To determine if two elastic tensors are similar I suggest you (like previous answers) to evaluate the distance between the given tensors (may be within the specific symmetry) and set a criteria on the "closeness".

I recommended some care on the choice of the distance function, as solutions from the Euclidean length are not invariant (ensured) under inversion, but Riemannian, log-Euclidean ,... distances are (as described in the literature).

For an online work in the literature that may interest to the problem, see e.g. https://arxiv.org/pdf/cond-mat/0608311.pdf Moakher and Norris, "The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry" (2018).

Hope it may help.

How about performing an Eigen Analysis? The principal components (principal stresses or principal strains) would be very close in case your tensors are similar. Does that make sense? You could compare your tensor on those basis.

I think that you need to make the comparison by considering both a "global" and a "local" measure:

a) as global metric, you can take a norm of a matrix

https://www.mathworks.com/help/matlab/ref/norm.html

b) a local measure is necessary to detect if two matrices have a large deviation in at least one element, regardless they have comparable global norms. As a local measure you can take the max absolute value of the element-by-element difference 

max (abs(Aij-Bij))

In the case of Euclidean norm you have to consider max abs(lambdai), where lambdai (i=1,2,3) is eigenvalue of matrix (A-B). It is nearly the same that Giovanni suggested.

Thanks for replies. Now I have three ideas:

After I deal with the work at hand, I will do a test about the above methods and update the results here.

What about energy.

I kink they must privude same energy on an arbitrary strain tensor.

Comparison of energies again returns us to the norm of matrix (A-B):

|1/2 ε·A·εT -1/2 ε·B·εT| ≤ 1/2||A-B|| ||ε||2.

To determine if two elastic tensors are similar I suggest you (like previous answers) to evaluate the distance between the given tensors (may be within the specific symmetry) and set a criteria on the "closeness".

I recommended some care on the choice of the distance function, as solutions from the Euclidean length are not invariant (ensured) under inversion, but Riemannian, log-Euclidean ,... distances are (as described in the literature).

For an online work in the literature that may interest to the problem, see e.g. https://arxiv.org/pdf/cond-mat/0608311.pdf Moakher and Norris, "The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry" (2018).

Hope it may help.

Thanks for replies.

The paper recommended by Miguel Matos Neves helps me a lot.