Problems for Chapter 4
Solution to Simple Problems
4.2. Axially and
Spherically Symmetric Solutions for Elastic-Plastic Solids
4.2.1. The figure shows a long hollow cylindrical shaft with
inner radius a and outer radius b, which spins with angular speed about its axis. Assume that the disk is made from an
elastic-perfectly plastic material with yield stress Y and density .
The goal of this problem is to calculate the critical angular speed that will
cause the cylinder to collapse (the point of plastic collapse occurs when the
entire cylinder reaches yield).
cylindrical-polar basis shown, list any stress or strain components that must
be zero. Assume plane strain
Write down the boundary conditions that the
stress field must satisfy at r=a and r=b
Write down the
linear momentum balance equation in terms of the stress components, the angular
velocity and the disk’s density. Use polar coordinates and assume axial
Using the plastic
flow rule, show that if the cylinder deforms plastically under
plane strain conditions
yield criterion, show that the radial and hoop stress must satisfy
Hence, show that
the radial stress must satisfy the equation
calculate the critical angular speed that will cause plastic collapse.
4.2.2. Consider a spherical pressure vessel subjected to
cyclic internal pressure, as described in Section 4.2.4. Show that a cyclic
plastic zone can only develop in the vessel if exceeds a critical magnitude. Give a formula for the critical value of ,
and find a (numerical, if necessary) solution for .
4.2.3. A long cylindrical pipe with internal bore a and outer diameter b is made from an elastic-perfectly
plastic solid, with Young’s modulus E,
Poisson’s ratio and uniaxial tensile yield stress Y is subjected to internal pressure. The (approximate) solution for a cylinder
that is subjected to a monotonically increasing pressure is given in Section
4.2.6. The goal of this problem is to
extend the solution to investigate the behavior of a cylinder that is subjected
to cyclic pressure.
Suppose that the
internal pressure is first increased to a value that lies in the range ,
and then returned to zero. Assume that
the solid unloads elastically (so the change
in stress during unloading can be calculated using the elastic solution). Calculate the residual stress in the cylinder
Hence, determine the critical internal pressure
at which the residual stresses cause the cylinder to yield after unloading
Find the stress
and displacement in the cylinder at the instant maximum pressure, and after
subsequent unloading, for internal pressures exceeding the value calculated in
4.2.4. The following technique is sometimes used to connect
tubular components down oil wells. As manufactured, the smaller of the two
tubes has inner and outer radii ,
while the larger has inner and outer radii ,
so that the end of the smaller tube can simply be inserted into the larger
tube. An over-sized die is then pulled through the bore of the inner of the two
tubes. The radius of the die is chosen
so that both cylinders are fully plastically deformed as the die passes through
the region where the two cylinders overlap.
As a result, a state of residual stress is developed at the coupling,
which clamps the two tubes together.
Assume that the tubes are elastic-perfectly plastic solids with Young’s
modulus E, Poisson’s ratio and yield stress in uniaxial tension Y.
Use the solution
given in Section 4.2.6 to calculate the radius of the die that will cause both
cylinders to yield throughout their wall-thickness (i.e. the radius of the
plastic zone must reach d).
effectively subjects to the inner bore of the smaller tube to a cycle of
pressure. Use the solution to the preceding problem to calculate the residual
stress distribution in the region where the two tubes overlap (neglect end
effects and assume plane strain deformation)
calculate the value of b that gives
the strongest coupling.
4.2.5. A spherical
pressure vessel is subjected to internal pressure and is free of traction on its outer
surface. The vessel deforms by creep,
and may be idealized as an elastic-power law viscoplastic solid with flow
where are material properties and is the Von-Mises equivalent stress. Calculate the steady-state stress and strain
rate fields in the solid, and deduce an formula for the rate of expansion of
the inner bore of the vessel. Note that
at steady state, the stress is constant, and so the elastic strain rate must