Continuous-fibre composites

11.3.1.2 Continuous-fibre composites

In mechanical terms, the prime function of the matrix is to transfer stresses to the fibres (item (4) above) because these are stronger and of higher elastic modulus than the matrix. The response of a composite to applied stress depends upon the respective properties of the fibre and matrix phases, their relative volume fractions, fibre length and the orientation of fibres relative to the direction of applied stress.

1 During the Battle of the Atlantic in World War II, Geoffrey Pyke proposed the construction of ocean-going aircraft

carriers from paper pulp/frozen sea water (Pykecrete). In 1985, ice/wood fibres (Icecrete) was proposed for wharf and off-shore oil platform construction in Norwegian waters.

362 Modern Physical Metallurgy and Materials Engineering

Figure 11.10 ‘Parallel’ (a) and ‘series’ (b) models of unidirectional filament alignment in composites .

Some basic principles of the elastic response to stress can be obtained from mechanical models in which continuous fibres are set unidirectionally in an isotropic, void-free matrix (Figures 11.10a and 11.10b). It will be assumed that the Poisson ratio for the fibre material is similar to that of the matrix. Using subscript letters c, f, m, l and t we can signify where property values refer respectively to composite, fibre, matrix, longitudinal and transverse directions.

Thus V f /V m is the ratio of volume fractions of fibre

f ⊳DV m . Certain longitudinal properties for a composite can be obtained by using the ‘parallel’ model shown in Figure 11.10a and applying the Rule of Mixtures. For this condition of isostrain, stresses are additive and the equations for stress (strength) and elastic modulus are:

It is now possible to derive the following relation:

Figure 11.11 illustrates this relation, showing that as the modulus ratio and/or the volume fraction of fibres increase, more and more stress is transferred to

Figure 11.11 Relation between modulus ratio and stress the fibres. If the modulus ratio is unity, the composite

ratio (continuous parallel fibres) . must contain at least 50% v/v fibres if the fibres are to carry the same load as the matrix. Three typical composites A, B and C with 50% v/v reinforcement

so that the equations for the ‘parallel’ and ‘series’ are superimposed on the graph to show the extent to

models express, in mathematical form, the dominant which two increases in modulus ratio raise the stress

effect of fibres on longitudinal properties and the ratio.

dominant effect of the matrix on transverse properties. An alternative arrangement of fibres relative to

If typical tensile stress versus strain curves for fibre applied tensile stress is shown in Figure 11.10b. The

and matrix materials (Figure 11.12a) are compared, transverse elastic modulus for the composite is given

it can be seen that the critical strain, beyond which by the equation:

the composite loses its effectiveness, is determined by the strain at fracture of the fibres, ε f . At this strain ⊲ 1/E ct ⊳ D ⊲V f /E ft ⊳ C ⊲V m /E m ⊳

value, when the matrix has usually begun to deform plastically and to strain-harden, the corresponding

This ‘series’ version of the Rule assumes a condition m 0 . Thus, in the related of isostress and is derived by adding strains: it is less

Figure 11.12b, it follows that the strength of the accurate than the ‘parallel’ version. Both versions can

f , depending

be used to calculate shear moduli and conductivities upon the volume fraction of fibres. When a few widely- (thermal, electrical). More refined mathematical

spaced fibres are present, the matrix carries more load treatments are available: they are particularly helpful

than the fibres. Furthermore, in accordance with the for transverse properties. Sometimes fibre properties

Rule of Mixtures, the strength of the composite falls are highly anisotropic and this feature influences the

as the volume fraction of fibres deceases. Construction corresponding value for the composite; for instance,

lines representing these two effects meet at a minimum

E ft − E fl for aramid (Kevlar, Nomex ) and carbon point, V min . Obviously, V f must exceed V crit if the

fl × m ,E fl × E m and E ft >E m ,

tensile strength of the matrix is to benefit from the

Plastics and composites 363

Figure 11.12 (a) Stress–strain curves for filament and matrix and (b) dependence of composite strength on volume fraction of continuous filaments .

presence of fibres. In practical terms, the upper limit length 1 if efficient transfer of stress is to take place. for V f is about 0.7–0.8. At higher values, fibres

With respect to diameter, fibre strength increases as the are likely to damage each other. The Rule is only

diameter of a brittle fibre is reduced. This effect occurs applicable when V f >V min .

because a smaller surface area makes it less likely that

c and V f D weakening flaws will be present.

V crit . From the Rule equation we derive: For the model, in which a matrix containing a short

V 0 crit 0 m m f m ⊳

fibre is subjected to a tensile stress (Figure 11.13a), it is assumed that the strain to failure of the matrix

In general, a low V crit is sought in order to minimize is greater than the strain to failure of the fibre. The problems of dispersal and to economize on the amount

differences in displacement between matrix and fibre of reinforcement. Very strong fibres will maximize the denominator and are clearly helpful. Strain-

interfaces toward each fibre end. A corresponding hardening of the matrix (Figure 11.12a) is represented approximately by the numerator of the above ratio.

end of the fibre, these stresses change over a distance Thus, a matrix with a strong tendency to strain-harden

known as the stress transfer length (l/2); that is, will require a relatively large volume fraction of fibres,

the tensile stress increases as the interfacial shear

a feature that is likely to be very significant for metallic stress decreases. In Figure 11.13a, for simplicity, we matrices. For example, an fcc matrix of austenitic

assume that the gradient of tensile stress is linear. If stainless steel (Fe-18Cr-8Ni) will tend to raise V crit

the length of the fibre is increased, the peak tensile more than a cph matrix of zinc.

stress coincides with fracture stress for the filament (Figure 11.13b). The total length of fibre now has a

11.3.1.3 Short-fibre composites critical value of l c and the transfer length becomes So far, attention has been focused on the behaviour

l c / 2. If the length is sub-critical, fibre failure cannot of continuous fibres under stress. Fabrication of

occur. At the critical condition, the average tensile these composites by processes such as filament-

f / 2. With any further winding is exacting and costly. On the other hand,

increase in fibre length, a plateau develops in the stress composites made from short (discontinuous) fibres

profile (Figure 11.13c). The average tensile stress enable designers to use cheaper, faster and more

on the fibre, which is stated beneath Figure 11.13c, versatile methods (e.g. injection-moulding, transfer-

f as the fibre length moulding). Furthermore, some reinforcements are only

increases beyond its critical value. In effect, the load- available as short fibres. At this point, it is appropriate

carrying efficiency of the fibre is approaching that of to consider an isolated short fibre under axial tensile

its direct-loaded continuous counterpart. Provided that stress and to introduce the idea of an aspect ratio

the shear stresses do not cause ‘pull-out’ of the fibre, ⊲ length/diameter D l/d⊳. It is usually in the order of

fracture will eventually occur in the mid-region of

10 to 10 3 for short fibres: many types of fibre and

the fibre.

whisker crystal with aspect ratios greater than 500 are The condition of critical fibre length can be available. For a given diameter of fibre, an increase

quantified. Suppose that an increment of tensile force, in length will increase the extent of bonding at the

, is applied to an element of fibre, υl. The balance fibre/matrix interface and favour the desired transfer

of working stresses. As will be shown, it is necessary for the length of short fibres to exceed a certain critical

1 A. Kelly introduced the concept of ‘critical fibre length’.

364 Modern Physical Metallurgy and Materials Engineering

between tensile force and interfacial shear force is:

f divided by the critical transfer length l c / 2. The critical length l c is

f d/ aspect ratio, the criterion for efficient stress transfer takes the form:

This relation provides an insight into the capabilities of short-fibre composites. For instance, for a given diameter of fibre, if the interfacial shear strength of the fibres is lowered, then longer fibres are needed in order to grip the matrix and receive stress. When the operating temperature is raised and the shear strength decreases faster than the fracture strength of the fibres, the critical aspect ratio increases. The presence of tensile and shear terms in the relation highlights the indirect nature of short-fibre loading: matrix strength and interfacial shear strength are much more crucial factors than in continuous-fibre composites where loading is direct. Interfacial shear strength depends upon the quality of bonding and can have an important effect upon the overall impact resistance of the composite. Ideally, the bond strength should be such that it can absorb energy by debonding, thus helping to inhibit crack propagation. Interfacial adhesion is particularly good in glass fibre/polyester resin and carbon fibre/epoxy resin systems. Coupling agents are used to promote chemical bonding at the interfaces. For example, glass fibres are coated with silane (size) which reacts with the enveloping resin. Sometimes these treatments also improve resistance to aqueous environments. However, if interfacial bonding is extremely strong, there is an attendant risk that an

impinging crack will pass into and through fibres with little hindrance.

Provided that V f >V crit , the tensile strength at fracture of a short-fibre reinforced composite can be calculated from the previous Rule of Mixtures equation by substituting the mean tensile stress term for the fracture stress, as follows:

c c / 2l⊳V f m 0 V m

⊲ 11.12⊳ and that fibres are perfectly aligned. First, the equation

shows that short fibres strengthen less than continuous ones. For example, if all fibres are ten times the critical length, they carry 95% of the stress carried by continuous fibres. On the other hand, if their length falls below the critical value, the strength suffers seriously. For instance, if the working load on the composite should cause some of the weaker fibres to fail, so that the load is then carried by a larger number of fibres, the effective aspect ratio and the strength of the composite tend to fall. The same type of equation can also be applied to the elastic modulus of a short-fibre composite; again the value will be less than that obtainable with its continuous counterpart. The equation also shows that matrix properties become more prominent as fibres are shortened.