Failure Rupture or Yield Theories

Ductile materials often have a stress-strain diagram similar to that of mild steel shown in Fig. 2.8 and can be approximated by a linear elastic-perfectly plastic material with a stress-strain diagram such as that given in Fig. 2.9(b). Failure for ductile materials is assumed to occur when stresses or strains exceed those at the yield point. Materials such as cast iron, glass, concrete and epoxy are very brittle and can often be approximated as perfectly linear elastic-perfectly brittle materials similar to that given in Fig, 2.9(a). Failure for brittle materials is assumed to occur when stresses or strains reach a value for which rupture (separation) will occur. 

The following are the simple statements and expressions for three well known and often used failure theories. They are described in terms of principal stresses, where Oi O2 C3, and a failure stress in a uniaxial tensile test, Ofl. , which is either the rupture stress or the yield stress as appropriate for the material. Typically, tensile and compression properties as found in a uniaxial test are assumed to be the same. 

Maximum normal stress theory (Lame-Navier) Failure occurs when the 

Maximum shear stress theory (Tresca) Failure occurs when the maximum shear stress at-an arbitrary point in a stressed body is equal to the maximum shear stress at failure (rupture or yield) in a uniaxial tensile test. 

Maximum distortion energy (or maximum octahedral shear stress) theory (von Mises) Failure occurs when the maximum distortion energy (or maximum octahedral shear stress) at an arbitrary point in a stressed medium reaches the value equivalent to the maximum distortion energy (or maximum octahedral shear stress) at failure (yield) in simple tension 

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