Introduction to Elasticity/Antiplane shear example 1

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Example 1[edit | edit source]

Given:

The body , is supported at and loaded only by a uniform antiplane shear traction on the surface , the other surface being traction-free.

A body loaded in antiplane shear

Find:

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 ]

Solution[edit | edit source]

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

or,

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

Solving,

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 ,

Therefore,

The approximate displacement field is