Introduction to Elasticity/Equilibrium example 1

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Example 1[edit | edit source]

Given:

Euler's second law for the conservation of angular momentum

The divergence theorem

The equilibrium equation (Cauchy's first law)

Show:

Solution[edit | edit source]

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.