Problems for Chapter 4
Solution to Simple Problems
4.1. Axially and Spherically Symmetric Solutions for Elastic Solids

4.1.1. A solid cylindrical bar with radius a and length L is subjected to a uniform pressure $p$ on its ends. The bar is made from a linear elastic solid with Young’s modulus $E$ and Poisson’s ratio $\nu $.
4.1.1.1. Write down the components of the stress in the bar. Show that the stress satisfies the equation of static equilibrium, and the boundary conditions ${\sigma}_{ij}{n}_{i}={t}_{j}$ on all its surfaces. Express your answer as components in a Cartesian basis $\{{e}_{1},{e}_{2},{e}_{3}\}$ with ${e}_{1}$ parallel to the axis of the cylinder.
4.1.1.2. Find the strain in the bar (neglect temperature changes)
4.1.1.3. Find the displacement field in the bar
4.1.1.4. Calculate a formula for the change in length of the bar
4.1.1.5. Find a formula for the stiffness of the bar (stiffness = force/extension)
4.1.1.6. Find the change in volume of the bar
4.1.1.7. Calculate the total strain energy in the bar.

4.1.2. Elementary calculations predict that the stresses in a
internally pressurized thinwalled sphere with radius R and wall thickness t<<R
are ${\sigma}_{\theta \theta}\approx {\sigma}_{\varphi \varphi}\approx pR/2t\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma}_{rr}\approx p/2$. Compare this estimate with the exact solution
in Section 4.1.4. To do this, set $a=R[1t/(2R)]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=R[1+t/(2R)]$ and expand the formulas for the stresses as a

4.1.3. A baseball can be idealized as a small rubber core with radius a, surrounded by a shell of yarn with outer radius b. As a first approximation, assume that the yarn can be idealized as a linear elastic solid with Young’s modulus ${E}_{s}$ and Poisson’s ration ${\nu}_{s}$, while the core can be idealized as an incompressible material. Suppose that ball is subjected to a uniform pressure p on its outer surface. Note that, if the core is incompressible, its outer radius cannot change, and therefore the radial displacement ${u}_{R}=0$ at $R=a$. Calculate the full displacement and stress fields in the yarn in terms of p and relevant geometric variables and material properties.
4.1.4. Reconsider problem 3, but this time assume that the core is to be idealized as a linear elastic solid with Young’s modulus ${E}_{c}$ and Poisson’s ration ${\nu}_{c}$. Give expressions for the displacement and stress fields in both the core and the outer shell.
4.1.5. Suppose that an elastic sphere, with outer radius $a+\Delta $, and with Young’s modulus $E$ and Poisson’s ratio $\nu $ is inserted into a spherical shell with identical elastic properties, but with inner radius $a$ and outer radius $b$. Assume that $\Delta <<a$ so that the deformation can be analyzed using linear elasticity theory. Calculate the stress and displacement fields in both the core and the outer shell.

4.1.6. A spherical planet with outer radius a has a radial variation in its density that can be described as
$\rho (R)={\rho}_{0}{a}^{2}/({a}^{2}+{R}^{2})$
As a result, the interior of the solid is subjected to a radial body force field
$b=\frac{ga}{R(1\pi /4)}\left(1\frac{a}{R}{\mathrm{tan}}^{1}\frac{R}{a}\right){e}_{r}$
where g is the acceleration due to gravity at the surface of the sphere. Assume that the planet can be idealized as a linear elastic solid with Young’s modulus $E$ and Poisson’s ratio $\nu $. Calculate the displacement and stress fields in the solid.
4.1.7. A solid, spherical nuclear fuel pellet with outer radius $a$ is subjected to a uniform internal distribution of heat due to a nuclear reaction. The heating induces a steadystate temperature field
$T(r)=\left({T}_{a}{T}_{0}\right)\frac{{r}^{2}}{{a}^{2}}+{T}_{0}$
where ${T}_{0}$ and ${T}_{a}$ are the temperatures at the center and outer surface of the pellet, respectively. Assume that the pellet can be idealized as a linear elastic solid with Young’s modulus $E$, Poisson’s ratio $\nu $ and thermal expansion coefficient $\alpha $. Calculate the distribution of stress in the pellet.

4.1.8. A long cylindrical pipe with inner radius a and outer radius b has hot fluid with temperature ${T}_{a}$ flowing through it. The outer surface of the pipe has temperature ${T}_{b}$. The inner and outer surfaces of the pipe are traction free. Assume plane strain deformation, with ${\epsilon}_{zz}=0$. In addition, assume that the temperature distribution in the pipe is given by
$T(r)=\frac{{T}_{a}\mathrm{log}(r/b){T}_{b}\mathrm{log}(r/a)}{\mathrm{log}(a/b)}$
4.1.8.1. Calculate the stress components ${\sigma}_{rr},{\sigma}_{\theta \theta},{\sigma}_{zz}$ in the pipe.
4.1.8.2. Find a formula for the variation of VonMises stress
${\sigma}_{e}=\sqrt{\frac{1}{2}\left\{{\left({\sigma}_{1}{\sigma}_{2}\right)}^{2}+{\left({\sigma}_{1}{\sigma}_{3}\right)}^{2}+{\left({\sigma}_{2}{\sigma}_{3}\right)}^{2}\right\}}$
in the tube. Where does the maximum value occur?
4.1.8.3. The tube will yield if the von Mises stress reaches the yield stress of the material. Calculate the critical temperature difference ${T}_{a}{T}_{b}$ that will cause yield in a mild steel pipe.