Introduction to 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?

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