Elasticity/Equilibrium example 2

Example 2

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Given: The displacement equation of equilibrium for an isotropic inhomogeneous linear elastic material can be written as

 

where

 

and   and   are the Lamé moduli.

Show:

Show that the displacement equation of equilibrium can be expressed as

 

Solution

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The skew part of the tensor   does not affect the stress because it leads to a rigid displacement field. Therefore, the displacement equation of equilibrium may be written as

 

where

 

In index notataion,

 

and

 

Therefore,

 

Now,

 

Hence,

 

Taking the divergence,

 

Recall that

 

Therefore,

 
 
 

Hence,

 

Therefore, the displacement equation of equilibrium can be expressed as required, i.e,