Materials Science on CDROM User Guide
Mechanics of Composite Materials
Version 2.1
Bill Clyne, University
of Cambridge
Boban Tanovic, MATTER
It is assumed that the student is familiar with simple concepts of mechanical
behaviour, such as the broad meanings of stress and strain. It would be an advantage for
the student to understand that these are really tensor quantities, although this is by no
means essential. All of the terms associated with the assumed preknowledge are defined in
the glossary, which can be consulted by the student at any time.
Most of the material in this package is based on a recently published book. This is:
"An Introduction to Composite Materials", D.Hull and T.W.Clyne, Cambridge
University Press (1996) Order!
This source should be consulted for background to the treatments in this module,
particularly mathematical details.
Most composites have strong, stiff fibres in a matrix which is weaker and
less stiff. The objective is usually to make a component which is strong and stiff, often
with a low density. Commercial material commonly has glass or carbon fibres
in matrices based on thermosetting polymers, such as epoxy or polyester
resins. Sometimes, thermoplastic polymers may be preferred, since they are
mouldable after initial production. There are further classes of composite in which the
matrix is a metal or a ceramic. For the most part, these are still in a
developmental stage, with problems of high manufacturing costs yet to be overcome.
Furthermore, in these composites the reasons for adding the fibres (or, in some cases,
particles) are often rather complex; for example, improvements may be sought in creep,
wear, fracture toughness, thermal stability, etc. This software package covers simple
mechanics concepts of stiffness and strength, which, while applicable to all composites,
are often more relevant to fibrereinforced polymers.
The module comprises three sections:
Brief descriptions are given below of the contents of these sections, covering both the
main concepts involved and the structure of the software.
Summary
This section covers basic ideas concerning the manner in an applied mechanical load is
shared between the matrix and the fibres. The treatment starts with the simple case of a
composite containing aligned, continuous fibres. This can be represented by the slab
model. For loading parallel to the fibre axis, the equal strain condition is imposed,
leading to the Rule of Mixtures expression for the Young's modulus. This is followed b y
the cases of transverse loading of a continuous fibre composite and axial loading with
discontinuous fibres.
What is meant by Load Transfer?
The concept of load sharing between the matrix and the reinforcing constituent (fibre)
is central to an understanding of the mechanical behaviour of a composite. An external
load (force) applied to a composite is partly borne by the matrix and partly by the
reinforcement. The load carried by the matrix across a section of the composite is given
by the product of the average stress in the matrix and its sectional area. The load
carried by the reinforcement is determined similarly. Equating the externally imposed load
to the sum of these two contributions, and dividing through by the total sectional area,
gives a basic and important equation of composite theory, sometimes termed the "Rule
of Averages".

(1) 
which relates the volumeaveraged matrix and fibre stresses (), in a composite containing a volume (or sectional area)
fraction f of reinforcement, to the applied stress s_{A}.
Thus, a certain proportion of an imposed load will be carried by the fibre and the
remainder by the matrix. Provided the response of the composite remains elastic, this
proportion will be independent of the applied load and it represents an important
characteristic of the material. It depends on the volume fraction, shape and orientation
of the reinforcement and on the elastic properties of both constituents. The reinforcement
may be regarded as acting efficiently if it carries a relatively high proportion of the
externally applied load. This can result in higher strength, as well as greater stiffness,
because the reinforcement is usually stronger, as well as stiffer, than the matrix.
What happens when a Composite is Stressed?
Figure 1
Consider loading a composite parallel to the fibres. Since they are bonded together,
both fibre and matrix will stretch by the same amount in this direction, i.e. they will
have equal strains, e (Fig. 1). This means that, since the fibres are stiffer (have
a higher Young modulus, E), they will be carrying a larger stress. This
illustrates the concept of load transfer, or load partitioning between
matrix and fibre, which is desirable since the fibres are better suited to bear high
stresses. By putting the sum of the contributions from each phase equal to the overall
load, the Young modulus of the composite is found (diagram). It can be seen that a "Rule
of Mixtures" applies. This is sometimes termed the "equal strain"
or "Voigt" case. Page 2 in the section covers derivation of the equation
for the axial stiffness of a composite and page 3 allows the effects on composite
stiffness of the fibre/matrix stiffness ratio and the fibre volume fraction to be explored
by inputting selected values.
What about the Transverse Stiffness?
Also of importance is the response of the composite to a load applied transverse to the
fibre direction. The stiffness and strength of the composite are expected to be much lower
in this case, since the (weak) matrix is not shielded from carrying stress to the same
degree as for axial loading. Prediction of the transverse stiffness of a composite from
the elastic properties of the constituents is far more difficult than the axial value. The
conventional approach is to assume that the system can again be represented by the
"slab model". A lower bound on the stiffness is obtained from the "equal
stress" (or "Reuss") assumption shown in Fig. 2. The value is an
underestimate, since in practice there are parts of the matrix effectively "in
parallel" with the fibres (as in the equal strain model), rather than "in
series" as is assumed. Empirical expressions are available which give much better
approximations, such as that of HalpinTsai. There are again two pages in the
section covering this topic, the first (page 4) outlining derivation of the equal stress
equation for stiffness and the second (page 5) allowing this to be evaluated for different
cases. For purposes of comparison, a graph is plotted of equal strain, equal stress and
HalpinTsai predictions. The HalpinTsai expression for transverse stiffness (which is not
given in the module, although it is available in the glossary) is:

(2) 
in which
Figure 2
The value of x may be taken as an adjustable parameter, but its magnitude is generally
of the order of unity. The expression gives the correct values in the limits of f=0
and f=1 and in general gives good agreement with experiment over the complete range
of fibre content. A general conclusion is that the transverse stiffness (and strength) of
an aligned composite are poor; this problem is usually countered by making a laminate
(see section on "composite laminates").
How is Strength Determined?
There are several possible approaches to prediction of the strength of a composite. If
the stresses in the two constituents are known, as for the long fibre case under axial
loading, then these values can be compared with the corresponding strengths to determine
whether either will fail. Page 6 in the section briefly covers this concept. (More details
about strength are given in the section on "Fracture Behaviour".) The treatment
is a logical development from the analysis of axial stiffness, with the additional input
variable of the ratio between the strengths of fibre and matrix.
Such predictions are in practice complicated by uncertainties about in situ
strengths, interfacial properties, residual stresses etc. Instead of
relying on predictions such as those outlined above, it is often necessary to measure the
strength of the composite, usually by loading parallel, transverse and in shear with
respect to the fibres. This provides a basis for prediction of whether a component will
fail when a given set of stresses is generated (see section on "Fracture
Behaviour"), although in reality other factors such as environmental degradation
or the effect of failure mode on toughness, may require attention.
What happens with Short Fibres?
Short fibres can offer advantages of economy and ease of processing. When the fibres
are not long, the equal strain condition no longer holds under axial loading, since the
stress in the fibres tends to fall off towards their ends (see Fig. 3). This means that
the average stress in the matrix must be higher than for the long fibre case. The effect
is illustrated pictorially in pages 7 and 8 of the section.
Figure 3
This lower stress in the fibre, and correspondingly higher average stress in the matrix
(compared with the long fibre case) will depress both the stiffness and strength of the
composite, since the matrix is both weaker and less stiff than the fibres. There is
therefore interest in quantifying the change in stress distribution as the fibres are
shortened. Several models are in common use, ranging from fairly simple analytical methods
to complex numerical packages. The simplest is the socalled "shear lag"
model. This is based on the assumption that all of the load transfer from matrix to fibre
occurs via shear stresses acting on the cylindrical interface between the two
constituents. The buildup of tensile stress in the fibre is related to these shear
stresses by applying a force balance to an incremental section of the fibre. This is
depicted in page 9 of the section. It leads to an expression relating the rate of change
of the stress in the fibre to the interfacial shear stress at that point and the fibre
radius, r.

(3) 
which may be regarded as the basic shear lag relationship. The stress distribution in
the fibre is determined by relating shear strains in the matrix around the fibre to the
macroscopic strain of the composite. Some mathematical manipulation leads to a solution
for the distribution of stress at a distance x from the midpoint of the fibre
which involves hyperbolic trig functions:

(4) 
where e_{1} is the composite strain, s is the fibre aspect ratio
(length/diameter) and n is a dimensionless constant given by:

(5) 
in which n_{m} is the Poisson ratio of the matrix. The variation of interfacial
shear stress along the fibre length is derived, according to Eq.(3), by differentiating
this equation, to give:

(6) 
The equation for the stress in the fibre, together with the assumption of a average
tensile strain in the matrix equal to that imposed on the composite, can be used to
evaluate the composite stiffness. This leads to:

(7) 
The expression in square brackets is the composite stiffness. In page 10 of the
section, there is an opportunity to examine the predicted stiffness as a function of fibre
aspect ratio, fibre/matrix stiffness ratio and fibre volume fraction. The other point to
note about the shear lag model is that it can be used to examine inelastic behaviour. For
example, interfacial sliding (when the interfacial shear stress reaches a critical value)
or fibre fracture (when the tensile stress in the fibre becomes high enough) can be
predicted. As the strain imposed on the composite is increased, sliding spreads along the
length of the fibre, with the interfacial shear stress unable to rise above some critical
value, t_{i*}. If the interfacial shear stress becomes uniform at t_{i*}
along the length of the fibre, then a critical aspect ratio, s_{*},
can be identified, below which the fibre cannot undergo fracture. This corresponds to the
peak (central) fibre stress just attaining its ultimate strength sf*, so that, by
integrating Eq.(3) along the fibre halflength:

(8) 
It follows from this that a distribution of aspect ratios between s_{*}
and s_{*}/2 is expected, if the composite is subjected to a large strain.
The value of s_{*} ranges from over 100, for a polymer composite with poor
interfacial bonding, to about 23 for a strong metallic matrix. In page 10, the effects of
changing various parameters on the distributions of interfacial shear stress and fibre
tensile stress can be explored and predictions made about whether fibres of the specified
aspect ratio can be loaded up enough to cause them to fracture.
Conclusion
After completing this section, the student should:
 Appreciate that the key issue, controlling both stiffness and strength, is the way in
which an applied load is shared between fibres and matrix.
 Understand how the slab model is used to obtain axial and transverse stiffnesses for
long fibre composites.
 Realise why the slab model (equal stress) expression for transverse stiffness is an
underestimate and be able to obtain a more accurate estimate by using the HalpinTsai
equation.
 Understand broadly why the axial stiffness is lower when the fibres are discontinuous
and appreciate the general nature of the stress field under load in this case.
 Be able to use the shear lag model to predict axial stiffness and to establish whether
fibres of a given aspect ratio can be fractured by an applied load.
 Note that the treatments employed neglect thermal residual stresses, which can in
practice be significant in some cases.
Summary
This section covers the advantages of lamination, the factors affecting choice of
laminate structure and the approach to prediction of laminate properties. It is first
confirmed that, while unidirectional plies can have high axial stiffness and strength,
these properties are markedly anisotropic. With a laminate, there is scope for tailoring
the properties in different directions within a plane to the requirements of the
component. Both elastic and strength properties can be predicted once the stresses on the
individual plies have been established. This is done by first studying how the stiffness
of a ply depends on the angle between the loading direction and then imposing the
condition that all the individual plies in a laminate must exhibit the same strain. The
methodology for prediction of the properties of any laminate is thus outlined, although
most of the mathematical details are kept in the background.
What is a Laminate?
High stiffness and strength usually require a high proportion of fibres in the
composite. This is achieved by aligning a set of long fibres in a thin sheet (a lamina
or ply). However, such material is highly anisotropic, generally being weak
and compliant (having a low stiffness) in the transverse direction. Commonly, high
strength and stiffness are required in various directions within a plane. The solution is
to stack and weld together a number of sheets, each having the fibres oriented in
different directions. Such a stack is termed a laminate. An example is shown in the
diagram. The concept of a laminate, and a pictorial illustration of the way that the
stiffness becomes more isotropic as a single ply is made into a crossply laminate,
are presented in page 1 of this section.
What are the Stresses within a Crossply Laminate?
The stiffness of a single ply, in either axial or transverse directions, can easily be
calculated. (See the section on Load Transfer). From these values, the stresses in a
crossply laminate, when loaded parallel to the fibre direction in one of the plies, can
readily be calculated. For example, the slab model can be applied to the two plies in
exactly the same way as it was applied in the last section to fibres and matrix. This
allows the stiffness of the laminate to be calculated. This gives the strain (experienced
by both plies) in the loading direction, and hence the average stress in each ply, for a
given applied stress. The stresses in fibre and matrix within each ply can also be found
from these average stresses and a knowledge of how the load is shared. In page 2 of this
section, by inputting values for the fibre/matrix stiffness ratio and fibre content, the
stresses in both plies, and in their constituents, can be found. Note that, particularly
with high stiffness ratios, most of the applied load is borne by the fibres in the
"parallel" ply (the one with the fibre axis parallel to the loading axis).
What is the OffAxis Stiffness of a Ply?
For a general laminate, however, or a crossply loaded in some arbitrary direction, a
more systematic approach is needed in order to predict the stiffness and the stress
distribution. Firstly, it is necessary to establish the stiffness of a ply oriented so the
fibres lie at some arbitrary angle to the stress axis. Secondly, further calculation is
needed to find the stiffness of a given stack. Consider first a single ply. The stiffness
for any loading angle is evaluated as follows, considering only stresses in the plane of
the ply The applied stress is first transformed to give the components parallel and
perpendicular to the fibres. The strains generated in these directions can be calculated
from the (known) stiffness of the ply when referred to these axes. Finally, these strains
are transformed to values relative to the loading direction, giving the stiffness.
Figure 4
These three operations can be expressed mathematically in tensor equations.
Since we are only concerned with stresses and strains within the plane of the ply, only 3
of each (two normal and one shear) are involved. The first step of resolving the applied
stresses, s_{x}, s_{y} and t_{xy}, into components parallel and
normal to the fibre axis, s_{1}, s_{2} and t_{12} (see Fig. 4),
depends on the angle, f between the loading direction (x) and the fibre axis (1)

(9) 
where the transformation matrix is given by:

(10) 
in which c = cosf and s = sinf. For example, the value of s_{1} would be
obtained from:

(11) 
Now, the elastic response of the ply to stresses parallel and normal to the fibre axis
is easy to analyse. For example, the axial and transverse Young’s moduli (E_{1}
and E_{2}) could be obtained using the slab model or HalpinTsai
expressions (see Load Transfer section). Other elastic constants, such as the shear
modulus (G_{12}) and Poisson’s ratios, are readily calculated in a
similar way. The relationship between stresses and resultant strains dictated by these
elastic constants is neatly expressed by an equation involving the compliance tensor,
S, which for our composite ply, has the form:

(12) 
in which, by inspection of the individual equations, it can be seen
that
Application of Eq.(12), using the stresses established from Eq.(9), now allows the
strains to be established, relative to the 1 and 2 directions. There is a minor
complication in applying the final stage of converting these strains so that they refer to
the direction of loading (x and y axes). Because engineering and tensorial
shear strains are not quite the same, a slightly different transformation matrix is
applicable from that used for stresses

(13) 
in which,
and the inverse of this matrix is used for conversion in the reverse direction,

(14) 
in which,
The final expression relating applied stresses and resultant strains can therefore be
written,

(15) 
The elements of  , the transformed compliance tensor, are obtained by concatenation
(the equivalent of multiplication) of the matrices  T '^{1},
 S  and  T . The following expressions are obtained
Figure 5
The final result of this rather tedious derivation is therefore quite straightforward.
Eq.(16), together with the elastic constants of the composite when loaded parallel and
normal to the fibre axis, allows the elastic deformation of the ply to be predicted for
loading at any angle to the fibre axis. This is conveniently done using a simple computer
program. The results of such calculations can be explored using pages 4 and 5 in this
section. As an example, Fig. 5 shows the Young's modulus for the an polyester50% glass
fibre ply as the angle, f between fibre axis and loading direction rises from 0° to 90°.
A sharp fall is seen as f exceeds about 510°.
How is the Stiffness of a Laminate obtained?
Once the elastic response of a single ply loaded at an arbitrary angle has been
established, that of a stack bonded together (i.e. a laminate) is quite easy to predict.
For example, the Young's modulus in the loading direction is given by an applied normal
stress over the resultant normal strain in that direction. This same strain will be
experienced by all of the component plies of the laminate. Since every ply now has a known
Young's modulus in the loading direction (dependent on its fibre direction), the stress in
each one can be expressed in terms of this universal strain. Furthermore, the force
(stress times sectional area) represented by the applied stress can also be expressed as
the sum of the forces being carried by each ply. This allows the overall Young's modulus
of the laminate to be calculated. The results of such calculations, for any selected
stacking sequence, can be explored using pages 4 and 5.
Are Other Elastic Constants Important?
There are several points of interest about how a ply changes shape in response to an
applied load. For example, the lateral contraction (Poisson ratio, n) behaviour may
be important, since in a laminate such contraction may be resisted by other plies, setting
up stresses transverse to the applied load. Another point with fibre composites under
offaxis loading is that shear strains can arise from tensile stresses (and vice versa).
This corresponds to the elements of S which are zero in Eq.(12) becoming nonzero
for an arbitrary loading angle (Eq.(16)). These socalled "tensileshear
interactions" can be troublesome, since they can set up stresses between
individual plies and can cause the laminate to become distorted. The value of , for example, represents the ratio between g_{12}
and s_{1}. Its value can be obtained for any specified laminate by using page 6 of
this section. It will be seen that, depending on the stacking sequence, relatively high
distortions of this type can arise. On the other hand, a stacking sequence with a high
degree of rotational symmetry can show no tensileshear interactions. When the
tensileshear interaction terms contributed by the individual laminae all cancel each
other out in this way, the laminate is said to be "balanced". Simple
crossply and angleply laminates are not balanced for a general loading angle,
although both will be balanced when loaded at f=0° (i.e. parallel to one of the plies for
a crossply or equally inclined to the +q and q plies for the angleply case). If the
plies vary in thickness, or in the volume fractions or type of fibres they contain, then
even a laminate in which the stacking sequence does exhibit the necessary rotational
symmetry is prone to tensileshear distortions and computation is necessary to determine
the layup sequence required to construct a balanced laminate. The stacking order in which
the plies are assembled does not enter into these calculations.
Conclusion
After completing this section, the student should:
 Appreciate that, while individual plies are highly anisotropic, they can be assembled
into laminates having a selected set of inplane properties.
 Understand broadly how the elastic properties of a laminate, and the partitioning of an
applied load between the constituent plies, can be predicted.
 Be able to use the software package to predict the characteristics of specified laminate
structures.
 Understand the meaning of a "balanced" laminate.
Summary
This section covers simple approaches to prediction of the failure of composites from
properties of matrix and fibre and from interfacial characteristics. The axial strength of
a continuous fibre composite can be predicted from properties of fibre and matrix when
tested in isolation. Failures when loaded transversely or in shear relative to the fibre
direction, on the other hand, tends to be sensitive to the interfacial strength and must
therefore be measured experimentally. An outline is given of how these measured strengths
can be used to predict failure of various laminate structures made from the composite
concerned. Finally, a brief description is given of what is meant by the toughness
(fracture energy) of a material. In composites the most significant contribution to the
fracture energy usually comes from fibre pullout. A simple model is presented for
prediction of the fracture energy from fibre pullout, depending on fibre aspect ratio,
fibre radius and interfacial shear strength.
How do Composites Fracture?
Figure 6
Fracture of long fibre composites tends to occur either normal or parallel to the fibre
axis. This is illustrated on page 1 of this section  see Fig. 6. Large tensile stresses
parallel to the fibres, s_{1}, lead to fibre and matrix fracture, with the
fracture path normal to the fibre direction. The strength is much lower in the transverse
tension and shear modes and the composite fractures on surfaces parallel to the fibre
direction when appropriate s_{2} or t_{12} stresses are applied. In these
cases, fracture may occur entirely within the matrix, at the fibre/matrix interface or
primarily within the fibre. To predict the strength of a lamina or laminate, values of the
failure stresses s_{1*}, s_{2*} and t_{12*} have to be determined.
Can the Axial Strength be Predicted?
Understanding of failure under an applied tensile stress parallel to the fibres is
relatively simple, provided that both constituents behave elastically and fail in a
brittle manner. They then experience the same axial strain and hence sustain stresses in
the same ratio as their Young's moduli. Two cases can be identified, depending on whether
matrix or fibre has the lower strain to failure. These cases are treated in pages 2 and 3
respectively.
Figure 7
Consider first the situation when the matrix fails first (e_{m*}<e_{f*}).
For strains up to e_{m*}, the composite stress is given by the simple rule of
mixtures:

(17) 
Above this strain, however, the matrix starts to undergo microcracking and this
corresponds with the appearance of a "knee" in the stressstrain curve. The
composite subsequently extends with little further increase in the applied stress. As
matrix cracking continues, the load is transferred progressively to the fibres. If the
strain does not reach e_{f*} during this stage, further extension causes the
composite stress to rise and the load is now carried entirely by the fibres. Final
fracture occurs when the strain reaches e_{f*}, so that the composite failure
stress s_{1*} is given by f s_{f*}. A case like this is
illustrated in Fig. 7, which refers to steel rods in a concrete matrix.
[FB2, RHS, real system data, mild steel fibres, concrete matrix, fibre fraction 40%,
"strength v. fraction of fibres" clicked]
Figure 8
Alternatively, if the fibres break before matrix cracking has become sufficiently
extensive to transfer all the load to them, then the strength of the composite is given
by:

(18) 
where s_{fm*} is the fibre stress at the onset of matrix cracking (e_{1}=e_{m*}).
The composite failure stress depends therefore on the fibre volume fraction in the manner
shown in Fig. 8. The fibre volume fraction above which the fibres can sustain a fully
transferred load is obtained by setting the expression in Eq.(18) equal to f s_{f*},
leading to:

(19) 
If the fibres have the smaller failure strain (page 3), continued straining causes the fibres
to break up into progressively shorter lengths and the load to be transferred to the matrix.
This continues until all the fibres have aspect ratios below the critical value (see
Eq.(8)). It is often assumed in simple treatments that only the matrix is bearing any load
by the time that breakup of fibres is complete. Subsequent failure then occurs at an
applied stress of (1f) s_{m*}. If matrix fracture takes place while
the fibres are still bearing some load, then the composite failure stress is:

(20) 
where s_{mf} is the matrix stress at the onset of fibre cracking. In principle,
this implies that the presence of a small volume fraction of fibres reduces the composite
failure stress below that of the unreinforced matrix. This occurs up to a limiting value f '
given by setting the right hand side of Eq.(20) equal to (1f) s_{m*}.

(21) 
The values of these parameters can be explored for various systems using pages 2 and 3.
Prediction of the values of s_{2*} and t_{12*} from properties of the
fibre and matrix is virtually impossible, since they are so sensitive to the nature of the
fibrematrix interface. In practice, these strengths have to be measured directly on the
composite material concerned.
How do Plies Fail under Offaxis Loads?
Failure of plies subjected to arbitrary (inplane) stress states can be understood in
terms of the three failure mechanisms (with defined values of s_{1*}, s_{2*}
and t_{12*}) which were depicted on page 1. A number of failure criteria
have been proposed. The main issue is whether or not the critical stress to trigger one
mechanism is affected by the stresses tending to cause the others  i.e. whether there is
any interaction between the modes of failure. In the simple maximum stress
criterion, it is assumed that failure occurs when a stress parallel or normal to the
fibre axis reaches the appropriate critical value, that is when one of the following is
satisfied:
For any stress system (s_{x}, s_{y} and
t_{xy}) applied to the ply, evaluation of these stresses can be carried
out as described in the section on Composite Laminates (Eqs.(9) and (10)).
Figure 9
Monitoring of s_{1}, s_{2} and t_{12} as the applied stress is
increased allows the onset of failure to be identified as the point when one of the
inequalities in Eq.(22) is satisfied. Noting the form of  T  (Eq.(10)), and
considering applied uniaxial tension, the magnitude of s_{x} necessary to
cause failure can be plotted as a function of angle f between stress axis and fibre axis,
for each of the three failure modes.
The applied stress levels at which these conditions become satisfied can be explored
using page 5. As an example, the three curves corresponding to Eqs.(23)(25) are
plotted in Fig. 9, using typical values of s_{1*}, s_{2*} and t_{12*}.
Typically, axial failure is expected only for very small loading angles, but the predicted
transition from shear to transverse failure may occur anywhere between 20° and 50°,
depending on the exact values of t_{12*} and s_{2*}.
In practice, there is likely to be some interaction between the failure modes. For
example, shear failure is expected to occur more easily if, in addition to the shear
stress, there is also a normal tensile stress acting on the shear plane. The most commonly
used model taking account of this effect is the TsaiHill criterion. This can be
expressed mathematically as

(26) 
This defines an envelope in stress space: if the stress state (s_{1}, s_{2}
and t_{12}) lies outside of this envelope, i.e. if the sum of the terms on the
left hand side is equal to or greater than unity, then failure is predicted. The failure
mechanism is not specifically identified, although inspection of the relative magnitudes
of the terms in Eq.(26) gives an indication of the likely contribution of the three modes.
Under uniaxial loading, the TsaiHill criterion tends to give rather similar predictions
to the Maximum Stress criterion for the strength as a function of loading angle. The
predicted values tend to be somewhat lower with the TsaiHill criterion, particularly in
the mixed mode regimes where both normal and shear stresses are significant. This can be
explored on page 6.
What is the Failure Strength of a Laminate?
The strength of laminates can be predicted by an extension of the above treatment,
taking account of the stress distributions in laminates, which were covered in the
preceding section. Once these stresses are known (in terms of the applied load), an
appropriate failure criterion can be applied and the onset and nature of the failure
predicted.
Figure 10
However, failure of an individual ply within a laminate does not necessarily mean that
the component is no longer usable, as other plies may be capable of withstanding
considerably greater loads without catastrophic failure. Analysis of the behaviour beyond
the initial, fully elastic stage is complicated by uncertainties as to the degree to which
the damaged plies continue to bear some load. Nevertheless, useful calculations can be
made in this regime (although the major interest may be in the avoidance of any
damage to the component).In page 7, a crossply (0/90) laminate is loaded in tension
along one of the fibre directions. The stresses acting in each ply, relative to the fibre
directions, are monitored as the applied stress is increased. Only transverse or axial
tensile failure is possible in either ply, since no shear stresses act on the planes
parallel to the fibre directions. The software allows the onset of failure to be predicted
for any given composite with specified strength values. Although the parallel ply takes
most of the load, it is commonly the transverse ply which fails first, since its strength
is usually very low.
In page 8, any specified laminate can be subjected to an imposed stress state and
the onset of failure predicted. An example of such a calculation is shown in Fig. 10.
What is the Toughness (Fracture Energy) of a Composite?
The fracture energy, G_{c}, of a material is the energy absorbed within
it when a crack advances through the section of a specimen by unit area. Potentially the
most significant source of fracture work for most fibre composites is interfacial
frictional sliding. Depending on the interfacial roughness, contact pressure and sliding
distance, this process can absorb large quantities of energy. The case of most interest is
pullout of fibres from their sockets in the matrix. This process is illustrated
schematically in page 9.
The work done as a crack opens up and fibres are pulled out of their sockets can be
calculated in the following way. A simple shear lag approach is used. Provided the fibre
aspect ratio, s (=L/r), is less than the critical value, s*
(=s_{f*}/2t_{i*}), see page 10 of the Load Transfer section, all of
the fibres intersected by the crack debond and are subsequently pulled out of their
sockets in the matrix (rather than fracturing). Consider a fibre with a remaining embedded
length of x being pulled out an increment of distance dx. The associated
work is given by the product of the force acting on the fibre and the distance it moves
dU = (2prxt_{i*}) dx

(27) 
where t_{i*} is the interfacial shear stress, taken here as constant along the
length of the fibre. The work done in pulling this fibre out completely is therefore given
by

(28) 
where x_{0} is the embedded length of the fibre concerned on the side of
the crack where debonding occurs (x_{0} = L). The next step is an
integration over all of the fibres. If there are N fibres per m^{2}, then
there will be (N dx_{0} / L) per m^{2} with an
embedded length between x_{0} and (x_{0} + dx_{0}).
This allows an expression to be derived for the pullout work of fracture, G_{c}

(29) 
The value of N is related to the fibre volume fraction, f, and the fibre
radius, r
N =

(30) 
Eq.(29) therefore simplifies to

(31) 
This contribution to the overall fracture energy can be large. For example, taking f=0.5,
s=50, r=10 µm and t_{i*}=20 MPa gives a value of about
80 kJ m^{2}. This is greater than the fracture energy of many metals.
Since s_{f*} would typically be about 3 GPa, the critical aspect ratio, s*
(=s_{f*}/2t_{i*}), for this value of t_{i*}, would be about 75.
Since this is greater than the actual aspect ratio, pullout is expected to occur (rather
than fibre fracture), so the calculation should be valid. The pullout energy is greater
when the fibres have a larger diameter, assuming that the fibre aspect ratio is the same.
In page 10, the cumulative fracture energy is plotted as the crack opens up and
fibres are pulled out of their sockets. The end result for a particular case is shown in
Fig. 11.
Figure 11
Conclusion
After completing this section, the student should:
 Appreciate that a unidirectional composite tends to fracture axially, transversely or in
shear relative to the fibre direction.
 Be able to use simple expressions for axial composite strength, based on fibre and
matrix fracturing similarly in the composite and in isolation.
 Understand what is meant by "mixed mode" failure and be able to use Maximum
Stress or TsaiHill criteria to predict how a unidirectional composite will fail under
multiaxial loading.
 Be able to use measured strength values for a unidirectional composite to predict how
ply damage will develop in a laminate.
 Understand the concept of the fracture energy of a composite and be able to use the
software package to predict the contribution to this from fibre pullout.
The student is referred to the following resources in this module:
Chavla, K.K., Ceramic Matrix Composites, Chapman and Hall,1993
Clyne, T.W., and Withers, P.J., An Introduction to Metal Matrix Composites,
Cambridge University Press, 1993
Hull, D. and Clyne, T.W., An Introduction to Composite Materials, Cambridge
University Press, 1996
Piggott, M.R. Load Bearing Fibre Composites, Pergamon Press, 1980
Chou, T.W., Microstructural Design of Fibre Composites, Cambridge University
Press, 1992
Harris, B., Engineering Composite Materials, Institute of Metals, 1986
Kelly, A.(Ed), Concise Encyclopaedia of Composite Materials, Pergamon Press,
1994
