The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on March 18, 2009
The Quarterly Journal of Mechanics and Applied Mathematics 2009 62(2):149-166; doi:10.1093/qjmam/hbp001
| ||||||||||||||||||||||||||||||||||||||||||||||||||
On obtaining effective orthotropic elasticity tensors
( Department of Mathematics and Statistics, Memorial University, St. John's NL A1C 5S7, Canada )
( Department of Earth Sciences, Memorial University, St. John's NL A1B 3X5, Canada )
mikhail{at}math.mun.ca
Corresponding author. mslawins{at}mun.ca
Received 16 July 2008.
Revise 24 December 2008.
 |
Abstract |
|---|
We consider the problem of obtaining the effective orthotropic tensor that corresponds to a given generally anisotropic one; herein, by ‘effective’, we mean the closest in the sense of the Frobenius norm, without a priori assuming the orientation of the orthotropic tensor. It is difficult to find the absolute minimum of the distance function since the minimization process is nonlinear, exhibiting several local minima. To find the effective orthotropic tensor, the minimization process must be performed on a three-dimensional manifold SO(3). In the case of monoclinic and transversely isotropic tensors, it can be performed on a two-dimensional sphere, which lends itself to an insightful plot that allows us to guide a numerical method. We use the orientation of the symmetry-plane normal of the effective monoclinic tensor to guide the method and obtain the effective orthotropic tensor—a two-step process.