Page 88 - Physical Principles of Sedimentary Basin Analysis

70 Linear elasticity and continuum mechanics

using the Jacobian J (3.143), where dV = da 1 da 2 da 3 .The volume V is a constant
0
volume in the Lagrangian coordinate system, because the particles are at rest there:
d ∂ ∂ J

JdV = J + dV
dt V 0 V 0 ∂t ∂t

∂
= + ∇· v JdV
V 0 ∂t

D
= + ∇· v dV. (3.166)
V (t) dt
Equation (3.166)usesthat ∂ J/∂t = J ∇· v in Lagrange coordinates, and that the par-
tial time derivative in Lagrange coordinates becomes the material derivative in Euler
coordinates. Since equation (3.166) is zero we get the expression for mass conservation
D
+ ∇· v = 0. (3.167)
dt
This is the generalization to any number of spatial dimensions of equation (3.161) for 1D
mass conservation. (See Exercise 3.24.) Mass conservation (3.167) can be used to derive
the following useful relationship:
d Df
fdV = dV (3.168)
dt V (t) V (t) dt
where f is any scalar function, see exercise 3.25.
Exercise 3.25 Verify equation (3.168).

Exercise 3.26 Show that mass conservation for an incompressible medium ( = constant)
is given by ∇· v = 0.
Exercise 3.27 Derive Reynolds transport theorem
d ∂ f
fdV = dV + f v · n dA. (3.169)
dt V (t) V (t) ∂t ∂V (t)
Hint: switch to Lagrangian coordinates, use equation (3.144) and then Gauss’s theo-
rem (3.171).

3.20 Momentum balance (Newton’s second law)

We will now derive an expression for Newton’s second law which says that the time rate
of change of momentum for a body is equal to the sum of all forces acting on it.Thislaw
is written as
d
v i dV = g δ i,z dV + σ ij n j dA (3.170)
dt V (t) V (t) ∂V (t)