Theories of Failure:

Different people have set different criteria for failure of a solid and therefore, a number of failure theories (also called as strength theories) have been formulated.

In what is known as Principal Stress Theory, if the maximum principal stress (or minimum principal stress in specific case of compression) reaches the similar value of maximum principal stress at failure in uniaxial strength test, then the yield of failure limit is considered as attained.

In according to Principal Strain Theory, a material is assumed to have reached the yield or failure limit while the maximum principal strain in the material has attained the value of the maximum principal strain at failure as observed in the uniaxial strength test.

According to Shear Stress Theory, a material is assumed to have reached the yield or failure limit whereas the maximum shear stress in the material has attained the value of the maximum shear stress at failure as observed in the uniaxial strength test.

In spite of the stress or strain limits as governing criteria for failure, the capacity of the material to store energy is also assumed as criteria for failure and theories have been formulated depend on these criteria.

According to Total Strain Energy Theory, a material is assumed to have reached the yield or failure limit while the total strain energy density (anywhere in the material) has reached the net strain energy density observed at failure in the case of uniaxial strength test.

Still another failure theory has been formulated, with the supposition that the distortion energy density, instead the total strain energy density, is important as failure criteria. According to this theory, known as Distortion Energy Theory, a material is assumed to have reached the yield or failure limit while the distortion energy density (anywhere within the solid) reaches the value of distortion energy density at failure as observed in the case of uniaxial strength test.

A few other theories are not so popular and therefore, are not dealt with here. Now, you might get a doubt, whether the different theories actually set different criteria for failure or are these theories just different ways of expressing the similar criteria and therefore, are essentially the similar? Let us verify a few simple cases and ascertain the answer.