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 $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ 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 thin-walled 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 $a=R\left[1-t/\left(2R\right)\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=R\left[1+t/\left(2R\right)\right]$ and expand the formulas for the stresses as a Taylor series in t/R.  Suggest an appropriate range of t/R for the thin-walled approximation to be accurate.

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 < 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

$\rho \left(R\right)={\rho }_{0}{a}^{2}/\left({a}^{2}+{R}^{2}\right)$

As a result, the interior of the solid is subjected to a radial body force field

$b=-\frac{ga}{R\left(1-\pi /4\right)}\left(1-\frac{a}{R}{\mathrm{tan}}^{-1}\frac{R}{a}\right){e}_{r}$

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 steady-state temperature field

$T\left(r\right)=\left({T}_{a}-{T}_{0}\right)\frac{{r}^{2}}{{a}^{2}}+{T}_{0}$

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

$T\left(r\right)=\frac{{T}_{a}\mathrm{log}\left(r/b\right)-{T}_{b}\mathrm{log}\left(r/a\right)}{\mathrm{log}\left(a/b\right)}$

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 Von-Mises 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.