The stored energy function
Consider a cubic element having unit edges is strained in such a way that in deformed state it is a cuboid having edges parallel to the axes of strain ellipsoid with lengths  and  respectively. Then due to incompressibility,
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(22.14) |
Then the work done in straining the material quasi-statically is  which can be worked out by considering the stresses which act on the deformed element  .
The element is subjected to three mutually perpendicular forces  ,  and  which are given by  ,  and  ,. Here  ,  and  are the areas on which the stresses  ,  and  act. Using the expression for , , and the incompressibility relation we have,
Work done in straining an element of volume to  is
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(22.17) |
Hence the work done in straining the material quasi-statically from dimension  to is
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(22.18) |
For an ideal rubber like material the elastic modulus  is given as  where  is the number of segments per unit volume,  is the Boltzmann’s constant and  is the absolute temperature.
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and 
, 
and 
