An orthotropic body has three orthogonal planes of symmetry; let a 1, a 2, and a 3 denote unit vectors perpendicular to these planes and
For an orthotropic elastic material, Eq. (5.111) holds for rotations through ? about the unit vectors a 1, a 2, and a 3. Zheng s (1993) representation theorem yields
where 
, ..., 
are functions of ? and invariants I 1, I 2, I 3 and K 1, ..., K 7 defined below.
Equations (5.132) 1 and (5.24) 2 give
where
Let the body be initially at a uniform temperature ? 0 and the temperature be kept constant during its deformations. In the reference configuration, F = C = 1,
where ?, ? and ? are constants for a homogeneous body but are functions of X for an inhomogeneous body. They satisfy
A nontrivial solution of these equations implies the existence of residual stresses in the reference configuration of an orthotropic body. For a homogeneous body, Eq. (5.137) 1 is identically satisfied, and Eq. (5.137) 3 gives ? = ? = ? = 0. Thus there can be no residual stress in a traction-free orthotropic, homogeneous body.
Equation (5.136) implies that the stress tensor in the reference configuration of an orthotropic body is composed of axial stresses along the axes of orthotropy.
Linear constitutive...
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