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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?


Francisco Lopez Jimenez's picture

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] 

Francisco Lopez Jimenez's picture

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. 

Francisco Lopez Jimenez's picture

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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