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

${\sigma }_{rr}=-p(1-\frac{a^3}{r^3});{\sigma }_{\theta \theta }={\sigma }_{\varphi \varphi }=-p(1+\frac{a^3}{r^3})$

What do you expect the values of ${\sigma }_{r\theta }$, ${\sigma }_{\theta \varphi }$, and ${\sigma }_{\varphi r}$ 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

$2\mu u_1=-{\phi }_{,1}+\alpha {\psi }_{,2};2\mu u_2=-{\phi }_{,2}+\alpha {\psi }_{,1}$

What is $\psi$ and what are the restrictions on it?