Hi everyone, recently i have a question about how to calculate the orthotropic elastic matrix of cancellous bone cube. I really need the help!!!!
I applied six stress states to the cancellous bone cube by ABAQUS and got an anisotropic stiffness matrix whose every element has value (because the material coordinate axis is not consistent with the local axis) as shown in figure.
I changed the anisotropic matix (6×6) to a fourth order tensor Cmnpq.
I really want to rotate Cmnpq to an orthotropic fourth tensor Cijkl by the following relation:
Cijkl = RimRjnRkpRlqCmnpq,
the Rim is an rotation (3×3) matrix as shown in figure. The α, β and γ is the angle of rotation ( euler angle) about z-, x- and y-axes.
Now my idea is that, we can calculate brute-force the all rotation angle α, β and γ by 1 degree increments. When sum of C14, C15, C16, C24, C25, C26, C34, C35, C36, C55, C56, C66 of matrix is minimum closely to ZERO, the Cijkl is orthotropic tensor. And my question is:
- what scope of α, β and γ should I give in Python procedure? I can get all rotation situation.
- If I can use any optimization algorithm? rather than brute-force calculating by 1degree increments. Such as POWELL algorithm and how to use?
Thank you very much!
Your question is a standard problem for which you can find a solution in a good source on continuum mechanics. But your figure 1.bmp is confusing. It mixes matrix and tensor notation of stiffness. The 4th order tensor is simply transformed by a standard rotation matrix with direction cosines. The matrix form is transformed twice by a matrix which represents already 2 rotations by entries in the form of products of sine and cosine so that in the end you have again 4 rotations. Where have you found the input for your figure?
Your method of performing each of the 4 rotations by already 3 rotations is discussed with some of its problems here but also more simple transformations with only 1 rotation: https://www.continuummechanics.org/rotationmatrix.html
Note, that rotations are no vectors so that the sequence in which the rotations are performed matters. We have discussed your problem already at https://www.researchgate.net/post/How_to_do_coordinate_transformation_of_a_6x6_stiffness_matrix_of_tilted_transverse_isotropic_medium
I hope that this discussion can help you.
If I understood your question correctly, you have an anisotropic matrix and you want find a rotation matrix that transforms your anisotropic matrix into an orthotropic stiffness matrix. In that case you are looking at a Procrustes problem. There is a Matlab (and Python) function available for this so you don't need to program an optimization routine for this.
However, I'm not sure if in general for every anisotropic stiffness matrix there is a rotation matrix that can transform it into an orthotropic stiffness matrix. But the solution of the Procrustes analysis will give you the closest possible.
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