The trace of the linearized strain tensor measures the relative change of volume, the determinant of the (left or right) Cauchy-Green tensor is the square of the ratio deformed volume to undeformed volume. Both invariants are used to split into volumetric and deviatoric tensors. What do the other invariants of symmetric second order tensors measure? What do the trace and the second principal invariant of the Cauchy-Green tensor measure, particularly if the determinant = 1?
The principal invariants as such do not have a readily relatable physical meaning to them except for specific classes of deformations. That's partly why we generally either resort to decomposing the strain tensors into volumetric and deviatoric parts or do a spectral decomposition (Ogden model). There are certain other invariants which are defined in such a way that they have a physical significance attached to them. Penn invariants, Lode invariants, Chen invariants are some examples.
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