Thin spinning disk edit

Problem 1: edit

A thin disk of radius   is spinning about its axis with a constant angular velocity  . Find the stress field in the disk using an Airy stress function and a body force potential.

 
An elastic disk spinning around its axis of symmetry

Solution: edit

The acceleration of a point ( ) on the disk is

 

The body force field is

 

Since there is no rotational acceleration, the body force can be derived from a potential  . The relations between the stresses, the Airy stress function and the body force potential are

 

where

 

From equations (2) and (6) , we have,

 

Integrating equation (7), we have

 

Substituting equation (9) into equation (8), we get

 

This constant can be set to zero without loss of generality. Therefore,

 

The spinning disk problem is a plane stress problem. Hence the compatibility condition is

 

where

 

Now, from equations (11) and (13)

 

Therefore, equation (12) becomes

 

Since the problem is axisymmetric, there can be no shear stresses, i.e.   and no dependence on  . From Michell's solution, the appropriate terms of the Airy stress function are

 

Axisymmetry also requires that  , the displacement in the   direction must be zero. However, if we look at Mitchell's solution, we see that   is non-zero if the term   is used in the Airy stress function. Hence, we reject this term and are left with

 

If we plug this stress function into equation (16) we see that  . Therefore, equation (18) represents a homogeneous solution of equation (16). The   that is a general solution of equation (16) is obtained by adding a particular solution of the equation.

One such particular solution is the stress function   since the biharmonic equation must evaluate to a constant. Plugging this into equation (16) we have

 

or,

 

Therefore, the general solution is

 

The corresponding stresses are (from equations (3, 4, 5)),

 

At  , the stresses must be finite. Hence,  . At  ,  . Evaluating   at   we get

 

Substituting back into equations (22) and (23), we get