Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 122

important type of non-linear elastic material model, the hyperelastic material, is that
the stress is the derivative of the strain energy with respect to strain, as in ( 5.26H ). In
the non-linear case the strain energy is not specified by an expression as simple as
( 5.27H ).
Problems

5.10.1. Consider the work done in a closed loading cycle applied to a unit cube of a
linear anisotropic elastic material. The loading cycle this cube will be
subjected to begins from an unstressed state and contains the following
four loading sequences: O ! A, the stress in the x 1 direction is increased
slowly from O to T A ;A B, holding the stress in the x 1 direction, T A

constant, the stress in the x 2 direction is increased slowly from O to T B ;
B

!

C, holding the stress in the x 2 direction constant, the stress in the x 1
direction is decreased slowly from T A to O; C

!

O, the stress in the x 2
direction is decreased slowly from T B to O. At the end of this loading cycle
the object is again in an unstressed state. Show that the work done on each
of these loading sequences is given by

!

1 1

2 2

2 S 11 ð T ð A Þ 2 S 22 ð T ð B Þ

þ S 21 T ð A Þ

T ð B Þ
2

W OA ¼ Þ ; W AB ¼ Þ

1 2 1

1
2 S 11 ð T ð A Þ

1
2 S 22 ð T ð B Þ

2 2

S 12 T ð A Þ
1

T ð B Þ
2

W BC ¼ Þ ; W OA ¼ Þ

1 2

and show that the work done around the closed cycle is given by

W OO ¼ðS 21 S 12 Þ T ð A Þ

T ð B Þ
2

:

1

Show that one may therefore argue that .

¼ S T implies C ¼ C T . 5.10.2 Show that S

5.11 Restrictions on the Coefficients Representing
Material Properties

In this section other restrictions on the four tensors of material coefficients are
considered. Consider first that the dimensions of the material coefficients contained
in the tensor must be consistent with the dimensions of the other terms occurring in
the constitutive equation. The constitutive equation must be invariant under
changes in gauge of the basis dimensions as would be affected, for example, by a
change from SI units to the English foot-pound system.
It will be shown here that all the tensors of material coefficients are positive
definite as well as symmetric except for the viscoelastic tensor function G ð s Þ .To

see that the permeability tensor H ( p ) is positive definite let ∇ p ¼ n ( ∂ p / ∂ w ) where