Example 1

edit

Given:

The compatibility equations in terms of the strains are (in index notation)

 

The stress-strain relations for a linear elastic material are

 

Show:

Substituting the stress-strain relations into the compatibility equations, show that the compatibility equation of stress can be expressed as

 

Solution

edit

Substituting the stress-strain relations into the left hand side of the compatibility equations and multiplying both sides by  , we get,

 

Now, the   rule states that

 

Therefore,

 

Recall that,

 

Therefore,

 

Hence,

 

Hence shown.