Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 111

Problem

5.3.1. Record a complete statement of Taylor's theorem in the case of three
independent variables for the function f ( X ) using the point X o as the point
about which the expansion occurs.

5.4 Invariance Under Rigid Object Motions

This guideline for the development of constitutive relations restricts the indepen-
dent variables and functional dependence of constitutive equations for material
behavior by requiring that the constitutive equations be independent of the motions
of the object that do not deform the object. The motions of the object that do not
deform the object are rigid object motions. This guideline requires that constitutive
equations for material behavior be independent of, that is to say unchanged by,
superposed rigid object motions. As an illustration consider the object shown in
Fig. 5.1 . If the object experiences a translation and a rigid object rotation such that
the force system acting on the object is also translated and rotated, then the state of
stress T ( X , t ) at any particle X is unchanged. As a second example recall that the
volume flow rate q is the flow rate relative to the solid porous matrix. It follows that
the volume flow rate q in a porous medium is unchanged by (virtual or very slow)
superposed rigid object motions.
The application of this guideline of invariance under rigid object motions is
illustrated by application to the three constitutive ideas involving stress. The two
constitutive ideas involving fluxes automatically satisfy this guideline because the
fluxes are defined relative to the material object and the rigid motion does not
change the temperature field or the pressure field. The constitutive idea for Hooke's
law ( 5.3H ) may be rewritten as

T

¼ T

ð u ð X ; t Þ; E ð X ; t Þ; Y ð X ; t Þ; X Þ; (5.4H)

Fig. 5.1 A rigid object rotation of an object, a rotation that includes the force system that acts
upon the object