Composites

The modulus of an unidirectional composite along the fiber direction can be calculated by the rule of mixtures:

E_c = E_fV_f + E_mV_m,

where E_f - the modulus of the fiber; E_m - is the modulus of the matrix; V_f, V_m - the volume fraction occupied by the fibers and the matrix respectively.

A cross-ply composite laminate [0^o/90^o], shown at the right, will warp under uniaxial tensile stress due to the difference in stiffness along and perpendicular to the fiber axis. The stiffness along the fiber axis is greater and therefore the ply with fibers parallel to the applied stress will be stretched less than the ply loaded across the fibers.

The elastic modulus of continuous fiber composites strongly depends on the angle between the fiber direction and the applied load. The maximum stiffness of the composite is achieved when the applied load is parallel to the fiber direction. The stiffness of cross-ply composites [+q/-q] is greater than that of the unidirectional composite due to additional restraint to shear deformation.

The greater the volume fraction Vf of fibrous unidirectional composites the higher their capability to sustain cyclic loading. The resistance to fatigue of composites strongly depends on the angle q between the direction of applied loading and the fiber axis. The fatigue strength decreases when the angle increases.

Composites with short fibers display a greater strength with an increase in the ratio of fiber length to diameter l/d. As the number of defects decreases along with the fiber diameter the strength increases. Additionally, for the same volume fraction the longer fibers can support a greater portion of the load transferred from the matrix.

Residual stresses are developed during the cooling of cross ply composite laminates due to anisotropy of thermal contraction parallel and perpendicular to the fiber direction. In glass fibers/epoxy composites the thermal expansion coefficient along the fiber direction is greater than that perpendicular to the fibers. Following the cooling the plies of 90^o are in tension and the plies of 0^o are in compression. Generally, cracks are formed perpendicularly to the tensile stress.

Composites are known to have a large specific strength, which is defined as the ratio of the ultimate tensile strength s_uts to the weight rg.
Specific strength of continuous fibers composites is much greater than that of such conventional materials as aluminium or steel.

The load applied to a fiber composite is supported by both the fibers and matrix. The relationship between the load carried by fibers P_f and matrix P_m depends on the ratio E_fV_f to E_mV_m:

P_f / P_m = E_fV_f / E_mV_m,

where E_f, E_m, V_f and V_m represent the elastic modulus and the volume fraction of the composite components. The higher the elastic modulus and the volume fraction, the greater portion of the applied load is carried by the fibers.

Assuming no slip between the fibers and matrix the stress acting on the composite is determined by the following:

s = s_fV_f + s_mV_m

where V_f and V_f - the volume fraction of the fibers and matrix respectively.

The stress in the matrix

s_m = s_f x E_m/E_f.

The stress in the fibers:

s_f = s / [V_f + (1-V_f) x E_m/E_f].

Assuming a constant applied stress, the greater the volume fraction of the fibers the less they are stressed.