CONTINUUM MECHANICS for ENGINEERS

In this equation, the first term on the right-hand side gives the change
occurring in the property at position x, known as the local rate of change; the
second term results from the particles changing position in space and is
referred to as the convective rate of change. Since by Eq 4.2-12 the velocity is
defined as v = dx/dt (or v k = dx k/dt), Eq 4.5-3 may be written as
d
∂
∂
[ Pij... ( x, t)] = [ Pij... ( x, t)]
+ [ Pij... ( x,
t)] v
dt ∂ t
∂ x
(4.5-4)
from which we deduce the material derivative operator for properties expressed
in the spatial description
d
= vk
dt t x
∂
∂ +
∂
∂
d
or = (4.5-5)
dt t
∂
+ v⋅ ∂
The first form of Eq 4.5-5 is for rectangular Cartesian coordinates, while the
second form is coordinate-free. The del operator ( )
will always indicate
partial derivatives with respect to the spatial variables unless specifically
stated.
Example 4.5-1
Let a certain motion of a continuum be given by the component equations,
x 1 = X 1e –t , x 2 = X 2e t , x 3 = X 3 + X 2(e –t – 1)
and let the temperature field of the body be given by the spatial description,
θ = e –t (x 1 – 2x 2 + 3x 3)
Determine the velocity field in spatial form, and using that, compute the
material derivative dθ/dt of the temperature field.
Solution
Note again here that the initial configuration serves as the reference configuration
so that Eq 4.2-9 is satisfied. When Eq 4.5-2 is used, the velocity
components in material form are readily determined to be
v 1 = –X 1e –t , v 2 = X 2e t, v 3 = –X 2e –t
Also, the motion equations may be inverted directly to give
k
X 1 = x 1e t , X = x 2e –t , X 3 = x 3 – x 2(e –2t – e –t )
k
k