Page 213 - Mechanical Behavior of Materials

214 Chapter 5 Stress–Strain Relationships and Behavior

5.4 ANISOTROPIC MATERIALS

Real materials are never perfectly isotropic. In some cases, the differences in properties for
different directions are so large that analysis assuming isotropic behavior is no longer a reasonable
approximation. Some examples of anisotropic materials are shown in Fig. 5.14.
Due to the presence of stiff fibers in particular directions, composite materials can be highly
anisotropic, and engineering design and analysis for these materials requires the use of a more
general version of Hooke’s law than was presented previously. In what follows, we will first discuss
Hooke’s law for anisotropic cases in general, and then we will apply it to in-plane loading of
composite materials. Anisotropic plasticity is not considered in this chapter or even in later chapters.
This advanced topic is important in some cases, but note that many composite materials fail prior to
the occurrence of large amounts of inelastic strain.

5.4.1 Anisotropic Hooke’s Law

In the general three-dimensional case, there are six components of stress: σ x , σ y , σ z , τ xy , τ yz , and
τ zx , as illustrated in Fig. 5.9. There are also six corresponding components of strain: ε x , ε y , ε z , γ xy ,
γ yz , and γ zx . In highly anisotropic materials, any one component of stress can cause strains in all six
components. The general anisotropic form of Hooke’s law is given by the following six equations,
here written with the coefficients shown as a matrix:

(a) (b)

Z
(c) (d)
Y

X

Figure 5.14 Anisotropic materials: (a) metal plate with oriented grain structure due to
rolling, (b) wood, (c) glass-fiber cloth in an epoxy matrix, and (d) a single crystal.