Elasticity/Antiplane shear example 1

Example 1

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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

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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