Page 341 - Introduction to Continuum Mechanics

Simple Shear of an incompressible Isotropic Elastic Rectangular Block 325

Since these stress components are constants, therefore the equations of equilibrium are clearly
satisfied. Also, from the boundary conditions that on the surface^ = h, T 22 — 0 and on the
surface x$ = c, 733 = 0, we obtain

2
everywhere in the bar. From these equations, we obtain ( noting that A^ = 1)

Thus, the normal stress TU needed to stretch the bar (which is laterally unconfined) in the
x\ direction is given by

5.36 Simple Shear of an Incompressible Isotropfc Elastic Rectangular Block
The state of simple shear deformation is defined by the following equations relating the
spatial coordinates */ to the material coordinates Xj:

The deformed configuration of the rectangular block is shown in plane view in Fig. 5.19, where
one sees that the constant K is the amount of shear
The left Cauchy-Green tensor B and its inverse are given by

The scalar invariants are

Thus, from Eq. (5.34.9), we have