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 $\omega$ about its axis.  Assume that the disk is made from an elastic-perfectly plastic material with yield stress Y and density $\rho$. 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).

4.2.1.1.            Using the cylindrical-polar basis shown, list any stress or strain components that must be zero.  Assume plane strain deformation.

4.2.1.2.             Write down the boundary conditions that the stress field must satisfy at r=a and r=b

4.2.1.3.            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 symmetry.

4.2.1.4.            Using the plastic flow rule, show that ${\sigma }_{zz}=\left({\sigma }_{rr}+{\sigma }_{\theta \theta }\right)/2$ if the cylinder deforms plastically under plane strain conditions

4.2.1.5.            Using Von-Mises yield criterion, show that the radial and hoop stress must satisfy $|{\sigma }_{\theta \theta }-{\sigma }_{rr}|=2Y/\sqrt{3}$

4.2.1.6.            Hence, show that the radial stress must satisfy the equation

$\frac{d{\sigma }_{rr}}{dr}=-\rho r{\omega }^{2}+\frac{2}{\sqrt{3}}\frac{Y}{r}$

4.2.1.7.            Finally, 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 $b/a$ exceeds a critical magnitude.  Give a formula for the critical value of $b/a$, and find a (numerical, if necessary) solution for $b/a$.

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 $\nu$ 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.

4.2.3.1.            Suppose that the internal pressure is first increased to a value that lies in the range $\left(1-{a}^{2}/{b}^{2}\right)<\sqrt{3}{p}_{a}/Y<2\mathrm{log}\left(b/a\right)$, 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 after unloading.

4.2.3.2.            Hence, determine the critical internal pressure at which the residual stresses cause the cylinder to yield after unloading

4.2.3.3.            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 4.2.3.2.

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 $\left(a,b\right)$, while the larger has inner and outer radii $\left(b,d\right)$, 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 $\nu$ and yield stress in uniaxial tension Y.

4.2.4.1.            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).

4.2.4.2.            The die 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)

4.2.4.3.            For $d/a=1.5$, calculate the value of b that gives the strongest coupling.

4.2.5.       A spherical pressure vessel is subjected to internal pressure ${p}_{a}$ 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 potential

$g\left({\sigma }_{e}\right)={\stackrel{˙}{\epsilon }}_{0}^{}{\left(\frac{{\sigma }_{e}}{Y}\right)}^{m}$

where $Y,m,{\stackrel{˙}{\epsilon }}_{0}$ are material properties and ${\sigma }_{e}$ 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 vanish.