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 17
Note that
DV Z
= ∂V Z
Dt ∂t .
Hence, we can rewrite Newton’s law in the Lagrangian coordinate system as
∫
Ω 0
ρ 0
∂V Z
∂t Ω 0 =
∫
( ∂PZX
Ω 0
∂X
+ ∂P ZY
∂Y
+ ∂P ZZ
∂Z
Note that the above equality holds for any Ω 0 . Thus we have
ρ 0
∂V Z
∂t
= ∂P ZX
∂X
+ ∂P ZY
∂Y
+ f 0Z
+ ∂P ZZ
∂Z + f 0Z,
)
dΩ 0 .
which is the equation of motion in the Z-direction. Then the equations of motion in the
Lagrangian coordinate system are given by
or, written concisely,
ρ 0
∂V X
∂t
ρ 0
∂V Y
∂t
ρ 0
∂V Z
∂t
= ∂P XX
∂X
= ∂P YX
∂X
= ∂P ZX
∂X
+ ∂P XY
∂Y
+ ∂P YY
∂Y
+ ∂P ZY
∂Y
+ ∂P XZ
∂Z + f 0X, (2.23a)
+ ∂P YZ
∂Z + f 0Y, (2.23b)
+ ∂P ZZ
∂Z + f 0Z, (2.23c)
∂V
ρ 0
∂t = ∇ · P + f 0.
Note that
V = ∂U
∂t .
Hence, the Lagrangian equations of motion in terms of displacement is given by
ρ 0
∂ 2 U
∂t 2 = ∇ · P + f 0. (2.24)
Remark 2.8. We note that the equations of motion (2.21) in the Eulerian (or moving)
coordinate system are inherently nonlinear independent of the constitutive law
assumptions (discussed in the next section) we might subsequently adopt. On the
other hand, the Lagrangian formulation (2.24) (relative to a fixed referential coordinate
system) will yield a linear system if a linear constitutive law is assumed. Thus,
there are obvious advantages to using the Lagrangian formulation in linear theory
(i.e., when a linear constitutive law is assumed).
3 Constitutive relationships: stress and strain
In the preceding discussions, we have focused on relationships between displacements
(and their rates) and the stress tensors. We have also related strain tensors