Two complementary methods of inferring elastic symmetry
Walter Tape and Carl Tape
Journal of Elasticity, 2022
An elastic map T describes the strain-stress relation at a particular point p in some material. A symmetry of T is a rotation of the material, about p, that does not change T. We describe two ways of inferring the group ST of symmetries of any elastic map T; one way is qualitative and visual, the other is quantitative. In the first method, we associate to each T its "monoclinic distance function" fT mono on the unit sphere. The function fTmono is invariant under all of the symmetries of T, so the group ST is seen, approximately, in a contour plot of fTmono. The second method is harder to summarize, but it complements the first by providing an algorithm to compute the symmetry group ST. In addition to ST, the algorithm gives a quantitative description of the overall approximate symmetry of T. Mathematica codes are provided for implementing both the visual and the quantitative approaches.
Suggested approach to the material:
1. check out these google slides
2. check out the paper
3. explore the mathematica notebooks that reproduce many of the equations and figures,
such as for finding the symmetry group or plotting the elastic map on the sphere
(pdf examples: sym group and contours)
4. check out the 2021 paper on the 6-ball basis
Reference shapes for 7 symmetry classes, from left to right: monoclinic, trigonal, orthorhombic, tetragonal, cubic, transverse isotropic, isotropic. The 8th symmetry class, trivial, can be represented by any irregular shape.
Figure 1. The zero contour (blue) of fTmono on the unit sphere for various elastic maps T. Each T here has as symmetry group one of either "reference" groups Utriv, ..., Uiso as indicated. The points of the zero contour fTmono are where the axes of the 2-fold symmetries of T intersect the unit sphere. Since the 2-fold rotations in any elastic symmetry group generate the group, the zero contour of fTmono determines the symmetry group of T. For any elastic map T, the zero contour of fTmono is one of the eight shown here, though probably reoriented.
Figure 2. A suggestion of the elastic symmetry class Ctet, showing only three of its infinitely many members.
Figure 3. Contour plots of fTmono for eight elastic maps T having symmetry groups Utriv, ..., Uiso, the same as in Fig. 1.
Figure 4. Four of the contour plots from Fig. 3 but seen looking down the z-axis. (The x-axis is to the right.) From left to right, the z-axis is a 2-fold, 3-fold, 4-fold, and regular axis, respectively, for the relevant elastic map. Except in the Umono diagram, the zero contour is not conspicuous in this view, since most of its points are in the xy-plane.
Figure 8. Contour plot of fTmono for the elastic map T = T3. The zero contour of fTmono on the sphere appears to be a great circle together with its poles, as if the symmetry of T were transverse isotropic. The contour plot as a whole, however, shows that the symmetry cannot be transverse isotropic, since the contours would then have to be concentric circles. Instead, it appears that the symmetry is tetragonal and that the symmetry group is one of the conjugates of Utet that have v1 as a 4-fold axis. The 2-fold axes, however, are not easy to discern from the contour plot alone.