Elasticity/Sample quiz5


 * An infinite body has a uniform state stress of pure hydrostatic pressure $$-p$$. This stress state is perturbed by a spherical void of radius $$a$$ which leads to a new stress state

\sigma_{rr} = -p\left(1 - \frac{a^3}{r^3}\right) ~; \sigma_{\theta\theta} = \sigma_{\phi\phi} = -p\left(1 + \frac{a^3}{r^3}\right) $$
 * What do you expect the values of $$\sigma_{r\theta}\,$$, $$\sigma_{\theta\phi}\,$$, and $$\sigma_{\phi r}\,$$ to be? What is the stress concentration factor at the void?


 * What are the displacement conditions for antiplane strain in rectangular co- ordinates? Give an example of a problem that can be approximated by antiplane strain.


 * What are the strain conditions for plane strain in rectangular co-ordinates? Give an example of a problem that can be approximated by plane strain.  What is the difference between plane strain and generalized plane strain?


 * What are the stress conditions for plane stress in rectangular co-ordinates? Give an example of a problem that can be approximated by plane stress.


 * Can the Airy stress function be used for three-dimensional problems? Write down the relation between the Airy stress function and stress in rectangular co-ordinates.  How does this relation change when you solve problems that involve a body force?  What additional restriction on the Airy stress function must be checked to see if compatibility is satisfied?


 * In the absence of body forces, the displacements (plane strain/stress) can be expressed as

2\mu u_1 = - \varphi_{,1} + \alpha \psi_{,2} ~; 2\mu u_2 = - \varphi_{,2} + \alpha \psi_{,1} $$
 * What is $$\psi\,$$ and what are the restrictions on it?