A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

16 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
We note that (2.20) is also called Cauchy’s equation of motion or Cauchy’s momentum
equation in some literature. Eq. (2.20) can be written in vector form as
( ∂v
)
ρ
∂t + (v · ∇)v = ∇ · σ + f,
where σ is the Cauchy stress tensor defined in (2.9). It is often desirable to express
these equations of motion in terms of displacements u. We find (because the Eulerian
velocity is given in terms of the displacement (2.4) by v = ∂u/∂t)
[ ∂ 2 u
( ∂u
) ∂u
]
ρ
∂t 2 + ∂t · ∇ = ∇ · σ + f. (2.21)
∂t
2.2.5 The Lagrangian equations of motion of a continuum
Next we will rewrite (2.20) in terms of a Lagrangian description, that is, we will derive
an equation of motion in the Lagrangian coordinate system (O-XYZ coordinate
system). Let Γ 0 denote the boundary of Ω 0 in the initial (undeformed/material) configuration,
and n 0 be the outer normal vector on Γ 0 . By Nanson’s formula [44] we
have
ndΓ = |A|(A −1 ) T n 0 dΓ 0 , (2.22)
where n 0 = (n 0X , n 0Y , n 0Z ) T , A is the configuration gradient defined by (2.5). Multiplying
both sides of (2.22) by σ we obtain
By (2.11), we have
σndΓ = |A|σ(A −1 ) T n 0 dΓ 0 .
σndΓ = Pn 0 dΓ 0 .
Let f 0 be the external body force acting on Ω 0 (f 0 = |A|f), let ρ 0 (X, Y, Z, t) be the material
density in the Lagrangian coordinate system (conservation of mass implies that
ρ 0 = |A|ρ), and V(X, Y, Z, t) be the velocity in the Lagrangian coordinate system.
Then we can rewrite the resultant force in the z-direction in the Eulerian coordinate
system as the resultant force in the Z direction in the Lagrangian coordinate system,
which is
∫
∫
F 0Z = (P ZX n 0X + P ZY n 0Y + P ZZ n 0Z )dΓ 0 + f 0Z dΩ 0 .
Γ 0 Ω 0
Then by Gauss’ Theorem and the above equation we find that
∫ ( ∂PZX
F 0Z =
Ω 0
∂X
+ ∂P ZY
∂Y
+ ∂P ) ∫
ZZ
dΩ 0 + f 0Z dΩ 0 .
∂Z
Ω 0
We can rewrite Reynolds transport theorem (2.18) in the Lagrangian coordinate system
and find
∫
∫
d
ρv z dΩ = ρ Dv ∫
z
dt Ω
Ω Dt dΩ = DV Z
ρ 0
Ω 0
Dt dΩ 0.