Airy stress function with body force

Airy stress function with body force potential

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If a body force exists, the Airy stress function ( ) has to be combined with a body force potential ( ). Thus,

 

or,

 

Is equilibrium still satisfied ?

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Recall the equilibrium equation in two dimensions:

 

or,

 

In terms of  ,

 

or,

 

Therefore,

 

or,

 

Hence,

 
  • Equilibrium is satisfied only if the body force field can be expressed as the gradient of a scalar potential.
  • A force field that can be expressed as the gradient of a scalar potential is called conservative.

What condition is needed to satisfy compatibility ?

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Recall that the compatibility condition in terms of the stresses can be written as

 

where,

 

or,

 

or,

 

which is the same as,

 

Plug in the stress potential and the body force potential in equation (18) to get

 

or,

 

Rearrange to get

 

which is the same as,

 

Therefore,

 

Compatibility is satisfied only if equation (24) is satisfied.

Equations for Airy stress function with body force potential

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The relation between the Cauchy stress and the Airy stress function is (in direct tensor notation)

 

The relation between the body force and the body force potential is

 

We also have to satisfy the compatibility condition for the Airy stress function to be a true stress potential, i.e.,

 

In rectangular Cartesian coordinates

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In rectangular Cartesian coordinates, the relation between the Cauchy stress components and the Airy stress function + body force potential can be written as

 

or,

 

The relation between the body force components and the body force potential are:

 

or,

 

The compatibility condition is written as

 

or,

 

In cylindrical coordinates

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In cylindrical coordinates, the relation between the Cauchy stresses and the Airy stress function + body force potential can be written as

 

The components of the body force are related to the body force potential via

 

The compatibility condition can be expressed as

 


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Introduction to Elasticity