what is stress strain invariants?

What does stress and strain Invariants mean? In simple terms I do unerstand that parameters that do not depend on cordinate system are called invariants. like principal stress and strains are invariants. But often i come across, First invariant(perhaps relating to volumetric strain) and second invariant (perhaps reltated to deviatoric strain). what are these quantities? what is the physical meaning of deviatoric strain? its invariant and volumetric strain invariant?

I would like to have a physical insight of this. would anyone please help me.


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This a series of good question. I have always wonder what deviatoric stress and strain were. From my understanding, deviatoric strain is the summation of the principal strains subtracted by the mean normal strain. What this means or is used for, I have no clue.

The first invariant is e11 +e22. The second invariant is e11 ee22 - e12 e12. From my understanding, the invariants are constants, meaning that for any axes, regardless of how they are rotated, the strain calculated from one axes will be the same for the new axes. So, for a set of strain eij, with respect to axes xi, the set of strains e'ij will be the same for axes x'i.

I hope this clears some of this up.

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Robert Smith

A change in a body's shape can be described as the sum of a volumetric strain and a distortional (volume conserving) strain.

Deviatoric strain subtracts out volumetric strains and represent net distortion.

Stress-strain relationships are fundamentally different for volumetric and distortional strains.

Volumetric strains are inherently limited. Squeeze hard enough and the atoms cannot get any closer together while maintaining the same 3 relative space geometry.

On the other hand, distortional strains (shears) are not limited in the same way; one could in principle stir a cup of tea forever and it would remain tea.


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