Problem

The problems for Sections 2.5-2.8 involve displacements, deformations, and strain states a...

The problems for Sections 2.5-2.8 involve displacements, deformations, and strain states at a point in a structural or machine member. These quantities, as with then stress counterparts, are important in design and failure criteria.

When solid circular torsion members are used to obtain material properties for finite strain applications, an expression for the engineering shear strain γzx is needed, where the (x, z) plane is a tangent plane and the z axis is parallel to the axis of the member as indicated in Figure P2.62. Consider an element ABCD in Figure P2.62 for the undeformed member. Assume that the member deforms such that the volume remains constant and the diameter remains unchanged. (This is an approximation of the real behavior of many metals.) Thus, for the deformed element A*B*C*D*, A*B* = AB, C*D* = CD, and the distance along the z axis of the member between the parallel curved lines A*B* and C*D* remains unchanged. Show that Eq. 2.71 gives the result γzx = tan α, where α is the angle between AC and A*C*, where γzx = 2ϵzx is defined to be the engineering shear strain.