Problems for Chapter 2
2.3. Equations of
motion and equilibrium for deformable solids
2.3.1. A prismatic concrete column of mass density supports its own weight, as shown in the figure. (Assume that the solid is subjected to a
uniform gravitational body force of magnitude g per unit mass).
Show that the
satisfies the equations of static equilibrium
and also satisfies the boundary conditions
on all free boundaries.
Show that the
traction vector acting on a plane with normal at a height is given by
Deduce that the
normal component of traction acting on the plane is
show also that
the tangential component of traction acting on the plane is
(the easiest way
to do this is to note that and solve for the tangential traction).
Suppose that the
concrete contains a large number of randomly oriented microcracks. A crack which lies at an angle to the horizontal will propagate if
where is the friction coefficient between the faces
of the crack and is a critical shear stress that is related to
the size of the microcracks and the fracture toughness of the concrete, and is
therefore a material property.
Assume that . Find the orientation of the microcrack that
is most likely to propagate. Hence, find
an expression for the maximum possible height of the column.
2.3.2. Is the stress field given below in static
equilibrium? If not, find the
acceleration or body force density required to satisfy linear momentum balance
Let be a twice differentiable, scalar function of
position. Derive a plane stress field
from by setting
that this stress field satisfies the equations of stress equilibrium with zero
The stress field
represents the stress in an infinite, incompressible
elastic solid that is subjected to a point force with components acting at the origin (you can visualize a
point force as a very large body force which is concentrated in a very small
region around the origin).
Verify that the
stress field is in static equilibrium
spherical region of material centered at the origin. This region is subjected to (1) the body
force acting at the origin; and (2) a force exerted by the stress field on the
outer surface of the sphere. Calculate
the resultant force exerted on the outer surface of the sphere by the stress,
and show that it is equal in magnitude and opposite in direction to the body
2.3.5. In this problem, we consider the internal forces in
the polymer specimen described in Problem 2.1.29 (you will need to solve 2.1.29
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
p is an indeterminate hydrostatic
pressure and is the viscosity.
for the Cauchy stress tensor, expressing your answer as components in
Assume steady, quasi-static
deformation (neglect accelerations).
Express the equations of equilibrium in terms of the angle
equilibrium equation, together with appropriate boundary conditions, to
Find the torque
necessary to rotate the external cylinder
acceleration of a material particle in the fluid
rotation rate where inertia begins to play a significant
role in determining the state of stress in the fluid