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 15
2.2.4 The Eulerian equations of motion of a continuum
We are now ready to use the above rules of calculus and the continuity of mass as
embodied in the Reynolds transport theorem to derive the equations of motion in
an Eulerian coordinate system. Throughout we have Ω = Ω t and Γ = Γ t (we will
suppress the subscripts) and we assume the coordinate system (x, y, z) is now moving
(changing with the volume element) with a velocity v = (dx/dt, dy/dt, dz/dt) T of the
deformation of the material. The resultant force F z in the z-direction on an arbitrary
volume Ω is
∫
∫
F z = T (n)
z dΓ + f z dΩ. (2.19)
Γ
Ω
By Cauchy’s formula (2.10) and Gauss’ theorem we have
∫
∫
T (n)
z dΓ =
Γ
Γ
∫
=
Ω
(σ xz n x + σ yz n y + σ zz n z
)
dΓ
( ∂σxz
∂x + ∂σ yz
∂y + ∂σ )
zz
dΩ.
∂z
Hence, by the above equality and (2.19), we obtain
∫
F z =
Newton’s law states that
∫
∫
d
ρv z dΩ =
dt
Ω
Ω
( ∂σxz
∂x + ∂σ yz
∂y + ∂σ zz
∂z + f z
Ω
)
dΩ.
( ∂σxz
∂x + ∂σ yz
∂y + ∂σ )
zz
∂z + f z dΩ.
Hence, by the Reynolds transport theorem we have
∫ ( ∂vz
ρ
Ω ∂t + v ∂v z
x
∂x + v ∂v z
y
∂y + v ∂v z )
∫
z dΩ =
∂z
Ω
( ∂σxz
∂x + ∂σ yz
∂y + ∂σ zz
∂z + f z
)
dΩ.
Note that because the above equality holds for an arbitrary domain Ω, the integrands
on both sides must be equal. Thus, we have
( ∂vz
ρ
∂t + v ∂v z
x
∂x + v ∂v z
y
∂y + v ∂v
) z
z = ∂σ xz
∂z ∂x + ∂σ yz
∂y + ∂σ zz
∂z + f z,
or written concisely as
ρ Dv z
= ∇ · σ •,z + f z ,
Dt
which is the equation of motion of a continuum in the z-direction. The entire set for the
equations of motion of a continuum in an Eulerian coordinate system is given as follows:
( ∂vx
ρ
∂t + v ∂v x
x
∂x + v ∂v x
y
∂y + v ∂v
) x
z = ∂σ xx
∂z ∂x + ∂σ yx
∂y + ∂σ zx
∂z + f x, (2.20a)
( ∂vy
ρ
∂t + v ∂v y
x
∂x + v ∂v y
y
∂y + v ∂v ) y
z = ∂σ xy
∂z ∂x + ∂σ yy
∂y + ∂σ zy
∂z + f y, (2.20b)
( ∂vz
ρ
∂t + v ∂v z
x
∂x + v ∂v z
y
∂y + v ∂v
) z
z = ∂σ xz
∂z ∂x + ∂σ yz
∂y + ∂σ zz
∂z + f z. (2.20c)