Elasticity/Equilibrium example 1

Example 1

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Given:

Euler's second law for the conservation of angular momentum

 

The divergence theorem

 

The equilibrium equation (Cauchy's first law)

 

Show:

 

Solution

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Let us first look at the first term of equation~(1) and apply the divergence theorem (2). Thus,

 

Plugging this back into equation~(1) gives

 

Therefore, bringing all terms to the left hand side,

 

Using the equilibrium equations~(3), equation~(5) reduces to

 

Since this holds for any  , we have

 

If you work this expression out, you will see that  . Hence, the stress tensor is symmetric.