Problems for Chapter 6

Analytical Techniques and Solutions for Plastic Solids

6.2.  Bounding Theorems

6.2.1.      The figure shows a pressurized cylindrical cavity.  The solid has yield stress in shear k.  The objective of this problem is to calculate an upper bound to the pressure required to cause plastic collapse in the cylinder

6.2.1.1.            Take a volume preserving radial distribution of velocity as the collapse mechanism.  Calculate the strain rate associated with the collapse mechanism

6.2.1.2.            Apply the upper bound theorem to estimate the internal pressure p at collapse.  Compare the result with the exact solution

6.2.2.      The figure shows a proposed collapse mechanism for indentation of a rigid-plastic solid.  Each triangle slides as a rigid block, with velocity discontinuities across the edges of the triangles.

6.2.2.1.            Assume that triangle A moves vertically downwards.  Write down the velocity of triangles B and C

6.2.2.2.            Hence, calculate the total internal plastic dissipation, and obtain an upper bound to the force P

6.2.2.3.            Select the angle $\theta$ that minimizes the collapse load.

6.2.3.      The figure shows a kinematically admissible velocity field for an extrusion process.   The velocity of the solid is uniform in each sector, with velocity discontinuities across each line.   The solid has shear yield stress k

6.2.3.1.            Assume the ram EF moves to the left at constant speed V. Calculate the velocity of the solid in each of the three separate regions of the solid, and deduce the magnitude of the velocity discontinuity between neighboring regions

6.2.3.2.            Hence, calculate the total plastic dissipation and obtain an upper bound to the extrusion force P per unit out-of-plane distance

6.2.3.3.            Select the angle $\theta$ that gives the least upper bound.

6.2.4.      The figure shows a kinematically admissible velocity field for an extrusion process.   Material particles in the annular region ABCD move along radial lines.  There are velocity discontinuities across the arcs BC and AD.

6.2.4.1.            Assume the ram EF moves to the left at constant speed V.  Use flow continuity to write down the radial velocity of material particles just inside the arc AD.

6.2.4.2.            Use the fact that the solid is incompressible to calculate the velocity distribution in ABCD

6.2.4.3.            Calculate the plastic dissipation, and hence obtain an upper bound to the force P.

6.2.5.      The purpose of this problem is to extend the upper bound theorem to pressure-dependent (frictional) materials.   Consider, in particular, a solid with a yield criterion and plastic flow rule given by

$\begin{array}{l}f\left({\sigma }_{ij}\right)={\sigma }_{e}+\frac{\mu }{2}{\sigma }_{kk}-Y=0\\ {\stackrel{˙}{\epsilon }}_{ij}^{p}=\frac{{\stackrel{˙}{\epsilon }}_{e}}{\sqrt{1+{\mu }^{2}/2}}\frac{\partial f}{\partial {\sigma }_{ij}}=\frac{{\stackrel{˙}{\epsilon }}_{e}}{\sqrt{1+{\mu }^{2}/2}}\left(\frac{3{S}_{ij}}{2{\sigma }_{e}}+\frac{\mu }{2}{\delta }_{ij}\right)\\ {S}_{ij}={\sigma }_{ij}-\left({\sigma }_{kk}{\delta }_{ij}\right)/3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{e}=\sqrt{3{S}_{ij}{S}_{ij}/2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˙}{\epsilon }}_{e}=\sqrt{2{\stackrel{˙}{\epsilon }}_{ij}^{p}{\stackrel{˙}{\epsilon }}_{ij}^{p}/3}\end{array}$

where $\mu <1$ is a friction coefficient like material parameter.  The solid is subjected to a traction ${t}_{i}^{*}$ on its exterior boundary $\partial R$ and a body force ${b}_{i}$ per unit volume in its interior.  The solid collapses for loading $\beta {t}_{i}^{*}$, $\beta {b}_{i}$, where $\beta$ is a scalar multiplier to be deterined.

6.2.5.1.            Show that the rate of plastic work associated with a plastic strain rate ${\stackrel{˙}{\epsilon }}_{ij}^{p}$ can be computed as $Y{\stackrel{˙}{\epsilon }}_{e}/\sqrt{1+{\mu }^{2}/2}$

6.2.5.2.            We need to understand the nature of the plastic dissipation associated with velocity discontinuities in this material.  We can develop the results for a velocity discontinuity by considering shearing (and associated dilatation) of a thin layer of material with uniform thickness h as indicated in the figure. Assume that the strain rate in the layer is homogeneous, and that the surface at ${x}_{2}=h$ has a uniform tangential velocity ${v}_{t}$ and normal velocity ${v}_{n}$.  Show that (i) the rate of plastic work per unit area of the layer can be computed as ${\sigma }_{22}{v}_{n}+{\sigma }_{12}{v}_{t}=Y\sqrt{2{v}_{n}^{2}+{v}_{t}^{2}}/\sqrt{3\left(1+{\mu }^{2}/2\right)}$, and (ii) to satisfy the plastic flow rule the velocities must be related by ${v}_{n}/{v}_{t}=\sqrt{3}\mu /2\sqrt{\left(1-{\mu }^{2}\right)}$.  Note that these results are independent of the layer thickness, and therefore (by letting $h\to 0$ ) also characterize the dissipation and kinematic constraint associated with a velocity discontinuity in the solid.

6.2.5.3.            To state the upper bound theorem for this material we introduce a kinematically admissible velocity field v, which may have discontinuities across a set of surfaces $\stackrel{^}{S}$ in the solid.  Define the strain rate distribution associated with v as

${\stackrel{^}{\stackrel{˙}{\epsilon }}}_{ij}^{}=\frac{1}{2}\left(\frac{\partial {v}_{i}}{\partial {x}_{j}}+\frac{\partial {v}_{j}}{\partial {x}_{i}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{^}{\stackrel{˙}{\epsilon }}}_{e}=\sqrt{2{\stackrel{^}{\stackrel{˙}{\epsilon }}}_{ij}^{}{\stackrel{^}{\stackrel{˙}{\epsilon }}}_{ij}^{}/3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

The velocity field must satisfy ${\stackrel{^}{\stackrel{˙}{\epsilon }}}_{kk}=3\mu {\stackrel{^}{\stackrel{˙}{\epsilon }}}_{e}/2\sqrt{1+{\mu }^{2}/2}$ in the interior of the solid, and must satisfy

$\begin{array}{l}〚{v}_{n}〛=〚{v}_{t}〛\mu \sqrt{3}\text{\hspace{0.17em}}/2\sqrt{1-{\mu }^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}〚{v}_{n}〛=\left({v}_{i}^{+}-{v}_{i}^{-}\right){m}_{i}\\ 〚{v}_{t}〛=|{v}_{i}^{+}-{v}_{i}^{-}-〚{v}_{n}〛{m}_{i}|\end{array}$

on $\stackrel{^}{S}$, where ${m}_{i}$ denotes a unit vector normal to $\stackrel{^}{S}$.  Define the plastic dissipation function as

$\Phi \left(v\right)=\underset{R}{\int }\frac{Y{\stackrel{^}{\stackrel{˙}{\epsilon }}}_{e}}{\sqrt{1+{\mu }^{2}/2}}dV+\underset{\stackrel{^}{S}}{\int }\frac{Y}{\sqrt{3\left(1-{\mu }^{2}\right)}}〚{v}_{t}〛dA-\underset{R}{\int }{b}_{i}{v}_{i}dV-\underset{\partial R}{\int }{t}_{i}^{*}{v}_{i}dA$

Show that (i) $\Phi \left(\stackrel{˙}{u}\right)=0$, where $\stackrel{˙}{u}$ denotes the actual velocity field in the solid at collapse, and (ii) $\Phi \left(v\right)\ge 0$

6.2.5.4.            Hence, show that an upper bound to the load factor at collapse can be calculated as

${\beta }_{L}\le \frac{\underset{R}{\int }\frac{Y{\stackrel{^}{\stackrel{˙}{\epsilon }}}_{e}}{\sqrt{1+{\mu }^{2}/2}}dV+\underset{\stackrel{^}{S}}{\int }\frac{Y}{\sqrt{3\left(1-{\mu }^{2}\right)}}〚{v}_{t}〛dA}{\underset{R}{\int }{b}_{i}{v}_{i}dV+\underset{\partial R}{\int }{t}_{i}^{*}{v}_{i}dA}$

6.2.6.      As an application of the results derived in the preceding problem, consider a soil embankment with vertical slope, as shown in the figure.   The soil has mass density $\rho$ and can be idealized as a frictional material with constitutive equation given in the preceding problem.   Using a collapse mechanism consisting of shearing and dilatation along the line AB shown in the figure (the angle $\theta$ for the optimal mechanism must be determined), calculate an upper bound to the admissible height h of the embankment.

6.2.7.      The figure shows a statically indeterminate structure.  All bars have cross-sectional area A, Young’s modulus E and uniaxial tensile yield stress Y.  The solid is subjected to a cyclic load with mean value $\overline{P}$ and amplitude $\Delta P$ as shown

6.2.7.1.            Select an appropriate distribution of residual stress in the structure, and hence obtain a lower bound to the shakedown limit for the structure.  Show the result as a graph of $\Delta P$ as a function of $\overline{P}$

6.2.7.2.            Select possible cycles of plastic strain in the structure, and hence obtain an upper bound to the shakedown limit for the structure.

You should be able to find residual stresses and plastic strain cycles that make the lower and upper bounds equal, and so demonstrate that you have found the exact shakedown limit.

6.2.7.3.            Calculate upper and lower bounds to the shakedown limit for a beam subjected to three point bending as shown in the figure.  Assume the applied load varies cyclically with mean value $\overline{P}$ and amplitude $\Delta P$.

6.2.7.4.            The stress state induced by stretching a large plate containing a cylindrical hole of radius a at the origin  is given by

$\begin{array}{l}{\sigma }_{11}={\sigma }_{0}\left(1+\left(\frac{3{a}^{4}}{2{r}^{4}}-\frac{{a}^{2}}{{r}^{2}}\right)\mathrm{cos}4\theta -\frac{3{a}^{2}}{2{r}^{2}}\mathrm{cos}2\theta \right)\\ {\sigma }_{22}={\sigma }_{0}\left(\left(\frac{{a}^{2}}{{r}^{2}}-\frac{3{a}^{4}}{2{r}^{4}}\right)\mathrm{cos}4\theta -\frac{{a}^{2}}{2{r}^{2}}\mathrm{cos}2\theta \right)\\ {\sigma }_{12}={\sigma }_{0}\left(\left(\frac{3{a}^{4}}{2{r}^{4}}-\frac{{a}^{2}}{{r}^{2}}\right)\mathrm{sin}4\theta -\frac{{a}^{2}}{2{r}^{2}}\mathrm{sin}2\theta \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$

Use these results to calculate lower and upper bounds to the shakedown limit for the solid (assume that ${\sigma }_{0}$ varies periodically between zero and its maximum value)