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MICHIG GANS

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1293 01020 4539

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This is to certify that the

dissertation entitled

A MI CROMECHANICS-BASED

INDEPENDENT MODE FAILURE CRITERION

FOR FIBER-REINFORCED COMPOSITES
presented by

Yong-Bae Cho

has been accepted towards fulfillment
of the requirements for

Master ' 3 degree in Mechanics

 

 

Magma

Major professor

Date 11/14/94

 

MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

 

 

LIBRARY
~Mlchlgan State
University

 

 

 

PLACE DI RETURN ”Xmmwombmumnywm
TO AVOID FINES return on or baton duo duo.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A MICROMECHANICS-BASED
INDEPENDENT MODE FAILURE CRITERION
FOR FIBER-REINFORCED COMPOSITES

By

Yong-Bae Cho

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Material Science and Mechanics

1994

ABSTRACT
AMICROMECHANICS-BASED
INDEPENDENT MODE FAILURE CRITERION
FOR FIBER-REINFORCED COMPOSITES
By

Yong-Bae Cho

Failure criteria for composite materials have been developed based on a
micromechanics elasticity solution for continuous fiber reinforced composites. The
criteria can be cast in several forms, including composite and independent mode
polynomial forms. Composite failure is predicted when stress or strain states in any of the
constituents due to a lamina (composite) load exceed a critical level as predicted by a
properly chosen failure criterion for each constituent. Thus, unidirectional strengths as
well as strengths under multi-axial loading of the lamina are determined. Failure
envelopes can be fit to a Tsai-Wu type lamina failure theory by using the predicted
unidirectional strengths and nonlinear regression to determine the interaction terms. It is
also possible to cast the failure model in a simple mathematical form with separate criteria
for each mode of failure as a function of the macro (composite) stress state. The current
model includes the option to extract average “in-situ” constituent strength properties using
unidirectional composite strength data, thus indirectly accounting for the variations in

geometry, stiffness, and strength in composite material system.

To My Parents

iii

ACKNOWLEDGMENT

I wish to express my deepest thanks to my advisor Dr. R. C. Averill for his kind
guidance and assistance throughout this study. His academic excellence and good
personality have provided a constant source of encouragement.

I also wish to express my appreciation to Dr. I.M. Jasiuk and Dr. D. Liu for serving on
my graduate committee.

Finally, I would like to thank my family for their loving support.

iv

TABLE OF CONTENTS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CHAPTER I INTRODUCTION .......................................................................................... l
1.1 Motivation ............................................................................................................................ l
1.2 Literature Review of Failure Criterion ................................................................................ 4
1.3 The Present Study .............................................. - ..... 12
CHAPTER II MICROMECHANICAL ELASTICITY ANALYSIS ................................ 13
2.1 Introduction ........................................................................................................................ 13
2.2 Analytical Model and Mathematical Formulation ............................................................. 13
2.3 Solution for Symmetric Loading ..... - ............................. 20
2.4 Solution for Transverse Shear Loading ............................................................................ .24
2.5 Solutions for Longitudinal Shear Loading ........................................................................ .26
2.6 Computation of Effective Composite Properties ............................................................... 28
CHAPTER III MICROMECHANICAL FAILURE THEORY ......................................... 31
3.1 Introduction -- ................... --31
3.2 An Example of Mieromechanieal Farlme .......................................................................... 31
3.3 Failure Criteria ........... - .......... 35
3 ..3 1 Failure Criteria for Isotropic Material ................................................................. 35
3.3.2 Failure Criteria for Orthotropic Material ............................................................. 38
3.4 Micromechaniml Failure Theory .............................................................................. 39
3.5 Curve Fitting of Failure Envelopes .................................................................................... 4 9
CHAPTER IV INDEPENDENT MODE FAILURE CRITERIA ..................................... 57
4.1 Introduction -- - -- -- 57
4.2 The Concept of Independent Mode Failure Criteria ......................................................... .57
4.3 Independent Mode Failure Criteria -- 58
CHAPTER V THE INDEPEDENT MODE FAILURE CRITERION BASED
ON “IN-SITU” CONSTITUENT STRENGTH .................................................... 61
5.1 Introduction -- ..................... - 61
5.2 The In-situ Constituent Strengths ........ - 63
5.3 The Independent Mode Failure Model Using “In-Sim” Properties- -- - 64
CHAPTER VI NUMERICAL IMPLEMENTATION AND DISCUSSION OF
RESULTS ............................................................................................................... 66
6.1 Numerical Algorithms - - ............. - -- - - --
6.2 The Numerical Results and Discussion ............................................................................. 6 9

CHAPTER VII CONCLUSIONS .................................................................................... 118

REFERENCES .................................................................................................... 121

TABLE 1.
TABLE 2.
TABLE 3.
TABLE 4.
TABLE 5.
TABLE 6.
TABLE 7.

LIST OF TABLES

Material properties of fiber used in calculation ......................................... 70
Material properties of matrix used in calculation ...................................... 71
Material properties of composites measured in experiment ...................... 72
Comparison of interaction strength parameter ........... '. ............................... 96
The strength parameters for independent failure criterion ....................... 109
The comparison between bulk- and in-situ strength ................................ 112
Material properties used in calculation .................................................... 117

vii

Figure 1.

Figure 2.
Figure 3.

Figure 4.

Figure 5.
Figure 6.

Figure 7.

Figure 8.
Figure 9.

Figure 10.
Figure 11.
Figure 12.
Figure 13.
Figure 14.
Figure 15. ~
Figure 16.

Figure 17.
Figure 18.
Figure 19.
Figure 20.
Figure 21.

Figure 22.

Figure 23.

LIST OF FIGURES

Cross section of a fiber reinforced composite material with fibers packed

in a diamond array ..................................................................................... 14
Diamond RVE used in the micromechanical model .................................. 16
The determination of boundary condition from congruency and symmetry
of repeating cells ........................................................................................ 25
The illustration of a simple composite body subjected to compressive

force and the free body diagram ................................................................ 32
The stress evaluation points in RVE .......................................................... 44
(a) The RVE subjected to an off-axis load (b) The reference axes and the
rotated axes ................................................................................................ 46
The variation of failure Envelopes due to change in the interaction term
F12 111 domain .............................................................................................. 53
The creep behavior of some materials used as matrix ............................... 62
The flow chart ............................................................................................ 67
The convergence of stresses according to the number of terms ................ 73
The stress evaluation position .................................................................... 74
The Stress distribution near boundary ....................................................... 75
The Variations of El and E2 with fiber volume fraction ........................... 77
The Variations of 612 and 023 with Fiber Volume Fraction .................... 78
The Variations of and with Fiber Volume Fraction .................................... 79
The Variation of maximum loads causing failure in transverse direction as
the applied angle changes .......................................................................... 80
The predicted tensile transverse strengths by micromechanical failure
theory ('1‘300/934) ...................................................................................... 82
The predicted compressive transverse strengths by micromechanics

failure theory (T300/934) ........................................................................... 83
The predicted longitudinal shear strengths by micromechanics failure
theory ('1'300/934) ...................................................................................... 84
The predicted tensile transverse strengths by micromechanics failure
theory (AS4I3501-6) .................................................................................. 85
The predicted compressive transverse strengths by micromechanics

failure theory (AS4I3501-6) ....................................................................... 86
The predicted longitudinal shear strengths by micromechanics failure
theory (AS4I3501-6) .................................................................................. 87
Failure envelope in stress space ('1'300/934) .............................................. 89

Figure 24.
Figure 25.
Figure 26.
Figure 27.
Figure 28.

Figure 29.

Figure 30.
Figure 31.
Figure 32.
Figure 33.

Figure 34.
Figure 35.
Figure 36.
Figure 37.
Figure 38.
Figure 39.
Figure 40.

Figure 41.
Figure 42.
Figure 43.
Figure 44.

Failure envelopes in 3 dimensional strain space ........................................ 91

Failure envelopes in strain space (TBOO/934) ............................................ 92
Failure envelopes in stress space (AS4I3501-6) ........................................ 93
Failure envelopes in strain space (AS4I3501-6) ........................................ 94

Fitting failure envelope to polynomial type failure envelope and the
failure envelopes using Tsai-Wu theory and experimental results
(T300/934) ................................................................................................. 95
Fitting failure envelope to polynomial type failure envelope and the
failure envelopes using Tsai-Wu theory and experimental results

(AS4/3501-6) ............................................................................................. 97
The changes in failure envelope in stress space due to the change in fiber
volume fraction .......................................................................................... 98
The changes in failure envelope in strain space due to the change in fiber
volume fraction .......................................................................................... 99
The failure envelopes in three dimensions when shear loading is

applied ...................................................................................................... 101
The failure envelopes in two dimensions when shear loading is

applied ...................................................................................................... 102
Independent mode failure envelope in stress space(T300/934) ............... 103
The zoom of the independent mode failure envelopes ............................ 104
The independent failure envelopes (Tresca criterion) .............................. 105
The independent failure envelopes in stress space (AS4I3501-6) ........... 107
The zoom of the independent mode failure envelopes ............................ 108

The curve fitting of independent mode failure envelopes (T300/934) l 10
The zoom of curve fitting of independent mode failure envelopes

('I‘300/934) ............................................................................................... 111
The failure envelope based on in-situ strength ........................................ 113
The independent mode failure envelope based on in-situ strength .......... 114
The composite under off-axis loading ..................................................... 115
The predicted strengths of ASI3501 under off-axis loading on the basis of

in-situ strength ......................................................................................... 116

CHAPTER I
INTRODUCTION

1.1 Motivation

Composite materials have been used for centuries, and a lot of natural composites
exist in nature. According to history, ancient Mongolians used animal tendons, wood and
silk to make laminated bows in BC. 7. Since the 1920’s, glass fiber reinforced resin has
been used, but its application was confined due to low stiffness of the glass fiber. It was
not until the 1960’s that the concept of composite materials was established and research
on composite materials began [ 1]. Since the early 1960’s, the space and aeronautics indus-
tries’ increasing demand on high performance materials that are stiffer, stronger, and
lighter has led to development of composite materials by combining different materials in
order to obtain specific characteristics and prOperties. In the 1970’s, advanced fibers of
extremely high modulus, for example, Boron, Aramid and SiC fibers, were developed.
They have been used as a reinforcing agent in the form of either fiber, whisker or particu-
late to reinforce the polymer (resin), metal and ceramic matrices. The fiber reinforced
composite materials have been more prevalent than any other form of composite, because
the fibrous form of a material is generally much stronger and stiffer than any other form.
In the composite materials where fibers are adopted as a reinforcing agent, there is typi-
cally no reinforcement in the transverse fiber direction. This problem can be resolved by

stacking layers (lamination) with different orientations.
1

2

In general, composite materials have properties which can not be achieved by any of
the constituents acting alone. They have better properties in stiffness, insulation, strength,
weight, damping, corrosion resistance and thermal expansion than the conventional mono-
lithic materials, and are able to meet design requirements with great advantage in aero-
space structures, aircraft, automobiles, sporting goods, marine structures and many others
[3]. On the other hand, the properties depend on many factors, like processing conditions,
the form of reinforcing agent, fiber volume fraction, fiber distribution, interface bonding
stress, void content, etc. [5].

Advanced fiber reinforced composites have outstanding strength-to-weight properties
and often have been used in strength-critical applications where the stiffness and strength
of the materials are the critical factors to be considered. Therefore, for the purpose of
designing composite structures efficiently, it is very important that we can predict the
strength of a composite material under the various loading conditions in service. The use
of a failure criterion for a composite material enables us to predict the strength under com-
bined stresses. Thus the efficient design of structures can be performed on the basis of the
predicted strengths.

A composite material is composed of matrix, interphase and fiber, and the perfor-
mance is achieved by binding the constituents. The fracture of composite materials results
from four possible major reasons: (i) fiber fracture, (ii) matrix cracking, (iii) fiber/matrix
interface debonding, and (iv) fiber buckling or kinking. These modes of failure don’t
occur simultaneously in all constituents but initiate as micro-cracks at the constituent
level, and unite or propagate as external loading (for example, force, heat and moisture) is
increased. Therefore, the strength of a composite material depends on the strength

properties of the constituents as well as their interactions. Generally, the existence of mul-

3
tiple failure modes complicates the assessment of the strength compared with isotropic
material.

To estimate the strength of composite materials, there are two different approaches for
failure or damage prediction - a phenomenological (macromechanics) approach which
tries to correlate experimental observations mathematically but does not necessarily
explain the mechanism of failure, and a mechanistic (micromechanics) approach where
micromechanical stresses in each constituent are used. Composite materials are anisotro-
pic and heterogeneous, so the mean stress at macro level is different from the actual stress
in the constituents. In the phenomenological analyses, the properties of all the constituents
are homogenized into an equivalent substance so that each lamina can be assumed to be
homogeneous and orthotropic. Then prediction of failure or damage is accomplished by
adopting a proper phenomenological failure criterion. The phenomenological failure crite-
ria are empirical in nature, and are basically an attempt to express the yield or failure
envelopes of a material in stress or strain space. Most of the phenomenological criteria are
able to predict only the failure load, but not to differentiate among the modes of failure.
Therefore, it would be more realistic and accurate to analyze stress and check the damage
in every point at the constituent level. However, it has been shown from the ultimate fail-
ure experiments of carbon-epoxy laminates that if a composite is laminated with plies, a
ply criterion can be presumably used for all laminates [6]. Thus, it is desirable to establish
a failure criterion at a lamina level.

In the present study, failure loads and modes are predicted by the evaluation of the
stress/strain states in each constituent using a micromechanical elasticity solution and
employing the phenomenological failure criteria in each constituent in order to evaluate

damage at the micromechanical level. In this way, the failure envelope for each constitu-

4
ent is represented in the stress or strain space. By superimposing failure envelopes of the
constituents and identifying their intersection, a failure envelope of a lamina is attained. If
the constituent strength properties are known, the strength of a lamina can be determined
by detecting the critical stress or strain states at the constituent level. Inversely, using the
experimentally measured unidirectional strengths of a lamina as in the solution, the
strengths of each constituent can be “backed out” by identifying the maximum stress in a
constituent. The resulting failure model thus requires less experimental strength data than
those models currently available, yet provides predictions of both the failure load and

mechanism.

1.2 Literature Review of Failure Criterion

As composite materials were introduced in structural members, a number of failure
criteria for composites were proposed from the late 1960’s to predict their uniaxial and
multiaxial strengths. Rowlands [7] reviewed the phenomenological failure criteria in
1985, and Nahas [8] surveyed the failure theories for laminated composites in 1986.
According to Nahas’s investigation, there exist at least 30 failure criteria, which can be
classified into four categories - the limit criteria, the interaction criteria, the tensor polyno-
mial criteria and the direct laminate criteria. Because failure of composites is complicated
by the existence of several mechanisms, including fiber fracture and micro-buckling, fiber/
matrix interface debonding, matrix cavitation and crack propagation, most of the available
failure criteria can predict the occurrence of failure but do not describe the failure mode.
Such criteria can thus be regarded as phenomenological in nature [15].

Azzi and Tsai [9] developed a failure criterion by applying Hill’s [10] failure criterion

5
for anisotropic materials to transversely isotropic materials in a plane stress state. Using
the stress transformation law, they proved that the failure criterion is applicable to the
most general case of a unidirectional composite subjected to a combined state of stress.
The theoretical predictions were in good agreement with the experimental results.

Hoffman [11] developed a failure criterion that was modified from the Hill’s failure
criterion for anisotropic materials by adding some linear terms to account for the different
tension and compression strengths. Only uniaxial tensile and compressive tests on vari-
ously oriented specimens were performed to verify the theory.

Tsai and Wu [12] proposed a failure criterion in the form of a scalar function of two
strength tensors on the assumption that there exists a failure surface in the stress space in
the scalar form. The linear terms in it take into account the difference in strengths due to
positive and negative stresses and the quadratic terms define an ellipsoid in stress space. It
also accounts for interactions between stress components as independent material proper-
ties. The strength components can be transformed to the other axes by the tensor transfor-
mation law.

Wu and Schueblein [13] proposed three basic approaches to establish a third order fail-
ure criterion for laminates and discussed the merit and shortcoming of each method. The
basic approaches are: (1) The determination of laminate failure from lamina failure crite-
rion, (2) Direct experimental determination of the laminate failure surface and (3) Hybrid
approach using a lamina failure criterion to guide testing of the laminate. Among the
approaches, the hybrid approach utilizing a quadratic polynomial type failure criterion for
lamina and classical laminated plate theory was adopted to develop their theory. Then the
third order failure criterion for laminate was obtained in terms of laminate loads. This

approach can provide the necessary critical experiments with the consequence of minimiz-

6
ing experimental permutation and maximizing the reliability of laminate failure.
Tennyson, et. a1. [14] developed the cubic tensor polynomial failure criterion to predict
failure stress more accurately, since the failure surface may not be ellipsoidal in shape.
They showed how to evaluate the strength parameters in plane stress state. The strength

parameters F i and F were determined by a hybrid method which utilized a biaxial

ijk
strength test and four constraint equations derived by establishing the cubic strength equa-
tion and its discriminant. The results of an extensive series of tests showed that the qua-
dratic polynomial failure criterion is too conservative and the cubic polynomial failure
criterion is quite accurate.

Jiang and Tennyson [15] developed a new approach to overcome the problem that the
failure surface based on a cubic tensor polynomial failure criterion is not always closed.
To ensure that the failure criterion yields a locally closed failure surface, the inherent con-
ditions in the cubic criterion were derived. The least square method with Lagrange multi-
pliers to incorporate the derived conditions was utilized to evaluate the least-squares best
fit curve to the available data of biaxial load tests. The quadratic and cubic interaction
strength parameters were obtained. They showed Tennyson’s cubic theory without con-
straints significantly overestimates the failure stress in the compression-compression
quadrant. Failure tests undertaken using a laminated tube agreed well with results of the
new criterion.

Yeh and Kim [16] proposed a failure criterion based on the Yeh-Stratton criterion [17]
for isotrOpic material. It is similar to Mohr’s criterion and is applicable for both ductile
and brittle materials. The criterion is composed of the first order terms of normal stresses,

the second order terms of shear stresses, and the interaction term. The unique feature of

the theory is the ability of its format to accommodate the different values of the coefficient

7
of interaction term by types of stress and types of materials. The predictions of the failure
by this theory were in good agreement with the experiments.

Wu [18] discussed the resolution of strength tensors and proposed the methods for the
critical determination of interaction terms in the Tsai-Wu failure criterion. It was found
out that the variation in the interaction component F12 is dependent on both the magni-
tude of the biaxial strength and the magnitude of the ratio of the stress components. He
also mentioned the optimal experiments and optimal biaxial stress ratio to provide good
resolution of F12.

Considering the influence of the interaction term on the shape of Tsai-Wu’s failure
envelope, which can vary from an ellipse to parallel lines, and to hyperbola depending on
the value of the interaction term, Tsai and Hahn proposed a new method to obtain interac—
tion terms [19]. The interaction terms are obtained by assuming that the Tsai-Wu’s failure
criteria is a generalization of the Von Mises criterion. They investigated the values of
interaction term, F12 , for several material, and showed Tsai-Wu failure theory using the
proposed interaction term is in good agreement with the experimental result.

Wu and Stachurski [21] suggested a different approach to derive the interaction term,
F12 , in the Tsai-Wu failure theory. They considered that the value of I"12 is related to the
rotation of the failure ellipse from the major strength axis, and proposed a way to deter-
mine [:12 different from the one proposed by Tsai and Hahn [19]. It was demonstrated that
F12 prOposed by them offered good prediction of failure for thermoplastic and paper
materials, while I"12 prOposed by Tsai and Hahn gave good failure predictions for fiber
reinforced composites.

Hashin [22] developed a three dimensional failure criterion of unidirectional fiber

composites in terms of quadratic stress polynomials by considering that unidirectional

8
fiber composites are transversely isotropic and the applied average stress state should be
invariant about the fiber axis. The criterion is composed of four distinct failure modes -
tensile and compressive modes of fiber and matrix. The theory was compared with the
experiment results of off-axis specimen for a plane stress state, and gave reasonable
strength predictions.

In order to make the classical lamination theory fit to a fully three dimensional lamina-
tion theory, Christensen [23] showed that the terms corresponding to out-of-plane strains
in stiffness matrix are independent of fiber orientation by introducing two restrictions, and
all the terms can be represented by three engineering properties. In connection with this
work, considering stress-strain relations, he developed a failure criterion which reduces
into direct fiber failure mode and fiber/matrix interaction failure mode, and involves four
parameters to be determined from experiments.

Zhu, et. al. [24] proposed a statistical tensile failure theory for unidirectional fiber
composites. Assuming that the strength of a fiber is a statistical quantity and that the fibers
around a broken fiber are subjected to local stress concentrations, they obtained the axial
strength of composites by correlating the number of broken fibers to the tensile stress. The
experimental results were in good agreement with the theory, however this theory is con-
fined to predicting the axial strength of unidirectional fiber composites.

Neale and Labossiere [25] developed a parametric failure criterion for lamina under
plane stress, considering that since the ultimate strength for any stress combination be
finite, the failure surface must be closed and must be symmetrical with respect to—the plane
112 = 0 . The failure surface is represented by the spherical coordinates, and the angles
and length from the origin to the failure surface are determined by the stress state. They

showed that this failure theory can encompass Tsai-Wu’s quadratic tensor polynomial and

9
Tennyson’s cubic tensor polynomial failure criteria. They demonstrated that the paramet-
ric failure envelope describes the failure data as efficiently as the quadratic and cubic fail-
ure criteria and less parameters were required.

Feng [26] developed a failure criterion as a function of strain energy invariants for uni-
directional fiber-reinforced Composites based on the transversely isotropic properties and
the strain invariants of finite elasticity. It was assumed that failure occurs when the strain
energy density reaches its maximum value. The distortional energy, the dilatational energy
and the differential between compressive and tensile strengths are the governing quantities
in the criterion. The criterion can be simplified into the fiber and matrix modes, and can be
derived into one for infinitesimal strain elasticity.

Hart-Smith [33, 34] pointed out the reason why it is incorrect to adopt the polynomial
type interaction failure theories whenever the failure mode of a composite changes with
the state of stress. He suggested that separate envelopes for every possible failure mode
should be obtained first, and then should be superimposed, not interacted.

Wakashima et. al. [27] derived micromechanically a failure criterion for unidirectional
composites composed of ductile matrix and plastically nondeforrnable inclusions. Assum-
ing that the matrix material obeys the Levy - Von Mises flow rule and the matrix plastic
strains are uniform, and using the potential energy function and Lagrange multipliers
method, they obtained a failure criterion which is formally identical with Tsai-Wu’s phe-
nomenological failure criterion. Adopting the modified Eshelby’s theory on the transfor—
mation and inhomogeniety problems of an ellipsoidal inclusion, they obtained the strength
parameters from the potential energy function. The numerical results were not compared
with the experimental results.

The strengths of unidirectional composites were predicted and the effects of thermal

10

residual stress on the strength was investigated by Ishikawa [28], utilizing the microme-
chanical elasticity solution and assuming that the composites see failure if the stress level
of one of the constituents reaches the failure criterion for the constituent. It was observed
that the thermal residual stresses do not have a primary effect on the composite strength if
the interface normal strength is large, and the failure prediction by Tsai - Wu criterion with
the minimum interaction coefficient I"12 is fit to that by maximum principal stress crite-
rion in the matrix. The thermal residual stress was verified by means of photoelasticity.

Dvorak and Bahei-El-Din [29] studied the elastic-plastic behavior of composites in
terms of constituent properties, their volume fractions and mutual constraints between
phases by assuming the model that each of the cylindrical fibers has a vanishingly small
diameter and that the fibers occupy a finite volume fiaction of the composite. By virtue of
the assumption, it was found that the local stress and strain fields are uniform, so the
microstructure of the composite was not explicitly accounted for. Elastic moduli, yield
conditions, and hardening rules are derived.

Realizing that the strength of composites depends on that of each constituent, Skudra
[30] developed a failure criterion to determine the polymer matrix composite strength on
the basis of the existing states of stress or strains in the constituent. After establishing the
strength criteria for each constituent by understanding the characteristics of material prop-
erties and how the constituents behave in a composite, he derived the suitable failure crite-
ria for each mode of failure for a unidirectional lamina. For matrix, since the ultimate
strain is not constant but depends on the loading duration, the energy criterion which
assumes that matrix fails with time if the work of stresses reaches an ultimate value was
chosen as the failure criterion. For fiber, it was assumed that the fiber fails if the strain in

the fiber is equal to or greater than the strain at the starting point of the avalanche fracture

1 1

of fibers (as predicted by statistical theory), and that the fibers are in a uniaxial state of
stress. For interface, the failure models were divided into two types - the abrupt transfer
model and the model of diffusion interlayer, and the failure criterion was derived sepa-
rately. The micromechanical stress analysis of the matrix was performed by taking a repre-
sentative volume element of the repeating cell. With the failure criterion of each
constituent and mechanical stresses, the failure theory for lamina was obtained. Then,
using the lamina failure criterion and lamination theory, the failure criteria for laminates
was derived. 1

A micromechanical failure criterion for unidirectional composites was suggested by
Aboudi [31]. It was assumed that failure of the composite occurs when one of the micro-
stresses reaches a micro-failure criterion, and that plane stress state loading is acting on
the composites. Maximum stress failure criterion was adopted for fiber and matrix. If
material strengths of constituents were not available, they were obtained from the micro-
mechnical model by checking the maximum stress in the constituent when the strength of
the composite was applied as a load. It was ascertained that the strength of matrix in com-
posites is larger than that of matrix material in bulk due to the constraint effect.

By means of homogenization of material properties and definition of a failure domain,
a strength criterion for unidirectional and multidirectional fiber composites was derived by
Landriani and Taliercio [32]. The periodic unit cell of composite was used, and homoge-
nized into an equivalent homogeneous medium. The strength domain of the homogenized
medium was defined first, and then the real strength domain was obtained by c6nsidering
the contribution of failure of each constituent. Some unavailable strengths of each constit-
uent were obtained from composite strength by making some reasonable assumption

about the behavior of the constituent material, and the obtained constituent strengths were

12

used in order to check the reliability of the theoretical model.

1.3 The Present Study

Based on a review of available failure models for fiber-reinforced composite materials,
it was concluded that there is Still a need for a failure criterion that is accurate, predicts
both failure loads and mechanisms, is easy to use, and requires limited experimental data,
The objective of the current study is to develop and assess a new model that attempts to
combine many useful aspects of other failure models and criteria into one that meets these
requirements. This model is based on a micromechnical model, though the final mathe-
matical form is cast in the form of convenient phenomenological models.

In Chapter 2, the micromechanics model used will be described briefly. Chapter 3-5
will discuss various aspects of the failure model. Numerical results will be presented in

Chapter 6, and Chapter 7 contains conclusions and recommendations for further study.

CHAPTER II
MICROMECHANICAL ELASTICITY ANALYSIS

2.1 Introduction

In this chapter, the analytical model for micromechanical elasticity analysis will be
reviewed briefly [35]. The constitutive relations will be set up first, and then the governing
equation for the model will be derived by writing the equilibrium equations in terms of
displacements. Considering the boundary conditions, the solution of the equations for sev-
eral loading cases will be discussed. Then the method of calculating effective composite

properties will be established.

2.2 Analytical Model and Mathematical Formulation

Since the fiber distribution in composites becomes more regular as the fiber volume
fraction increases, it is usually assumed that the fiber packing is uniform, even though
fibers are irregularly. distributed throughout the cross-section. The frequently adopted fiber
distributions are the square and hexagonal arrays which can be regarded as special cases
of the diamond arrays (see Figure l).

The fundamental assumptions of the analytical model are made as follows:

a) The fibers have infinite length and a uniform circular cross-section along the length of

the fiber.

13

   

 

/\

. g .
. - a ;»
we)“ . ..

We

   

 

Figure 1. Cross section of a fiber reinforced composite material with
fibers packed in a diamond array

15
b) The fibers are continuous, straight, perfectly aligned with the Xl-axis, and arranged in a
periodic diamond array.
c) Fiber coatings or interfaces are represented by concentric circular cylinders around the
fiber, and are perfectly bonded with the other constituents (fiber and matrix).
(1) All materials are homogeneous and linearly elastic.
e) The outermost constituent (matrix) is transversely isotropic, and the other phases (fiber
and interfaces) are cylindrically orthotropic.
f) Mechanical loads are applied at infinity, and do not vary along the X1 -direction.

Based on the assumptions a) - d), it is secured that many planes of geometric symme-
try exist in the composite material. If the region of interest is sufficiently far away from the
point of load application or geometric constraint, each fiber and its encompassing phases
in that region will see the same deformation due to the applied loads. Therefore, only a
single fiber and its surrounding constituents need to be analyzed. Thus, the behavior of the
lamina can be investigated by considering only a small region, or a unit cell (representa-
tive volume element, (RVE)) (See Figure 2).

Since the overall composite properties are dominated by the properties of each constit-
uent, it is of crucial importance to establish realistic constitutive relationships for each
material in any micromechanical model. Therefore, in the present model, all stifl‘ness
properties are functions of temperature and moisture content. On the other hand, the coef-
ficients of thermal expansion depend only on temperature and the coefficients of expan-
sion due to moisture content depend only on the current moisture level.

The cylindrical coordinates shown Figure 2 are chosen to get the analytical solution,
therefore x, 9, r are utilized as subscripts. In this notation, the constitutive equations for a

material with cylindrically-orthotropic properties take the form:

l6

 

 

 

 

 

 

 

Figure 2. Diamond RVE used in the micromechanical model

17

r \

 

 

 

 

_ (n) (n) (n)-
OI") C11 C12 C13 €§"’-¢:n) (T) -‘I’x(n) (M)
n) = (I!) (II) (n) I P
06 C12 C22 C23 eg"’_¢§"’ (n—wé'” (M) (2.2.1)
(n)
W 9%" 6.3;" city .e:">—d>.<"><n—‘I'.S"><M> .

(n) (n) (n) (n) (n) (n)
1:61;!) = C44 Yer ’ 1x9!) = C55 Yxr ’ 1x91.) = C66 7x0
where the superscript (n) ranges from 1 to N, and N is the number of constituents in the

composite system. In Eq. (2.2.1),
trim = [:‘otimdr, (i = x,0,r) (2.2.2)

is the thermal strain, T0 and T1 are the reference (stress free) and final (current) tempera-
tures respectively, and or, is the coefficient of thermal expansion in the r"h -direction. Sim-

ilarly,

WM) = [SBAMMM (i = x.e.r) (2.2.3)

is the strain due to moisture, Mo and M 1 are the reference (stress free) and final(current)
moisture levels respectively, and [3‘- (M) is the coefficient of expansion in the r"h ~direction
due to moisture permeation.

For transversely isotropic materials, the material stiffness coefficients Ci]. (T, M) have

the following relationships:

18

The assumptions (3 - c, 1‘) ensure that the strains are in a state of generalized plane strain,
therefore the displacements take the form:
a?” (x, 0, r) = u‘") (9, r) +xéx

ué") (x, 0, r) = v(") (6, r) (2.2.5)
WW (9. r)

u,"" (x, 9, r)

where Ex is the uniform strain in the lamina in the fiber direction (along X, axis). The

strain - displacement relations are as follows:

 

 

(n) (n)
.. 3v v‘") law
I ) (n)
2(a) _ 13" n Wm (n) -.- a“
9 r55 r x' r
( ) ( )
8.0:) 3‘” n 7(a) _ la“ n
8’ 8r ’9 r 0

The equilibrium equations in cylindrical coordinates for generalized plane stress are given

as:

(n) ( ) ( )
latxa a192+102 = 0 (227)
PT +5—0r r . .
13600!) 313:) 210!)

 

 

r39 +5 + r =0
1+31.x) 80’ (n) GEM—0'3") 0
:56 “a: ““7“-

The governing equations in terms of displacements are obtained by substituting Eq.

(2.2.1) and Eqs. (2.2.5) - (2.2.6) into Eq. (2.2.7):

2 (n) (n) (n)
C‘§’[ rza—l; “$5 ]+ c‘” ’[3-2—2- “ ]= 0
3r r 302

l9

 

n 32 (n) Och (n) a“, (n)
r __§ _ x _ x + C( ) r W _ __r_ _ r
39 as 39 23 3739'" 39 a?

 

r 2 ,, ,, ( ) (n)
+C(n) a v( )+8w( )_ra¢on _ra‘¥o ’0]
22 (392 BE '55 6
2 (n) (n) (n)
(n) 3 av (n)+ raw +aw _
* C44 \ a? ’37 " ”W a" J ' 0
(n) ~ on (n) (n) av”) (n) (n) (n)
C12 r(—ex+rl>x +‘I’x )‘I’sz -a—9 —w +rd>x +r‘l’1c (2.2.8)

aé‘ ad") arm

(n) (n) (n) x x x
-t-C13 {£— -<1>x -‘I’x +rE—r—ar —r—ar ]

(n) (n)
(n) av (n) (n) (n) (n)_ 3% We
+C23r r—z—[am ae —<r> ure +<I>, +\r ra—r 4.5—;
2 (n) (n) ad) (I!) a? (n)
.cgg).[.:_r_‘; .gt; —<I>,‘"’—\P,‘"’—r$’ 457'

b.) av av‘") 32w”) _
”’44 (+8861? +52— ’0

In the Eqs. (2.2.8), the first one governs the deformation due to longitudinal shear, and the
latter two govern the behavior due to normal, transverse shear, and hygrotherrnal loading.
The first one is decoupled from the other ones. Moreover, considering the number of
planes of geometric symmetry in the fiber geometry, the solution to the last two of Eq.
(2.2.8) can be divided into a symmetric part (due to normal and hygrotherrnal loads) and
an antisymmetric part (due to transverse shear loads). The solution to the first equation
may be separated into a response due to shear in the X1 - X2 plane and a response due to

shear in the X1 - X3 plane.

20

2.3 Solution for Symmetric Loading

The solution to the last two of Eq. (2.2.8) for deformations due to applied normal

strains (t:1 = €1,82 = €2,83 = é3)and hygrotherrnal loads (AM, AT) are:

v‘"’(e, r) = 2 vi(")(r)sin(2i9)
‘j‘ (2.3.1)
w‘”’(e, r) = Z w,‘"’(r)eos(2te)

i=1

and:

ej"’(e, r) = 2¢:‘")(r)cos(2i9), ‘11:")(9, r) = 2?:‘n)(r)cos(2i9) (2.3.2)

1: 0 I: 0
¢§"’(e, r) = 2 egf’(r)eos (2:9), v§"’(e, r) = 2 ‘11:" (r)cos (2:9)
i= 0 i8 0
(Doom _ .. (n) . (n) _ .. (n) .
r , r) — 2 d)" (r)cos (219), ‘1’, (0, r) .. Z ‘1’” (r)cos (210)
i= 0 r= 0
eye, r) = 2 exp) cos (2:9)
t'= 0

The following equations are obtained by substituting Eq. (2.3.1) and Eq. (2.3.2) into the

last two of Eq. (2.2.8).

.. ( ) 2 ( ) ( ) ( ) dwm
. . n . n . n u . i
.Eosrn (210) C22 (- 41 vi — 2rwi )+ C23 -2rr$
”‘ (2.3.3)
2 (n) (n) (n)
d v. dv. dw. }
(n) 2 t r (n) . t . (n) _
'1' C44 7' 'd? +r'd—r- —V'- -21r$ -21Wi — 0

21

.. . (n) . (n) Won (:0 .- -
.20 cos (216) {C22 (— 21vi )+ C33 r -—2- + r-d—r (2.3.4)
3:

(n) (n)
dv. dv.
+ Cg) [2ir—dr' ]+ C3) [Zir—dr' - 2ivim — 41' 2mm]

(n) (n) .. (n) (n) (n) (n)
+r(C12 -Cl3 (—£xr+d)xr +‘I’x‘ )+r(C22 —C33 )

. ((1);?) + if" ) + r( cg" — egg) )(eff’ + \Pff’) } = 0

When i = 0, Eq. (2.3.3) vanishes, and the solution to Eq. (2.3.4) for orthotropic materials

can be obtained as follows:

wé") = 33:5“; 33;" r” +H‘"’r (2.3.5)

where Bx) and 35;) areconstants, kg) = JCSVCS), kg) = -,/C$)/C3(") ,and
(n) ___ 1 (n) (n) . (n) (n)

H {W(Cl3 -C12 )(-Exo+¢xo +1130 ) (2..36)

+(Cz(3")- C‘"’)(d>2+§"’+\ré"’) (C‘"’- *C("))(¢r(n)+‘*’r(n))}
0 0 0 0

For transversely isotropic or isotropic materials: H (n) = 0, 3.3:) = 1, kg) = -

When 1' >0, the solution to Eqs. (2.3.3) and (2.3.4) is written in the form:

4
v“) = 2A‘”’r "' (2.3.7)

22

where A5") and B 5.") are constants and 1.3.") are the eigenvalues of the solution. By sub-

stitution of Eq. (2.3.7) into Eqs. (2.3.3) and (2.3.4), the solution can be written as:

v,""(e r) = 2 2:9“)3‘5" msin(2i9)

 

 

 

 

 

 

 

 

 

 

 

”‘1“ (2.3.8)
1(a)
w,""(e r) = B‘"’r° (”Nagy 1” +H‘”’r + 2 23.9%” "' eos(2ie)
i=lj=l
where:
(u) (n) (n) (n)
" —i,22..‘.'"(c‘"’+c‘"’)— 2iIc:”+C‘”’)
2.“) __ + _ brm 1 (btn))2 —4a‘"’e‘"’
2'1 - 20.90.1200!) 5 i
(n)
b 2
If,” _ I m 1(a) (bite) Jar-MCI")
+aitIz 2a.
(2.3.10)
(n)
b 2
(n) ___1___ (n) (n) (n)
793 ="’ (n) (n) (bi I’4ai Ci
2a 2a a,
(n)
(n) _ I b.- l ( (n))2 (n) (n)
2a,. 20‘.
(n) (n) (n)
“1’ = C33 C44
(n) .2 W (n) (n) (n) (n) (n) .2 (n) 2
(2.3.11)
+ 8i 2C” )c‘”

(n) (n) (n) .2 2
C, = C22 C44 (41 —1)

23

B 1.5.") are the unknown coefficients to be determined as follows.
To eliminate displacement and stress singularity in fiber, some coefficient should be

set to zero. That is:

(l) (1) (1)
302 =Bi3 =Bi4 =0

At each constituent interface r" , displacements and normal stress components must be

continuous:

v”) (0, r”) = v(M 1) (9. 7,.)

WW (0, r") = w(M I) (0, r")

( ) ( 1) (2.3.12)
II n +
or (6’ r") = a (e, r")

1'

t3? (9, r") = 1332+” (e, r")

In the diamond array, the outer boundary conditions of a cell are determined, when a
cell is deformed, by assuming that [36, 37, 38]:
1. a cell must retain the symmetry with respect to both axes (two fold symmetry),
2. the deformed shape must remain typical of all the other cells,
3. the repeating adjacent cells are congruent.
In addition, the displacements and stresses must be continuous across the repeating cell

boundaries, so the following conditions are obtained:

(142),,4- (112),” = é2d2

é3‘13
(0,), = (0,9,.

(1...), = (1,9,.

(u3),+ ((43),. (2 313)

24
where (on) P and (12",) P are the normal and shearing stresses at the boundary AB, and
the £2 and (3.3 are the applied strains in the X2 and X3 directions, respectively. The points
P and P” are located on boundary AB as shown in Figure 3, and the locations are defined
as follows:
If the point P has the polar coordinates (r,, 0,.) , then due to the two fold symmetry

and congruency of the adjacent repeating cells, the coordinate of point P” will be:

 

r,. = J(d2—r,,cosep)2+ ((13-1',,sin01,)2
. 2. .14
6 _ amn[d3—rpsmep] ( 3 )
P" - d -r cost)
2 P P

 

2.4 Solution for Transverse Shear Loading

Since the last two of Eqs. (2.2.8) are also the governing equations for the deformations
caused by applied transverse shearing strain (723 ), and the solution procedure is the same
as the aforementioned symmetric loading, only the final form of solution will be given

here. The solution can be written as follows:

"' 4 (I)
(n) __ (n) ('01 (n) (n) 2., .
v (0, r) — D01 r+D02 ;+ 2 2;]. Di]. r cos (219)

”12"” (2.4.1)
(It) 4 (n) (n) 1""
w (9,r) = 2 2g]. Di]. r" sin(2i0)

i=1j=l

where 2.5.") are defined in Eq. (2.3.10) and Eq. (2.3.11). and

25

 

 

 

 

 

 

x3 I

T B

d3 ‘1’

i >
' A x2
|< .. >|

P,

Figure 3. The determination of boundary condition
from congruency and symmetry of repeating cells

26

(,0 = 2 W’Icz‘g’ +6”) 2 (C?) +C‘”’)

‘7 ( ) ( ) 3(3 ) ( ) (2'42)
'1 n n n

 

D 1.3.") are constants, which are the unknown coefficients to be determined as follows, as
in the symmetric loading case. In order for the solution to be bounded at r = 0 , some of

D II") are set to zero:

DI? _ D_(31) _ Duo) = 0

The rest of coefficients are determined by the continuity condition Eq. (2.3.12) at each
interface and the boundary conditions at the boundary of the repeating cell. In this case,

the boundary conditions at the edge AB are:

(“2)P'I’ (”2) pa = %?23d3
1
(2(3) ,+ (143) P. = 512342 (2.4.3)

(0,), = (0,9,.

(2,9, = (0,9,.

where points P and P” are defined in Eq. (2.3.14), and 723 is the applied shearing stain in

the XZ‘Xg plane.

2.5 Solutions for Longitudinal Shear Loading

The deformation under longitudinal shear loading is governed by the first of Eqs.
(2.2.8). In this case, the solution is divided into one for applied loading in the X, - X3

plane and one for applied loading in the X1 - X2 plane.

27
The solution for longitudinal shear loading in the X, - X3 plane (713 = 7:3, 712 = 0)

is:

u x0) 10')
a”) (e, r) = 2 (F35 " +F,f;"r ‘2 )sinUcG) (2.5.1)

Ic=l

(n) (n) (n) (n) (n) (n) (n) (n)
where 7L“ = kJC66 /C55 ,1” = ’kn/Cso /C55 ,and F,l ,sz areconstants. To

eliminate the singular contribution, the following coefficient are set to zero:
Féé’ = 0
The rest of coefficient F X ) and F3) are determined by the similar procedure like previ-

ous cases. The continuity conditions are at each interface are:

not) (6, r") = u“H 1) (9, r")

(n) (n + 1) (2.5.2)
In (9, r") = I” (9, r")
The boundary condition at the boundary AB are:
(“2) + (“2) = ?13d
P P" 3 (2.5.3)

(11;)P = (111),,»

where 713 is the applied shearing strain in the X,-X3 plane, and 1 indicates X, direction.
The solution for longitudinal shear loading in the X, - X2 plane (712 = 7:2, 713 = 0)
is:

” 10') (l)
a“) (e, r) = Z (63% *' +03)?“ )cos(k9) (2.5.4)
k = l
(n) (n) . (n) (n)
where A.“ , A.” are the same ones as grven above, and (3,1 , Gm are constants. To

eliminate the singular contribution, the following coefficient are set to zero:

28
0;,” = 0
The rest of coefficient 0,: f) and 03) are determined by the similar procedure like previ-

ous cases. The continuity conditions are at each interface are:

a“) (e,r,,) = u‘"*” «mg

(n) (n + 1) (2.5.5)
I” (9, r") = 1x, (6, r”)
The boundary condition at the boundary AB are:
(u ) + (u ) = 712d
2 p 2 P 3 (2.5.6)

(111)!) = (11‘) Pp

where 712 is the applied shearing strain in the X,-X3 plane, and 1 indicates X, direction.

2.6 Computation of Effective Composite Properties

For the composite material with a hexagonal array of the fibers, the composite consti-

tutive relations in material coordinates are:

 

 

61 C11 C12 C12 51" ¢r ‘ “’1
0'2 = C12 C22 C23 82 ‘ 4’2 ’ V2
03 _C12 C23 C22 33 “ 4’2 ' ‘V2
C22 ‘ C23
123 = T723, t13= 55713, ”‘12: 55712 (2-6-1)

The effective composite properties are calculated by applying a single unit strain in one
direction, setting all other strains and hygrothermal loads to zero, and calculating the

resulting average stresses. For example, if t»:l = l and all other strains are zero, then:

29

Cu=ovcu=62 as»

Likewise the other effective composite properties can be determined in this way. The aver-

age stresses are calculated as follows:

an»
a, = diz O’o,(0,r)dr
1:13 — i 021:“ (O, r) dr
where
.. M _
He) .. 42$in (“H e)’ a) _. atan(./§) (2.6.4)

If the C i J. are obtained at a certain temperature and moisture level, the coefficients of ther-
mal expansion and moisture expansion can be calculated by considering only constant
thermal expansion or constant moisture expansion over their ranges (with all other strains
zero). The procedures are very similar to that of getting Ci]. . Then the coefficients of ther-

mal expansion are obtained by setting AT = l with all other strains set to zero:

{a} = -[C1“{o) (2.6.5)

The coefficient of moisture expansion are obtained by setting AM = l with all other

strains set to zero:

30

{M} = —[C] ‘1 {a} (2.6.6)

where AM is a small change in moisture content.

CHAPTER III
MICROMECHANICAL FAILURE THEORY

3.1 Introduction

The failure theory developed here utilizes the micromechanical elasticity solution
obtained in the previous chapter, and the well-known failure criteria for isotropic as well
as orthotropic materials to predict the failure in a given constituent.

In this chapter, a brief illustration will be given of a simple example of micromechan-
ics based failure predictions. Detailed explanations will be given concerning the concept
of the micromechanical failure theory and how the micromechanical elasticity solution
and failure criteria are utilized here. Finally, the procedure for curve fitting of failure enve-
lopes obtained by the micromechanical failure theory to polynomial type failure envelopes

will be discussed.

3.2 An Example of Micromechanical Failure

Consider the following example where a compressive force is acting on composite
material through a rigid body as shown in Figure 4. The composite material is composed
of material A with cross sectional area S A and material B with cross sectional area 8;. The
total cross sectional area and length are S, and L, respectively. The Young’s modulus of

composite, material A and material B are E,, EA and EB respectively. From the equilibrium

31

32

Material A

  
 

h Load P = CtA,

Rigid Body
T Material B
Base
PA #

Figure 4. The illustration of a simple composite body subjected
to compressive force andl the free body diagram

 

33

of horizontal forces, we can get:

2PA+PB = P (3.2.1)

The strain and displacement of each constituent are:

for constituent A, 8A = — = _ = _

 

5, = (3.2.2)

for constituent B, EB = — = — = ——

a, = — (3.2.3)

Since material A and B are compressed through a rigid body, it can be assumed that the

displacements are same. Therefore, by equating Eq. (3.2.2) and Eq. (3.2.3):

EASA
P = —P (3.2.4)
A EBSB B

The substitution of Eq. (3.2.4) into Eq. (3.2.1) leads to:

EBSB

P8 = ZEASA+EBSB

P (3.2.5)

 

34

Substituting of Eq. (3.2.5) into Eq. , PA is obtained:

P = 5‘5“ P
A ZEASA+EBSB

 

(3.2.6)

Then, the stress state of each constituent and the combined material, respectively, when

only the compressive load P is acting is obtained as follows:

 

a, = g (3.2.7)
P E P
A A
0' = — = (3.2.8)
A A ZEASA-t-EBSB
P E P
a, = —B = 3 (3.2.9)

 

If the applied load is increased, the stress in constituent A and/or constituent B will even-
tually reach its ultimate value. Failure of the composite may be defined as failure of either

constituent A or B. The stress in the composite at that time is:

 

of = P: (3.2.10)
and we have either
P“ E P“
u A A
o - — (3.2.11)
A SA ZEASA-t-EBSB

Ol'

35

2
"U
a:

PP“
OB=— B

= .212
SB 2EASA+EBSB (3 )

 

where of, P", 0;, Pg, 6;, and P; are the stresses and loads in the composite material
and its constituents, respectively, when the failure occurs. In this case, the stress in each
constituent can be calculated, and a proper failure criterion for each constituent has to be
chosen to predict failure. Here, it can be observed that the (average) stress state of the
composite material is different from that of each constituent from Eq. (3.2.10), Eq.
(3.2.10) and Eq. (3.2.12), when the failure occurs in some constituent.

In order adopt this concept to failure of fiber reinforced composite materials, we need
to 1) set up an appropriate RVE (Representative Volume Element), 2) perform stress anal-
ysis in RVE for a given set of external loads, and 3) choose the proper failure criterion for
each constituent to predict the onset of failure. Steps 1 and 2 were discussed in the previ-

ous chapter. Below, some common failure criteria are discussed.

3.3 Failure Criteria

Failure criteria can be roughly divided into those for isotropic materials and those for
orthotropic materials. Failure criteria play a very important role in the micromechanical
failure theory, because failure criteria predict whether or not failure occurs at the selected
points in the R.V.E. Thus, the several failure criteria useful for prediction of the failure in

fiber, matrix and fiber/matrix interface will be described briefly.

3.3.1 Failure Criteria for Isotropic Material

36
Maximum Principal Stress Theory, or Rankine Theory
The maximum principal stress criterion assumes that failure occurs when one of the prin-
cipal stresses is equal to or greater than the value of the uniaxial yield stress in tension,
6:, or is equal to or less than the value of the uniaxial yield stress in compression, of.

This criterion asserts that yielding will occur when any one of the following conditions is

reached:
I . 2 T > T 2 T
ntensron, 61 a“, or 62.6", or 0'3 on
I . 2 C > C 2 C
ncompressron, (31 a“, or 02.0“, or 03 a,

This theory assumes that only the maximum principal stress causes failure, and the contri-

bution of stress interaction is disregarded [39].

Maximum Principal Strain Theory, or Saint- Venant Theory

The maximum principal strain criterion assumes that failure occurs when one of the prin-
cipal strains is equal to or greater than the value of the uniaxial yield strain in tension, 8:,
or is equal to or less than the value of the uniaxial yield strain in compression, sf . The

1' C .
a“ and an are related to the ultrrnate stresses by:

where E is the elastic modulus of the material. This criterion asserts that yielding will

occur when any one of the following conditions is reached:

T

. T T
In tensron, £1 2 a“, or 82 2 e“, or £3 2 e“

. C C C
In compressron, e, 2 a“, or £2 2 an, or 83 2 e,

37

In this theory, the role of the interaction of the principal stress is accounted for failure

[39].

Maximum Shear Stress Criterion, or Tresca Criterion

The maximum shear stress criterion (sometimes called the Coulomb theory) assumes that
failure occurs when the maximum shear stress is equal to or greater than the value of the

maximum shear stress, 1:“ = - whrch rs occumng under unraxral tensron. The man-

2°u’
mum shear stress is equal to half of the difference between the maximum and minimum
principal stresses. This criterion asserts that failure will occur when any one of the follow-

ing conditions is reached:
'01 " c’2' = c’u .

lo2 — 63] o

“9
[03—01]: a

In this theory, some interaction of the principal stresses is included, but this theory is not

applicable for the material whose tensile and compressive strengths are difi‘erent [39].

Distortion Energy Criteria, or the Von Mises Criterion

The distortion energy theory assumes that failure begins when the distortion energy is
equal to or greater than the distortion energy at failure in uniaxial tension. This criterion

asserts that yielding will occur when the following condition is reached:
1 2 2 2
§[(°1 -02) + (oz-03) +(03’01)2] = on

The distortional energy criterion accounts for interaction of the principal stresses [39].

38

3.3.2 Failure Criteria for Orthotropie Material

Independent Maximum Stress Criterion
The independent maximum stress criterion assumes that failure will occur when any of the
stress components referred to the principal material axes is equal to or greater than its cor-
responding allowable strength in that direction. Thus, this criterion asserts that failure

would occur when any one of the following conditions is reached:

c"11 = X,T. “611 = Xi:
022 = X; ”322 = X:
(’33 = X; ‘033 = X:
c’23 = X23, c’31 = X311 (’12 = X12

where 0,]. are the stress components referred to the material axes, and X? , X? and Xijare
the tensile, compressive and shear strengths, respectively, of the material in its i"I princi-

pal direction. This theory does not account for stress interaction [40].

Independent Maximum Strain Criterion

The independent maximum strain criterion assumes that failure will occur when any of the
strain components referred to the principal material axes is equal to or greater than its cor-
responding allowable strain in that direction. Thus, this criterion asserts that failure would

occur when any one of the following condition is reached:

_ T _ C
811' e,. “311 " er

_ T __ C
322 " ‘32, ’322 ' 32

823 = £23, 531 = 331’ 512 = 912

. . T C
where 8,]. are the strarn components referred to the materral axes, and e, , e, and e ,1. are
. . . . . . .th . .
the tensrle, compressrve and shear strengths, respectrvely, of the materral 1n rts l prrncr-
pal direction. This theory accounts for the stress interaction but disregards any strain inter-

action [40].

3.4 Micromechanical Failure Theory

Failure criteria for composite materials have been deve10ped on the basis of the afore-
mentioned micromechanics elasticity solution and failure criteria for isotropic and ortho-
tropic materials. They can be cast in several forms in either stress- or strain- space, and
require only stiffness properties and uniaxial strength data of each constituent

The failure of composite materials is predicted when stress or strain states of a point
in any of the constituents (e.g., fiber, matrix and interface) exceed a critical level as pre—
dicted by a properly chosen failure criterion for each constituent. When a lamina (compos-
ite) load gives rise to a micromechanical stress state that causes failure in some
constituent, then that load state is assumed to lie on the failure envelope for the composite
material. When a failure criterion for each constituent is chosen, the mechanical properties
of each constituent and the failure characteristics of each constituent in the composite
material should be taken into account. Since the primary emphasis here was on polymer
matrix composite materials, the details to be discussed are most suitable for these material

systems.

40

Fiber

Fibers in composite materials are used as a reinforcing agent. In polymer matrix com-
posite materials, the longitudinal Young’s modulus of the fibers is often 10 times greater
than that of the matrix [4]. If a unidirectional lamina is subjected to a tensile or a compres-
sive longitudinal load, the failure of the lamina is usually a result of the fracture of fiber in
tension or fiber buckling/kinking in compression, and the failure modes and the failure
stresses for each case are quite different. When a tensile longitudinal load is acting on a
lamina, the axial strain of each constituent in the lamina is the same, since it was assumed

that fibers are perfectly bonded to the matrix through fiber/matrix interface:
5’ = e’" = a’ (3.4.1)

where 8;, a: and 8:! are the longitudinal strains of fibers, matrix and lamina, respec-
tively. However, generally, the fibers have a lower ultimate strain than matrix, and lamina
fracture occurs at the fiber ultimate strain [41]. When a lamina is subjected to a compres-
sive longitudinal load, the failure of a lamina is often caused by buckling or kinking of
fibers in out-of phase mode or in-phase [42]. But, according to Hashin [22], the depen-
dence of both fiber failure modes on the axial shear stresses is not identified clearly.
Hence, it is concluded that axial tensile and compressive strains of fiber dominate the
composite longitudinal strength. Thus, the maximum strain criterion was chosen as the
fiber failure criterion in this research. Failure modes such as fiber buckling/kinking are
assumed to initiate at some critical level of axial strain, and so are accounted for in indi-
rectly. In addition, it is assumed that the fiber never fails due to transverse and shear

strains (82, 83, 723, 731 and 712 ), therefore, the much bigger values than that of

41

longitudinal strength are assigned to transverse and shear strength components.

F iber/matrix interface

Generally, the fiber/matrix interface is a vanishingly thin region in the composite
between the fiber and matrix, and is formed during the composite manufacturing process.
It transfers the load between the fiber and matrix in composites, so that it affects the
mechanical response of composite materials.

If the interface region has a small but finite thickness, it is often referred to as an inter-
phase. The interphase may have distinct elastic properties, and it was shown in references
[43, 44] that the elastic moduli and the thickness of the interphase have an influence on the
effective elastic properties. But the elastic moduli and the thickness are often not available
for many composite systems. Thus, in the current study, an interphase region was not con-
sidered, though the model is sufficiently general to have considered one.

Furthermore, its normal and shearing strengths are difficult to measure experimentally.
So, in the current research, the normal and shearing strengths were determined using the
following procedure [45]. With all other applied loads being zero, the lamina ultimate
transverse strength was applied to the micromechanical model. Then, normal and shearing
stresses were computed in the matrix at the fiber/matrix interface region, and principal
stresses were computed elsewhere in the matrix. If the maximum principal stress in the
matrix due to the load was less than the tensile strength of the matrix material in the bulk
state, then it was assumed that failure occurred at the fiber/matrix interface and the maxi-
mum normal stress at the fiber/matrix interface region represents the normal strength of
interface. Otherwise, it was assumed that failure occurred in the matrix, and the normal

strength of interface is equal to or greater than the computed maximum normal stress at

42
the interface. The shear strength of interface was calculated in a similar way by applying
the lamina ultimate inplane shearing stress. It was assumed that interface was perfectly
bonded to both fiber and matrix, and failed due to tensile transverse normal stress or shear
stresses (are, on ). Hence independent maximum stress criterion was applied to predict

the failure in the interface. [28, 32]

Matrix:

When a lamina is subjected to a longitudinal load, the matrix plays the role of both
protecting the fibers and distributing load between and among the fibers. However, when a
transverse tensile load is applied to the lamina, the fibers do not serve as a load-carrying
constituent in the lamina, but act as solid inclusions in matrix. Due to the existence of
solid inclusions in the matrix, the local stresses and strains in the matrix are higher than
the applied stress [4]. In addition, even though the region with maximum stress in the
composites is often located in some other phase, failure generally occurs in the matrix
first. Furthermore, a uniaxial load acting on a composite material in any direction can
cause a three dimensional stress state in the matrix, which causes the stress components to
interact. So, it is assumed that the transverse and shear strength of composite material is
dominated by the strength of the matrix. The suitability of a matrix failure criterion
depends on the properties of the matrix - ductile, brittle, etc. In this work, the following
criteria are considered: maximum principal stress, maximum principal strain, Von-Mises
and Tresca criterion, wherein the Tresca failure criterion is applicable to matrices of which
the tensile strength and the compressive strength are the same.

The stress evaluation points are carefully selected as shown Figure 5 in order not to

miss the locations of high stress. The stress evaluation points at the interface region and

43
near the interface region in the matrix should be densely distributed, because not only do
the highest stresses often exist in that region, but also the stress gradient in that region is
very large. In the current research, 249 points in the RVE were chosen at which to evaluate
the stresses as shown in Figure 5. The interface is treated as a finite region, but is so thin
that the stress evaluation points in that region appear to overlap with those in the other

regions.

Lamina Strength Components

The uniaxial normal strengths of a lamina are determined if a uniaxial lamina load act-
ing along a principal material of a lamina creates a micromechanical stress state that
causes some constituent to fail, and the lamina load is assumed to be the unidirectional
strength of the lamina in that direction. Constituent failure due to the micromechanical
stress state caused by a lamina load is predicted by the failure criterion chosen for the con-
stituent. The shear strengths of a lamina are obtained in the same way as the longitudinal
normal strength are found.

But more attention and insight should be paid to determine the transverse normal
strength and the longitudinal shear strength. In reality, the arrangement of fibers in contin-
uous fiber-reinforced composite materials is random, therefore these materials can be
regarded to be transversely isotropic in both stiffness and strength. The assumption of hex-
agonal packing of fibers, as used in the current elasticity solution, gives a rise to transverse
isotropy of elastic moduli, but not strength. Thus, the transverse normal load and the lon-
gitudinal shear load causing failure of some constituent should be applied and averaged
over a range of loading directions from 0 degrees loading direction to 360 degrees (or 0

degrees to 30 degrees due to symmetry) to obtain the transverse normal and the

r
9
"m
f
+

: interface

0 : matrix

 

Figure 5. The stress evaluation points in RVE

45
longitudinal shear strength of the lamina. To perform these processes, a uniaxial stress
load acting on the model at a certain angle should be transformed to the reference axes by
using the stress transformation law.

When a stress load is acting on the lamina at a certain angle as shown in Figure 6, by
stress transformation laws, the stress tensor 6 in the Cartesian coordinate PX,XZX3 , of
which components are 6,]. i, j = 1, 2, ..., 6 , can be expressed in terms of the stress tensor
0' in the Cartesian coordinate PX1X2X3 , which has 6,}. i, j = l, 2, ..., 6 as components,

as follows [47]:

o = ATOA (3.4.2)

where A is a proper transformation matrix, and AT is the transpose matrix of A. The A

T . .
and A matrices are wrrtten as:

l 0 0
A = [0,1] = 0 cos (--6) sin (—9) (3'43)
0 -sin (-6) cos (-6)

T 1 o o
T [4.1] = 0 cos (-6) -sin (-6) (3.4.4)
0 sin (-9) cos (-6)

>-
II

The tensors 0', 6 are:

011 0’12 031 311 312 831
O = [0:1] = c’12 c’22 “23 ' 6 = [31,] = 312 322 323 (3°45)
c’31 023 033 631 623 t’33

46

 

 

 

 

 

X3
x, A
X2
P , x2
x, x,
(b)

Figure 6. (a) The RVE subjected to an off-axis load
(b) The reference axes and the rotated axes

47
Substituting Eq. (3.4.3), Eq. (3.4.4) and Eq. (3.4.5) into Eq. (3.4.2), stress tensor 6 in the
Cartesian coordinate PX1X2X3 is transformed into the stress tensor 0' in the Cartesian

coordinate PX 1X2X3 as follows:

[ail = 21.-Hal [a]

1 O 0 611 512 (’31 1 0 0
= 0 cos (—6) —sin (-9) 3,2 022 523 0 cos (-6) sin (—6)
_0 sin (-6) cos (-6) 331 523 533 0 -sin (—0) cos (-6)

 

011 = 1"11
0‘12 = 612c036+63lsrn8

622 = {5‘22cosfi2 + 2623 sinOcosB + 633 sin 62 (3.4.6)

623 = —622cosesin9 + 623( cos62 — sinez) + 633cos9sin0

0’33 = 622 sine2 — 2623 cos6sin6 + 633 cost)2
Eq. (3.4.6) gives us the relationships between the stresses in reference axes and those in
certain rotated axes. Therefore, an off-axis stress load can be transformed into the refer-
ence axes (the Cartesian coordinate PX1X2X3 shown in Figure 6) by using Eq. (3.4.6).
The transformed stresses are taken as a stress load acting on the model, and cause a micro-
mechanical stress state in each constituent. If the failure of some constituent is predicted
by the failure criterion chosen for that constituent, then the stress load is taken as the
strength of lamina at that angle, and all the lamina loads at each angle from 0 degrees to
360 degrees are obtained. The lamina transverse normal and longitudinal shear strengths

are obtained by averaging the obtained lamina loads from 0 degrees to 360 degrees.

Failure Envelopes

48

The failure envelopes in X, - X2 plane, i.e., when the biaxial loads in X, and X2 direc-
tions are acting together, are obtained in the following way. First, the longitudinal lamina
load is fixed at some value which is less than or equal to the longitudinal normal strength
of the lamina, then the transverse lamina load giving rise to failure in some constituent is
sought and averaged from 0 degrees to 360 degrees. Accordingly, the lamina fails at the
biaxial load, and the longitudinal and transverse normal loads are assumed to lie on the
failure envelope. Then, the longitudinal lamina load is incremented, and the transverse
normal lamina load that causes failure is again obtained. After the above processes are
repeated over the range from the tensile longitudinal strength to compressive longitudinal
strength of the lamina, the failure envelope for biaxial loading in stress space is obtained.
The failure envelope in strain space can be obtained by converting the stress states in fail-

ure envelope to strain states by means of the constitutive law for an orthotropic body [48]:

l v12 V13
811 - forr‘fazz'faaa
1 1 1
l v12 v23
822 - Eazz-EGn-E—zoaa
(3.4.7)
1 V13 v23
8 = —o -—o -—o
33 33 11 2
E3 51 52
28 --—-—lo 28 -—lo 28 -——lo
23 " 23» 13 ‘ 13' 12 ' 12
023 613 012

where 0,]. is Cauchy stress tensor, 2,} is infinitesimal strain tensor, and E, and 6,1. are
Young’s moduli and shear moduli, respectively. When converting a failure envel’ope from
stress space to strain space, it should be noted that a biaxial stress state creates a tri - axial

strain state due to Poisson’s effect. This phenomenon can be easily verified from Eq.

(3.4.7).

49

If we expand the concept for obtaining the failure envelope for biaxial loading, we can
get the failure envelope when the longitudinal normal, transverse normal and inplane
shear loads are acting on the lamina at the same time. In this case, the maximum loads in
longitudinal and transverse directions should be obtained first. Then, similar to the biaxial
case, the longitudinal shear load is applied, with the longitudinal normal load fixed at
some value equal to or less than the maximum load in the longitudinal direction. Then the
transverse load causing failure in some constituent is averaged from 0 degrees to 360
degrees. Thus the transverse load is determined by averaging the obtained transverse loads
from 0 degrees to 360 degrees as before. The longitudinal normal and shear loads, and the

averaged transverse load are assumed to be on the 3 dimensional failure surface.

3.5 Curve Fitting of Failure Envelopes

The failure envelope obtained by the micromechanical failure theory for biaxial lam-
ina loading can be fit to a polynomial type failure theory by using the predicted unidirec-
tional strengths and nonlinear regression. Among the polynomial type failure criteria for
orthotropic materials, Tsai - Wu’s failure criterion is widely used, and includes interaction
among the stress components analogous to the Von - Mises criterion for isotropic materi-

als [21]. The general form of Tsai - Wu’s failure criterion is [12]:

Fifi-”yap,- = 1 131' = 1,2.....6 (3.5.1)

where summation on the repeated subscripts is implied, a, are the components of the
Cauchy stress tensor and F i , F i]. are strength parameters. If we expand the Eq. (3.5.1),

then we have:

50

F161 + F262 + F303 + F464 + F565 + F666 + Pnof + F226: (3.5.2)
+ F336: + F440: + F550: + [765635 4» 217120162 + 2F130'10'3

+ 2F1 40104 + 217150105 + 2F1651°62F2392°3 + 217240204

+ 2F250'20'5 + 217266266 + 217346304 + 217350365 + 2F360’3O'6

+ 217456465 + 217460406 + 217560566 = 1

In the above equation, the linear stress terms provide for strength difference and sign
reversal of normal suesses in tension and compression, respectively, which is extremely
important for a lamina. But the sign reversal is unsubstantial for the (shear stresses,
because the strength of a lamina should not be affected by the direction or the sign of the
shear stress components. So the terms pertaining to the first order shear stresses must van-
ish. The relevant terms are:
F464, F505, F606, 217146164, 2F150'los, 217160106, 217240204, 217256265
2F260206, 2F34o3o4, 217350365, 217360306, 217450405, 217460406, 2F560506

Since the stress components are in general not zero, the only way to make the relevant

terms vanish is to set the strength parameters zero:

F4=F5=F6=Fr4=Frs=F16=F24=F25

(3.5.3)
=F26=F34=F35=F36= 45=F46=F56=0
Now the Eq. (3.5.2) can be rewritten as:
2 2 2 2
Flo1 + F202 + F363 + F1101 + F2262 + F3303 + F4464 (3.5.4)

2 2
+ F5565 + F6606 + 217120162 + 217130103 + 2F230'20’3 = 1

 

 

1 l l 1 1 l
F=—--—— F=—-— F=-————
1 , 2 r 3 '
XIT Xf x: Xf x; x:
1 l 1
F =—— F = F =
11 C. 22 T c, 33 c, (3.5.5)
Xfxr X2X2 3X3
1 l l
F44=—, F55=—, F66:—

The other strength parameters will be discussed later.
For the case of biaxial stress state, (i, j = 1, 2), the only non - zero stress components

are 61 and 62. Consequently, the Eq. (3.5.4) is simplified to:
2 2
F191 + F262 + F1 lo1 + F2202 + 217126162 = 1 (3.5.6)

The non - zero strength parameters F 1 and F :1 can be determined in terms of strength
measured by simple tension or compression or shear tests. The non - zero strength param-
eters of Fij (i ¢j) i.e., F12, F23 and F31 give an account of the interaction between the
two normal stress components. For example, among the interaction terms, F12 can be
obtained from Eq. (3.5.6):

F - 1 1F 1 61F 021? 357
12' ‘ _1+—F2+—11+;l' 22 (..)

l
2 0'2 e1 02

The Eq. (3.5.7) shows us that F1] are functions of the ratio of a, to of . The only way that
the interaction parameter can be evaluated experimentally is to perform the biaxial tests
with every possible ratio of o". to O’j . However, this experimental task is unhappily not as
easy and simple as the uniaxial test or shear test. In addition, even if F1} are very small,

they play an important role in the failure criterion in that small changes in F”. can signifi-

52
cantly affect the predicted strength and its failure envelope. In Figure 7, the influence of
F12 among the interaction terms is shown. To ensure that the failure envelope is closed and

finite, Tsai and Wu suggest [12]:

F1,“ = _L (3.5.8)
F ,,F j,
where —1 $17,)?“ $1.
Tsai and Hahn suggest [19]:
F..* = -O.5 (3.5.9)

'1

to make the failure enve10pe similar to that of Von Mises failure criterion.

In this study, the problematical interaction strength parameters, F ,j , are obtained by
making the failure envelope obtained by micromechanical failure theory fit to the polyno-
mial type failure theory by means of the nonlinear regression method[15, 49]. For exam-
ple, the interaction strength parameter F12 is obtained as follows.

Let the left hand side of Eq. (3.5.6) be Y, and the right hand side of Eq. (3.5.6) be y, :

2 2
Y, = F,c,, + F262,- + F11°11+ F2202: + ”1261,62.- (3.5.10)

yi=1

F12 which is the regression coefficient in this nonlinear regression procedure can be cal-

culated by minimizing a deviation function S defined as follows:

N
s = X (Y,-—y,)2 (3.5.11)

i=1

SIGMA2

53

 

0.5 '

 

 

l

 

 

Figure 7. The variation of failure Envelopes due to change
in the interaction term 1",, in a, — 02 domain

X10

54

2
= 2(F1611+F2621+F11°ii+F22°§1+2171291552," 1) (3-5-12)

i=1

where N is the number of data points. Using the method of calculus, we can minimize the

function given by Eq. (3.5.11) as shown below:

as _ (3.5.13)
'apj - 0

Substituting Eq. (3.5.11) into Eq. (3.5.13) gives:

N

as a 2 2 2
a!"12 = 3F,2(F161i+F2°2i+F1161:+F22°21+2F12°11°21' 1) = 0 (3-5-14)
1

If we take the partial derivative of Eq. (3.5.14), we have:

N
8S 2 2
5F” = 2 {46“02‘(F1°li+F2°2i+F11011+F22°2FI) (35.15)

i=1

2
+8F12(oli62i) } = 0

If we solve for the regression coefficient, F12, from Eq. (3.5.15), then we obtain the coeffi-

cient as follows:

(3.5.16)

:1
N
u
«I ><

where

N N N N N
2 2 3 3
X = 2 olio2i- 2 FloliOZi— 2 FZGIIOZi" 2 Fllclioli- 2 F220,,0'2,

i=1 i=1 i=1 i=1 i=1

N
2 2
Y: 2201302:

i=1

55
Hence, the problematical interaction terms are determined without the need for any biaxial
experiments.

With the determined strength parameters and the uniaxial strengths of lamina, it is pos-
sible to plot the polynomial type failure envelope in two ways, when a biaxial stress state
is applied to the lamina. One of the approaches is to find a set of a, and 02 to satisfy Eq.
(3.5.6) by trial and error method. That is, a, (say) is fixed at some value first which is
equal to or less than the ultimate strength in X, or X2 direction, and then 0'2 is sought to
satisfy the Eq. (3.5.6) by varying it from its uniaxial tensile strength to its compressive
strength. By repeating the above process, the set of points on the failure envelope is calcu-
lated.

The other method is to obtain an equation for a, or 0'2 from Eq. (3.5.6) by means of
the discriminant formula for a quadratic equation. Rearranging Eq. (3.5.6) with regard to

0'2 in the descending order gives:
2 2
F2262 + (F2 + 2F12crl)o'2 + Flo, + Flo, — 1 = 0 (3.5.17)

Solving Eq. (3.5.17) for 0'2 gives:

4L

2 I 2
a, = 21, (3.5.18)
22

 

A

2 I 2
— (F2 + 2171261) - Jar, + 217126,) — 4172411716l + F161 -1)
62 = 2F (3.5.19)
22

 

At a value of a, , Eq. (3.5.18) represents points in one half of the failure envelope, and Eq.

(3.5.19) represents points in the other half of the failure envelope. Therefore, a set of 62

56
and a, on the failure envelope, which satisfy Eq. (3.5.6), is obtained by varying a, from

the tensile strength to the compressive strength of the lamina.

CHAPTER IV
INDEPENDENT MODE FAILURE CRITERIA

4.1 Introduction

A lot of researches have been carried out to predict the failure envelope more realisti-
cally, on the assumption that the failure envelope is a piecewise smooth curve, since the
mechanical behavior of composite material is anisotropic. Some researchers established
separate failure criteria in each quadrant of stress space by defining a difl‘erent functions,
in order to obtain piecewise smooth curve [23, 16]. In this chapter, it will be discussed
how to obtain a more realistic and applicable failure criterion by considering the failure

mode of each constituent [22].

4.2 The Concept of Independent Mode Failure Criteria

The stress distribution in each constituent is affected by the magnitude of applied load
and/or the ratio of the applied loads in each direction. The change in the magnitude of
applied loads causing failure in some constituent leads to changing only the magnitude of
all the existing stress components at the same rate, but does not afl‘ect the relative magni-
tude of all the existing stress components. Thus it makes a contribution to the first occur-
rence of failure in the same constituent. But the change in the ratio of applied loads

causing failure changes the distribution of stress components in the constituents, and may

57

the

8111

Eli):

SUE

Em}

58
induce a change in the failure mode. This fact implies that the failure mode of all the
points on the failure envelopes are not the same. In other words, the failure envelope of a
composite material is represented by theintersected area of the failure envelopes of each
constituent. Moreover, it would be unreasonable that the entire failure envelope can be
represented by a single equation, which gives a mathematically smooth curve, since each
failure mode is entirely different. Thus, it is very useful to establish a separate failure cri-
terion for each constituent in terms of lamina loads by supposing that failure takes place in
only the constituent under consideration, rather than using a single equation for entire fail-

ure envelope.

4.3 Independent Mode Failure Criteria

The failure envelope of each constituent in a given composite material system can be
written as function of macro (lamina) load. This function is obtained via the microme-
chanical model by allowing failure to occur in only the constituent under consideration.
All other constituents are assigned strength values that are orders of magnitude larger than
the actual strength to ensure that failure will not be predicted to occur in these constitu-
ents. Then the failure envelopes are predicted in terms of lamina loads, using the micro-
mechanical failure theory. By repeating this process for each constituent, the failure
envelope of each constituent can be calculated as a function of the macro (composite)
stress state. The inner intersected area of the failure envelopes is regarded as the failure
envelope for the lamina.

Since both the failure load and the mode of failure are predicted in the current scheme,

it is possible to cast the failure model in a simple mathematical form with separate criteria

59

for each mode of failure. For example, if a polynomial form is desired, the criteria can be
written as follows:

For fiber:
1 t l
Ffo, + Fgopj = 1 (4.3.1)
If the lamina loading is biaxial, Eq. (4.3.1) is expanded to:

Pfia’1 +P26’2+P{ (a ,2) +P§2(a‘2) +2P§2a" 02 = 1 (4.3.2)
For matrix:

1","6, +F'."o.o (4.3.3)

rji

If the lamina load is biaxial, Eq. (4.3.3) is expanded to:

2 2
Fret, + F2012 + Ff,(o',) + F’2"2(ol2) + 2F'l"2o',ol2 = 1 (4.3.4)
For fiber/matrix interface:
'11 n
P“, 6’.+ P’Uo’a , (4.3.5)

If the lamina load is biaxial, Eq. (4.3.5) is expanded to:
F',"o 1+ F'2no 2+ 173(01): + F’2’2(o2)2 + 2F‘1'2011012- = 1 (4.3.6)

where the superscripts m, f, in and I refer to matrix, fiber, interface and lamina, respec-

tively, Ff and Ff are the strength parameters in terms of the lamina load as shown rn Eq.

60
(3.5.5), and 0': represents the components of stress in the lamina. The strength parameters
of each constituent in Eqs. (4.3.1 — 4.3.6) should be computed as functions of the lamina
loads using the micromechanical failure theory. Then, the strength parameters (i.e.,
F, and F ,,) can be calculated as in Eq. (3.5.5).
Alternatively, other mathematical forms of the failure criterion may be used. For

example, an appropriate form for fiber failure may be:

in tension:
a?
_I_ = 1 (4.3.7)
slur

in compression:
; = 1 (4.3.8)
slur

Once the shape of the failure envelope for each constituent is known, an appropriate

mathematical expression to describe each surface can be chosen.

CHAPTER V
THE INDEPEDENT MODE FAILURE CRITERION
BASED ON “IN-SITU” CONSTITUENT STRENGTH

5.1 Introduction

The accuracy of the micromechanical failure theory depends largely on the degree of
error in the material prOperties of each constituent in the composite system. The geometry,
stiffness and strength of each constituent in a composite system are inherently randomly
varied during the manufacturing processes of a composite system, and are usually differ-
ent from those of each constituent in bulk state [31]. For this reason, the use of bulk
mechanical properties in the micromechanical failure theory may lead to poor failure pre-
dictions in composite systems. Especially, the mechanical properties of polymers which
are commonly used as matrix are time - dependent, so that the mechanical properties of
polymers in a composite system are possibly different from its bulk mechanical properties.

For example, the creep function of an epoxy matrix is given by the four-parameter

model [52]:

1 1( 4:) t
J t = —+— l—e +— 5.1.1
() E0 Er “o ( )

where E0, E,, h and “o are constants, and t is time. A plot of the creep function for 934

resin and the epoxy matrix used in glass/epoxy composites is shown in Figure 8. If the

61

applied loading, a , is constant with time, the strain will be [53]:

62

 

 

 

 

3 I I I I I
934
2.5 - _____________ . ............................. . ............ ,
./ . I
2 - .
a?
E
5 EPOXY
1.5 " d
:;
1 I
0.5 - .
o I I I I I
o 20 40 so so 100 120
TIM E (min.)

Figure 8. The creep behavior of some materials used as matrix

63

£(t) = O’J(t) (5.1.2)

This visco-elastic behavior of matrix leads to poor prediction of the transverse strength of
a lamina, since matrix failure dominates the transverse strength of a lamina.

Another obstacle to using micromechanical models is that some mechanical properties
of constituents are not commonly available. For example, the compression strength of a
matrix material, which is sometimes different from the tension strength, is often not
readily available. Hence, the current model includes the option to extract average “in-situ”
constituent strength properties using unidirectional composite strength data, so that it indi-
rectly accounts for the above variations as well as thermal residual stresses, voids, and

other processing effects.

5.2 The In-situ Constituent Strengths

To obtain the in-situ strength of each constituent, a failure mode must be assumed for
a given applied uniaxial loading state. For example, transverse lamina strength is often
dominated by matrix strength, while failures due to longitudinal loads usually occur in the

fiber. Thus, the processes to obtain the in-situ strength of each constituent are different.

Fiber Strength
Since the longitudinal tension and compression strengths of a composite material are
dominated by the tension and compression strengths of fiber, the longitudinal strength of a

composite material is applied as a load. The resulting maximum strain in the fiber at that

64

load is taken as the ultimate strain of the fiber.

Matrix Strength

The transverse tensile and compressive strength of a composite are dominated by the
tensile and compressive strengths of the matrix or by the strengths of the interface.
According to Kominar and Wagner [50], the failure of unidirectional composites with very
stiff resins can take place without significant preliminary interfacial debonding but as the
result of matrix crack nucleation and growth. Moreover, Folias [51] found that if the deb-
onding at the interface does not occur, there exists a stress magnification factor in the
matrix which attains a maximum between the fibers. If a crack does initiate at the inter-
face, then it commonly propagates through the matrix to connect with cracks at other
interfaces, eventually leading to lamina failure. When a transverse load is applied to a uni-
directional lamina, it is difficult to distinguish between the load at initial failure and the
ultimate load, because the cracks often propagate so rapidly. Consequently, in the current
model, the transverse strength of the composite is applied as a load in the model at each
angle from 0 degrees to 360 degrees, and the maximum principal stress in the matrix is
determined for each angle, and averaged from 0 degrees to 360 degrees. The averaged
maximum principal stress is taken as the ultimate strength of the matrix. Note that this
value of matrix strength may also be related to the interface strength, depending upon the
mode of failure in the experimental tests employed for mechanical characterization of the

lamina.

65

5.3 The Independent Mode Failure Model Using “In-Situ” Properties

The obtained in-situ strengths are used as constituent strengths in the failure criterion
for each constituent. In this independent mode failure criterion, the interface mode of fail-
ure is excluded, because the transverse strength of the composite was assumed to be dom-
inated by matrix strengths. In polynomial form, the failure criteria can be written as:

Fiber

Pfof +P’o’o’. (5.3.1)

rjio

where Ff and Ff are obtained in terms of macro by substituting‘ ‘in- -situ” strengths of the
matrix into Eq. (3.5.5) and the interaction terms F; are obtained by nonlinear regression
like Eq. (3.5.16).

Matrix

F76, +F’."oo (5.3.2)

'1‘]:

where 15’"I and F" are obtained by substituting‘ ‘in-situ” strengths of the matrix into Eq.
(3.5.5) and the interaction terms P1; are obtained by nonlinear regression like Eq.
(3.5.16).

This technique enables us to match the predicted and experimental unidirectional
strengths as well as to obtain more realistic failure envelope under multi-axial loading

conditions, because much of the inherent randomness and processing effects are efl‘ec-

tively accounted for.

' CHAPTER v1
NUMERICAL IMPLEMENTATION
AND DISCUSSION OF RESULTS

6.1 Numerical Algorithms

The numerical procedure used in this research consists of conversion of macro stress
loading to macro strain loading, micromechanical stress analysis, micromechanical failure
prediction, prediction of uniaxial strengths, and the failure envelope for multiaxial load-
ing, and curve fitting of the failure envelopes. The used algorithm is depicted in Figure 9
in detail and is described as follows:

1. Using the micromechanical elasticity solutions, compute the effective lamina material
properties and store the micromechanical stresses due to each of the six single components
of unit macro strain (lamina strain).

2. Convert 1 MPa of unidirectional lamina load to the corresponding lamina strain state
and apply the strain state to the micromechanical model. Compute stresses at the desired
points within each constituent (i.e., fiber, matrix, or interface) of the material system due
to the uniaxial lamina load.

3. Evaluate the chosen failure criterion at each point within each constituent of the mate-
rial, store the failure information, and determine the maximum failure index (mum) for all
the points.

4. Check for failure (tymx ). If the failure criterion is not Von - Mises criterion among

66

67

START

 

l. 0 Compute the lamina effective moduli.

- Calculate and store micromechanical stresses and
strains due to unit macro strains.

 

 

 

 
   
   

Yes

 

2. 0 Convert lamina load(s) to
lamina strains.
0 Apply the strains to the model.
V 0 Compute stresses at the desired
points within each constituent.

l

in-situ properties 9

 

 

 

 

7. - Apply the longitudinal or transverse
strength as a load.

 

 

 

 

 

 

 

 

 

. OCheckthemaximumstressorstrain —
3. 2:33:16 31:1: $033?“ {allure within the related constituent.
- Determine the maximum failure 0‘
index (trim) for all points.
t . Increase or decrease
the lamina load.

   
 

 

 

 

4. 0 Check for failure.
0.99 zwms l

 
        

 

Applied lamina load

Yes 5 Failure Index

- Uniaxial Strength

I

5. 0 Fix longitudinal stress at some value less than the tensile strength.
-Choose 1MPaasatransversestless.

 

 

 

 

 

 

 

 

 

 

 

 

 

olncreaseordecrease

 

 

 

 

transverse stress.

 

 

 

 

 

 

'The longitudinal and the transverse stress

are assumed to be on the failure envelope. ' F“ the @1013 envelope (0 polynomial
- Repeat this process until the longitudinal "7 type failure envelope or Other form.

 

 

 

 

 

 

 

stress reaches the longitudinal strength. 5‘

Figure 9. The flow chart

 

 

68

the aforementioned failure criteria, the uniaxial strength is determined as:

 

Applied lamina load (6.1.1)

U'Nax‘al Strength = Failure Index

If Von — Mises criterion is chosen as the failure criterion, the acting lamina load is
regarded as the uniaxial strength, when 1 - e < wmax < 1 + e, where e is a failure toler-
ance. If Wm“ < 1 —e or wmax> 1 + a, increase or decrease the lamina load, and return to
step 2. Thus, the uniaxial lamina strength is determined in an iteration fashion.

5. On the basis of the uniaxial strengths of the lamina, calculate the biaxial strength as
follows. First, vary the longitudinal stress of the lamina from the value of tensile strength
to the compressive strength. Then find the transverse stress for the fixed longitudinal stress
thatsatisfies 1-e<\vmax< 1 +e,using steps 2, 3 and 4. If 1—a<\ymax< l +e,thelon-
gitudinal stress and the transverse stress are considered to be on the failure envelope.

6. Using a nonlinear regression procedure and the uniaxial strengths of the lamina, the
failure envelope is fit to a polynomial type failure envelope or some other mathematical
form.

7. To get the in-situ strengths of each constituent, perform step 1 first. Then apply the
longitudinal or transverse strengths as a load to the model, and check the maximum princi-
pal stress or maximum strain for the load among each related constituent.

The failure envelope and uniaxial strength of the lamina based on the in-situ strengths
of each constituent are obtained by inputting the in-situ strengths as the strength of each

constituent and performing steps 1 - 6 again.

69

6.2 The Numerical Results and Discussion

The Verification of the Micromechanics Solution

In the simulation, the composite material T300/934 system was used, and the material
prOperties are listed in the Tables 1 - 3.

The solution for each loading case consists of a series summation, so that it is very
important to see how many terms are needed for the solution to converge. From Figure 10,
it can be observed that 9 terms are not sufficient for 03 to converge. But, using 19 terms,
all stress components converge with acceptable errors. Here, the convergence of the solu-
tion is checked along line 2 in Figure 11 in terms of stresses, not displacements, because
the stress is of greatest interest. The stresses at or near the interface also converge within
about 19 terms.

The point matching technique or collocation was applied to obtain some constant
terms in the solution for each loading case, at some selected points along the diagonal
boundary AB of the RVE. Therefore, it needs to be assessed whether or not the behavior
of stress components along the diagonal boundary AB changes abruptly or becomes sin-
gular. Figure 12 shows that the predicted principal stresses on the diagonal boundary and
lines parallel to the diagonal boundary as shown Figure 11 are changing very smoothly as
expected.

The effective material properties are determined by Eqs. (2.6.1) - (2.6.6), based on the
average stresses. The obtained effective material properties are compared with those pro-

posed by Chamis [2]. The formulae used by Chamis are:

En = EflVf-t- Emu --V,) (6.2.1)

TABLE 1. Material properties of fiber used in calculation

70

 

 

 

 

 

 

 

T300 AS4
E, (GPa) 0 233.000 235.0000
E2 (GPa) 23.1000 14.0000
F4 (GPa) 23.1000 14.0000
nun 0.4000 0.250
nu,3 0.2000 0.2000
nun 0.2000 0.2000
023 (GPa) 8.2700 5.5000
6,3 (GPa) 8.9600 28.0000
on (GPa) ’ 8.9600 28.0000
alphal (lldeg. °C) -5.4000E-07 -3.6000E-07
alpha2 (lldeg. °C) 1.0080E-05 1.80005-05
alpha3 (lldeg. °C) 1.0080E-05 1.8000E-05
allow. axial strain (tension) 0.0118 0.0153
allow. axial strain (comp.) 0.0094 0.0145

 

T300, AS4: as used in [46]

 

TABLE 2. Material properties of matrix used in calculation

71

 

 

 

 

 

 

 

934 3501-6
E, (GPa) 4.6500 4.3000
E, (GPa) 4.6500 4.3000
E, (GPa) 4.6500 4.3000
nu23 0.3630 0.3400
nu,3 0.3630 0.3400
nun 0.3630 0.3400
623 (GPa) 1.7000 1.6000
(3,, (GPa) I 1.7000 1.6000
6,2 (GPa) 1.7000 1.6000
alphal (lldeg. °C) 0.414E-04 040005-04
alpha2 (lldeg. °C) 0.414E—04 0.4000E-04
alpha3 (lldeg. °C) 0.414E-04 0.4000E-4
ult. tensile stress (MPa) 58.8000 83.0000
ult. comp. stress (MPa) 58.8000 207.0000

 

934, 3501-6: as used in [46]

 

72

TABLE 3. Material properties of composites measured in experiment

 

 

 

 

 

 

 

 

AS4/3501-6
E, (GPa) 148.0000 145.0000
E, (GPa) 9.6500 10.6000
nun 0.3000 0.2700
0,; (GPa) 4.5500 7.6000
tens. long. strength (MPa) 1314.0000 2090.0000
comp. long. strength (MPa) 1220.0000 1440.0000
tens. trans. strength (MPa) 43.0000 64.0000
comp. trans. strength (MPa) 168.0000 228.0000
long. shear strength (Mpa) 48.0000 71.0000
fiber vol. fraction (%) 60.0 65.0

 

T300/934 [54]

AS4/3501-6 [56]

 

SIGMA (PA)

73

 

............ : TermS‘: 9
-------- :Terms= 19
----- : Terms: 29

: Terms: 39

 

 

 

4.0 5D
A N G L E (DEG.)

20 30 60 7O 80

..

Figure 10. The convergence of stresses according to the number of terms

   
   
   
    

 

74

1.732

 

 

 

>X2

0'8” |L>1.0 :Linel
0.98:Line2

~—> 0.96:1.ine3
——->0.94:Line4

 

 

Figure 11. The stress evaluation position

SIGMA (PA)

75

 

 

 

 

 

20 30 40 50 60 7O 80 90
A N G L E (056.)

Figure 12. The Stress distribution near boundary

76

E E E'"
22' 33-1—JVIU-Em/Ef2)
G :0 = "'
‘2 ‘3 l—JVfU—Gm/Gfl)
Gm
1— Eu — rim/0,23)

v12 = v13 = vfl2Vf+vm (l - Vf)

 

 

 

G23

V23 = 522-: -- 1

where E, G and v are Young’s modulus, shear modulus and Poisson’s ratio, the arabic
subscripts stand for the material directions, and the subscript f and m mean fiber and
matrix, respectively.

It is shown in Figure 13 that the prediction of longitudinal and transverse Young’s
modulus, E1 and E2 , respectively, by the current model has a better agreement with
experiments than that by Chamis. It is observed in Figure 14 and Figure 15 that the longi-
tudinal shear modulus and Poisson’s ratio, 612 and v12 , respectively, predicted by the
current model have some deviation with the experimental results. But the transverse shear

modulus and Poisson’s ratio, G23 and v23 are difficult to measure reliably, so the source

of the difference is difficult to identify.

Results using the Micromechanical Failure Theory

In Figure 16, the lamina loads leading to failure in some constituent applied in the
direction varying from 0 degrees to 360 degrees are plotted by adopting the maximum
principal stress criterion for the matrix. It can be observed that there exists an axes of sym-
metry every 30 degrees, and the axes of symmetry for transverse normal strengths coin-

cides with that for longitudinal shear strength. The absolute maximum value of transverse

77

x 1011 (a)

2 I I I I I I I I I I

 

—— : micromechanics

15’ ---- :Chamis

 
   

E1 (Pa)

 

 

 

 

 

 

 

0 I I I I I 4 I I I

10 15 20 25 30 35 4o 45 50 55
Fiber Volume Fraction (%)

 

8%

o : Tsai [54]

x : Chang-Kutlu [55]

Figure 13. The Variations of El and E2 with fiber volume fraction

78

1(109 . (a)

8 I I I I I I I j I 1

 

: micromechanics
5' ---- :Chamis

 

 

 

 

 

 

 

 

: micromechanics , - -— "

    

'
fl
'
'
v
’
‘
mi‘
1’

- - - - :Chamis

   

"
-
fl
—
'
”
-
—

 

623 (Pa)

 
 

 

I I I I I I I I I

O I
10 15 20 25 30 35 40 45 50 55 60
Fiber Volume Fraction (%)

o : Tsai [54]

x : Chang-Kutlu [55]

Figure 14. The Variations of 612 and 623 with Fiber Volume Fraction

79

 

 

 

 

 

 

 

 

 

0.5 . a . . . . . . . . a 1
0.4 - -
g 0.3 - _____________________ I_ _ _ -
0'2 ' : micromechanics '
0.1 - - — - - :Chamis -
010 1'5 2'6 55 3.0 35 4'6 45 53 55 66
(b)
0.5 .. . a . . . . . . r 1 .
0.4 :_ ____________________________________ 1
g 0.3 - 0 -
0'2 ' :micromechanics '
0.1' ---- :Chamis '
0 . l L . . . . .

 

 

 

10 15 20 25 30 35 40 45 50 55 60
Fiber Volume Fraction (%)

o : Tsai [54]

x : Chang-Kutlu [55]

Figure 15. The Variations of v12 and v23 with Fiber Volume Fraction

 

SIGMA (Pa)

80

x 107 T300/934

 

 

 

 

 

2" _"‘3 X2T II
---- : X2C
------- : X12&X13
0" d
-2. I
.4 \ ’/\\ ’~\ I"\ l—.‘ ”~\ [a
\\’l \\”l \\v’/ \\v’/ \\\’/ \\ll
-6' n n 1 n n n J
0 50 100 150 200 250 300 350

A N G L E (DEG)

Figure 16. The Variation of maximum loads causing failure in
transverse direction as the applied angle changes

81
tensile and compressive lamina loads causing failure are found out at 30, 90, 150...
degrees, but the maximum value of longitudinal shearing lamina load bringing about fail-
ure are found at 0, 60, 120... degrees. The axes of symmetry can be expected from the
assumed arrangement of fibers in the composite.

The predicted unidirectional strengths are compared with the experimental results in
Figures 17 - 22. To determine the unidirectional strengths of a lamina, the maximum strain
criterion was adopted for fiber, the maximum stress criterion was used for interface, and
one of the following failure criteria were used for matrix as discussed in Chapter 3: maxi-
mum principal strain criterion, maximum principal stress criterion, Tresca criterion, and
Von - Mises criterion. Hereafter, the only failure criterion for matrix will be mentioned,
since the failure criteria for the other constituents are fixed. In each case, bulk matrix prop-
erties used.

As shown in Figures 17 and 20, the predicted transverse tensile strengths for both
T300/934 and AS4/3501-6 by maximum principal stress criterion, Tresca criterion and
Von - Mises criterion were in good agreements with experimental results, but by maxi-
mum principal strain criterion were relatively poor. The maximum principal stress crite-
rion rendered the strength closest to the experimental results for both composite systems.
For AS4I3501-6, Tresca criterion is not applicable, because the tensile strength of matrix
is different from the compressive strength. It was observed that all the predicted transverse
tensile strengths were lower than the those of matrices. This phenomenon comes from the
fact that the fibers are acting as solid inclusions and give rise to higher stress than the
applied stress load, when loading is applied in the transverse direction.

No matter what criterion was used, the prediction of transverse compressive strength

as shown in Figures 18 and 21 was very poor, when compared with the experimental

82

“or x 2 T (T300/934)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. Maximum principal strain criterion 2. Maximum principal stress criterion
3. Tresca criterion 4. Von - Mises criterion
5. Experimental result [54]

Figure 17 . The predicted tensile transverse strengths
by micromechanical failure theory ('I‘300l934)

Pa

83

“or x 2 c (1'300/934)

 

 

 

 

 

 

 

 

 

 

 

 

 

-14 '-

-16 -

 

 

 

 

 

1. Maximum principal strain criterion 2. Maximum principal stress criterion
3. Tresca criterion 4. Von - Mises criterion
5. Experimental result [54]

Figure 18. The predicted compressive transverse strengths
by micromechanics failure theory ('I‘300/934)

84

“or x 1 2 (ram/934)

 

4.5

 

 

 

I

3.5

a 2.5

1.5-

 

 

 

 

 

 

 

 

 

 

 

 

1. Maximum principal strain criterion 2. Maximum principal stress‘criterion
3. Tresca criterion 4. Von - Mises criterion
5. Experimental result [54]

Figure 19. The predicted longitudinal shear strengths
by micromechanics failure theory ('1‘300/934)

85

“or x 2 T (AS4I3506)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0.5 1 1.5

2.5 3 3.5 4 4.5

1. Maximum principal strain criterion 2. Maximum principal stress criterion

3. Von - Mises criterion 4. Experimental result [56]

Figure 20. The predicted tensile transverse strengths
by micromechanics failure theory (AS4I350 1 -6)

Pa

86

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0,5108 x 2 C (AS4I3506)
-0.5 -
-1 .-
-1.5 -
-2 .
'2 5 0:5 I 1.5 2 2:5 3 3.5 4 415

1. Maximum principal strain criterion 2. Maximum principal stress criterion
3. Von - Mises criterion 4. Experimental result [56]

Figure 21. The predicted compressive transverse strengths
by micromechanics failure theory (AS4I3501-6)

87

“07 X1 2 (AS4I3506)

 

 

I

 

 

 

 

 

 

 

 

 

I I j I I I I

 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

1. Maximum principal strain criterion 2. Maximum principal stress criterion
3. Von - Mises criterion 4. Experimental result [56]

Figure 22. The predicted longitudinal shear strengths
by micromechanics failure theory (AS4I3501-6)

88
results. This may originate from the fact that the compressive strength of the matrix in a
composite is different from that in bulk state. For T300l934, Von - Mises criterion pre-
dicted better transverse compressive strength than the other criteria. For AS4l3501-6, the
prediction by maximum principal suess criterion was much better than any other criteria,
because the absolute compressive strength of matrix is greater than the absolute tensile
strength.

In Figures 19 and 22, the predicted longitudinal shear strength is depicted. The
obtained longitudinal shear strengths by maximum principal strain criterion, maximum
principal stress criterion and Tresca criterion are exactly same, but the shear strength by
Von - Mises criterion is slightly bigger than those by the other method. When the predicted
shear strength was compared with the experimental results for both cases, the error was
less than about 15 percent.

Since it was supposed that the longitudinal strength of a lamina is dominated by the
fibers, and fibers are brittle, the ultimate longitudinal strain of the fiber was taken to be the
same as that of the lamina. Therefore, the longitudinal strength predicted from the micro-
mechanics failure theory is exactly the same as the experimental result.

In the process of obtaining unidirectional strengths, it was found that the matrix
strength can influence the axial strength of composite. When the axial load is applied, ten-
sile or compressive stress is developed in the matrix due to the Poisson’s effect, and a lam-
ina may fail by failure of matrix, not in the fiber. This implies that, to obtain higher
strength of composite, the strengths of all constituents should be considered. Thus, this
micromechanical failure criterion can be used in designing composite systems.

The failure envelopes for T300l934 in stress space predicted by the micromechanical

failure theory are shown in Figure 23, and the failure envelope by maximum principal

N

SIGMA2 (Pa)
0

x10

89

T300l934

 

 

 

 

 

 

-2 . .
-4 - -
—6 I I
—2 -1 .5 -1 —O.5 0 0.5 1 1 .5 2
S'GMAI (Pa) X109
— : Maximum principal strain criterion - - - :Maximum principal stress criterion
_._ :Trescacriterion ----- :Von-MisesCriterion

Figure 23. Failure envelope in stress space (T300l934)

90

strain criterion is most conservative among the predicted failure envelopes. Considering
the different scales of 0', axis and 62 axis in the plot, it is noticed that the failure enve-
lopes are long and thin along the 6, -axis. The conversion of the failure envelopes in biax-
ial stress state to strain space gives rise to three-dimensional strain state due to Poisson’s
effect by stresses in X, and X2 directions. The failure envelopes in 3 dimensional strain
space are shown in Figure 24, and are inclined to 83 axis. If they are replotted in the
8, and 82 space, the enve10pes will be the same as shown in Figure 25.

For AS4/3501-6, the failure envelopes in stress space are shown in Figure 26, and the
failure envelopes in the two-dimensional strain space are shown in Figure 27. Since the
compressive strength of the matrix in this composite system is much larger than its tensile
strength, the maximum principal stress criterion gives better results than any other criteria
in this research, while the maximum principal strain failure criterion provided poor
results.

In Figure 28, the failure envelope for T300l934 was plotted by choosing the Von-
Mises failure criterion for the matrix, then it was fit to the polynomial type failure enve-
lope by using nonlinear regression and unidirectional strengths of the lamina as discussed
in Chapter 3, Section 5. In this process, the interaction strength parameter, F12, was
obtained. The failure envelopes obtained by the micromechanical failure theory and the
nonlinear regression are compared with the Tsai-Wu failure envelope which is obtained
using experimental results and the interaction strength parameter suggested by Tsai-Hahn
as in Eq. (3.5.9). The interaction strength parameters predicted by the two methods are

shown in Table 4 below:

EPSILON3

91

T300l934

3‘

2~

 

0.01
0.01

  
  
 

'°'°°5 —o.005
-°-°1 —0.01

EPS'LONi EPSILON2

Figure 24. Failure envelopes in 3 dimensional strain space

EPSILON2

92

 

 

 

 

 

 

T300/934

0.01 '

0-005l‘ ~—~ .......
o b
-0.005 I'
-0.01 '-

-0.01 —0.005 0 0.005 0.01
E P S I L O N 1

— :Maximumprincipalstraincriterion --- :Maximumprincipalsuesscriterion

_.- :Treseacriterion ----- :Von-MisesCriterion

Figure 25. Failure envelopes in strain space (T‘300/934)

93

 

 

 

 

 

 

 

 

x m" AS4/3501—6
2 I I I I I I I I
1.5 - .
1 " II
A 0.5 I f... I ----------------------- I
m r——
0- l
V I
N O ' i '
< i
2 I- . , . .-
(D 0.5 l ........... 1
I- .............................
(D —1 I I II
I
I
-1 .5 I I ____________________________ u
—2 I 1
-2.5 I I I I I I J I
-2 -1.5 -‘I -0.5 0 0.5 1 . 1.5 2 2.5
SIGMA1 (Pa) ms

— : Maximum principal strain criterion - - - : Maximum principal stress criterion

--- :Von-MisesCriterion

Figure 26. Failure envelopes in stress space (AS4I3501-6)

94

AS4/3501 -6

 

0.01 r

EPSILON2

 

—0.02 '

 

 

 

-0.01 0.01 0.02

0
EPSILON1
-— :Maximumprincipalstraincriterion --- :Maximum principal stresscriterion

--- :Von-MisesCriterion

Figure 27. Failure envelopes in strain space (AS4I3501-6)

SIGMA2 (Pa)

95

 

 

 

 

 

 

 

“on T300l934
2 I I I I I I I I I
1.5 - '-
1 ' I
0.5 '- -
0 . . .
_O.5 I I". I
I" ,l'
/ .1
_1 I '/' .l '
a, o’.’
-1.5 - ! ,-" -
\ ,1
.\. .I".
-2P \'\ .................. d
-225 I I
-3 I I J J I I I I I
-3 —2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
SIGMA1 (Pa) X109
— :Von-MisesCriterion --- :Cuve fitting ofmicromodel
prediction
-'- :Tsai-Wu

Figure 28. Fitting failure envelope to polynomial type failure
envelope and the failure envelopes using Tsai-Wu
theory and experimental results (T300l934)

96

TABLE 4. Comparison of interaction strength parameter

 

Nonlinear regression

 

Strength Parameter -0.235339655944E-l7 -O.46462603093E- 17

 

In tension - tension space the failure envelope obtained by the micromechanical failure
theory is very close to the Tsai-Wu’s failure envelope obtained using experimental values.
The deviation between the computed Tsai-Wu type and the experimental Tsai-Wu enve-
lopes mainly comes from the differences in interaction term F,2 for each envelope and the
transverse compressive strength.

Figure 29 shows the failure envelopes for AS413501-6 where the maximum principal
stress failure criterion was used for the matrix to get the failure envelope from the micro-
mechanical failure theory. In tension - tension space, the failure envelopes from the micro-
mechanical failure theory are in good agreement with the Tsai-wu’s failure envelope using
experimental results.

Thus, unidirectional strengths as well as strengths under multiaxial loading of the lam-
ina are determined. The predicted strengths of the lamina by the model are in reasonably
good agreement with experimental results.

The fiber volume fraction in composites has an influence on the strengths of compos-
ites, so the failure envelope varies as the fiber volume fraction changes. The variations in
failure envelope due to the changes in fiber volume fraction are shown in Figures 30 and
31 for T300l934. From Figure 30, it can be observed that the longitudinal strength of a
lamina is proportional to the fiber volume fraction, that is, the longitudinal strength
increases as fiber volume fraction increases. But the transverse strength is not influenced

as much as the longitudinal strength as shown in Figure 30. The transverse strength

SIGMA2 (Pa)

O

I
..h

97

I I I I I I

 

 

 

 

 

 

 

 

..3 —2 —1 O 1 2
SIGMA1 (P8) x10

”0.1

— :Maximum principal stress criterion --- :Curve fitting ofmicromodel
predictions

"" :Tsai-Wu

Figure 29. Fitting failure envelope to polynomial type failure
envelope and the failure envelopes using Tsai-Wu theory
and experimental results (AS4I3501-6)

98

T300/934

x107

 

 

 

1.5

0.5

0
SIGMA1 (Pa)

-0.5

-1.5

 

 

-6-

4 2 0 2 .4

_
32229.6.

x10“

due to the change in fiber volume fraction

Figure 30. The changes in failure envelope in stress space

EPSILON2

99

T300l934

 

00015 i I I U i

0.01 '-

0.005 '-

 

 

 

—0.01 -

 

 

 

.0015 - -
-0.015 -0.01 -0.005 0 0.005 0.01

EPSILON1

Figure 31. The changes in failure envelope in strain
space due to the change in fiber volume fraction

0.015

100

increases a little bit, and then decreases slightly as fiber volume fraction increases. The
failure strain in the transverse direction decreases as fiber volume fraction increases.

When a lamina is subjected to a plane stress state, the failure envelope is plotted in
three-dimensional space which takes longitudinal, transverse, and longitudinal shear
stresses as axes. In this case, since the strength of a lamina is finite, the failure surface
should be closed. It is shown in Figure 32 that the failure surface is closed and symmetric
about the plane in which the longitudinal shear stress is zero. The failure surface is
obtained by using the maximum principal stress criterion for the matrix. When the shear-
ing load is applied on the lamina, the influence of shearing stress on lamina strength is
shown in Figure 33. The longitudinal strength is not affected up to about 20 MPa of longi-
tudinal shear stress, but when the longitudinal shearing suesses exceed about 20 MPa, the
longitudinal strength decreases very fast. On the other hand, the transverse strength

decreases steadily as the shearing load increases.

The Results of Independent Mode Failure Criteria

The failure envelope of each constituent in a lamina is predicted in terms of lamina
loads by using the micromechanical failure theory and allowing failure to occur in only
the constituent under consideration. A typical result is shown in Figure 34, which was
obtained by choosing the Von - Mises criterion for the matrix. The intersected part of each
constituent is regarded as the failure envelope for the composite (T300l934). To see more
clearly, the failure envelope is replotted in Figure 35, and it can be ascertained that fiber
dominates the longitudinal strength, while interface and/or matrix govern the transverse
strength. If Tresca failure criterion is used for matrix, only the matrix dominates the trans-

verse strength as shown in Figure 36. For AS4I3SOl-6, the independent failure envelopes

101

T300l934

 

‘ \

 

 

 

 

 

 

 

 

,a. <<llllh> ,,____

c3; 2‘ I /\ g"

f: 0. ;::::::;, 4‘,¢——-—".____.——r <“IIIII’

5 ‘ / -
_4 M

 

 

 
  

x107

SIGMA2 (Pa) '5 ‘15 SIGMA1 (Pa)

Figure 32. The failure envelopes in three dimensions
when shear loading is applied.

x 107 T300l934

102

 

SIGMA2 (Pa)
O N

l
N

 

 

 

 

 

 

 

 

-1 .5 -1 -0.5 O 0.5 1 1.5
Shear load: L1: 0 MPa L2: 10 MPa L3: 20 MPa
L4: 30 MPa L5: 35 MPa L6: 38.5 MPa

Figure 33. The failure envelopes in two dimensions
when shear loading is applied.

SIGMA2 (Pa)

 

-12
.3

103

7 T300l934

 

 

 

 

 

Figure 34. Independent mode failure envelope in stress space(T300/934)

j-
’I
I
/ u
I
I
,z’ :fiber
II '*
I,’ ---- :interface
-—- :matrix 1
-2 -1 0 1 2 3 4 5 6 7
SIGMA1 (P8) x1010

104

 

 

 

 

 

 

 

 

x107 T300l934
8- H
6- d
[ .. - .. ..-: '.".'.'_'.:.:.:..-.'.-..' 2‘: ""~ _________
4.. ’./"’ \\ .
w" \o
75 I ’ \-
9: 2” I' \ +
cl. ‘
N i
< O- I d
2 ,'
(D .I'
—-2_ 'I -.
CD \ I I
.\. I'l-
-4r- \-\.‘ O’ I. d
-6, j _
..3. q
‘2 -1-5 -1 -0.5 0 0.5 1 1.5 2
SIGMA1 (Pa) x10.
— :Ftber --- :Interfaoe
--— :Matrix

Figure 35. The zoom of the independent mode failure enve10pes

x107 T300l934
8- .
6~ [ q
——————————— .-;,‘-_.'.".": :.T..’_T_".:.:.._-_._.__\
4r l/ -\. -
.’ is
75 -’ '\
Q; 2” .1" '\ "
.’ \
N l-’ .\
< O- I' .' _,
2 .\ _/"
\ Ll
c_D_2- \ I-’ ..
(D \ ‘l'
\ .1
0“ .’
.4 - -\ -------------------------- I.’ -
-6 - I .
-aL .
-2 -1.5 -1 -O.5 O 0.5 1 1.5 2
SIGMA1 (Pa) X10“
-— :Fibel’ "" :Interfaoe
- - — :Matrix

105

 

 

 

 

 

 

 

 

Figure 36. The independent failure envelopes (Tresca criterion)

106
are shown in Figures 37 and 38 by adopting maximum principal stresses failure criterion
for matrix, and it is also confirmed that the matrix dominates the transverse strength of the
composite.

Making use of the uniaxial strengths of each constituent in terms of lamina load, and
fitting the failure envelope of each constituent to a polynomial type failure criterion, the
strength parameters are obtained for the independent mode failure criteria as functions of
macro loads. The obtained suength parameters of each constituent for a plane stress state
are listed in Table 5. But the strength parameters for matrix can be changed by the choice
of failure criterion for the matrix. If the strength parameters are substituted into Eqs.
(4.3.2), (4.3.4) and (4.3.6), the failure envelope of each constituent are obtained as shown
in Figures 39 and 40. In the figures, as expected, the fiber dominates the longitudinal
strength of lamina, and interface and/or matrix dominates the transverse strength. The pre-
diction in tension and tension space has good agreement with the Tsai - Wu failure enve-

lope based on experimental results.

The independent mode failure criterion based on “in-situ ” constituent strengths

Thus far, the AS4I3501-6 and T300l934 composite systems were simulated to assess
theory, but the prediction of transverse strengths did not have good agreement with the
experimental results. This may be due to the reasons discussed in Section 5.1. Thus, to
make the failure envelopes more realistic, the in-situ matrix strength can be used instead
of the bulk strength. Table 6 shows the comparison between the in-situ strength and the
bulk strength of 3501-6 and 934 resins. In Figure 41, the failure envelope based on in-situ
strength of matrix are shown and compared with Tsai-Wu failure enve10pe using experi-

mental results. In the tension - tension region, the current failure envelope based on in-situ

 

107

 

 

 

 

 

 

 

1.5 r j 1 t I l r
A 0.5- - 3.
(U .
e. l
N ...
< O- -=======-_====-1=.- :============== -l
2
(D
(”—05% —— fiber -
—-- :lnterface
------ :matrix
-1- -+
_1.5 r r r r r r r
-4 -3 -2 -1 O 1 2 3 4
SIGMA1 (Pa) mm

Figure 37. The independent failure envelopes in stress space (AS4I3501-6)

SIGMA2 (Pa)

108

 

 

 

 

 

 

3 l I I l I
2- -
1- .
L-.-..-. =. -: .-_.-..- ...-. =.1. .-.-_.-. :. : .: .—_—_". :. : ’— ."..T_-. :. 2 ZT‘T‘ “““““
O'- "-\'-t
i
l
i
-1" g.
!
.............. - — —.— —.-.—.—.- -.—.—.-.--—.—.—.—.—o-o—o—-n'
-2» _

-3 . . . . l

- -2 -1 O 1 2
SIGMA1 (Pa) “0.

— :Frber --- :Interface

Figure 38. The zoom of the independent mode failure envelopes

109

TABLE 5. The strength parameters for independent failure criterion

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AS4/350l-6
F: 0.586385108445310 0.215978453212309
pg 1 0.623829307604318 0332273910265318
I“; 0.230417004521308 0.529573725026310
ng 0.248058914102315 0.199772075123519
qu 0.242261961673318 0.818557039821319
F26 0.910411205632315 0.568826873132323
Fin 0.201801474273310 0.0000000000005400
FY; 0.512993473221321 0.865685248847321
F2" 0.104603918702307 0.835030761877308
F; 0.201807946708315 0.845712457806316
Flt; 0.225731392658318 0.623613958904519
I72; 0.604565155100315 0.258876386798316
Fin 0.000000000me 0.210148107289309
[fl 0.318131154099318 0.493464634521519
F: 0.00000000000E+00 0.935305955771308
F’znz 0.353761427543315 0.977490324149316
Finz 0.360073205736317 0.455226648330318
’7'; 0.9104079092589355 0.227530749253360

 

 

 

 

 

 

 

 

110
no“ T300l934
1 I U ' i U ,W
,”' ‘\
,” \
,, : f
0.5- ,’ - ‘j-
/
I
/
OI- I, I
I
A I’
E; c.
—0.5- ,r - i"
N ,’
< I’
2 ' x’
(D -1- \\ ”” I
(.5 _____
-‘LS- II
-2. u
_2.5 l l j l l l l
—6 -4 -2 0 2 4 6 8 10
SIGMAt (Pa) “010
— :F‘rber --- :Interface
_ . ._ :Matrix ----- :Tsai-Wu failure envelope

Figure 39. The curve fitting of independent mode failure envelopes
. (T300l934)

111

 

x 100 T300l934
2 . : fiber '
- - - — : interface
1'5 ' ------- : matrix I
1 .- .......... : Tsai-Wu (experiment) .

)

.0
tn

SIGMA2 (Pa
.L . .5
01 d 0| 0

I
N
I

 

 

..
.0
. .U
a. o.
'''''
'''''''''

 

 

 

in

Figure 40. The zoom of curve fitting of independent mode
failure envelopes (T300l934)

O
SIGMA1 (Pa)

‘0)

x10

112
strength is very close to the Tsai - Wu failure envelope. In Figure 42, the independent
mode failure envelopes are shown. In the tension - tension region, the matrix fails at a

lower transverse stress as the longitudinal load increases.

TABLE 6. The comparison between bulk- and in-situ strength

 

     
 

Tension (MPa) Compression (MPa)

 

Bulk strength In-situ strength Bulk strength In-situ strength

 

 

     

 

 

Verification of the independent mode failure theory and in-situ strengths

In order to examine the proposed micromechanical approach for the prediction of the
strength of composites under combined loading, the unidirectional lamina (AS/3501)
under off—axis loading as shown in Figure 43 was simulated. The unavailable properties
were determined by backing out “in-situ” properties. As shown in Figure 44, the predicted
strengths are in good agreement with the experimental results in both tension and com-
pression loading cases. The composite strength was predicted as follows. First, the failure
envelopes and unidirectional strengths were predicted by the micromechanical theory, and
the interaction parameters were obtained by fitting to the Tsai-Wu type polynomial failure.
Then using the tensor transformation law, the strength parameters were transformed into
the off-axis coordinate. The independent mode failure curves were obtained by applying
Eqs. 4.3.7 and 4.3.8 to fiber and Eq. (5.3.2) to matrix. As the loading angle varies, it is
shown that the failure mode changes from the fiber to the matrix. The used material prop-

erties are listed in TABLE 1.

 

113

 

 

x10“ T300/934
2 I f I I I 1 I I I
1.5 - —- : maximum principal stress r
._ - - - - :Tsai-Wu based on experiment
1 - ‘
0.5 ' r , .— ‘‘‘‘‘ m -\ I
g 0 p I , ’ - ’ ’ ..\ I
N ./".
< .415 :- /°’ I u
2 .I. I
l I
(2 -1 " I I." '
CD I .I.
-1.5 " l I I —J 'I
\ I .’
.\. ’ a " '
-2 - "~ .................. 4
_2.5 I It
.3 J j l l l I J l A
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
SIGMA1 (Pa) X109

 

 

 

 

 

Figure 41. The failure enve10pe based on in-situ strength

 

SIGMA2 (Pa)

1
P
or

1.5

.0
or

O

I
d

-1.5

114

T300l934

 

 

 

 

 

 

x10
' -— :flber
- - - :matrix
I ‘ 5
I
l
I
' I
l
I
_ I
I
I
. ------------------------------- 'l
-1.5 -1 -0.5 0 0.5 1 1.5
SIGMA1 (Pa)

Figure 42. The independent mode failure envelope

based on in-situ strength

 

 

////AL. I

ooooooooooooo

 

 

 

116

AS/ 3501

 

 

I I I I

— :eomposite

- - - :fiber

~--- : matrix

0 : experimental
+ c

 

 

4.0 5.0 6.0 7.0
A N G L E (DEG.)

 

Figure 44. The predicted strengths of AS/3501 under off-axis
loading on the basis of in-situ strength

80

 

 

117

TABLE 7. Material properties used in calculation

 

 

 

 

 

 

AS 3501
E1 (GPa) 213.000 3.45000
32 (GPa) 13.8000 3.45000
E3 (GPa) 13.8000 3.45000
nu23 0.2500 0.3500
nun 0.2000 0.3500
nun 0.2000 0.3500
623 (GPa)*** 5.5200 1.3000
GB (GPa) 13.800 1.3000
6,; (GPa) 13.800 1.3000
Tensile Strength (Pa)* 0.01017321'" 0.7009612E+08
Compression Strength (Pa)* 0.01017321‘” -0.2792998E+09

 

I ASI3501

XlT (MPa) 1447.000
x1c (MPa) 1447.000
X2T (MPa) , 51.7000
xzc (MPa) 206.000

 

 

 

*: The in-situ strengths, “z The maximum allowable strain

E
***Z The assumed property from 623 = W
23

AS, 3501: [31], ASI3501: [20]

CHAPTER VII
CONCLUSIONS

A model for predicting failure in fiber-reinforced composites has been developed and

assessed. The following conclusions have been reached:

0 On the basis of micromechanics elasticity solutions, the material properties of the

lamina can be obtained with good agreement to the experimental results.

0 The predicted unidirectional strengths of the lamina based on the micromechanical
failure theory are in good agreement with experimental results except the u'ansverse com-

pressive strength.

- To estimate the strength of a lamina for the purpose of designing composite systems,
the micromechanical failure theory needing only the fiber volume fraction and the mate-

rial properties in bulk state as input can be employed.

0 The interaction terms in the Tsai-Wu type failure theory can be determined by non-
linear regression and fitting the failure envelope from the current (or other) micromechan-

ics failure theory to a Tsai-Wu type failure criterion.

0 When the compressive strength of matrix is different from the tensile strength, the
maximum principal stress criterion for matrix gives the best prediction of transverse ten-

sile strength and compressive strength of composites.

118

119
° To reflect the difference in tensile and compressive strengths of constituent materials
in the micromechanical failure theory, it is better to adopt a failure criterion that can

accommodate such a difference in strength.

0 For a plane stress state, it was shown that the failure surface is closed and symmetric
about the plane where shear load is zero. The longitudinal strength was not affected by a

small shear load.

0 It is possible to cast the failure model in a simple mathematical form with separate
criteria for each mode of failure as a function of lamina load. The strength parameters can

be obtained by the micromechanical failure theory.

0 The independent mode failure criterion based on “in-situ” constituent strength tech-
nique shows excellent promise for inexpensive and efficient characterization of compos—

ites under multi-axial loading conditions.

0 In the current work, primary attention has been given to the case of plane stress load-
ing. However, the concepts used are sufficiently general to allow extension of the models

to fully three-dimensional failure analysis.

0 To predict the failure envelope more realistically, it is recommended that the thermal

residual stress be considered.

0 To verify and develop the suggested theory, more experimental results of multiaxial

strength are needed.

0 Additional mathematical forms of the independent-mode criteria need be considered.

120
0 Further assessment of the theory is needed. In particular, first-ply and progressive
failure analysis of laminates should be performed and the results compared with experi-

mental data.

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2. Ibid, pp. 183

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