This theory states that the failure occurs when the maximum shear strain energy component for the complex state of stress system is equal to that at the yield point in the tensile test.
Hence the criterion for the failure becomes
As we know that a general state of stress can be broken into two components i.e,
(i) Hydrostatic state of stress ( the strain energy associated with the hydrostatic state of stress is known as the volumetric strain energy )
(ii) Distortional or Deviatoric state of stress ( The strain energy due to this is known as the shear strain energy )
As we know that the strain energy due to distortion is given as
This is the distortion strain energy for a complex state of stress, this is to be equaled to the maximum distortion energy in the simple tension test. In order to get we may assume that one of the principal stress say ( s1 ) reaches the yield point ( syp ) of the material. Thus, putting in above equation s2 = s3 = 0 we get distortion energy for the simple test i.e