A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

6 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
Figure 1: Deformation of a body.
for every particle in the body. Thus the (Eulerian) displacement of the particle relative
to x is given by
u(x) = x − X(x). (2.4)
To relate deformation with stress, we must consider the stretching and distortion of
the body. For this purpose, it is sufficient if we know the change of distance between
any arbitrary pair of points.
Consider an infinitesimal line segment connecting the point P(X 1 , X 2 , X 3 ) to a
neighboring point Q(X 1 + dX 1 , X 2 + dX 2 , X 3 + dX 3 ) (see Fig. 1). The square of the
length of PQ in the original configuration is given by
|dX| 2 = (dX) T dX = (dX 1 ) 2 + (dX 2 ) 2 + (dX 3 ) 2 .
When P and Q are deformed to the points P ′ (x 1 , x 2 , x 3 ) and Q ′ (x 1 + dx 1 , x 2 + dx 2 , x 3 +
dx 3 ), respectively, the square of length of P ′ Q ′ is
|dx| 2 = (dx) T dx = (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 .
Definition 2.2. The configuration gradient (often, in something of a misnomer, referred to as
deformation gradient in the literature) is defined by
⎡
⎤
∂x 1 ∂x 1 ∂x 1
A = dx ∂X
dX = 1 ∂X 2 ∂X 3
⎢ ∂x 2 ∂x 2 ∂x 2 ⎥
⎣ ∂X 1 ∂X 2 ∂X 3 ⎦ . (2.5)
The Lagrangian strain tensor
∂x 3 ∂x 3 ∂x 3
∂X 1 ∂X 2 ∂X 3
The Lagrangian strain tensor is measured with respect to the initial configuration (i.e.,
Lagrangian description). By the definition of configuration gradient, we have dx =
AdX and
|dx| 2 − |dX| 2 =(dx) T dx − (dX) T dX
=(dX) T A T AdX − (dX) T dX
=(dX) T (A T A − I)dX.