Introduction to Elasticity/Polar coordinates

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The Edge Dislocation Problem[edit | edit source]

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[edit | edit source]

Problem 1[edit | edit source]

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:[edit | edit source]

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).