@melly

The below one is principal invariants expressed i terms of squares of principal stretches in hyper elasticity

the below one is the stress invariant is expressed in terms of summation of principal stresses in elasticity

Why is the difference so

Well, this is pretty straight forward.Â

In the first case, they are known as 'strain invariants' because they are based on the Cauchy-Green deformation tensors (both the left and the right result to the same values) whereas the second case are known as 'stress invariants' and are based on the Cauchy stress tensor.Â

From the attached file, you can see how to get the strain invariants. The same method applies to stress invariants.Â

Hope it helps.Â

~Melly