Review: finite element analysis of stress transfer in short-fibre composite materials

Abstract

This paper addresses the application of finite element (FE) analysis for understanding the various processes that can occur and contribute to stress transfer between the matrix and fibres in short-fibre composite materials. These processes are elastic stress transfer, plastic stress transfer, matrix yielding (mode α), interfacial debonding (mode β), matrix cracking (mode γ), fibre pull-out and fibre fragmentation. In this paper, the discussion will cover the theory underlying each process, description of FE models, and analysis of stress transfer.

Full text

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Composites Science and Technology 64 (2004) 1091–1100
Review : finite element analysis of stress transfer in
short-fibre composite materials
K. L. Goh1, R. M. Aspden2, D. W. L. Hukins3
1 Biomaterials Division, Department of Optometry & Vision Sciences, Cardiff
University, Redwood Building, King Edward VII Ave, Cathays Park, Wales, CF10
3NB, UK.
2Department of Orthopaedic Surgery, University of Aberdeen, Foresterhill,
Aberdeen AB25 2ZD, UK.
3School of Engineering, Mechanical Engineering, University of Birmingham,
Edgbaston, Birmingham B15 2TT, UK.


Correspondence: Dr K. L. Goh
Telephone: 029 20870204
Fax: 029 20874859
E-mail:gohk@cf.ac.uk

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Abstract:
This paper addresses the application of finite element (FE) analysis for understanding the
various processes that can occur and contribute to stress transfer between the matrix and
fibres in short-fibre composite materials. These processes are elastic stress transfer,
plastic stress transfer, matrix yielding (mode α), interfacial debonding (mode β), matrix
cracking (mode γ), fibre pull-out and fibre fragmentation. In this paper, the discussion
will cover the theory underlying each process, description of FE models, and analysis of
stress transfer.
(79 words)

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Key words: (A) short fibre composites, (B) fibre/matrix bond, (C) elastic properties, (C)
stress transfer, (C) finite element analysis

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1
Introduction
In general, the stress-strain curve characterizing a short-fibre composite material, when
strained in tension parallel to a fibre direction, consists of four stages [1,2]. These stages
are (1) elastic deformation of fibres and the matrix; (2) plastic deformation of the matrix
whilst the fibres deform elastically; (3) plastic deformation of fibres and the matrix and
(4) fracture of fibres followed by fracture of the composite material.
This paper addresses the application of finite element (FE) analysis for understanding the
various processes that can occur during the four stages and contribute to stress transfer
between the matrix and fibres in short-fibre composite materials. To date, the processes
that have been studied by FE analysis are elastic stress transfer, plastic stress transfer,
matrix yielding (mode α), interfacial debonding (mode β), matrix cracking (mode γ),
fibre pull-out and fibre fragmentation. These processes are involved in the four stages as
follows. In stage 1, elastic stress transfer occurs from an elastic matrix to an elastic fibre
between a bonded interface during initial loading. During stage 2, the deforming matrix
adjacent to the bonded interface may yield and become plastic (mode α). When
debonding occurs at the fibre-matrix interface (mode β), sliding of the matrix over the
fibre results in frictional stress transfer. If frictional stress transfer occurs at the interface
adjacent to the deforming plastic matrix, then the combination of mode α and β lead to
plastic stress transfer. Cracks may also occur in the matrix from the debonded fibre ends
(mode γ). It follows that the combination of mode β and γ may lead to fibre pull-out; in
the absence of mode β, if the cracks propagate to neighbouring intact fibres, then these
neighbouring fibres become responsible for bridging the matrix cracks and hence fibre

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pull-out still occurs. During stage 4, fibre fragmentation may occur in fibres that are
bridging cracks in the matrix. Subsequently, the entire composite material fractures.
Many fibre composite materials consist of brittle fibres in a ductile matrix. Since
composites with brittle fibres fracture at the fracture strain of the fibre, it follows that
stage 3 is not then observed [1,2].
The remainder of the paper consists of four main sections. Section 2 presents an overview
of the models for studying the processes of stress transfer. Section 3 focuses on the
contributions of the various stress transfer processes to a deforming fibre composite
material. Section 4 presents the conclusions of this paper. Throughout this paper, we shall
also highlight insights gained from understanding the response of stresses in the model to
variation in material and geometric properties which would not be gained from an
analysis that relies on specific values [3,4,5,6].
The following symbols used repeatedly throughout this paper are now explained. They
refer to a model in which the composite consists of a single fibre surrounded by a coaxial
cylinder of matrix. For fibres, we have the fibre axial ratio, q (= L/ro ); here L and ro are
the half-length of the fibre and the radius at the center of the fibre, respectively. Lm and rm
denote the half-length and radius of the matrix, respectively. For material properties, we
have Ef, Em, νf and νm. Here, E and ν denote Young’s modulus and Poisson’s ratio,
respectively; subscripts f and m in these symbols refer to the fibre and matrix,
respectively. We also have σu, σy and τy; these refer to the fibre fracture stress, matrix
tensile yield stress and interfacial yield stress in shear, respectively; a Coulomb frictional
model enables τy to be related to the coefficient of static friction, µ, and to the normal

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(perpendicular) pressure, N [7]. Stress components, σz and τ, denote the fibre axial stress
and interfacial shear stress; σc denotes the applied tensile stress in the direction of the
fibre axis. Stress distributions are described with respect to the distance along the fibre
axis, Z (= z/L ); here, z is the axial coordinate of the cylindrical polar coordinate system.
2
Description of models
2.1 Overview
Models for studying stress transfer in fibre composite materials can be grouped into two
basic designs. The first is illustrated in Figure 1 which shows an axisymmetric FE model
of a cylindrical fibre embedded in a cylindrical matrix [8,9,10,11,12,13]. The second
design is similar to the first except that DEF’C (Figure 1) has been excluded
[14,15,16,17,18,19]. In both designs, the axis of symmetry defines the z-axis of the
cylindrical polar coordinate system. In many studies, the center of the model defines the
origin of the coordinate system; for exceptions see Rauchs et al. [17] (section 2.6) and
fibre pull-out models (section 2.7). In some studies, an annulus and an outer layer of
material were introduced to describe fibre coating [16] and average composite properties
[19], respectively; for the mathematical theory leading to these models, see Rosen [20].
It then follows that models for studying stress transfer processes are derived from one of
these two basic designs and are distinguished by having different boundary conditions.
Axisymmetry is implemented by constraining OD (Figure 1). In many models mirror-
symmetry is implemented by constraining OB (Figure 1). However, mirror symmetry
along OB cannot be invoked for fibre pull-out models (section 2.7) or when the origin of

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the coordinate system is not at the centre of the fibre [17] (see also section 2.6). It
followed that models which featured mirror and axisymmetry are loaded in tension by
prescribing an applied tensile force acting at CD in the direction of the fibre (Figure 1) or,
in the case of the second basic design, at FF’; alternatively, these may be replaced by
prescribing displacement. For a discussion of loading approaches for models in which
mirror symmetry cannot be invoked, e.g. fibre pull-out models, or when the origin of the
coordinate system is not at the center of the fibre see section 2.7 and 2.6, respectively.
The model shown in Figure 1 can also be modified to investigate stress transfer to fibres
which are non-cylindrical in shape [10,21]. Revolution of the model about OD produces a
half fibre which possess a conical, paraboloidal or ellipsoidal shape. These models were
prompted by observations of taper at the ends of collagen fibrils in connective tissues
[22,23] but the results are also applicable to synthetic fibre composite materials. In this
paper, all results are for the cylindrical fibre study unless otherwise stated.
In the model shown in Figure 1, values assigned to rm and Lm are determined by a ‘far-
field’ analysis to ensure that any edge effects, i.e. arising from BC and CD, do not perturb
the stresses in the fibre [8,10,11,15,24]. Note that the term ‘far-field’ has also been used
for analytical studies of stress in fibre composite materials [25,26]. For the model
described in Figure 1, it follows that by increasing rm/ro, stress distributions converge
(Figure 2) and eventually one arrives at a sufficiently large matrix which is equivalent to
an infinite matrix. Note that varying Lm/L has little effect on stresses along the fibre and
across CD. Thus, the value of rm/ro at convergence corresponds to a model which is then
useful for FE analysis.

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2.2 Elastic stress transfer
The elastic stress transfer process involves the transfer of stress from an elastic matrix to
an elastic fibre via a bonded interface. This predominates during initial loading (stage 1).
Bonding between the fibre and matrix, which is an important factor influencing the
overall material properties of a fibre composite material [27], is imposed by node-sharing
along the interface, AF. The assumption of an unbonded fibre end has been adopted as in
Cox’s theory [28]; physically, this has little effect on the overall stress distribution
because of the small area of fibre which could adhere to the matrix [8].
In a typical FE analysis [8,10,11], the fibre and matrix materials are assumed to be
isotropic. For an elastic material that is isotropic, it follows that out of a total of twenty-
one material constants, only two are independent, i.e., E and ν [29]. Hence, the material
constants to be considered in the FE analysis are the Young’s moduli and Poisson’s ratios
of the fibre and matrix.
It is convenient to compare the effects of Ef and Em on stresses together as a ratio of
moduli, Ef/Em; Ef/Em quantifies the elastic mismatch between a fibre and a matrix. A
sensitivity analysis on varying Ef/Em showed that increasing Ef/Em (from 50 to 10000) for
a fixed value of q (200 and 1000) had little effect on the shape of the stress distribution
along the fibre axis [9]. Stress transfer between the matrix and fibre was more effective
for larger values of Ef/Em as this resulted in greater values for σz/σc, the stress produced
in the fibre by a given applied stress. However, the presence of high stresses may lead to
an increased risk of fibre fracture, i.e. when the maximum value of σz reaches the fibre
fracture stress (section 2.8, 3.1, 3.6). The magnitude of the axial stress in the fibre is

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much less sensitive to changes in q, and, as before, the stress distributions are all very
similar [10]. Poisson’s ratios, νf and νm, are often specified as single values [8,15]; they
have negligible effect on the stresses in the model [10]. Although the effects of Ef/Em, νf
and νm were demonstrated for a cylindrical fibre [10], similar trends were also observed
with tapered fibres [30].
2.3 Matrix yielding (mode α)
Mode α is said to occur when the deforming matrix yields and turns plastic adjacent to
the bonded interface. In the case of uniform cylindrical fibres, stress concentrations in the
matrix around the end, EF (Figure 1) [11] may lead to matrix yielding; for tapered fibres
the lower stress concentrations in the matrix around the ends [31] may make them less
susceptible to mode α. Results from an experiment suggested that stress distributions at
the ends of a carbon fibre embedded in an epoxy matrix can be attributed to mode α [12].
In a typical FE model described by Figure 1, the matrix is modeled as an elastic-plastic
material [11,12,16], i.e. the stress-strain curve is characterized by an initial (elastic)
region described by a rapidly increasing stress with increasing strain and followed by a
plastic region, described by little or no increase in stress with increasing strain [32]. The
change from elastic to plastic properties at any point in the deforming matrix may be
determined by comparing the stress state with σy [16]; by applying incremental
displacements to the model, the change takes effect if the stress state exceeds σy.

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2.4 Matrix cracking (mode γ)
Mode γ is said to occur when a crack at the debonded fibre end, EF (Figure 1), propagates
into the matrix but not along the fibre-matrix interface [13]. A mode I crack, i.e. parting
of two surfaces, implies that stress transfer will not occur across the crack planes and
subsequently reduces the effectiveness of stress transfer between the matrix and fibre; a
mode II crack, i.e. shear failure, enables stress transfer via friction at the crack surfaces
[12]. It has been suggested [13] that matrix cracks described in experiments on carbon
fibres embedded in epoxy matrix materials [33,34] are attributed to mode γ.
A modified Rice and Tracey micro-void nucleation, growth and coalescence model was
used to predict the crack propagation in an FE analysis [13]. The model accounts for how
a matrix crack is formed and how the crack propagates owing to voids present in the
region ahead of the crack which nucleate, grow and coalesce due to the presence of high
stress at the fibre corner. The modeling process was implemented by considering the
matrix as an elastic-plastic material (section 2.3) and, under incremental displacement
applied to the model at CD (Figure 1), the deforming matrix changes from elastic to
plastic. A critical fracture strain was defined which depends on the void size and the
stresses around the region. If the strain state at a point in the matrix was greater than this
critical fracture strain then a crack was considered to have formed.
2.5 Interfacial debonding (mode β)
Mode β is said to occur when a crack initiates in the interface at the debonded fibre end,
EF (Figure 1), and propagates along the interface, AF; frictional stress transfer occurs as

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the deforming matrix slides over the fibre surface [12]. It has been pointed out that there
is a relationship between the rate of debonding and Ef/Em: the greater the Ef/Em, the higher
the rate of failure [9]. This implies that the greater the elastic mismatch between the fibre
and matrix the higher the rate of interfacial failure. Interfacial debonding observed in
experiments on carbon fibre embedded in epoxy matrix material, measured using laser
Raman spectroscopy [33], have been attributed to mode β [26].
FE modeling on debonding and sliding at the fibre-matrix interface has been well
documented [9,12,17]. Typically this has been implemented using a Coulomb friction
model [12]. A model of this kind describes the increase in τ (to the value τy), as a result
of an incremental load, when there is bonding at the fibre-matrix interface. Here τy
depends on the frictional stress, µN, and a cohesive sliding resistance. When τ exceeds τy
the two surfaces are deemed to have ‘debonded’ and will slide relative to one another
when deformed further; this is referred to as the ‘sliding’ state. During the sliding state,
the value of τ is constant throughout the interface; Rauchs et al. [17] have assigned τ = τy;
clearly τ during sliding would be related to kinetic frictional stress and would thus be
smaller than τy, which is related to static frictional stress.
2.6 Plastic stress transfer
Simultaneous occurrence of mode β and α at the interface, AF (Figure 1), and in the
adjoining matrix region, respectively, leads to plastic stress transfer. Plastic stress transfer
has been reported in mechanical test experiments combined with x-ray strain
measurements using synchrotron radiation [17] or Raman spectroscopy [12].

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Goh et al. [21] have considered an FE model consisting of an elastic fibre in tension
under the application of a constant shear stress, τ, on its surface. This study showed that τ
also acts as a scaling constant for σz; this is consistent with the constant shear model of
Kelly & Tyson [36].
In contrast, Rauchs et al. [17] considered an FE model consisting of an elastic-plastic
matrix in which an elastic fibre of the same length as the matrix was embedded. This
model corresponds to the second basic design described in section 2.1; in this case, the
load in the direction of the fibre was applied through a prescribed axial displacement at
EF’ while OB was constrained to ensure no axial displacement could take place along it.
This model represented conditions similar to an experiment that was conducted to study
fibre fragmentation. By making assumptions about the different steps through which the
modeling process should proceed (section 2.8), predictions from the model were
compared with results obtained from experiment to yield an understanding about the
stress transfer processes that took place. These steps progress from elastic stress transfer,
mode α, β and finally to plastic stress transfer. Thereafter, the implementation of fibre
fragments made it possible to analyse the stress distributions in short fibres as a
consequence of plastic stress transfer.
It is useful to have an estimate of τ which may provide some information about the
interfacial strength [35]. In the study carried out by Rauchs et al. [17], when mode β
occurred, the value of τ was assigned equal to τy (section 2.5). It follows that the value of
τ could be derived from τy = [Efro/2]dεz/dz, adapted from the constant shear model of

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Kelly & Tyson [36]; here, the fibre axial strain gradient, dεz/dz, is derived from the strain
distribution curve obtained experimentally [17].
2.7 Fibre pull-out
Fibre pull-out may occur when fibres are drawn out from the faces of a matrix crack. In
section 1, it was pointed that this may arise from the combination of modes β and γ; in
the absence of mode β, the matrix crack arising from mode γ may propagate to
neighbouring fibres so that these fibres are responsible for bridging cracks in the matrix
and hence fibre pull-out may also occur. The ability of a fibre to bridge a matrix crack
depends on the interface (section 2.2), the fibre strength and fibre modulus [22]. If
bonding is present at the interface, then the ability of a fibre to bridge the crack depends
on τy; if debonding has occurred, then this depends on friction as a result of sliding
[27,37]. There are two possible outcomes of fibre pull-out arising from fibres bridging a
matrix crack and they may also occur in combination [27]. The first is that matrix crack
propagation is ‘deflected’ by the fibre and is led to propagate along the interface (mode
β); the second is that the fibre fractures when attempts to bridge the crack causes it to be
stressed beyond its fracture stress (section 2.8). Fibre pull-out has been observed in
mechanical tests on synthetic materials [27,38] and in biological tissues [39].
The bridging of fibres across a crack to enable stress transfer can be investigated by fibre
pull-out tests [37]; these tests are intended also to study the strength of the interface as a
result of bonding and to investigate friction arising from sites where debonding has
occurred [27]. A typical FE analysis [18] on fibre pull-out can be implemented by using
the second basic design described in section 2.1 by loading the fibre at OA (Figure 1). In

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order to model a matrix crack plane, AB is not constrained. In addition, FF’ is then
constrained to ensure no axial displacement can take place along it.
2.8 Fibre fragmentation
During the process of deformation, stress transfer from the matrix to a fibre may cause
the fibre to break at the site where the fracture stress is attained. Fragments are generated
by this process. It eventually stops when the fragments are so short (less than the critical
length) that the mid-lengths of the fibre fragments fail to reach σu, i.e. not enough stress
can be transferred to the fibre fragment to cause further fragmentation [17,40]. Fibre
fragmentation has been observed in several experiments [33,41,42].
Rauchs et al. [17] performed an FE analysis to investigate the stresses in fibre fragments
using the model described in section 2.6. Cracks were introduced at locations along the
fibre by an element removal technique; the locations of these cracks were based on
experimental results. The modeling process was divided into four steps. The first step
corresponded to elastic fibre deformation; here the matrix changes from elastic to plastic
during deformation, i.e. initially mode α, and subsequently, when mode β occurs, plastic
stress transfer takes effect. Thereafter, the other steps involve mainly increasing the
number of fragments in accordance with the sequence observed experimentally.

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3
Stress transfer analysis
3.1 Elastic stress transfer
Figure 3 shows how the axial tensile stress in a cylindrical fibre, σz/σc, depends on Ef/Em
and q; this result is taken from our previous work [10] but similar results, at restricted
values for Ef/Em and q, had been published previously [8,11]. In all cases, σz/σc is greatest
at the centre of the fibre and decreases non-linearly to zero at its end. A similar
dependence is predicted by shear lag models [11,28] and it has been shown that the
predictions of FE and shear lag models are closely similar [10,11]. Similar results are also
obtained experimentally [44]. Note that changing q has little effect on σz/σc.
A similar FE analysis has also been applied to non-cylindrical fibres [10]. In complete
contrast to the cylindrical case, axial stress in a fibre with a conical taper is a minimum at
the fibre centre; the stress rises and peaks near the fibre end before dropping to zero at the
end. Stress distributions in ellipsoidal and parabolloidal fibres lie between the extremes
of the cylinder and cone, leading to more uniform stress distributions. For low values of
Ef/Em (around 50), the stress distributions in conical, ellipsoidal and parabolloidal fibres
are very similar and there is little to distinguish them. The predictions for an ellipsoidal
fibre are consistent with the theoretical results of Eshelby [43]. However, cylindrical
fibres have a substantial region near the end which is not being maximally stressed and
which, therefore, contributes little to reinforcing the composite. Once again, changing q
has little effect on the stress distributions in the cylindrical, the paraboloidal and
ellipsoidal fibres. However, when Ef/Em is large, reducing q has the effect of