Problems for Chapter 2
Governing Equations
2.4. Work and
Energy, the Principle of Virtual Work
2.4.1. A solid with volume V is subjected to a distribution of traction on its surface. Assume that the solid is in
static equilibrium. By considering a virtual velocity of the form ,
where is a constant symmetric tensor, use the
principle of virtual work to show that the average stress in a solid can be
computed from the shape of the solid and the tractions acting on its surface
using the expression
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2.4.2. The figure shows a cantilever beam that is subjected
to surface loading per unit length. The state of stress in the beam can be
approximated by ,
where is the area moment of inertia of the beam’s
cross section and is an arbitrary function (all other stress
components are zero). By considering a
virtual velocity field of the form
where
is an arbitrary function satisfying at ,
show that the beam is in static equilibrium if
By integrating the first integral expression by parts
twice, show that the equilibrium equation and boundary conditions for are
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2.4.3. The figure shows a plate with a clamped edge that is
subjected to a pressure on its surface. The state of stress in the plate can be
approximated by
where the subscripts can have values 1 or 2, and is a tensor valued function. By considering a virtual velocity of the form
where
is an arbitrary function satisfying on the edge of the plate, show that the beam
is in static equilibrium if
By applying the divergence theorem appropriately, show
that the governing equation for is
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2.4.4. The shell shown in the figure is subjected to a radial
body force ,
and a radial pressure acting on the surfaces at and .
The loading induces a spherically symmetric state of stress in the shell, which
can be expressed in terms of its components in a spherical-polar coordinate
system as . By considering a virtual velocity of the
form ,
show that the stress state is in static equilibrium if
for all w(R). Hence, show that the stress state must
satisfy
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2.4.5. In this problem, we consider the internal forces in
the polymer specimen described in Problem 2.1.29 and 2.3.5 (you will need to
solve 2.1.29 and 2.3.5 before you can attempt this one). Suppose that the
specimen is homogeneous, has mass density in the reference configuration, and may be
idealized as a viscous fluid, in which the Kirchhoff stress is related to
stretch rate by
where
p is an indeterminate hydrostatic
pressure and is the viscosity.
2.4.5.1.
Calculate the
rate of external work done by the torque acting on the rotating exterior
cyclinder
2.4.5.2.
Calculate the rate
of internal dissipation in the solid as a function of r.
2.4.5.3.
Show that the
total internal dissipation is equal to the external work done on the specimen.