124
3 Continuum Mechanics and Nonlinear Elasticity
This relation can be simplified somewhat by manipulating the velocity gradient
term. First, the third formula in Table 2.2 (page 28) and symmetry of the Cauchy
stress tensor yield
σ :
∇v
= σ
T
:
∇v
T = σ :
∇v
T
.
Hence,
σ :
∇v
=
1
2
σ :
∇v
+ σ :
∇v
=
1
2
σ :
∇v
+ σ :
∇v
T
= σ :
1
2
∇v
+
∇v
T
= σ : D,
where the definition for the rate-of-deformation tensor D has been used [see
Eq. (3.96)]. With this result, Eq. (3.183) takes the alternate form
ρ ˙
u = σ : D − ∇ · q + r.
(3.184)
Equations (3.183) and (3.184) are valid in any coordinate system. In spatial
Cartesian coordinates x i , these equations yield
ρ
∂u
∂t
= σ ij
∂v j
∂x i
−
∂q i
∂x i
+ r = σ ij D ij −
∂q i
∂x i
+ r,
(3.185)
which reduces to Eq. (3.173) in 1D.
Material Form The procedure for deriving the material form of the energy balance
equation is similar to that used for the spatial form. Here, rather than going through
the full analysis, we outline the changes that are needed.
First, spatial quantities are replaced by their material counterparts in the initial
configuration:
A → A
0 ,
V → V
0 ,
n → N,
∇ → ∇,
ρ → ρ 0
T → T
0 ,
σ → P,
b → b 0 ,
q → q 0 ,
r → r 0 .
In terms of these variables, Eqs. (3.174)–(3.177) become
K =
1
2
V 0
v · v ρ 0 dV
0
U =
V 0
u ρ 0 dV
0