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**Failure Theories Why do parts fail? What kind of stresses?**

What else should be considered?

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**Consider a tensile test specimen under load**

σ τ

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Plane Stress

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**Maximum Shear Stress Theory or Tresca Yield Criterion**

Yielding begins whenever the maximum shear stress in any element of a structural member or machine part becomes equal to the maximum shear stress in a tensile test specimen of the same material when that specimen begins to yield.

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**Maximum Distortion Energy Theory or von Mises – Hencky Theory**

When materials are deformed they store energy internally. The energy, u, stored per unit volume when a material undergoes a uniaxial stress, σ, that causes a strain, ε, can be defined in general as follows: By using Hooke’s law for a part subjected to a plane stress state, it can be shown that the amount of energy, ud, required to deform the part is as follows: For a uniaxial tensile test, the energy required to yield or deform a test specimen made from the same material as the part is as follows: By equating the energies, an equation can be developed that shows the relationship between the plane stress state defined by σ1 and σ2 in the part with the yield strength of a test specimen made from the same material:

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**Maximum Distortion Energy Theory (continued)**

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