A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 7
The Lagrangian (finite) strain tensor E is defined by
E = 1 2 (AT A − I). (2.6)
The strain tensor E was introduced by Green and St. Venant. Accordingly, in the literature
E is often called the Green’s strain tensor or the Green-St. Venant strain tensor. In
addition, from (2.6) we see that the Lagrangian strain tensor E is symmetric.
If the strain satisfies A T A − I = 0, then we say the object is undeformed; otherwise,
it is deformed. We next explore the relationship between the displacement and strain.
By (2.3) and (2.5) we have the true deformation gradient given by
or
∇U =
⎡
⎢
⎣
∂x 1
∂X 1
− 1
∂x 2
∂X 1
∂x 2
∂X 2
− 1
∂x 1
∂X 2
∂x 1
∂X 3
∂x 2
∂X 3
∂x 3
∂X 1
∂x 3
∂X 2
∂x 3
∂X 3
− 1
⎤
⎥
⎦ = A − I,
∂U i
∂X j
= ∂x i
∂X j
− δ ij , j = 1, 2, 3, i = 1, 2, 3,
where I is the identity matrix, and U = (U 1 , U 2 , U 3 ) T . Thus, because
A = ∇U + I,
the relationship between Lagrangian strain (2.6) and displacement is given by
E ij = 1 2
[ ∂Ui
+ ∂U j
+ ∂U k ∂U
]
k
,
∂X j ∂X i ∂X i ∂X j
where E ij is the (i, j) component of strain tensor E.
The Eulerian strain tensor
The Eulerian strain tensor is measured with respect to the deformed or current configuration
(i.e., Eulerian description). By using dX = A −1 dx, we find
|dx| 2 − |dX| 2 =(dx) T dx − (dx) T (A −1 ) T A −1 dx
=(dx) T (I − (A −1 ) T A −1 )dx,
and the Eulerian (finite) strain tensor e is defined by
e = 1 2(
I − (A −1 ) T A −1) . (2.7)
The strain tensor e was introduced by Cauchy for infinitesimal strains and by Almansi
and Hamel for finite strains; e is also known as Almansi’s strain in the literature. In
addition, we observe from (2.7) that Eulerian strain tensor e is also symmetric.