Introduction to Elasticity/Plate with hole in shear

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Circular hole in a shear field[edit | edit source]

Elastic plate with circular hole under shear

Given:

  • Large plate in pure shear.
  • Stress state perturbed by a small hole.

The BCs are

at
at

We will solve this problem by superposing a perturbation due to the hole on the unperturbed solution. The effect of the perturbation will decrease with increasing distance from the hole, i.e. the effect will be proportional to .

Unperturbed Solution[edit | edit source]

Therefore,

Integrating,

Since is a potential, we can neglect the integration constants (these do not affect the stresses - which are what we are interested in). Hence,

or,

Note that we have arranged the expression so that it has a form similar to the Fourier series of the previous section.

Perturbed Solution[edit | edit source]

For this we have to add terms to in such a way that

  • The unperturbed solution continues to be true as .
  • The terms have the same form as the unperturbed solution,i.e., terms.
  • The new leads to stresses that are proportional to .

Recall,

where,

So the appropriate stress function for the perturbation is

or,

Hence, the stress function appropriate for the superposed solution is

We determine and using the boundary conditions at .

The stresses are

Hence,

or,

Solving,

Back substituting,

Example homework problem[edit | edit source]

Consider the elastic plate with a hole subject to pure shear.

Elastic plate with a circular hole under pure shear

The stresses close to the hole are given by

  • Show that the normal and shear traction boundary conditions far from the hole are satisfied by these stresses.
  • Calculate the stress concentration factors at the hole, i.e., () (shear) and () (normal).
  • Calculate the displacement field corresponding to this stress field (for plane stress). Plot the deformed shape of the hole.


Solution[edit | edit source]

Far from the hole, . Therefore,

To rotate the stresses back to the coordinate system, we use the tensor transformation rule

Setting and , we get the simplified set of equations

Plugging in equations (32-34) in the above, we have

Hence, the far field stress BCs are satisfied.


The stresses at the hole () are

The maximum (or minimum) hoop stress at the hole is at the locations where . These locations are and . The value of the hoop stress is

The maximum shear stress is given by

Therefore, the stress concentration factors are

The stress function used to derive the above results was

From Michell's solution, the displacements corresponding to the above stress function are given by

or,

For plane stress, . Hence,

At ,

Now . Hence, we have

The deformed shape is shown below

Displacement field near a hole in plate under pure shear