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Approximately how many types of stress are known/defined? Ex. Piola stress, Cauchy stress ...

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Your examples aren't really "types" of stress, but different ways to measure stress. The real world structure just responds the way it responds - it doesn't worry about what sort of math you used when you tried to model its behaviour.

https://en.wikipedia.org/wiki/Stress_measures lists a handful different stress measures that have acquired "names". Cauchy stress ("true stress") and Second Piola-Kirchhoff stress are probably the most commonly used.

It doesn't really make sense to talk about a measure of stress in isolation, without also considering how to measure strain, and how to model the constitutive equations that give the relationship between stress and strain.

For example 2-PK stress works nicely with Green strain, which is a convenient way to define the behavior of a body which has "small" elastic deformations superimposed on arbitrary large rigid body motions - in particular, large rigid body rotations, where approximations like $\sin \theta \approx \theta$ are not appropriate.

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  • $\begingroup$ More importantly, what motivates the need for these different stress (or strain) measures? Why wouldn't one stress be sufficient to handle all potential linear elasticity situations? $\endgroup$
    – Paul
    Aug 24 '16 at 3:23

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