Elasticity/Sample midterm3

Sample Homework Problem 3

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For an isotropic material with   GPa and  , find the stress tensor and strain energy density at a point in a body if the components of the strain tensor are given by

 

Solution

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The shear modulus ( ) is given by

 

The Lamé modulus ( ) is given by

 

The stress-strain relation for isotropic materials is

 

Therefore, (after converting   and   into KPa so that the   term in the strain cancels out),

 

In 3 3 matrix form (after converting into MPa from KPa)

 

The strain energy density is given by

 

Therefore,

 

The strain energy density is