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 13
Then the rate of change of Φ(t) with respect to t is given by (suppressing the multiple
integral notation here and below when it is clearly understood that the integral is a
volume or surface integral)
∫
∫
dΦ
dt = ∂ϕ
Ω t ∂t dΩ +
where on the boundary Γ t of Ω t , v = v(t) is the velocity
v(t) =
Eq. (2.15) can be written concisely as
Γ t
(ϕv x n x + ϕv y n y + ϕv z n z )dΓ, (2.15)
( dx
dt , dy
dt , dz ) T.
dt
∫
∫
dΦ
dt = ∂ϕ
Ω t ∂t dΩ +
Γ t
ϕv · ndΓ.
The first term on the right side corresponds to rate of change in a fixed volume, and
the second term corresponds to the convective transfer through the surface. By Gauss’
theorem, Eq. (2.15) can also be written as
or more concisely as
dΦ
dt = ∫
( ∂ϕ
Ω t ∂t + ∂ϕv x
∂x
+ ∂ϕv y
∂y
∫
dΦ
( ∂ϕ
)
dt = Ω t ∂t + ∇ · (ϕv) dΩ.
+ ∂ϕv )
z
dΩ, (2.16)
∂z
This rate, called the material derivative of Φ, is defined for a given set of material particles
in a moving volume. We note that when Ω t = Ω 0 for all t (i.e., the boundary Γ is
not moving so that v = 0), this becomes simply
∫
∫
d
∂ϕ
ϕ(x, y, z, t)dxdydz = (x, y, z, t)dxdydz.
dt Ω 0 Ω 0
∂t
2.2.2 The equation of continuity
We next derive the equation of continuity for an arbitrary mass of particles that may
be changing in time. The mass contained in a domain Ω t at time t is
∫
m(t) = ρ(x, y, z, t)dxdydz.
Ω t
Conservation of mass requires that dm/dt = 0 and thus we have from (2.16)
Hence, we obtain
dm
dt = ∫
∫
[ ∂ρ
Ω t ∂t + ∂ρv x
∂x
[ ∂ρ
Ω t ∂t + ∂ρv x
∂x
+ ∂ρv y
∂y
+ ∂ρv y
∂y
+ ∂ρv ]
z
dΩ.
∂z
+ ∂ρv ]
z
dΩ = 0.
∂z