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Homogenization - If materials in the model are isotropic, is it possible to get truly anisotropic resulting material?


I read that "In general, even if the materials on the micro-level are isotropic, the effective 

material can show anisotropic behavior. A general anisotropic linear elastic material 

may have twenty one independent material parameters.''


If I understand my results correctly then simple structures like ''ball in the unit cell'' result in orthotropic material.

I am a bit puzzled - what would be the simplest structure that would result in anisotropic material behaviour?


A composite with isotropic matrix and fibers, all of them parallell, would do.

 For balls you'd need to know the spatial arrangement of the balls.

Ok-ey I run test with fiber and this is the effective tensor I got. It is hard to interpret it - zeros are not really zeros, but it is hard to tell if that is anisotropic behaviour or they simply have not converged.

[img_assist|nid=16713|title=effective tensor 2|desc=|link=none|align=left|width=640|height=222] 

I don't know which values are you using (sitffnesses and concentrations, for example), or how are you getting that matrix.

However, if you consider the case of carbon fibers (E ~ 200 GPa) in an epoxy matrix (E ~ 5 GPa), with the fibers extending only in the x direction, with a reasonable fiber volume fraction (something between 30% and 60% would do), you'll get quite different values for, say, Exx and Eyy in the homogenized material (you can do the math, but I think it's clear just by intuition alone).

Note: carbon fibers are anisotropic themselves, but even if you consider isotropic fibers, you get an anisotropic homogenized material.

Let me rephrase my question: Is it possible to get clearly non-zero matrix entries in top right and buttom left corners?

 I already have anisotropy in the sense of different values for Exx and Eyy. 

Those 3x3 matrices are zero for ortothropic materials, i.e. materials with three planes of symmetry. So I guess you could have them be non-zero, but I don't know how you'd homogenize something without symmetry...

nicoguaro's picture

For me, your stiffness tensor looks like just transverse isotropic.

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