Problem $3$

A long bar has its long axis aligned along the $z$ direction as shown below. A uniform normal compressive traction $p$ is applied on the top and bottom sides of the bar as shown. The bar has Young's modulus $E$ and Poisson's ratio

$u.$

a$.$ Starting with the $3$D form of Hooke's law, determine all nonzero components of the strain tensor as a functions of the variables given.

b$.$ Given that the bar is long and that there is no variation in the $z$ direction, plane strain can also be used to solve this problem. Draw a sketch showing the superposition required to use plane strain. $($One part of the superposition must be a state of plane strain.$)$

c$.$ Use the superposition you drew in part b to determine all nonzero components of the strain tensor. $($For the plane strain part of the superposition, you should use the plane strain form of Hooke's law.$)$

Hint: Your answers to parts a and c should be the same.