Background

Layer property prediction 

This section contains information on the prediction of 

Volume fraction and weight fraction

Composite density

Layer thickness

Elastic properties for a unidirectional layer

Fibre direction Young's modulus

Transverse Young's modulus

In-plane shear modulus

Major Poisson's ratio

Elastic constants for other layer forms

Strengths of a unidirectional layer

Longitudinal tensile strength

Longitudinal compressive strength

Transverse tensile strength

Transverse compressive strength

In-plane shear strength

Volume fraction and weight fraction

The fibre and matrix properties are typically very different. The properties of the composite are therefore strongly dependent on the relative proportion of fibres and matrix. This is commonly characterised by the use of volume or weight fractions. The volume fraction of a particular component (e.g. fibre, matrix or filler) is given simply as the fraction of the total composite volume made up by that component. Likewise, the weight fraction of a particular component is given simply as the fraction of the total composite weight made up by that component.

Volume fraction is commonly used in micro-mechanics type calculations (as applied to estimating layer properties) because it is mathematically convenient. However, for other types of calculations, e.g. to estimate quantities for manufacturing, it may be more convenient to work with weight fraction. The volume and weight fractions are related as follows

where V_i, W_i and r _i represent the volume fraction, weight fraction and density respectively of the i^th constituent.

The volume (or weight) fraction is determined during manufacture by the relative amounts of constituent materials used. It is governed, however, to a large extent by the form of the fibre reinforcement used. When purely uni-directional material is used, a relatively high volume fraction can be achieved (typically >55%) since this allows geometric arrangement of the fibres so as to maximise packing. The fibre volume fraction is reduced in woven fabrics where the geometry inherently makes for resin rich spaces in between fibre bundles. The fibre volume fraction is further reduced in random mat materials, again because the geometry makes for spaces between individual fibre bundles. The method of manufacture also plays a role in determining volume fraction - processes that use high pressures to ensure laminate consolidation (e.g. autoclave curing of prepreg) allow for higher volume fractions while processes relying of hand lamination to ensure full wet out will tend to have lower volume fractions.

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Composite density

The density of the composite can be estimated from the rule of mixtures equation, i.e.

Note that for a two component (fibres - as indicated by the subscripted f ) and matrix (as indicated by the subscript m) the above is simply

Layer thickness

The thickness attributable to a material having mass m_i per unit area is given by

The thickness of a composite layer made up of fibres with areal mass m_f and matrix with areal mass m_m is therefore given by

Further useful relationships are

The above relationships can be used in determining the number of layers and quantities of materials necessary in achieving a certain thickness.

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Elastic properties for a unidirectional layer

Several simple approximations to the elastic constants for a single unidirectional layer, as shown in the figure below, are presented here.

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Fibre direction Young's modulus

The fibre direction (or longitudinal) Young's modulus is given by the rule of mixtures equation, namely

where E_f is the Young's modulus of the fibres (in their longitudinal direction) and E_m is the Young's modulus of the matrix. Note that, since the modulus of commonly used fibres is likely to be much higher than that of most polymer matrix systems, the longitudinal Young's modulus is dominated by the fibre modulus and also depends heavily on fibre volume fraction.

Transverse Young's modulus

The transverse Young's modulus can be estimated from the following equation

This equation predicts that the transverse Young's modulus is determined mostly by the matrix modulus in the range of practical (i.e. < 65% fibre volume fractions). Hence, as expected the fibres do not offer significant enhancement to transverse stiffness compared to that of the matrix alone. Note that in the above equation the fibre modulus is that transverse to the fibre - this can be significantly lower than the longitudinal modulus for anisotropic fibres such as aramid or carbon.

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In-plane shear modulus

The in-plane shear modulus is the ratio of shear stress in the 1-2 plane to the corresponding shear strain. It can be estimated from the following equation

For most unidirectional composites the in-plane shear modulus will be determined mainly by the matrix stiffness.

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Major Poisson's ratio

The major Poisson's ratio is determined by consideration of the transverse strain response to a longitudinal load. It can be estimated from a rule of mixtures equation considering the fibre and matrix Poisson's ratios, i.e.

Note that the equations presented above are approximate and based on certain assumptions regarding materials behaviour, geometry etc that have varying degrees of limitation. They can provide a basis for making estimates comparing the relative merits of different materials. In most detailed design situations however the designer would seek greater confidence in knowledge and accuracy of the material properties used.

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Considerations for other layer forms

The equations presented above relate to unidirectional materials and cannot be applied directly to random mat or woven fabric layers. The more complex structure of these materials mean that the derivation of the equations is not straightforward. Some simple, semi-empirical, equations are presented below.

For random (chopped strand) mat type materials the elastic constants are given approximately by

where E₁ and E₂ are as calculated for a unidirectional layer of the same material constituents and fibre volume fraction as the mat composite. These can be calculated by the equations for a unidirectional layer.

An alternative approach is the use of efficiency factors applied to the equations for a unidirectional material. For example, the Young's modulus of a random mat material can be estimated from

where the efficiency factor h is usually taken as 0.3 for a random mat material.

The elastic properties for a random mat can also be estimated assuming that it is made up of individual unidirectional layers arranged so as to produce a quasi-isotropic laminate. The elastic constants for the unidirectional layer are first calculated using the equations presented above and then properties for the quasi-isotropic composite determined by application of lamination theory. This will tend to overestimate the stiffness of the mat material since it does not account for the fact that fibres in the mat can lie at an angle to the 1-2 plane.

The properties of woven fabric layers can also be calculated by using lamination theory to consider a laminate made up of unidirectional layers with proportions in the 0 and 90 degree directions determined according to the fabric construction. This approach will tend to overestimate the stiffnesses since it ignores the fibre waviness introduced by the fabric geometry. The effect will be most marked for fabrics with a tight weave pattern (e.g. plain weave) and smaller for loose weave fabrics (e.g. 8-harness satin).

The efficiency factor approach can also be applied to fabrics - for plain weave balanced fabrics the efficiency factor is usually taken as 0.5.

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Strengths of a unidirectional layer

This section presents some simple equations that can be used to obtain estimates of the strengths of a unidirectional composite layer

Longitudinal tensile strength

The nature of longitudinal tensile strength is determined by the strains to failure of the matrix (e _m^* ) and fibres (e _f^* ). Four cases need to be considered

When e _f^* > e _m^*

(i) The matrix fails first and the fibres continue to carry load up until they fail.

This situation occurs where the fibre volume fraction exceeds some limit.

(ii) The matrix fails first and the fibres can not support the load

This situation occurs where the fibre volume fraction is less than a certain limit.

When e _f^* < e _m^*

(iii) The fibres fail first but the matrix can continue to carry load

This situation occurs where the fibre volume fraction is less than a certain limit.

(iv) The fibres fail first and the matrix cannot support the load

This situation occurs where the fibre volume fraction exceeds a certain limit.

The volume fractions which determine which equation should be used are typically lower than would be encountered in practice hence only equations (i) and (iv) are likely to be applied.

For most glass reinforced plastic composites e _f^* > e _m^* hence (i) will usually apply.

For most carbon fibre reinforced plastic composites e _f^* < e _m^* hence (iv) will usually apply.

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Longitudinal compressive strength

Failure in longitudinal compression is typically determined by fibre local stability considerations. Equations giving reasonable estimates of longitudinal compressive strength (X_c ) are available but these tend to be very complex and depend on materials property data that is not always readily available. The designer will therefore depend heavily on test data to obtain values for the longitudinal compressive strength. For most commonly used glass and carbon fibre reinforced plastics the longitudinal compressive strength will be in the range of 500 to 1000 MPa. It is worth noting that it will often be difficult in practice to load a composite structure in compression to the point that its failure is determined by the compressive strength of the material. Failure in compression is more commonly by buckling or instability of the laminate at a stress much lower than the compressive strength.

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Transverse tensile strength

The transverse tensile strength ( Y_t ) can be estimated from

where s _m^* is the matrix tensile strength. It is worth noting that the transverse tensile strength for a single isolated layer (as estimated using the above equation) is actually governed to a large extent by fracture mechanics type considerations where local variations in strain play a large role. In a multi-directional laminate the strain field is controlled to a great extent by stiffer layers in other directions and the failure stress for the transverse layers is determined more by strength of materials considerations. Hence the observed strength for a 90 degree layer within a laminate will often be higher than when the layer is considered in isolated. Determination of strength in the laminate is complex and depends on a number of additional factors including the overall laminate construction and layer thickness.

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Transverse compressive strength

The transverse compressive strength (Y_c) can be estimated by the same equation as that used for the transverse tensile strength except that the matrix tensile strength is replaced by the matrix compressive strength.

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In plane shear strength

The in plane shear strength (S ) can be estimated from

where t _m^* is the matrix shear strength.

The designer should use the above strength equations for scoping purposes only. Layer properties used for detail design purposes should be verified for the particular material under consideration.