Page 84 - Physical Principles of Sedimentary Basin Analysis

66 Linear elasticity and continuum mechanics

and the material derivative becomes
DJ(x, t) ∂ J(a, t)
= = J(a, t) ∇· v(x, t). (3.144)
dt ∂t
The material derivative (3.144) is first shown in 1D where the Jacobian is J(a, t) = ∂x/∂a,
and the time derivative is
∂ J ∂ ∂x
∂ ∂x
∂v
(a, t) = (a, t) = (a, t) = (a, t)
∂t ∂t ∂a ∂a ∂t ∂a
∂v
= (x(a, t), t)
∂a
∂v ∂x
= (x, t) (a, t)
∂x ∂a
∂v
= (x, t) J(a, t). (3.145)
∂x
Note 3.6 A little more work is needed to show expression (3.144) for the general case of
any number of spatial dimensions. First we need to recall the definition of the determinant
function before we can undertake this task. A determinant is defined as a scalar function
d = d(b 1 ,..., b n ) of n vectors with the following properties:

(1) It is linear in each slot:

(1) (2) (1) (2)
d ..., c 1 b + c 2 b ,... = c 1 d ..., b ,... + c 2 d ..., b ,...
k k k k
(2) The determinant vanishes if any two vectors are equal:
d (b 1 ,..., b n ) = 0, if b i = b j for i = j
(3) It is unity for the identity matrix:
d (e 1 ,..., e n ) = 1, where e i is the unit vector in direction i.
Equation (3.144) is now shown for 3D, but the same approach applies equally well for a
different number of spatial dimensions. Using the definition of a determinant the Jacobian
in 3D is
J = d(b 1 , b 2 , b 3 ) (3.146)
where the vector b i is

∂x i ∂x i ∂x i
b i = , , . (3.147)
∂a 1 ∂a 2 ∂a 3
We then get
∂ J ∂b 1
∂b 2
∂b 3
= d , b 2 , b 3 + d b 1 , , b 3 + d b 1 , b 2 , (3.148)
∂t ∂t ∂t ∂t
where the components of ∂b i /∂t are

∂ ∂x i ∂ ∂x i ∂
∂v i ∂x s
= = v i x(a, t), t = . (3.149)
∂t ∂a j ∂a j ∂t ∂a j ∂x s ∂a j