𝑚𝑏2 −6𝐶𝑚
𝑏4−
2𝐷𝑚
𝑏2) − (2𝐴𝑐 −
6𝐶𝑐
𝑏4−
2𝐷𝑐
𝑏2) = 0 (2-129)
Continuity of tangential displacement gradient must be ensured along the
fiber/matrix interface
𝜕𝑢𝜃𝑓
𝜕𝜃(𝑎) =
𝜕𝑢𝜃𝑚
𝜕𝜃(𝑎) (2-130)
1
𝐸𝑓′ (𝐴𝑓𝑎(1 + 𝜈𝑓
′ ) + 𝐵𝑓𝑎3(3 + 𝜈𝑓′ ))
− 1
𝐸𝑚′
(𝐴𝑚𝑎(1 + 𝜈𝑚′ ) + 𝐵𝑚𝑎3(3 + 𝜈𝑚
′ ) +𝐶𝑚
𝑎3(1 + 𝜈𝑚
′ ) +𝐷𝑚
𝑎(𝜈𝑚
′ − 1))
= 0 (2-131)
Similarly, considering the continuity of shear stress along matrix/composite
interface we get the following equation:
1
𝐸𝑚′
(𝐴𝑚𝑏(1 + 𝜈𝑚′ ) + 𝐵𝑚𝑏3(3 + 𝜈𝑚
′ ) +𝐶𝑚
𝑏3(1 + 𝜈𝑚
′ ) +𝐷𝑚
𝑏(𝜈𝑚
′ − 1))
−1
𝐸𝑐′
(𝐴𝑐𝑏(1 + 𝜈𝑐′) +
𝐶𝑐
𝑏3(1 + 𝜈𝑐
′) +𝐷𝑐
𝑏(𝜈𝑐
′ − 1))
= 0 (2-132)
Constants 𝐴𝑓 , 𝐵𝑓 , 𝐴𝑚, 𝐵𝑚, 𝐶𝑚𝐷𝑚, 𝐴𝑐, 𝐶𝑐 and 𝐷𝑐 can be found from the above
derived continuity equations. The stresses in each phase can be calculated from the
below mentioned equations:
Fiber Equations
𝜎𝑟𝑓 = −2𝐴𝑓𝐶𝑜𝑠2𝜃
𝜎𝜃𝑓 = (2𝐴𝑓 + 12𝐵𝑓𝑟2)𝐶𝑜𝑠2𝜃
𝜏𝑟𝜃𝑓 = (2𝐴𝑓 + 6𝐵𝑓𝑟2)𝑆𝑖𝑛2𝜃 (2-133)
42
𝑢𝑟𝑓 =𝐶𝑜𝑠2𝜃
𝐸𝑓′ (−2𝐴𝑓𝑟(1 + 𝜈𝑓
′ ) − 4𝜈𝑓𝐵𝑓𝑟3)
𝜕𝑢𝜃𝑓
𝜕𝜃=
4𝐶𝑜𝑠2𝜃
𝐸𝑓′ (𝐴𝑓𝑟(1 + 𝜈𝑓
′ ) + 𝐵𝑓𝑟3(3 + 𝜈𝑓′ ))
Matrix Equations
𝜎𝑟𝑚 = − (2𝐴𝑚 +6𝐶𝑚
𝑟4+
4𝐷𝑚
𝑟2) 𝐶𝑜𝑠2𝜃
𝜎𝜃𝑚 = (2𝐴𝑚 + 12𝐵𝑚𝑟2 +6𝐶𝑚
𝑟4) 𝐶𝑜𝑠2𝜃
𝜏𝑟𝜃𝑚 = (2𝐴𝑚 + 6𝐵𝑚𝑟2 −6𝐶𝑚
𝑟4−
2𝐷𝑚
𝑟2) 𝑆𝑖𝑛2𝜃 (2-134)
𝑢𝑟𝑚 =𝐶𝑜𝑠2𝜃
𝐸𝑚′
((−2𝐴𝑚𝑟(1 + 𝜈𝑚′ ) − 4𝜈𝑚
′ 𝐵𝑚𝑟3 +2𝐶𝑚
𝑟3(1 + 𝜈𝑚
′ ) +4𝐷𝑚
𝑟))
𝜕𝑢𝜃𝑚
𝜕𝜃=
4𝐶𝑜𝑠2𝜃
𝐸𝑚′
(𝐴𝑚𝑟(1 + 𝜈𝑚′ ) + 𝐵𝑚𝑟3(3 + 𝜈𝑚
′ ) +𝐶𝑚
𝑟3(1 + 𝜈𝑚
′ ) +𝐷𝑚
𝑟(𝜈𝑚
′ − 1))
Composite Equations
𝜎𝑟𝑐 = − (2𝐴𝑐 +6𝐶𝑐
𝑟4+
4𝐷𝑐
𝑟2) 𝐶𝑜𝑠2𝜃
𝜎𝜃𝑐 = (2𝐴𝑐 +6𝐶𝑐
𝑟4) 𝐶𝑜𝑠2𝜃
𝜏𝑟𝜃𝑐 = (2𝐴𝑐 −6𝐶𝑐
𝑟4−
2𝐷𝑐
𝑟2) 𝑆𝑖𝑛2𝜃 (2-135)
𝑢𝑟𝑐 =𝐶𝑜𝑠2𝜃
𝐸𝑐′
((−2𝐴𝑐𝑟(1 + 𝜈𝑐′) +
2𝐶𝑐
𝑟3(1 + 𝜈𝑐
′) +4𝐷𝑐
𝑟))
𝜕𝑢𝜃𝑐
𝜕𝜃=
4𝐶𝑜𝑠2𝜃
𝐸𝑐′
(𝐴𝑐𝑟(1 + 𝜈𝑐′) +
𝐶𝑐
𝑟3(1 + 𝜈𝑐
′) +𝐷𝑐
𝑟(𝜈𝑐
′ − 1))
Substituting 𝑟 = ∞ in Equations (2-135) i.e. composite phase we get,
𝜎𝑟𝑐 = −(2𝐴𝑐)𝐶𝑜𝑠2𝜃
43
𝜎𝜃𝑐 = (2𝐴𝑐)𝐶𝑜𝑠2𝜃 (2-136)
𝜏𝑟𝜃𝑐 = (2𝐴𝑐)𝑆𝑖𝑛2𝜃
Transforming the stresses into Cartesian co-ordinates yields
[
𝜎𝑦
𝜎𝑧
𝜏𝑦𝑧
] = [𝐶𝑜𝑠2𝜃 𝑆𝑖𝑛2𝜃 − 𝑆𝑖𝑛2𝜃𝑆𝑖𝑛2𝜃 𝐶𝑜𝑠2𝜃 𝑆𝑖𝑛2𝜃𝐶𝑜𝑠𝜃𝑆𝑖𝑛𝜃 − 𝐶𝑜𝑠𝜃𝑆𝑖𝑛𝜃 𝐶𝑜𝑠2𝜃
] [−𝐶𝑜𝑠2𝜃𝐶𝑜𝑠2𝜃𝑆𝑖𝑛2𝜃
] 2𝐴𝑐
[
𝜎𝑦
𝜎𝑧
𝜏𝑦𝑧
] = [−2𝐴𝑐
+2𝐴𝑐
0] (2-137)
It can be observed in Equation (2-137), the shear in the transverse plane is zero
and the only non-zero stresses are 𝜎𝑦 and 𝜎𝑧 which are equal but are acting in opposite
directions which is equivalent to shear in 45 degrees.
Transverse Shear Equations
In this section, the stress equations for the case of transverse shear yz are
derived. This two-dimensional problem can be dealt by assuming a suitable stress
function [12] ∅ which is shown below
∅ = − (𝐴𝑟2 + 𝐵𝑟4 +𝐶
𝑟2+ 𝐷) 𝑠𝑖𝑛2𝜃 (2-138)
Since plane strain is assumed, the three fundamental stresses pertaining to a 2-
D problem [ 𝜎𝑟 𝜎𝜃 𝜏𝑟𝜃] are derived from the Airy stress function. The radial stress at any
point in the model can be derived using the elasticity equation as follows:
𝜎𝑟 =1
𝑟
𝜕∅
𝜕𝑟+
1
𝑟2
𝜕2∅
𝜕𝜃2 (2-139)
𝜎𝑟 = − (2𝐴 + 4𝐵𝑟2 −2𝐶
𝑟4) 𝑐𝑜𝑠2𝜃 + (4𝐴 + 4𝐵𝑟2 +
4𝐶
𝑟4+
4𝐷
𝑟2) 𝑠𝑖𝑛2𝜃
𝜎𝑟 = (2𝐴 +6𝐶
𝑟4+
4𝐷
𝑟2) 𝑠𝑖𝑛2𝜃 (2-140)
44
The tangential stress derivation is shown below:
𝜎𝜃 =𝜕2∅
𝜕𝑟2→ 𝜎𝜃 = − (2𝐴 + 12𝐵𝑟2 +
6𝐶
𝑟4) 𝑠𝑖𝑛2𝜃 (2-141)
The transverse shear in the plane is derived as follows:
𝜏𝑟𝜃 = −𝜕
𝜕𝑟(
1
𝑟
𝜕∅
𝜕𝜃) (2-142)
𝜏𝑟𝜃 = −𝜕
𝜕𝑟(−
2
𝑟(𝐴𝑟2 + 𝐵𝑟4 +
𝐶
𝑟4+ 𝐷) 𝑐𝑜𝑠2𝜃)
𝜏𝑟𝜃 = (2𝐴 + 6𝐵𝑟2 −6𝐶
𝑟4−
2𝐷
𝑟2) 𝑐𝑜𝑠2𝜃 (2-143)
The radial strain at any point is given by the following equations:
휀𝑟 =1
𝐸′(𝜎𝑟 − 𝜗′𝜎𝜃) 𝑜𝑟 휀𝑟 =
𝜕𝑢𝑟
𝜕𝑟 (2-144)
Here, 𝐸′ is the plane strain modulus and 𝜈′ is the plane strain Poisson’s ratio.
The radial displacement is calculated from the Equation (2-144) as follows
𝑢𝑟 = ∫(𝜎𝑟 − 𝜈′𝜎𝜃)𝑑𝑟 (2-145)
Substituting 𝜎𝜃 from Equation (2-141) in the above equation we get,
𝑢𝑟 =1
𝐸′∫ ((+2𝐴 +
6𝐶
𝑟4+
4𝐷
𝑟2) 𝑠𝑖𝑛2𝜃 − 𝜈′ (− (2𝐴 + 12𝐵𝑟2 +
6𝐶
𝑟4)) 𝑠𝑖𝑛2𝜃) 𝑑𝑟
𝑢𝑟 =𝑠𝑖𝑛2𝜃
𝐸′∫ ((+2𝐴 +
6𝐶
𝑟4+
4𝐷
𝑟2) + 𝜈′ (2𝐴 + 12𝐵𝑟2 +
6𝐶
𝑟4)) 𝑑𝑟 (2-146)
𝑢𝑟 =𝑠𝑖𝑛2𝜃
𝐸′((2𝐴𝑟 −
2𝐶
𝑟3−
4𝐷
𝑟) + 𝜈′ (2𝐴𝑟 + 4𝐵𝑟3 −
2𝐶
𝑟3))
The radial displacement at any point on the surface can be obtained by the
following equation:
45
𝑢𝑟 =𝑠𝑖𝑛2𝜃
𝐸′((2𝐴𝑟(1 + 𝜈′) + 4𝜈′𝐵𝑟3 −
2𝐶
𝑟3(1 + 𝜈′) −
4𝐷
𝑟)) (2-147)
The tangential strain at any point is given by the following equations:
휀𝜃 =1
𝐸′(𝜎𝜃 − 𝜈′𝜎𝑟) → 휀𝜃 =
𝑢𝑟
𝑟+
1
𝑟
𝜕𝑢𝜃
𝜕𝜃 (2-148)
Equation (2-148) can be modified to obtain the angular displacement gradient
with respect to 𝜃 as follows:
𝜕𝑢𝜃
𝜕𝜃= 𝑟 (
1
𝐸′(𝜎𝜃 − 𝜈′𝜎𝑟) −
𝑢𝑟
𝑟) (2-149)
Substituting 𝜎𝜃 and 𝜎𝑟 from Equations (2-140) and (2-141) we get the following
relations:
𝜕𝑢𝜃
𝜕𝜃= 𝑟 (
1
𝐸′(− (2𝐴 + 12𝐵𝑟2 +
6𝐶
𝑟4) 𝑠𝑖𝑛2𝜃 − 𝜈′ (2𝐴 +
6𝐶
𝑟4+
4𝐷
𝑟2) 𝑠𝑖𝑛2𝜃) −
𝑢𝑟
𝑟)
𝜕𝑢𝜃
𝜕𝜃= (
𝑠𝑖𝑛2𝜃
𝐸′(−2𝐴𝑟(1 + 𝜈′) − 12𝐵𝑟3 −
6𝐶
𝑟3(1 + 𝜈′) −
4𝐷𝜗
𝑟)
−𝑠𝑖𝑛2𝜃
𝐸′((2𝐴𝑟(1 + 𝜈′) + 4𝜈′𝐵𝑟3 −
2𝐶
𝑟3(1 + 𝜈′) −
4𝐷
𝑟)))
The obtained angular displacement gradient is as follows:
𝜕𝑢𝜃
𝜕𝜃= (
4𝑠𝑖𝑛2𝜃
𝐸′(−𝐴𝑟(1 + 𝜈′) − 𝐵𝑟3(3 + 𝜈′) −
𝐶
𝑟3(1 + 𝜈′) +
𝐷
𝑟(1 − 𝜈′))) (2-150)
In the above equations 𝐴, 𝐵, 𝐶 and 𝐷 are constants specific to each phase and
vary with applied stresses.
Since the stresses in fiber at 𝑟 = 0 are finite 𝐶𝑓 𝑎𝑛𝑑 𝐷𝑓 are zero.
Continuity of 𝜎𝑟 , 𝜏𝑟𝜃, 𝑢𝑟 , 𝜕𝑢𝜃/𝜕𝜃 must be satisfied at 𝑟 = 𝑎 (fiber/matrix interface)
and 𝑟 = 𝑏 (matrix/composite interface)
46
Continuity of radial stress along fiber/matrix interface yields the following
equation:
𝜎𝑟𝑓(𝑎) = 𝜎𝑟𝑚(𝑎) (2-151)
2𝐴𝑓 − (2𝐴𝑚 +6𝐶𝑚
𝑎4+
4𝐷𝑚
𝑎2) = 0 (2-152)
Similarly, incorporating continuity of radial stress along the matrix/composite
interface we get,
𝜎𝑟𝑚(𝑏) = 𝜎𝑟𝑐(𝑏) (2-153)
(2𝐴𝑚 +6𝐶𝑚
𝑏4+
4𝐷𝑚
𝑏2) − (2𝐴𝑐 +
6𝐶𝑐
𝑏4+
4𝐷𝑐
𝑏2) = 0 (2-154)
Continuity of radial displacement must be ensured along the fiber/matrix interface
𝑢𝑟𝑓(𝑎) = 𝑢𝑟𝑚(𝑎) (2-155)
1
𝐸𝑓′ (2𝐴𝑓𝑎(1 + 𝜈𝑓) + 4𝜈𝑓𝐵𝑓𝑎3)
−1
𝐸𝑚′
((2𝐴𝑚𝑎(1 + 𝜈𝑚) + 4𝜈𝑚𝐵𝑚𝑎3 −2𝐶𝑚
𝑎3(1 + 𝜈𝑚) −
4𝐷𝑚
𝑎))
= 0 (2-156)
Similarly, considering the continuity of radial displacement along the
matrix/composite interface we get the following equation:
1
𝐸𝑚′
((2𝐴𝑚𝑏(1 + 𝜈𝑚) + 4𝜈𝑚𝐵𝑚𝑏3 −2𝐶𝑚
𝑏3(1 + 𝜈𝑚) −
4𝐷𝑚
𝑏))
−1
𝐸𝑐′
((2𝐴𝑐𝑏(1 + 𝜗𝑐) −2𝐶𝑐
𝑏3(1 + 𝜈𝑐) −
4𝐷𝑐
𝑏))
= 0 (2-157)
Continuity of shear stress must be ensured along the fiber/matrix interface
𝜏𝑟𝜃𝑓(𝑎) = 𝜏𝑟𝜃𝑚(𝑎) (2-158)
47
(2𝐴𝑓 + 6𝐵𝑓𝑎2) − (2𝐴𝑚 + 6𝐵𝑚𝑎2 −6𝐶𝑚
𝑎4−
2𝐷𝑚
𝑎2) = 0 (2-159)
Similarly, considering the continuity of shear stress along matrix/composite
interface we get the following equation:
(2𝐴𝑚 + 6𝐵𝑚𝑏2 −6𝐶𝑚
𝑏4−
2𝐷𝑚
𝑏2) − (2𝐴𝑐 −
6𝐶𝑐
𝑏4−
2𝐷𝑐
𝑏2) = 0 (2-160)
Continuity of tangential displacement gradient must be ensured along the
fiber/matrix interface
𝜕𝑢𝜃𝑓
𝜕𝜃(𝑎) =
𝜕𝑢𝜃𝑚
𝜕𝜃(𝑎) (2-161)
1
𝐸1𝑓(−𝐴𝑓𝑎(1 + 𝜗𝑓) − 𝐵𝑓𝑎3(3 + 𝜗𝑓))
−1
𝐸1𝑚(−𝐴𝑚𝑎(1 + 𝜈𝑚) − 𝐵𝑚𝑎3(3 + 𝜈𝑚) −
𝐶𝑚
𝑎3(1 + 𝜈𝑚) +
𝐷𝑚
𝑎(1 − 𝜈𝑚))
= 0 (2-162)
Similarly, considering the continuity of shear stress along matrix/composite
interface we get the following equation:
1
𝐸𝑚′
(−𝐴𝑚𝑏(1 + 𝜗𝑚′ ) − 𝐵𝑚𝑏3(3 + 𝜗𝑚
′ ) −𝐶𝑚
𝑏3(1 + 𝜗𝑚
′ ) +𝐷𝑚
𝑏(1 − 𝜗𝑚
′ ))
−1
𝐸𝑐′
(−𝐴𝑐𝑏(1 + 𝜗𝑐′) −
𝐶𝑐
𝑏3(1 + 𝜗𝑐
′) +𝐷𝑐
𝑏(1 − 𝜗𝑐
′))
= 0 (2-163)
Constants 𝐴𝑓 , 𝐵𝑓 , 𝐴𝑚, 𝐵𝑚, 𝐶𝑚𝐷𝑚, 𝐴𝑐, 𝐶𝑐 and 𝐷𝑐 can be found from the above
derived continuity equations. The stresses in each phase are calculated as follows
Fiber Equations
𝜎𝑟𝑓 = 2𝐴𝑓𝑠𝑖𝑛2𝜃
48
𝜎𝜃𝑓 = −(2𝐴𝑓 + 12𝐵𝑓𝑟2)𝑠𝑖𝑛2𝜃
𝜏𝑟𝜃𝑓 = (2𝐴𝑓 + 6𝐵𝑓𝑟2)𝑐𝑜𝑠2𝜃 (2-164)
𝑢𝑟𝑓 =𝑠𝑖𝑛2𝜃
𝐸𝑓′ (2𝐴𝑓𝑟(1 + 𝜈𝑓
′ ) + 4𝜈𝑓′ 𝐵𝑓𝑟3)
𝜕𝑢𝜃𝑓
𝜕𝜃=
4𝑠𝑖𝑛2𝜃
𝐸𝑓′ (−𝐴𝑓𝑟(1 + 𝜈𝑓
′ ) − 𝐵𝑓𝑟3(3 + 𝜈𝑓′ ))
Matrix Equations
𝜎𝑟𝑚 = (2𝐴𝑚 +6𝐶𝑚
𝑟4+
4𝐷𝑚
𝑟2) 𝑠𝑖𝑛2𝜃
𝜎𝜃𝑚 = − (2𝐴𝑚 + 12𝐵𝑚𝑟2 +6𝐶𝑚
𝑟4) 𝑠𝑖𝑛2𝜃
𝜏𝑟𝜃𝑚 = (2𝐴𝑚 + 6𝐵𝑚𝑟2 −6𝐶𝑚
𝑟4−
2𝐷𝑚
𝑟2) 𝑐𝑜𝑠2𝜃 (2-165)
𝑢𝑟𝑚 =𝑠𝑖𝑛2𝜃
𝐸𝑚′
((2𝐴𝑚𝑟(1 + 𝜈𝑚′ ) + 4𝜈𝑚
′ 𝐵𝑚𝑟3 −2𝐶𝑚
𝑟3(1 + 𝜈𝑚
′ ) −4𝐷𝑚
𝑟))
𝜕𝑢𝜃𝑚
𝜕𝜃=
4𝑠𝑖𝑛2𝜃
𝐸𝑚′
(−𝐴𝑚𝑟(1 + 𝜈𝑚′ ) − 𝐵𝑚𝑟3(3 + 𝜈𝑚
′ ) −𝐶𝑚
𝑟3(1 + 𝜈𝑚
′ ) +𝐷𝑚
𝑟(1 − 𝜈𝑚
′ ))
Composite Equations
𝜎𝑟𝑐 = (2𝐴𝑐 +6𝐶𝑐
𝑟4+
4𝐷𝑐
𝑟2) 𝑠𝑖𝑛2𝜃
𝜎𝜃𝑐 = − (2𝐴𝑐 +6𝐶𝑐
𝑟4) 𝑠𝑖𝑛2𝜃
𝜏𝑟𝜃𝑐 = (2𝐴𝑐 −6𝐶𝑐
𝑟4−
2𝐷𝑐
𝑟2) 𝑐𝑜𝑠2𝜃 (2-166)
𝑢𝑟𝑐 =𝑠𝑖𝑛2𝜃
𝐸𝑐′
((2𝐴𝑐𝑟(1 + 𝜈𝑐′) −
2𝐶𝑐
𝑟3(1 + 𝜈𝑐
′) −4𝐷𝑐
𝑟))
𝜕𝑢𝜃𝑐
𝜕𝜃=
4𝑠𝑖𝑛2𝜃
𝐸𝑐′
(−𝐴𝑐𝑟(1 + 𝜈𝑐′) −
𝐶𝑐
𝑟3(1 + 𝜈𝑐
′) +𝐷𝑐
𝑟(1 − 𝜈𝑐
′))
49
Substituting 𝑟 = ∞ in Equation (2-166) i.e. composite phase we get,
𝜎𝑟𝑐 = (2𝐴𝑐)𝑠𝑖𝑛2𝜃
𝜎𝜃𝑐 = −(2𝐴𝑐)𝑠𝑖𝑛2𝜃 (2-167)
𝜏𝑟𝜃𝑐 = (2𝐴𝑐)𝑐𝑜𝑠2𝜃
Transforming the stresses into Cartesian co-ordinates yields
[
𝜎𝑦
𝜎𝑧
𝜏𝑦𝑧
] = [𝑐𝑜𝑠2𝜃 𝑠𝑖𝑛2𝜃 − 𝑠𝑖𝑛2𝜃𝑠𝑖𝑛2𝜃 𝑐𝑜𝑠2𝜃 𝑠𝑖𝑛2𝜃𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃 − 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃 𝑐𝑜𝑠2𝜃
] [𝑠𝑖𝑛2𝜃
−𝑠𝑖𝑛2𝜃𝑐𝑜𝑠2𝜃
] 2𝐴𝑐
We get,
[
𝜎𝑦
𝜎𝑧
𝜏𝑦𝑧
] = [00
2𝐴𝑐
] (2-168)
It can be observed in Equation (2-168), the shear in the transverse plane is the
only non- zero stress present in the plane whereas 𝜎𝑦 and 𝜎𝑧 are zero. So, the applied
transverse shear can be equated to 2𝐴𝑐.
The micro stresses in fiber and matrix from all the five-decomposed cases are
superimposed. Principal stresses are calculated from the resultant stresses in both the
phases. Depending on the failure criterion used, failure envelopes for multiaxial stresses
can be developed and various strengths can be predicted.
A plane strain case along the transverse plane of the composite is considered to
validate the analytical model using finite element analysis which will be discussed in
Chapter 3.
50
CHAPTER 3 FINITE ELEMENT ANALYSIS AND COMPARISON
Modelling and analysis of Hexagonal RVE
In this section, finite element modelling of a rectangular RVE (representative
volume element) of a composite with hexagonal unit-cell is discussed. The results will
be used in section 3.2 for comparing with analytical results. Only a planar hexagonal
array model is considered for the current analysis.
Figure 3-1. Representative volume element of a hexagonal unit cell
The benefit of using a rectangular RVE is that periodic boundary conditions can
be applied with least effort in the finite element analysis. Although models with
rectangular array of fibers is easier to analyze, the hexagonal unit-cell is a better
approximation of the three-phase model used in the present study. It is also a better
idealization of unidirectional fiber composites [13]
The commercial finite element software ABAQUS is used in the present study.
The coordinate system used in ABAQUS is depicted in figure 3-2. The conventional
principal material coordinate system 1-2-3 is designated as y-z-x in ABAQUS.
51
Figure 3-2. Coordinate system used in ABAQUS and principal coordinate system
A 2D deformable planar shell type element is chosen for the micromechanical
analysis. Due to symmetry, a quarter model was considered for the analysis. The
section sketch is depicted in the figure 3-3. For a fiber volume ratio 𝑉𝑓 of 0.6, the
dimensions of the rectangle are calculated as 15X8.66.
Figure 3-3. Sectional view and dimensions of the RVE
Multiple sections were created to obtain a uniform and regular mesh in both the
fiber and matrix phases. Fiber and matrix materials were assumed to be isotropic. The
following table 3-1 provides the list of properties used in the FEA.
52
Table 3-1. Properties of Kevlar/Epoxy used in the FEA
Property Fiber Matrix
Young’s modulus 130 GPa 3.5 GPa
Poisson’s ratio 0.3 0.35
Tensile strength 2.8 GPa 0.07 GPa
The meshing was fashioned differently for the fiber and matrix phases. Figure 3-4
highlights the different regions as well as the meshing pattern. The region enclosed in a
red border is the fiber and rest of the region is matrix. A 4-node bilinear plane strain
quadrilateral element (CPE4R) was chosen as an element type for both the phases.
More information regarding meshing can be observed in the figure 3-5. Reduced
integration was selected to reduce the computational time. A total of 756 elements were
created of which 144 belong to the matrix and 612 belong to the fiber. All the elements
in the fiber and matrix phase are quadrilaterals.
Figure 3-4. Meshed RVE, red bounded regions represent fiber and green unbounded region represents matrix
53
Figure 3-5. Element type used for meshing and analysis
Linear analysis was performed and micro stresses were calculated at the
centroid of each element. From the stress matrix, the maximum principal stress and von
Misses stresses were calculated in each element. Apart from the stresses, volume of
each element is also extracted to compute the macro stresses. Three different plane
strain analysis were performed on the RVE explicitly considering unit strain elongation
in y and z direction as well as unit strain elongation in both y and z directions together.
Figure 3-7 depicts a schematic which summarizes the whole procedure of obtaining the
coefficients of stiffness matrix 𝐶22, 𝐶33, 𝐶23, and 𝐶32 from the unit strain finite element
analysis.
54
Figure 3-6. Boundary conditions and loading in unit strain analysis (A) Direction 2 (B) direction 3
Since we are using plane strain analysis coefficients of stiffness matrix,
𝐶12 and 𝐶13 can also be obtained from the data. The coefficient 𝐶11 can be found using
the Rule of Mixtures
𝐶11 = 𝐶11𝑓𝑉𝑓 + 𝐶11𝑚(1 − 𝑉𝑓) (3-1)
where, 𝑉𝑓 = (𝑎
𝑏)
2
is the volume fraction of the fiber, which is assumed to be 0.6
for the analysis and individualistic stiffness coefficients 𝐶11𝑓 and 𝐶11𝑚 can be calculated
from the material properties of fiber and matrix.
The stiffness matrix 𝐶 can be populated by considering symmetry of [C].
𝐶 = [
𝐶11 𝐶12 𝐶13
𝐶12 𝐶22 𝐶23
𝐶13 𝐶23 𝐶33
]
l l
b
(A) (B)
b
55
Figure 3-7. Schematic of the procedure followed to obtain stiffness matrix
Unit strain analysis
Extract S11, S22, and S33 at the
centroid of each element
Multiply the stresses of each element
with their respective volumes
Summation of the products divided by
the total volume gives the macro
stress
Analysis along 2 and 3 directions yields 2nd and 3rd Columns of the
composite stiffness matrix respectively
56
Table 3-2. Coefficients of stiffness matrix obtained from unit strain analysis
Composite 𝐶22 𝐶33 𝐶23 𝐶32
Kevlar/Epoxy 1.63 1.63 0.730 0.730
E-glass/Epoxy 1.62 1.62 0.680 0.680
After the micromechanical analysis of the hexagonal RVE one can notice that
𝐶22 = 𝐶33 as well as 𝐶23 = 𝐶32. From table 3-2 it is apparent that analysis along one
direction of the transverse axis is sufficient to determine all the coefficients of the
stiffness matrix due to symmetry.
Figure 3-8 depicts deformed RVE from different unit strain analysis, the unit
strain difference can be discerned by overlapping the initial and the deformed bodies.
(A)
Figure 3-8. Initial and deformed hexagonal RVE under unit strain in A) 2nd Direction B) 3rd direction C) 2nd and 3rd direction
57
(B)
(C)
Figure 3-8 Continued
58
Unit strain analysis in both 2 and 3 directions, i.e. figure 3-8 C, is solely done for
comparison of maximum stresses with the analytical model and will be discussed later
in this chapter.
From the micro stresses obtained using the FEA unit strain analysis as well as
individual constituent material strengths and failure criteria, failure envelopes can be
plotted in the 2-3 plane. A flowchart explaining the procedures is given in Fig. 3-9.
The stiffness matrix for the case of Kevlar/epoxy composite has been found out to be
[𝐶] = [84.4 7.50 7.507.50 16.3 7.307.50 7.30 16.3
] 𝐺𝑝𝑎
Considering the tensile strengths mentioned in table 3-1 for fiber and matrix, we
determine the load factors in each element for maximum stress criterion and a quadratic
failure criterion. Identifying the maximum load factor from among the element load
factors will determine the strength of the composite for the given applied stress. Figure
3-10 shows the comparison of maximum principal stress theory and quadratic theory
(von Mises theory) in the transverse direction. The difference in the transverse strength
is attributed to the hexagonal RVE used for unit strain analysis. Figure 3-10 shows the
failure envelopes obtained for Kevlar/epoxy from unit strain analysis in 2 and 3
directions. Table 3-3 contains strengths calculated at different stress conditions. The
difference in the values of 𝑆𝑇+ in direction 2 and 3 can be attributed to the shape of the
hexagonal RVE. The strengths obtained are in a good agreement with strengths from
Zhu, Marrey, and Sankar [15] finite element model.
59
Figure 3-9. Schematic of procedure to plot a failure envelope in 2-3 plane
Inputs: Stiffness Matrix and Strengths of the materials
Calculating compliance matrix and composite properties
Apply a given set of macro-stresses on the composite and calculate corresponding macro-
strains
Calculating the micro stresses and principal stresses in fiber and
matrix
Calculating the load factor for each element using the strength
of the material
Finding out the maximum load factor and dividing it by the applied stress will result in the maximum stress that can be applied
60
A three-letter notation [15] has been used to refer the type of failure criterion
used for fiber and matrix in this section as well as in chapter 4. The first letter refers to
the failure criterion used for fiber and the second letter refers to matrix failure criterion.
The last letter denotes the type of interface failure incorporated and N denotes that the
interface was not considered in the micromechanical analysis. The letters M and Q refer
to maximum stress theory and quadratic failure theory, respectively. For example, QQN
means that quadratic failure criteria were used for both fiber and matrix and interface
failure was not considered.
Figure 3-10. Failure envelopes of Kevlar/Epoxy in transverse direction obtained through unit strain analysis
Table 3-3. Transverse strengths at various points for Kevlar/Epoxy (plane strain)
Failure Criteria 𝑆𝑇+ (Direction 2) 𝑆𝑇
+ (Direction 3) 𝜎2 = 𝜎3 𝜎2 = −𝜎3
MMN 54 38 58.2 28
QQN 71 59 126 34
61
Comparison with analytical model
In this section, comparison of results from finite element model and analytical
model are presented. The validation of the analytical model is done by comparing the
unit strain analysis results with finite element unit strain analysis.
Direct Micro-Mechanics method (DMM) is used in both models to obtain the micro
stresses which are used to calculate the load factors (inverse of factor of safety) to
develop the failure envelopes. ADMM refers to Analytical Direct Micromechanics
Method and FEA refers to Finite Element Analysis. Figure 3-11 shows the comparison
of maximum principal stress failure envelopes in the transverse plane. As one can
notice the envelopes compare fittingly.
Figure 3-11. Comparison of analytical and finite element model failure envelopes using
maximum stress theory in the transverse plane (2-3 plane)
Figure 3-12 portrays the comparison of the failure envelopes considering
quadratic failure theory which also compare favorably. Since the fiber is considered
isotropic, von Mises criteria is used.
62
Figure 3-12. Comparison of analytical and finite element model failure envelopes using
quadratic theory in the transverse plane (2-3 plane)
Table 3-4 shows the comparison of strengths at different locations. One can note
the values compare satisfactorily for (𝜎2 = −𝜎3) pure shear considering quadratic failure
for fiber and matrix. Tables 3-5 to 3-10 contains detailed comparison of various stresses
i.e. maximum principal stress, maximum von Mises stress. Since failure of an element
does not cause catastrophic failure of the composite, 10th percentile stress in each case
is also compared. An additional case for 휀2 = 휀3 = 1 is simulated specifically to compare
the stresses at various points.
In this case the results compared positively. Relatively noticeable difference can
be observed for the case of 휀3 = 1 which can be attributed to the shape of the
hexagonal RVE as discussed in the previous section.
63
Table 3-4. Comparison of various transverse strengths for Kevlar/Epoxy (plane strain)
Failure Criteria 𝑆𝑇+ (Direction 2) 𝑆𝑇
+ (Direction 3) 𝜎2 = 𝜎3 𝜎2 = −𝜎3
FEA DMM FEA DMM FEA DMM FEA DMM
MMN 54 53 38 53 58.2 64 28 44
QQN 71 75 59 75 126 137 34 39
Table 3-5. Comparison of maximum principal stress for Kevlar/Epoxy (plane strain)
Case Fiber Matrix % Difference
FEA DMM FEA DMM Fiber Matrix
휀2 = 1 20.6 20.2 19.4 20.6 2 -6.1
휀3 = 1 25 20.2 25.2 20.5 19 18.5
휀2 = 휀3 = 1 27 26.4 27 26.4 2 2.2
Table 3-6. Comparison of maximum von Mises stress for Kevlar/Epoxy (plane strain)
Case Fiber Matrix % Difference
FEA DMM FEA DMM Fiber Matrix
휀2 = 1 14.4 13.2 11.3 11.5 8.4 -1.4
휀3 = 1 15.1 13.2 14.4 11.4 12.7 20.5
휀2 = 휀3 = 1 10.5 10.6 12.3 11.9 -0.2 3.3
Table 3-7. Comparison of average of top 10% maximum principal stresses for
Kevlar/Epoxy (plane strain)
Case Fiber Matrix % Difference
FEA DMM FEA DMM Fiber Matrix
휀2 = 1 19.4 20.2 19.1 20.4 -3.8 -6.7
휀3 = 1 22.1 20.2 23.5 20.4 8.8 13.2
Table 3-8. Comparison of average of top 10% von Mises stresses for Kevlar/Epoxy
(plane strain)
Case Fiber Matrix % Difference
FEA DMM FEA DMM Fiber Matrix
휀2 = 1 12.3 13.2 11 11.1 -7.1 -0.7
휀3 = 1 14.4 13.2 12.8 11.1 8.5 13.0
64
Table 3-9. Comparison of 10th percentile maximum principal stress for Kevlar/Epoxy (plane strain)
Case Fiber Matrix % Difference
FEA DMM FEA DMM Fiber Matrix
휀2 = 1 19.2 20.2 18.9 20.2 -5.2 -6.9
휀3 = 1 21.2 20.2 22.8 20.2 4.7 11.4
휀2 = 휀3 = 1 26.6 26.0 26.6 26.0 2.3 2.3
Table 3-10. Comparison of 10th percentile maximum von Mises stress for Kevlar/Epoxy
(plane strain)
Case Fiber Matrix % Difference
FEA DMM FEA DMM Fiber Matrix
휀2 = 1 11.9 13.2 10.8 10.8 -10.9 0
휀3 = 1 13.9 13.2 12.0 10.8 5 10
휀2 = 휀3 = 1 10.4 10.5 12.2 12.0 -1 1.6
To summarize, reasonable comparison of the two models validates the analytical
model for further analysis and prediction of composite strengths currently used in
industry which will be discussed in Chapter 4.
65
CHAPTER 4 ANALYTICAL MODEL RESULTS AND DISCUSSION
In this section, the results obtained from Analytical Direct Micro-mechanics
(ADMM) will be discussed. The results are divided into four parts: (1) Failure envelopes
for composites with isotropic fibers, e.g., Kevlar/epoxy composite; (2) Failure envelopes
for composites with transversely isotropic fibers, e.g., carbon/epoxy composite; (3)
Interface effects on the above composites; (4) Volume fraction analysis for the above-
mentioned composites; and (5) Strengths for several composites used currently in
industries.
Results for Kevlar/Epoxy
Two failure criteria, maximum principal stress and quadratic criteria have been
compared in this chapter. The maximum principal stress theory is shown below
𝑆𝐿− < 𝜎1 < 𝑆𝐿
+
𝑆𝑇− < 𝜎2, 𝜎3 < 𝑆𝑇
+ (4-1)
𝜏12 < 𝑆𝐿𝑇
Where, 𝜎1, 𝜎2, 𝜎3 are stresses in the principal material directions.
Strengths 𝑆𝐿+, 𝑆𝑇
+, 𝑆𝐿𝑇 and 𝑆𝑇𝑇 have been tabulated at the end of each analysis.
𝑆𝑇𝑇 Refers to the shear strength in the 2-3 plane. Longitudinal compressive strength
𝑆𝐿− has not been studied since the failure due to compressive stress is due to buckling of
the fibers and is considered more of an instability phenomenon rather than failure of the
material. The notations such as MMN, QQY are defined as follows. The letters M and Q
refer to maximum stress theory and quadratic failure theory, respectively. For example,
QQN means that quadratic failure criteria was used for both fiber and matrix and
interface failure was not considered.
66
In this section, the effective strength properties of Kevlar/Epoxy will be studied
using analytical based micromechanics. The fiber and matrix materials were assumed
to be isotropic. Table 4-1 lists the constituent properties used
Table 4-1. Properties of Kevlar/Epoxy
Property Fiber Matrix
Young’s modulus 130 GPa 3.5 GPa
Poisson’s ratio 0.3 0.35
Tensile strength 2.8 Gpa 0.07 Gpa
Figure 4-1. Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy on 𝜎1 − 𝜎2
plane
67
Figure 4-2. Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy in the 𝜎2 − 𝜎3 plane
Figure 4-3. Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy for
longitudinal shear in the 1-2 or 1-3 plane
68
Figure 4-4. Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy subjected
to both longitudinal and transverse shear stresses
Figure 4-5. Comparison of MMN and QQN failure envelopes of Kevlar/Epoxy for shear
in longitudinal direction and stress in fiber direction
69
Figures 4-1 through 4-5 shows the failure envelopes of Kevlar/Epoxy for various
combinations of normal and shear stresses. The strength properties of Kevlar/epoxy
have been extracted from the above plots and are tabulated in table 4-1. The strengths
obtained show good comparison with the properties obtained through finite element
micromechanics by Zhu, Marrey and Sankar [16].
Table 4-2. Strengths at various points for Kevlar/Epoxy (MPa)
Failure Criteria 𝑆𝐿+ 𝑆𝑇
+ 𝑆𝐿𝑇 𝑆𝑇𝑇
MMN 1573 53 56 44
QQN 1592 75 32 39
Carbon/Epoxy plots
In this section, the effective strength properties of carbon/epoxy will be studied
using analytical micromechanics. A transversely isotropic fiber is considered for the
current analysis. Table 4-3 lists the properties used in detail.
Since a transversely isotropic fiber failure is governed generally by Hashin type
failure theories [17], quadratic failure criterion for transversely isotropic composites is
incorporated for carbon with few modifications.
The quadratic failure can be summarized into four cases as shown below:
Tensile fiber mode
22 2
12 1311
21
L LTS S
(4-2)
Compressive fiber mode
11 LS (4-3)
Tensile matrix mode
70
2 2 2
22 33 23 22 33 12 23( )2 2 2
1 1 1( ) ( ) ( ) 1
T TT LTS S S
(4-1)
Compressive matrix mode
2
2
22 33 22 332
2 2 2
23 22 33 12 232 2
1 11 ( ) ( )
2 4
1 1( ) ( ) 1
T
T TT TT
TT LT
S
S S S
S S
(4-2)
Here 𝑆𝑇𝑇 refers to the transverse shear strength in the composite.
Table 4-3. Properties of Carbon-T300/Epoxy-5208
Property Fiber
Carbon T300
Matrix
Epoxy 5208
Young’s modulus 𝐸1 = 231 𝐺𝑃𝑎 𝐸2 = 13.7 𝐺𝑃𝑎 𝐸𝑚 = 3.5 𝐺𝑃𝑎
Poisson’s ratio 𝜗12 = 0.27 𝜗23 = 0.00125 𝜗𝑚 = 0.35
Tensile strength 𝑆𝐿+ = 2400 𝑀𝑃𝑎 𝑆𝑇
+ = 650 𝑀𝑃𝑎 𝑆 = 60 𝑀𝑃𝑎
Figure 4-6. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 1-2
plane
71
Figure 4-7. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy in 2-3
plane
Figure 4-8. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for shear
in longitudinal directions
72
Figure 4-9. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy subjected
to both longitudinal and transverse shear stresses
Figure 4-10. Comparison of MMN and QQN failure envelopes of Carbon/Epoxy for
longitudinal shear and normal stress in fiber direction
Figures 4-6 to 4-10 shows the failure envelopes of Carbon/Epoxy for various
combinations of normal and shear stresses. The strength properties of Carbon/epoxy
73
have been extracted from the above plots and are tabulated in table 4-4. The strengths
obtained showed excellent comparison with the properties from principles of composite
material mechanics by Gibson [18].
Table 4-4. Predicted strengths of T300/5208/Carbon/Epoxy
Failure Criteria 𝑆𝐿+ 𝑆𝑇
+ 𝑆𝐿𝑇 𝑆𝑇𝑇
MMN 1456 49 57 43
QQN 1456 62 33 31
Effects of Interface
In this section, effects of interface on different strength properties and envelopes
will be discussed. Fiber/ matrix interface plays a pivotal role in determining the strength
of the composites. Two failure criteria are used to study the effects of interface on
strengths. It is assumed that compressive stresses will not affect the interface.
Maximum stress criteria used is discussed as follows:
The normal stress acting at the interface is 𝜎𝑟 i.e. radial tensile stress acting at
the interface.
Interfacial tensile (𝜎𝑟 > 0)
𝜎𝑟 < 𝑆𝐼𝑇 (4-4)
Interfacial shear
√(𝜏𝑟𝜃2 + 𝜏𝑟𝑥
2 ) < 𝑆𝐼𝑆 (4-5)
Where 𝑆𝐼𝑇 is interfacial tensile strength and 𝑆𝐼𝑆 is interfacial shear strength
74
The quadratic interface theory is explained below:
𝜎𝑟2
𝑆𝐼𝑇2 +
(𝜏𝑟𝜃2 + 𝜏𝑟𝑥
2 )
𝑆𝐼𝑆2 = 1 (4-6)
Interface strength determination through experiments is inaccurate and difficult
as very limited data is available. Significant amount of work was done by Huges [19] on
carbon/Epoxy properties. Many factors were considered and there was no simple
conclusion found about the nature of Carbon/Epoxy interface. It was found out that the
interface has a very high stress concertation. The following plots demonstrates the
effect of interface on failure envelopes.
Figure 4-11. Interface effects on failure envelopes for Kevlar/Epoxy in 1-2 plane using
maximum stress theory
75
Figure 4-12. Interface effects on failure envelopes for Kevlar/Epoxy in 1-2 plane using
quadratic theory
Figure 4-13. Interface effects on failure envelopes for Kevlar/Epoxy in 2-3 plane using
maximum stress theory
76
Figure 4-14. Interface effects on failure envelopes for Kevlar/Epoxy in 2-3 plane using
quadratic theory
Figure 4-15. Interface effects on failure envelopes subjected to both longitudinal and
transverse shear stresses using maximum stress theory
77
Figure 4-16. Interface effects on failure envelopes subjected to both longitudinal and
transverse shear stresses using quadratic theory
Figure 4-17. Interface effects on failure envelopes for Carbon/Epoxy in 1-2 plane using
maximum stress theory
78
Figure 4-18. Interface effects on failure envelopes for Carbon/Epoxy in 1-2 plane using
quadratic theory
Figure 4-19. Interface effects on failure envelopes for Carbon/Epoxy in 2-3 plane using
maximum stress theory
79
Figure 4-20. Interface effects on failure envelopes for Carbon/Epoxy in 2-3 plane using
quadratic theory
Figure 4-21. Interface effects on failure envelopes for Carbon/Epoxy for envelopes
subjected to both longitudinal and transverse shear stresses using maximum stress theory
80
Figure 4-22. Interface effects on failure envelopes for Carbon/Epoxy longitudinal shear
and normal stress in fiber direction using maximum stress theory
Figures 4-11 to 4-22 shows the failure envelopes of Carbon/Epoxy for various
combinations of normal and shear stresses. The strength properties of Kevlar/Epoxy
and Carbon/epoxy have been extracted from the above plots and are tabulated in tables
4-5 and 4-6. The strengths obtained showed acceptable comparison with the properties
obtained through finite element micromechanical analysis by Zhu, Marrey and Sankar
[20].
Table 4-5. Comparison of strengths for Kevlar/epoxy including interface failure obtained using ADMM
Failure Criteria 𝑆𝐿+ 𝑆𝑇
+ 𝑆𝐿𝑇 𝑆𝑇𝑇
MMN 1573 53 56 44
QQN 1592 75 32 39
MMM 1573 23.4 22.55 20.38
QQQ 1592 23.4 18.15 20.38
81
Table 4-6. Comparison of strengths for Carbon/Epoxy including interface failure obtained using ADMM
Failure Criteria 𝑆𝐿+ 𝑆𝑇
+ 𝑆𝐿𝑇 𝑆𝑇𝑇
MMN 1456 49 57 43
QQN 1456 62 33 31
MMM 1456 38.6 36 36.5
QQQ 1456 39 37 37
Volume fraction analysis
In this section, the variation of strengths with volume fraction will be discussed for
composites Kevlar/Epoxy and Carbon /Epoxy. Figures 4-23 to 4-28 depicts the variation
of 𝑆𝐿+, 𝑆𝑇
+, 𝑆𝐿𝑇 , 𝑆𝑇𝑇 with 𝑉𝑓.
Interface failure strengths are also considered and as explained in Chapter 3,
MMM refers to maximum stress theory for fiber, matrix and interface.
Figure 4-23. Variation of longitudinal shear strength with volume fraction in
Kevlar/Epoxy
82
Figure 4-24. Variation of Transverse tensile strength with volume fraction in
Kevlar/Epoxy
Figure 4-25. Variation of Transverse shear strength with volume fraction in Kevlar/Epoxy
83
Figure 4-26. Variation of longitudinal shear strength with volume fraction in
Carbon/Epoxy
Figure 4-27. Variation of transverse tensile strength with volume fraction in
Carbon/Epoxy
84
Figure 4-28. Variation of transverse shear strength with volume fraction in
Carbon/Epoxy
Summary
In this section, a detailed comparison of strengths obtained from the analytical
model with reference strengths extracted from Gibson [21] will be discussed.
Table 4-7 shows the comparison of properties obtained from maximum principal stress
theory and quadratic failure theory with the properties available in literature. von Mises
theory was used for an isotropic fiber and Hashin’s [22] criteria was used for
transversely isotropic fibers.
Table 4-8 shows the percentage difference with reference strengths. It can be
noticed that the model showed an excellent comparison with 𝑆𝐿+ in both MMN and QQN
except for boron/aluminum and E-glass/epoxy. In the case of boron/aluminum, yielding
plays a crucial role in determining the strengths which the present model will not
consider.
85
Table 4-7. Comparison of strengths for several composites with analytical model strengths
Reference MMN QQN
Composite 𝜗𝑓 𝑆𝐿+ 𝑆𝑇
+ 𝑆𝐿𝑇 𝑆𝐿+ 𝑆𝑇
+ 𝑆𝐿𝑇 𝑆𝐿+ 𝑆𝑇
+ 𝑆𝐿𝑇
Boron/5505 Boron/epoxy
0.5 1586 62.7 82.7 1593 46 65 1593 63 38
AS/3501 Carbon/epoxy
0.6 1448 48.3 62.1 1456 49 57 1456 62 33
T300/5208 Carbon/Epoxy
0.6 1488 44.8 62.1 1456 48 57 1456 62 33
IM7/8551-7 Carbon/Epoxy
0.6 2578 75.8 --- 2492 67 69 1492 86 42
AS4/APC2 carbon/PEEK
0.58 2060 78 157 2169 63 68 2169 81 41
B4/6061 Boron/Aluminum
0.5 1373 118 128 400 96 95 400 115 57
Kevalr49/epoxy aramid/epoxy
0.6 1379 27.6 60 1332 38 50 1332 45 31
Scocthply1002 E-glass/epoxy
0.45 1100 27.6 82.7 440 35 36 440 44 23
E-glass/470-36 E-glass/Vinyl ester
0.3 584 43 64 478.5 43 40 478.5 50 24
Table 4-8. %Difference of strengths for several composites relative to reference
strengths MMN QQN
Composite 𝜗𝑓 𝑆𝐿+ 𝑆𝑇
+ 𝑆𝐿𝑇 𝑆𝐿+ 𝑆𝑇
+ 𝑆𝐿𝑇
Boron/5505 Boron/epoxy
0.5 0 1 21 0 -31 54
AS/3501 Carbon/epoxy
0.6 0 -1 8 0 -28 47
T300/5208 Carbon/Epoxy
0.6 2 -7 8 2 -38 47
IM7/8551-7 Carbon/Epoxy
0.6 3 10 --- 3 -13 ---
AS4/APC2 carbon/PEEK
0.58 -5 18 57 -5 -5 74
B4/6061 Boron/Aluminum
0.5 67 9 26 67 -7 55
Kevalr49/epoxy aramid/epoxy
0.6 3 -38 17 3 -63 48
Scothply1002 E-glass/epoxy
0.45 60 -37 56 60 -59 72
E-glass/470-36 E-glass/Vinyl ester
0.3 18 0 38 18 -16 63
86
𝑆𝑇+ comparison has been acceptable for all the composites and maximum
principal stress theory is found out to be more suitable to calculate transverse tensile
strength. On the other hand, there is a lot of discrepancy for 𝑆𝐿𝑇 but maximum stress
theory proved to be best considering the cases of carbon fibers. Predicted strengths for
carbon-IM7/epoxy have also been presented in table 4-7. Overall maximum principal
stress theory overlooked quadratic theories for predicting strengths through Direct
Micromechanics Method (DMM).
87
CHAPTER 5 CONCLUSIONS AND FUTURE WORK
The three-phase composite model for unidirectional composites which is
originally proposed for homogenization is extended for a method called Analytical Direct
Micro-Mechanics (ADMM) to predict the strength properties of composites. In the three-
phase model the fiber is surrounded by an annular region of matrix and the fiber-matrix
assemblage is embedded in an infinite medium of composite. The elastic constants of
the composite are evaluated using a modified Halpin-Tsai type equations. In the ADMM
the micro-stresses are calculated in the fiber and matrix phases for a given macro-
stress applied to the composite. The micro-stresses in conjunction with failure theories
for fiber, matrix, and fiber/matrix interface are used to determine the failure of the
composites.
The ADMM is evaluated by comparing the results with that of finite element
based micromechanics. The ADMM results compare reasonably well with FEA based
DMM results for failure envelopes. Then the ADMM is extended to various composite
systems and compared with available results for strength values. The ADMM is able to
predict the strength reasonably well in majority of the cases. The strength in the fiber
direction compares well. The transverse strength properties are different in the 2 and 3
directions in the FEA model because of the hexagonal unit cell. However, the three-
phase model being axisymmetric predicts the same strength in the 2 and 3 directions,
which is closer to practical fiber composites. Since it is based on elasticity solution, it is
much faster than FEA based micro-mechanics. Due to its speed, ADMM can be used in
understanding the effects of variability of constituent properties on the composite
88
strength. The model will be very much suited for non-deterministic design of composite
structures.
Although present failure theories have been partially successful, many theories
have to be modified to keep up with the growing composite technology. The need for
unprecedented methods for predicting failure strengths is everlasting.
The normal strengths obtained from Direct Micromechanics Method (DMM) were
reasonable and are in good agreement with the properties of the composites currently
used in the industry. The modified Halpin-Tsai formulations played a productive role in
calculating the effective composite properties for the three-phase model. Although there
were slight discrepancies while comparing ADMM and FEA in principal direction 2 and
3, it still validates the analytical model since symmetry is not observed in a hexagonal
RVE. The unit strain finite element analysis of a hexagonal RVE validated the analytical
model. Interface strength has no effect on longitudinal tensile strength.
Future work. The newly developed Analytical Direct Micromechanics Method
(ADMM) has a lot of potential for future research. Since a preliminary model was
constructed, modifications can be done to consider buckling of fibers under
compression. The ADMM model with correction response surfaces can be a useful
analytical tool wherein a polynomial function of design variables is used to rectify the
difference from the experimental strength. It is also amenable to probabilistic
micromechanics, in which the effect of variability in fiber and matrix properties on the
stiffness and strength properties can be studied. The model can be extended to fracture
analysis (crack propagation in matrix and interface). Since there was a relatively
noticeable difference in longitudinal shear with properties available in literature, more
89
modifications to the model should be made. Thus, the ADMM paves the way for detailed
analysis of the interface.
90
LIST OF REFERENCES
[1] Nam Ho Kim., Course material for EML5526 Finite Element Analysis, University of Florida, Gainesville.
[2] Voigt, , “Theoretischestudien ber die elasticit tsverh ltnisse der krystalle”,
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BIOGRAPHICAL SKETCH
Sai Tharun Kotikalapudi was born in Hyderabad, India, in 1991. He grew up in
various southern states of India. He received a Bachelor of Technology in the field of
mechanical engineering from SASTRA University, India. From there he proceeded to
work for a year at Tata Consultancy Services Engineering and Infrastructure Services,
in Bangalore, India. He has worked under the guidance of personnel from various
internationally recognized companies such as Johnson & Johnson and Beckman
Coulter. His work involved developing and redesigning medical equipment in
SolidWorks. He defended his M.S. thesis in October 2017.