Elasticity/Polar coordinates

The Edge Dislocation Problem

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Stress due to an edge dislocation

Assume that stresses vanish at   and that   is the radius of an undeformed cylindrical hole. Also stresses vanish at  . Relative displacement   is prescribed on each face of the cut.

The edge dislocation problem is a plane strain problem. However, it is not axisymmetric.

It is probable that   and   are symmetric about the   plane. Similarly, it is probable that   is symmetric about the   plane.

These probable symmetries suggest that we can use a stress function of the form

 

In cylindrical co-ordinates, the gudir beta Airy stress function leads to

 
 

and

 

Proceeding as usual, after plugging the value of   in to the biharmonic equation, we get

 

Applying the stress boundary conditions and neglecting terms containing  , we get

 

Next we compute the displacements, in a manner similar to that shown for the cantilever beam problem. The displacement BCs are   at   and   at  . We can use these to determine   and hence the stresses.

Rigid body motions are eliminated next by enforcing zero displacements and rotations at   and  . The final expressions for the displacements can then be obtained.

Sample homework problems

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Problem 1

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Consider the Airy stress function

 
  • Show that this stress function provides an approximate solution for a cantilevered triangular beam with a uniform traction   applied to the upper surface. The angle   is the angle subtended by the free edges of the triangle.
 
A cantilevered triangular beam with uniform normal traction
  • Find the value of the constant   in terms of   and  .

Solution:

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Given:

 

Using a cylindrical co-ordinate system, the stresses are

 

At  ,  ,  ,  . Therefore,   and  .

 

Hence, the shear traction BC is satisfied and the normal traction BC is satisfied if

 

At  ,  ,  ,  . Therefore,   and  . Both these BCs are identically satisfied by the stresses (after substituting for  ). Hence, equilibrium is satisfied.

 

To satisfy compatibility,  . Use Maple to verify that this is indeed true.

The remaining BC is the fixed displacement BC at the wall. We replace this BC with weak BCs at  . The traction distribution on the surface   are   and  . The statically equivalent forces and moments are

 

You can verify these using Maple.

Hence, the given stress function provides an approximate solution for the cantilevered beam (in the St. Venant sense).