Introduction to Elasticity/Antiplane shear example 1
Example 1 
The body , is supported at and loaded only by a uniform antiplane shear traction on the surface , the other surface being traction-free.
Find the complete stress field in the body, using strong boundary conditions on and weak conditions on .
[Hint: Since the traction is uniform on the surface , from the expression for antiplane stress we can see that the displacement varies with . The most general solution for the equilibrium equation for this behavior is ]
Step 1: Identify boundary conditions
The traction boundary conditions in terms of components of the stress tensor are
Step 2: Assume solution
Assume that the problem satisfies the conditions required for antiplane shear. If is to be uniform along , then
The general form of that satisfies the above requirement is
where , , are constants.
Step 3: Compute stresses
The stresses are
Step 4: Check if traction BCs are satisfied
The antiplane strain assumption leads to the and BCs being satisfied. From the boundary conditions on , we have
This gives us the stress field
Step 5: Compute displacements
The displacement field is
where the constant corresponds to a superposed rigid body displacement.
Step 6: Check if displacement BCs are satisfied
The displacement BCs on and are automatically satisfied by the antiplane strain assumption. We will try to satisfy the boundary conditions on in a weak sense, i.e, at ,
This weak condition does not affect the stress field. Plugging in ,
The approximate displacement field is