CONTINUUM MECHANICS for ENGINEERS

u 2 = – x 2 – (x 1 + x 2)e –t /2 + (x 1 – x 2)e t /2
and by using the material derivative operator (dui/dt =
∂ui / ∂ t + vj∂ui/ ∂ x j
), verify the velocity and acceleration components
calculated in Problem 4.1.
4.13 The Lagrangian description of a deformation is given by
x 1 = X 1 + X 3(e 2 – e –2 )
x 2 = X 2 – X 3(e 2 – 1)
x 3 = X 3e 2
Determine the components of the deformation matrix FiA and from it
show that the Jacobian J does not vanish. Invert the mapping equations
to obtain the Eulerian description of the deformation.
Answer: J = e2 X1 = x1 – x3 (1 – e –4 )
X2 = x2 + x3 (1 – e –2 )
X3 = x3 e –2
4.14 A homogeneous deformation has been described as one for which all
of the deformation and strain tensors are independent of the coordinates,
and may therefore be expressed in general by the displacement
field ui = AijXj where the Aij are constants (or in the case of a motion,
functions of time). Show that for a homogeneous deformation with
the Aij constant
(a) plane material surfaces remain plane
(b) straight line particle elements remain straight
(c) material surfaces which are spherical in the reference configuration
become ellipsoidal surfaces in the deformed configuration.
4.15 An infinitesimal homogeneous deformation ui = AijXj is one for which
the constants Aij are so small that their products may be neglected.
Show that for two sequential infinitesimal deformations the total displacement
is the sum of the individual displacements regardless of
the order in which the deformations are applied.
4.16 For the homogeneous deformation defined by
x 1 = αX 1 + βX 2
x 2 = – βX 1 + αX 2
x 3 = µX 3