22 January 2021 2 9K Report

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!

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