Identifying the symmetry class and determining the closest symmetry class of an elasticity tensor

Diner, Çağri (2009) Identifying the symmetry class and determining the closest symmetry class of an elasticity tensor. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In this thesis, two main problems are solved and then applied to a geophysical problem. First, we identify the symmetry class of an elasticity tensor, if it exhibits a symmetry. Second, if the tensor is generally anisotropic, we find the closest symmetric elasticity tensor of a given class among all possible orientations of the coordinate systems. Using these results, we suggest a method to find the symmetry class that is the “best choice” to represent the given anisotropic elasticity tensor. -- For an application of these methods and results, we investigate the closest symmetry class of a medium that is obtained by combining two differently oriented planar structures. More precisely, we combine two transversely isotropic (TI) media that may correspond to layering and cracks in a subsurface, and whose rotation axes are neither parallel nor perpendicular to each other. Then, we find the closest symmetry class of the combined medium as the angle varies between their orientations. We give several examples for the case of two TI media where the angle between their rotation axes are small (< 15°), intermediate (40° - 65°) and large (> 75°). We see that if the angle is large enough (> 75°), then the combination can be approximated as an orthotropic medium. If the angle is small (< 15°) then the resultant medium is close to TI symmetry. However, intermediate angles between the structures may or may not give the symmetry of the combined medium close to a higher symmetry class than monoclinic. -- Moreover, we investigate the velocities of the waves propagating in a combined medium that has more than one planar structure. We find that the fastest velocity direction of the waves is not aligned with any of the orientations of the TI media that composes the medium. -- To measure closeness in the space of elasticity tensors, we use the Euclidean norm for defining a distance function. The nonlinearity and existence of several extrema makes it difficult to find the absolute minimum of the distance function among all coordinate systems. Fortunately, in the case of monoclinic and TI symmetry, the parameters of the distance function reduce to two; there are three for other symmetry classes. Thus, one can plot the monoclinic- and TI-distance functions on the surface of the two-dimensional sphere. -- We prove that the symmetry of the elasticity tensor is also a symmetry of the monoclinic-distance function and vice versa. Furthermore, we show that the monoclinic-distance function vanishes along the normals of the mirror planes of the medium. Therefore, by observing the plot one can infer the symmetry of a given elasticity tensor. The plot also allows us to guide a search for finding the absolute minimum of the monoclinic-distance function. -- We prove that the value of the orthotropic-distance function for any coordinate system is half of the value of the sum of onoclinic-distance functions along particular directions. These directions correspond to three mutually perpendicular vectors that are the normals of the mirror planes of orthotropic symmetry. This relation allows us to use the plot of the monoclinic-distance function to determine the closest orthotropic elasticity tensor. -- Furthermore, we present examples of other symmetry classes, namely trigonal, tetragonal and cubic. We see that the orientations of the closest tetragonal, trigonal and cubic elasticity tensors are either the same as, or as close as 1°, to the orientation of the minimum of the sum of monoclinic-distance functions along some particular directions. These particular directions are aligned with the normals of the mirror planes of the corresponding symmetry class. Thus, one can use the plot of the monoclinic-distance function to infer about the closeness to any symmetry class.

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