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 thin-walled sphere with radius R and wall thickness t<<R are ${\sigma }_{\theta \theta }\approx {\sigma }_{\varphi \varphi }\approx pR/2t{\sigma }_{rr}\approx p/2$.  Compare this estimate with the exact solution in Section 4.1.4.  To do this, set $a=R[1-t/(2R)]b=R[1+t/(2R)]$ 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 $\Delta <<a$ 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

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

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

$T(r)=\left(T_a-T_0\right)\frac{r^2}{a^2}+T_0$

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

$T(r)=\frac{T_a\log (r/b)-T_b\log (r/a)}{\log (a/b)}$

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 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.