CONTINUUM MECHANICS for ENGINEERS

5.2 Material Derivatives of Line, Surface, and Volume Integrals
Let any scalar, vector, or tensor property of the collection of particles occupying
the current volume V be represented by the integral
*
Pij…(t) = P x,
t dV
(5.2-1)
*
where P represents the distribution of the property per unit volume and
ij
has continuous derivatives as necessary. The material derivative of this property
is given in both spatial and material forms, using Eq 4.11-8, by
Since V° is a fixed volume in the referential configuration, the differentiation
and integration commute, and the differentiation can be performed inside
the integral sign. Thus, from Eq 4.11-6, using the notation [] to indicate
differentiation with respect to time,
•
and converting back to the spatial formulation
∫
V
(5.2-2)
With the help of the material derivative operator given in Eq 4.5-5, this
equation may be written (we omit listing the independent variables x and t
for convenience),
ijL
∫ ∫
( )
˙ d * d *
PijL()= t PijL( x, t) dV = PijL[ x( X,
t) , t] JdV
dt V
dt o
V
•
[ ijL(
) ] = ( ijL ijL
)
o o
V
V
∫ ∫
* o
* *
P X,
t J dV P˙ J+P J˙ dV
= ( ˙ * *
∫ ijL k,k ijL)
o
V
P + v P JdV
[ ]
˙ ˙ * *
PijL()= t PijL( x, t ) +vk,k PijL( x,
t) dV
∫V ⎡
P˙ Pij
P
ijL()=
t
+ v ∫ ⎢
V ⎣⎢
t
∂ ∂
∂
⎡
∫ ⎣⎢
∂
V ∂ t
⎤
⎦⎥
* *
L
k
ijL
*
+vk,k PijL⎥dV ∂ x k
⎤
vP k ij , k⎥dV
⎦⎥
*
PijL
*
= ⎢ +( L)
o
o
o