Elasticity/Principal stresses

Principal Stresses in Two and Three Dimensions

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The principal stresses are the components of the stress tensor when the basis is changed in such a way that the shear stress components become zero. To find the principal stresses in two dimensions, we have to find the angle   at which  . This angle is given by

 

Plugging   into the transformation equations for stress we get,

 

Where are the shear tractions usually zero in a body?

The principal stresses in three dimensions are a bit more tedious to calculate. They are given by,

 

where,

 

The quantities   are the stress invariants.

Note: Be careful while implementing above relations in a solver, as the value of:

 

can be out of range of  , which is (-1, 1).

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Introduction to Elasticity

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