Example 6

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Given:

For an isotropic material

 

Verify:

  1.  
  2.  

where  .

Solution

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From the second equation that has been given

 

or,

 

Therefore,

 

or,

 

or,

 

To find out whether the plus or the minus sign should be placed before   in the above equation, we put everything in terms of   and  . Thus,

 

Plugging these into the equation for  , multiplying both sides by   and dividing by  , we get

 

The limiting value of   is 0.5. Plugging this value into the above equation, we get,

 

The above can be true only if the sign is positive. Therefore, the correct relation is

 

Plugging this relation into the first of the given equations, we have,