Introduction to Elasticity/Disk with hole

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Disk with a central hole[edit | edit source]

An elastic disk with a central circular hole

Under general loading, for the stresses and displacements to be single-valued and continuous, they must be periodic in , e.g., .

An Airy stress function appropriate from this situation is

In the absence of body forces,

Plug in .

or,

Therefore,

To satisfy the compatibility condition , we need

The general solution of these Euler-Cauchy type equations is

We can use either to determine . Thus,

or,

The homogeneous and particular solutions of this equation are

Hence, the general solution is

This form is valid for . If , alternative forms are obtained. Thus,

Terms in are chosen according to the specific problem of interest.

Traction BCs[edit | edit source]

at
at

Express in Fourier series form.

Terms in are chosen according to the specific problem of interest.