Page 218 - Mechanical Behavior of Materials

Section 5.4 Anisotropic Materials 219

5.4.4 Elastic Modulus Parallel to Fibers

Consider a uniaxial stress σ x parallel to fibers aligned in the X-direction, as shown in Fig. 5.15(a).
Let the fibers (reinforcement) be an isotropic material with elastic constants E r , ν r , and G r , and let
the matrix be another isotropic material, E m , ν m , G m . Assume that the fibers are perfectly bonded to
the matrix so that fibers and matrix deform as a unit, resulting in the same strain ε X in both. Further,
let the total cross-sectional area be A, and let the areas occupied by fibers and by matrix be A r and
A m , respectively. Then

A = A r + A m (5.49)
Since the applied force must be the sum of contributions from fibers and matrix, we have

σ X A = σ r A r + σ m A m (5.50)

where σ r , σ m are the differing stresses in fibers and matrix, respectively. The definitions of the
various elastic moduli require that

σ X = E X ε X , σ r = E r ε r , σ m = E m ε m (5.51)

Note that the strain in the composite is the same as that in both fibers and matrix:

ε X = ε r = ε m (5.52)

Figure 5.15 Composite materials with various combinations of stress direction and
unidirectional reinforcement. In (a) the stress is parallel to fibers, and in (b) to sheets of
reinforcement, whereas in (c) and (d) the stresses are normal to similar reinforcement.