CONTINUUM MECHANICS for ENGINEERS

(a) Show that the Jacobian determinant J does not vanish, and solve
for the inverse equations X = X(x, t).
(b) Calculate the velocity and acceleration components in terms of
the material coordinates.
(c) Using the inverse equations developed in part (a), express the velocity
and acceleration components in terms of spatial coordinates.
Answer: (a) J = cosh2 t – sinh2 t = 1
X 1 = (x 1 + x 2)e –t + (x 1 – x 2)e t
X 2 = (x 1 + x 2) e –t – (x 1 – x 2) e t
X 3 = x 3
(b) v 1 = (X 1 + X 2)e t – (X 1 – X 2)e –t
v 2 = (X 1 + X 2)e t + (X 1 – X 2)e –t
v 3 = 0
a 1 = (X 1 + X 2)e t + (X 1 – X 2)e –t
a 2 = (X 1 + X 2)e t – (X 1 – X 2)e –t
a 3 = 0
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2 1
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(c) v1 = x2, v2 = x1, v3 = 0
a1 = x1, a2 = x2, a3 = 0
4.2 Let the motion of a continuum be given in component form by the
equations
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2
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2 1
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x 1 = X 1 + X 2t + X 3t 2
x 2 = X 2 + X 3t + X 1t 2
x 3 = X 3 + X 1t + X 2t 2
(a) Show that J ≠ 0, and solve for the inverse equations.
(b) Determine the velocity and acceleration
(1) at time t = 1 s for the particle which was at point (2.75, 3.75,
4.00) when t = 0.5 s.
(2) at time t = 2 s for the particle which was at point (1, 2, –1)
when t = 0.
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