Failure theory:

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Maximum Principal Strain Theory (St. Venant’s Theory):

Strain is an actual physical quantity related to the change in dimensions, but stress is an abstract concept, that is, internal resistance per unit area. Even for the determination of stresses, strain gauges are used to measure the strains and the principal stresses are indirectly measured using the Young’s modulus and Poisson’s ratio of the material. In this theory, it is assumed that failure by yielding of a material takes place when the maximum principal strain in the material subjected to principal stresses is equal to the strain at the yield point in a simple tension or compression test.

If p1 > p2 > p3 are the principal stresses, the maximum principal strain is

where E is the Young’s modulus and v is the Poisson’s ratio of the material.

Maximum shear stress theory:

This theory states that the failure can be assumed to occur when the maximum shear stress in the complex stress system is equal to the value of maximum shear stress in simple tension.

The criterion for the failure may be established as given below :

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For a simple tension case

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Maximum shear strain energy per unit volume theory:

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

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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 Deviatory 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

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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

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