# Elasticity/Sample quiz5

• An infinite body has a uniform state stress of pure hydrostatic pressure ${\displaystyle -p}$. This stress state is perturbed by a spherical void of radius ${\displaystyle a}$ which leads to a new stress state
${\displaystyle \sigma _{rr}=-p\left(1-{\frac {a^{3}}{r^{3}}}\right)~;~~\sigma _{\theta \theta }=\sigma _{\phi \phi }=-p\left(1+{\frac {a^{3}}{r^{3}}}\right)}$
What do you expect the values of ${\displaystyle \sigma _{r\theta }\,}$, ${\displaystyle \sigma _{\theta \phi }\,}$, and ${\displaystyle \sigma _{\phi r}\,}$ 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
${\displaystyle 2\mu u_{1}=-\varphi _{,1}+\alpha \psi _{,2}~;~~2\mu u_{2}=-\varphi _{,2}+\alpha \psi _{,1}}$
What is ${\displaystyle \psi \,}$ and what are the restrictions on it?