Theories of Failure:If maximum principal stress is the design criteria then the maximum principal stress must not exceed the working stress for the material. For brittle materials, maximum principal theory is considered to be satisfactory because they do not fail by yield stress.
There are some theories to predict the failure of materials at certain amount of applied load. Theories are checked experimentally and they are supported by the experimental results too.(1) Maximum Principal stress Theory or Rankine's Theory:
(2) Maximum Principal Strain Theory or St. Venant's Theory:This theory says that a ductile material begins to yield when maximum principal strain reaches the strain at which yielding occurs in simple tension or when the minimum principal strain equals the yield point strain in simple compression.
This theory is satisfactory for the brittle materials but it has its drawbacks.
(3) Maximum Shear stress Theory or Guests theory:
The maximum shear stress is equal to the half of the difference between the maximum and minimum principal stresses.
This theory gives fairly justified for ductile materials however doesn't give good results for the materials subjected to the hydro-static pressure.
(4) Maximum strain energy theory or Haigh's theory:
This theory says that inelastic action at any point in a body due to any state of stress begins only when energy per unit volume absorbed at any point in a body due to any state of stress begins only when energy per unit volume absorbed at that point is equal to energy absorbed per unit volume of the material when subjected to elastic limit under a uni-axial state of stress as occurs in a simple tensile test. This theory has considerable experimental support for ductile material specially thick cylinders.
(5) Maximum shear strain energy theory or Mises Hensky theory:
The portion of the energy producing change in shape of the element is assumed to be completely responsible for the failure of material by yielding. This theory is in good agreement with experimental results of ductile materials.
(Reference: Made Easy, Strength of Materials by Er. R.K.Rajput)