∇¯ denotes the gradient with respect to the material coordinates x¯, and where the symmetric Green deformation tensor C may be written as
The Eulerian and Lagrange strain tensors are defined by
so that (2.85) may also be written as
On using the relation u=x−x¯ for the displacement vector and (2.84), the Lagrangian strain tensor may be written in terms of the displacement vector as follows
where I is the symmetric fourth-order identity tensor (see (2.15)). The quantity E is the strain tensor that is used in finite deformation theory where there is no restriction on the degree of deformation provided that the deformation is continuous and the condition (2.18) is satisfied at all points in the system.
The invariants of the strain tensors in terms of principal stretches are given by the relations
It is clear that
It will be very useful to introduce here the principal values CJ, J = 1, 2, 3, of Green’s deformation tensor defined using the following relations
such that the symmetric tensor C may be written in the form
The quantities νJ,J=1,2,3, are orthogonal unit vectors defining the directions of the principal values. They have the following properties
The polar decomposition principle (see, for example, [2, Section 1.5]) states that the deformation gradient may be expressed in the following forms (dyadic and tensor)
