Two Complementary Methods of Inferring Elastic Symmetry

Abstract

AbstractAn elastic map $\mathbf {T}$ T describes the strain-stress relation at a particular point $\mathbf {p}$ p in some material. A symmetry of $\mathbf {T}$ T is a rotation of the material, about $\mathbf {p}$ p , that does not change $\mathbf {T}$ T . We describe two ways of inferring the group $\mathcal {S} _{ \mathbf {T} }$ S T of symmetries of any elastic map $\mathbf {T}$ T ; one way is qualitative and visual, the other is quantitative. In the first method, we associate to each $\mathbf {T}$ T its “monoclinic distance function” "Equation missing" on the unit sphere. The function "Equation missing" is invariant under all of the symmetries of $\mathbf {T}$ T , so the group $\mathcal {S} _{ \mathbf {T} }$ S T is seen, approximately, in a contour plot of "Equation missing". The second method is harder to summarize, but it complements the first by providing an algorithm to compute the symmetry group $\mathcal {S} _{ \mathbf {T} }$ S T . In addition to $\mathcal {S} _{ \mathbf {T} }$ S T , the algorithm gives a quantitative description of the overall approximate symmetry of $\mathbf {T}$ T . Mathematica codes are provided for implementing both the visual and the quantitative approaches.