Page 335 - Introduction to Continuum Mechanics

Constitutive Equation for an Elastic Medium under Large Deformation. 319

Thus, from the result of the last example, we have

i.e, in a change of frame

Thus, the left Cauchy-Green deformation tensor is frarne-mdifferent (i.e., it is objective).

We note that it can be easily proved that the inverse of an objective tensor is also objective
and that the identity tensor is obviously objective. Thus both the left Cauchy Green deforma-
tion tensor B and the Eulerian strain tensor e = -(I- B~ *) are objective, while the right Cauchy

Green deformation tensor C and the Lagrangian strain tensor E = —(C—V) are non-objective.

We note also that the material time derivative of an objective tensor is in general non-ob-
jective,

5.33 Constitutive Equation for an Elastic Medium under Large Deformation.
As in the case of infinitesimal theory for an elastic body, the constitutive equation relates
the state of stress to the state of deformation. However, in the case of finite deformation, there
are different finite deformation tensors (left Cauchy-Green tensor B, right Cauchy-Green
tensor C, Lagrangian strain tensor E, etc.,) and different stress tensors (Cauchy stress tensor
and the two Piola-Kirchhoff stress tensors) defined in Chapter 3 and Chapter 4 respectively.
It is not immediately clear what stress tensor is to be related to what deformation tensor. For
example, if one assumes that

where T is the Cauchy stress tensor, and C is the right Cauchy-Green tensor, then it can be
shown [see Example 5.33.1 below] that this is not an acceptable form of constitutive equation
because the law will not be frame-indifferent. On the other hand if one assumes

then, this law is acceptable in that it is independent of observers, but it is limited to isotropic
material only (See Example 5.33.3).
The requirement that a constitutive equation must be invariant under the transformation
Eq. (5.32.1) (i.e., in a change of frame), is known as the principle of material frame indif-
ference. In applying this principle, we shall insist that force and therefore, the Cauchy stress
tensor be frame-indifferent. That is in a change of frame