Elasticity/Disk with hole

Disk with a central hole

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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

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at  
 
at  
 

Express   in Fourier series form.

 

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