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3 Continuum Mechanics and Nonlinear Elasticity
As mentioned in Sect. 3.6.2, p is analogous to a pressure, and many authors refer
to it as a “hydrostatic pressure.” Clearly, internal pressure in an unloaded block
would be zero, but this simple analysis gives p = 0. Hence, p is not a physical
pressure.
Problems
3.1 The motion of a continuum is described by the relations
x 1 = X 1 + X 2 (1 − e
t )
x 2 = X 2 − X 3 t
2
x 3 = X 3 ,
where the X i and x i are material and spatial Cartesian coordinates, respectively.
(a) Compute the velocity field in terms of material coordinates and in terms
of spatial coordinates.
(b) Compute the acceleration field using both the Lagrangian and Eulerian
forms of dv i /dt. Show that both forms produce the same results.
3.2 In Cartesian coordinates, the motion of a particle is described by
r = (X 1 + X 2 t)e 1 + (X 2 − t
2 )e 2 + X 3 e 3 .
If the temperature distribution is
θ = x
2
1 x
2
2 + x 2 x 3 + x 3 t
3 ,
determine dθ/dt in terms of the spatial coordinates x i .
3.3 In spherical coordinates, the velocity field in a continuum is given by v =
v r (r)e r , where r is the radial coordinate. Compute ∇ · v using Eqs. (2.54) and
∇ = e r
∂
∂r
+ e θ
1
r
∂
∂θ
+ e φ
1
r sin θ
∂
∂φ
.
3.4 Show that ∇ = F T · ∇, where ∇ and ∇ are the gradient operators in
undeformed and deformed coordinates, respectively.