R is the orthogonal rigid rotation tensor having the properties R.RT=RT.R=I with det(R)=±1, and where U and V are positive-definite symmetric right and left stretch tensors.
It follows from (2.84) and (2.96) that in tensor form
so that in dyadic form
The symmetric tensors U and V have common eigenvalues λJ but different mutually orthogonal eigenvectors νJ such that
The eigenvalues λJ, J = 1, 2, 3, are the principal stretches and
It then follows that the principal values CJ of the C may be written in terms of the principal stretches λJ as follows
It follows from (2.94) that
and on multiplying by νK using the properties (2.95) it can be shown that
In terms of the principal values C1, C2 and C3 of the tensor C and E1, E2 and E3 of the tensor E, the corresponding invariants may be written as (see [1, Section 1.10])
It can be shown that
where ρ0 is the uniform mass density before deformation has occurred at some reference temperature T0 and reference pressure p0.
2.11 Field Equations for Infinitesimal Deformations
Continuum mechanics is based upon conservation laws that lead to the basic field equations that are independent of the properties of material to which the laws are applied. The mathematical statement of these laws is now given for the case of infinitesimal deformations where there is no practical distinction between the use of so-called material coordinates and spatial coordinates.
For many practical applications the deformation gradients are sufficiently small for quadratic terms to be neglected when compared with linear terms leading to an infinitesimal deformation theory. The expression (2.88) for the Lagrangian strain tensor may then be approximated by υ≅ε where
is known as the infinitesimal strain tensor and where ∇ denotes the gradient with respect to the coordinates x. In addition, there is no need to distinguish between the initial and deformed states of the medium so that x≅x¯. Relation (2.106) may be written in component form so that
