 Problems for Chapter 5

Analytical Techniques and Solutions for Linear Elastic Solids

5.1.  General Principles

5.1.1.      A spherical shell is simultaneously subjected to internal pressure, and is heated internally to raise its temperature at $r=a$ to a temperature ${T}_{a}$, while at $r=b$ its surface is traction free, and temperature is  ${T}_{b}$.  Use the principle of superposition, together with the solutions given in Chapter 4.1, to determine the stress field in the sphere. 5.1.2.      The stress field around a cylindrical hole in an infinite solid, which is subjected to uniaxial tension ${\sigma }_{11}={\sigma }_{0}$ far from the hole, is given by

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

Using the principle of superposition, calculate the stresses near a hole in a solid which is subjected to shear stress ${\sigma }_{12}={\sigma }_{0}$ at infinity.  5.1.3.      The stress field due to a concentrated line load, with force per unit out-of-plane distance P acting on the surface of a large flat elastic solid are given by

${\sigma }_{11}=-\frac{2P}{\pi }\frac{{x}_{1}^{3}}{{\left({x}_{1}^{2}+{x}_{2}^{2}\right)}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{22}=-\frac{2P}{\pi }\frac{{x}_{1}^{}{x}_{2}^{2}}{{\left({x}_{1}^{2}+{x}_{2}^{2}\right)}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{12}=-\frac{2P}{\pi }\frac{{x}_{1}^{2}{x}_{2}^{}}{{\left({x}_{1}^{2}+{x}_{2}^{2}\right)}^{2}}$

The stress field due to a uniform pressure distribution acting on a strip with width 2a is

$\begin{array}{l}{\sigma }_{22}=-\frac{p}{2\pi }\left(2\left({\theta }_{1}-{\theta }_{2}\right)+\left(\mathrm{sin}2{\theta }_{1}-\mathrm{sin}2{\theta }_{2}\right)\right)\\ {\sigma }_{11}=-\frac{p}{2\pi }\left(2\left({\theta }_{1}-{\theta }_{2}\right)-\left(\mathrm{sin}2{\theta }_{1}-\mathrm{sin}2{\theta }_{2}\right)\right)\\ {\sigma }_{12}=\frac{p}{2\pi }\left(\mathrm{cos}2{\theta }_{1}-\mathrm{cos}2{\theta }_{2}\right)\end{array}$

where $0\le {\theta }_{\alpha }\le \pi$ and ${\theta }_{1}={\mathrm{tan}}^{-1}{x}_{1}/\left({x}_{2}-a\right)$ ${\theta }_{2}={\mathrm{tan}}^{-1}{x}_{1}/\left({x}_{2}+a\right)$

Show that, for $\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}>>a$ the stresses due to the uniform pressure become equal to the stresses induced by the line force (you can do this graphically, or analytically). 5.1.4.      The stress field in an infinite solid that contains a spherical cavity with radius a at the origin, and is subjected to a uniform uniaxial stress ${\sigma }_{33}={\sigma }_{0}$ far from the sphere is given by

$\begin{array}{l}\frac{{\sigma }_{ij}}{{\sigma }_{0}}=\frac{3{a}^{3}}{2\left(7-5\nu \right){R}^{3}}\left(3-5\nu -4\frac{{a}^{2}}{{R}^{2}}\right){\delta }_{ij}+\frac{3{a}^{3}{x}_{i}{x}_{j}}{2\left(7-5\nu \right){R}^{5}}\left(6-5\nu -5\frac{{a}^{2}}{{R}^{2}}+10\frac{{x}_{3}^{2}}{{R}^{2}}\right)\\ \text{\hspace{0.17em}}+\frac{{\delta }_{i3}{\delta }_{j3}}{\left(7-5\nu \right)}\left(\left(7-5\nu \right)+5\left(1-2\nu \right)\frac{{a}^{3}}{{R}^{3}}+3\frac{{a}^{5}}{{R}^{5}}\right)-\frac{15{a}^{3}{x}_{3}\left({x}_{j}{\delta }_{i3}+{x}_{i}{\delta }_{j3}\right)}{\left(7-5\nu \right){R}^{5}}\left(\frac{{a}^{2}}{{R}^{2}}-\nu \right)\end{array}$

Show that the hole only influences the stress field in a region close to the hole.