Problems for Chapter 3
Constitutive Models: Relations between Stress and Strain
3.7. Small Strain Plasticity

3.7.1. The stress state induced by stretching a large plate containing a cylindrical hole of radius a at the origin is given by
$\begin{array}{l}{\sigma}_{11}={\sigma}_{0}\left(1+\left(\frac{3{a}^{4}}{2{r}^{4}}\frac{{a}^{2}}{{r}^{2}}\right)\mathrm{cos}4\theta \frac{3{a}^{2}}{2{r}^{2}}\mathrm{cos}2\theta \right)\\ {\sigma}_{22}={\sigma}_{0}\left(\left(\frac{{a}^{2}}{{r}^{2}}\frac{3{a}^{4}}{2{r}^{4}}\right)\mathrm{cos}4\theta \frac{{a}^{2}}{2{r}^{2}}\mathrm{cos}2\theta \right)\\ {\sigma}_{12}={\sigma}_{0}\left(\left(\frac{3{a}^{4}}{2{r}^{4}}\frac{{a}^{2}}{{r}^{2}}\right)\mathrm{sin}4\theta \frac{{a}^{2}}{2{r}^{2}}\mathrm{sin}2\theta \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$
Here, ${\sigma}_{0}$ is the stress in the plate far from the hole. (Stress components not listed are all zero)
3.7.1.1. Plot contours of vonMises equivalent stress (normalized by ${\sigma}_{0}$ ) as a function of $r/a$ and $\theta $, for a material with $\nu =0.3$ Hence identify the point in the solid that first reaches yield.
3.7.1.2. Assume that the material has a yield stress Y .Calculate the critical value of ${\sigma}_{0}/Y$ that will just cause the plate to reach yield.

3.7.2. The stress state (expressed in cylindricalpolar
coordinates) in a thin disk with mass density ${\rho}_{0}$ that spins with angular velocity $\omega $ can be shown to be
$\begin{array}{l}{\sigma}_{rr}=\left(3+\nu \right)\frac{{\rho}_{0}{\omega}^{2}}{8}\left\{{a}^{2}{r}^{2}\right\}\\ {\sigma}_{\theta \theta}=\frac{{\rho}_{0}{\omega}^{2}}{8}\left\{(3+\nu ){a}^{2}(3\nu +1){r}^{2}\right\}\end{array}$
Assume that the disk is made from an elasticplastic material with yield stress Y and $\nu =1/3$.
3.7.2.1. Find a formula for the critical angular velocity that will cause the disk to yield, assuming VonMises yield criterion. Where is the critical point in the disk where plastic flow first starts?
3.7.2.2. Find a formula for the critical angular velocity that will cause the disk to yield, using the Tresca yield criterion. Where is the critical point in the disk where plastic flow first starts?
3.7.2.3. Using parameters representative of steel, estimate how much kinetic energy can be stored in a disk with a 0.5m radius and 0.1m thickness.
3.7.2.4. Recommend the best choice of material for the flywheel in a flywheel energy storage system.

3.7.3. An isotropic, elasticperfectly plastic thin film with Young’s Modulus $E$, Poisson’s ratio $\nu $ , yield stress in uniaxial tension Y and thermal expansion coefficient $\alpha $ is bonded to a stiff substrate. It is stress free at some initial temperature and then heated. The substrate prevents the film from stretching in its own plane, so that ${\epsilon}_{11}={\epsilon}_{22}={\epsilon}_{12}=0$, while the surface is traction free, so that the film deforms in a state of plane stress. Calculate the critical temperature change $\Delta {T}_{y}$ that will cause the film to yield, using (a) the Von Mises yield criterion and (b) the Tresca yield criterion.
3.7.4. Assume that the thin film
described in the preceding problem shows so little strain hardening behavior
that it can be idealized as an elasticperfectly plastic solid, with uniaxial
tensile yield stress Y. Suppose the film is stress free at some
initial temperature, and then heated to a temperature $\beta \Delta {T}_{y}$,
where $\Delta {T}_{y}$ is the yield temperature calculated in the
preceding problem, and $\beta >1$.
3.7.4.1.
Find the stress in the film at this
temperature.
3.7.4.2.
The film is then cooled back to its original
temperature. Find the stress in the film
after cooling.

3.7.5. Suppose that the thin film described in the preceding
problem is made from an elastic, isotropically hardening plastic material with
a Mises yield surface, and yield stressvplastic strain as shown in the
figure. The film is initially stress
free, and then heated to a temperature $\beta \Delta {T}_{y}$,
where $\Delta {T}_{y}$ is the yield temperature calculated in problem
1, and $\beta >1$.
3.7.5.1.
Find a formula
for the stress in the film at this temperature.
3.7.5.2.
The film is then
cooled back to its original temperature.
Find the stress in the film after cooling.
3.7.5.3. The film is cooled further by a temperature change $\Delta T<0$. Calculate the critical value of $\Delta T$ that will cause the film to reach yield again.

3.7.6. Suppose that the thin film described in the preceding
problem is made from an elastic, linear
kinematically hardening plastic
material with a Mises yield surface, and yield stressvplastic strain as shown
in the figure. The stress is initially
stress free, and then heated to a temperature $\beta \Delta {T}_{y}$,
where $\Delta {T}_{y}$ is the yield temperature calculated in problem
1, and $\beta >1$.
3.7.6.1.
Find a formula
for the stress in the film at this temperature.
3.7.6.2.
The film is then
cooled back to its original temperature.
Find the stress in the film after cooling.
3.7.6.3. The film is cooled further by a temperature change $\Delta T<0$. Calculate the critical value of $\Delta T$ that will cause the film to reach yield again.

3.7.7. A thinwalled tube of mean radius a and wall thickness t<<a is subjected to an axial load P which exceeds the initial yield load by 10% (i.e. $P=1.1{P}_{Y}$ ). The axial load is then removed, and a torque Q is applied to the tube. You may assume that the axial load induces a uniaxial stress ${\sigma}_{zz}=P/(2\pi at)$ while the torque induces a shear stress ${\sigma}_{z\theta}=Q/(2\pi {a}^{2}t)$. Find the magnitude of Q to cause further plastic flow, assuming that the solid is
3.7.7.1. an isotropically hardening solid with a Mises yield surface
3.7.7.2. a linear kinematically hardening solid with a Mises yield surface
Express your answer in terms of ${P}_{Y}$ and appropriate geometrical terms, and assume infinitesimal deformation.

3.7.8. A cylindrical, thinwalled pressure vessel with initial radius R, length L and wall thickness t<<R is subjected to internal pressure p. The vessel is made from an isotropic elasticplastic solid with Young’s modulus E, Poisson’s ratio $\nu $, and its yield stress varies with accumulated plastic strain ${\epsilon}_{e}$ as $Y={Y}_{0}+h{\epsilon}_{e}$. Recall that the stresses in a thinwalled pressurized tube are related to the internal pressure by ${\sigma}_{zz}=pR/(2t)$, ${\sigma}_{\theta \theta}=pR/t$
3.7.8.1. Calculate the critical value of internal pressure required to initiate yield in the solid
3.7.8.2. Find a formula for the strain increment $d{\epsilon}_{rr},d{\epsilon}_{\theta \theta},d{\epsilon}_{zz}$ resulting from an increment in pressure $dp$
3.7.8.3. Suppose that the pressure is increased 10% above the initial yield value. Find a formula for the change in radius, length and wall thickness of the vessel. Assume small strains.
3.7.9. Write a simple program that will compute the history of stress resulting from an arbitrary history of strain applied to an isotropic, elasticplastic vonMises solid. Assume that the yield stress is related to the accumulated effective strain by $Y={Y}_{0}{(1+{\overline{\epsilon}}^{p}/{\epsilon}_{0})}^{n}$, where ${Y}_{0}$, ${\epsilon}_{0}$ and n are material constants. Check your code by using it to compute the stress resulting from a volume preserving uniaxial strain ${\epsilon}_{11}=\lambda ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\epsilon}_{22}={\epsilon}_{33}=\lambda /2$, and compare the predictions of your code with the analytical solution. Try one other cycle of strain of your choice.
3.7.10. In a classic paper, Taylor, G. I., and Quinney, I., 1932, “The Plastic Distortion of Metals,” Philos. Trans. R. Soc. London, Ser. A, 230, pp. 323$\u2013$362 described a series of experiments designed to investigate the plastic deformation of various ductile metals. Among other things, they compared their experimental measurements with the predictions of the vonMises and Tresca yield criteria and their associated flow rules. They used the apparatus shown in the figure. Thin walled cylindrical tubes were first subjected to an axial stress ${\sigma}_{zz}={\sigma}_{0}$. The stress was sufficient to extend the tubes plastically. The axial stress was then reduced to a magnitde $m{\sigma}_{0}$, with $0<m<1$, and a progressively increasing torque was applied to the tube so as to induce a shear stress ${\sigma}_{z\theta}=\tau $ in the solid. The twist, extension and internal volume of the tube were recorded as the torque was applied. In this problem you will compare their experimental results with the predictions of plasticity theory. Assume that the material is made from an isotropically hardening rigid plastic solid, with a Von Mises yield surface, and yield stressvplastic strain given by $Y={Y}_{0}+h{\overline{\epsilon}}^{p}$.
3.7.10.1. One set of experimental results is illustrated in the figure to the right. The figure shows the ratio $\tau /{\sigma}_{0}$ required to initiate yield in the tube during torsional loading as a function of m. Show that theory predicts that $\tau /{\sigma}_{0}=\sqrt{\left(1{m}^{2}\right)/\sqrt{3}}$
3.7.10.2. Compute the magnitudes of the principal stresses $({\sigma}_{1},{\sigma}_{2},{\sigma}_{3})$ at the point of yielding under combined axial and torsional loads in terms of ${\sigma}_{0}$ and m.
3.7.10.3. Suppose that, for a given axial stress ${\sigma}_{zz}=m{\sigma}_{0}$, the shear stress $\tau $ is first brought to the critical value required to initiate yield in the solid, and is then increased by an infinitesimal increment $d\tau $. Find expressions for the resulting plastic strain increments $d{\epsilon}_{zz},d{\epsilon}_{\theta \theta},d{\epsilon}_{rr},d{\epsilon}_{r\theta}$, in terms of m, ${\sigma}_{0}$, h and $d\tau $.
3.7.10.4. Hence, deduce expressions for the magnitudes of the principal strains increments $(d{\epsilon}_{1},d{\epsilon}_{2},d{\epsilon}_{3})$ resulting from the stress increment $d\tau $.
3.7.10.5. Using the results of 11.2 and 11.6, calculate the socalled “Lode parameters,” defined as
$\nu =2\frac{(d{\epsilon}_{2}d{\epsilon}_{3})}{(d{\epsilon}_{1}d{\epsilon}_{3})}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu =2\frac{({\sigma}_{2}{\sigma}_{3})}{({\sigma}_{1}{\sigma}_{3})}1$
and show that the theory predicts $\nu =\mu $
3.7.11. The Taylor/Quinney experiments show that the constitutive equations for an isotropically hardening VonMises solid predict behavior that matches reasonably well with experimental observations, but there is a clear systematic error between theory and experiment. In this problem, you will compare the predictions of a linear kinematic hardening law with experiment. Assume that the solid has a yield function and hardening law given by
$f=\sqrt{\frac{3}{2}({S}_{ij}{\alpha}_{ij})({S}_{ij}{\alpha}_{ij})}{Y}_{0}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{\alpha}_{ij}=\frac{2}{3}cd{\epsilon}_{ij}^{p}$
3.7.11.1. Assume that during the initial tensile test, the axial stress ${\sigma}_{zz}={\sigma}_{0}$ in the specimens reached a magnitude ${\sigma}_{0}=\beta {Y}_{0}$, where ${Y}_{0}$ is the initial tensile yield stress of the solid and $\beta >1$ is a scalar multiplier. Assume that the axial stress was then reduced to $m{\sigma}_{0}$ and a progressively increasing shear stress was applied to the solid. Show that the critical value of $\tau /{\sigma}_{0}$ at which plastic deformation begins is given by
$\frac{\tau}{{\sigma}_{0}}=\frac{1}{\sqrt{3}}{\left\{\frac{1}{{\beta}^{2}}{(m1+{\beta}^{1})}^{2}\right\}}^{1/2}$
Plot $\tau /{\sigma}_{0}$ against m for various values of $\beta $.
3.7.11.2. Suppose that, for a given axial stress ${\sigma}_{zz}=m{\sigma}_{0}$, the shear stress is first brought to the critical value required to initiate yield in the solid, and is then increased by an infinitesimal increment $d\tau $. Find expressions for the resulting plastic strain increments, in terms of m, $\beta $, c ${Y}_{0}$ and $d\tau $.
3.7.11.3. Hence, deduce the magnitudes of the principal strains in the specimen $(d{\epsilon}_{1},d{\epsilon}_{2},d{\epsilon}_{3})$.
3.7.11.4. Compute the magnitudes of the principal stresses $({\sigma}_{1},{\sigma}_{2},{\sigma}_{3})$ at the point of yielding under combined axial and torsional loads in terms of m, $\beta $, and ${Y}_{0}$.
3.7.11.5. Finally, find expressions for Lode’s parameters
$\nu =2\frac{(d{\epsilon}_{2}d{\epsilon}_{3})}{(d{\epsilon}_{1}d{\epsilon}_{3})}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu =2\frac{({\sigma}_{2}{\sigma}_{3})}{({\sigma}_{1}{\sigma}_{3})}1$
3.7.11.6. Plot $\nu $ versus $\mu $ for various values of $\beta $, and compare your predictions with Taylor and Quinney’s measurements.
3.7.12. An elastic nonlinear kinematic hardening solid has Young’s modulus $E$, Poisson’s ratio $\nu $, a VonMises yield surface
$f({\sigma}_{ij},{\alpha}_{ij})=\sqrt{\frac{3}{2}\left({S}_{ij}{\alpha}_{ij}\right)\left({S}_{ij}{\alpha}_{ij}\right)}Y=0$
where Y is the initial yield stress of the solid, and a hardening law given by
$d{\alpha}_{ij}=\frac{2}{3}cd{\epsilon}_{ij}^{p}\gamma {\alpha}_{ij}d{\overline{\epsilon}}^{p}$
where $c$ and $\gamma $ are material properties. In the undeformed solid, ${\alpha}_{ij}=0$. Calculate the formulas relating the total strain increment $d{\epsilon}_{ij}$ to the state of stress ${\sigma}_{ij}$, the state variables ${\alpha}_{ij}$ and the increment in stress $d{\sigma}_{ij}$ applied to the solid
3.7.13. Consider a rigid nonlinear kinematic hardening solid, with yield surface and hardening law described in the preceding problem.
3.7.13.1. Show that the constitutive law implies that ${\alpha}_{kk}=0$
3.7.13.2. Show that under uniaxial loading with ${\sigma}_{11}=\sigma $, ${\alpha}_{22}={\alpha}_{33}={\alpha}_{11}/2$
3.7.13.3. Suppose the material is subjected to a monotonically increasing uniaxial tensile stress ${\sigma}_{11}=\sigma $. Show that the uniaxial stressstrain curve has the form $\sigma =Y+(c/\gamma )[1\mathrm{exp}(\gamma \epsilon )]$ (it is simplest to calculate ${\alpha}_{ij}$ as a function of the strain and then use the yield criterion to find the stress)
3.7.14. Suppose that a solid contains a large number of randomly oriented slip planes, so that it begins to yield when the resolved shear stress on any plane in the solid reaches a critical magnitude k.
3.7.14.1. Suppose that the material is subjected to principal stresses ${\sigma}_{1},{\sigma}_{2},{\sigma}_{3}$. Find a formula for the maximum resolved shear stress in the solid, and by means of appropriate sketches, identify the planes that will begin to slip.
3.7.14.2. Draw the yield locus for this material.
3.7.15. Consider a rate independent plastic material with yield criterion $f({\sigma}_{ij})=0$. Assume that (i) the constitutive law for the material has an associated flow rule, so that the plastic strain increment is related to the yield criterion by $d{\epsilon}_{ij}^{p}=d{\overline{\epsilon}}^{p}\partial f/\partial {\sigma}_{ij}$; and (ii) the yield surface is convex, so that
$f\left[{\sigma}_{ij}^{*}+\beta ({\sigma}_{ij}{\sigma}_{ij}^{*})\right]f\left[{\sigma}_{ij}^{*}\right]\ge 0$
for all stress states ${\sigma}_{ij}$ and ${\sigma}_{ij}^{*}$ satisfying $f({\sigma}_{ij})=0$ and $f({\sigma}_{ij}^{*})\le 0$ and $0\le \beta \le 1$. Show that the material obeys the principle of maximum plastic resistance.
3.7.16. The yield strength of a frictional material (such as sand) depends on hydrostatic pressure. A simple model of yield and plastic flow in such a material is proposed as follows:
Yield criterion $F({\sigma}_{ij})=f({\sigma}_{ij})+\mu {\sigma}_{kk}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f({\sigma}_{ij})=\sqrt{\frac{3}{2}{S}_{ij}{S}_{ij}}$
Flow rule $d{\epsilon}_{ij}^{p}=d{\overline{\epsilon}}^{p}\frac{\partial f}{\partial {\sigma}_{ij}}$
Where $\mu $ is a material constant (some measure of the friction between the sand grains).
3.7.16.1. Sketch the yield surface for this material in principal stress space (note that the material looks like a Mises solid whose yield stress increases with hydrostatic pressure. You will need to sketch the full 3D surface, not just the projection that is used for pressure independent surfaces)
3.7.16.2. Sketch a vector indicating the direction of plastic flow for some point on the yield surface drawn in part (3.8.3.1)
3.7.16.3. By finding a counterexample, demonstrate that this material does not satisfy the principle of maximum plastic resistance
$({\sigma}_{ij}{\sigma}_{ij}^{*})d{\epsilon}_{ij}^{p}\ge 0$
(you can do this graphically, or by finding two specific stress states that violate the condition)
3.7.16.4. Demonstrate that the material is not stable in the sense of Drucker $\u2013$ i.e. find a cycle of loading for which the work done by the traction increment through the displacement increment is nonzero.
3.7.16.5. What modification would be required to the constitutive law to make it satisfy the principle of maximum plastic resistance and Drucker stability? How does the physical response of the stable material differ from the original model (think about compaction under combined shear and pressure).