Study On Symmetry Classification Of High-order Tensors In Mechanics

For uniform crystal materials, at a point, the material property is general different in distinct direction, but in some special directions, the material property is the same for the material symmetry. The tensor can be used as a description of the physical property, so we can define its symmetry by it invariance under the orthogonal transformation. The space of tensors is divided into equivalence classes by a relation stating that such tensors are equivalent when the share the same type of symmetry, or, more precisely, when there exist a rotation transformation make their point group equal. e.g., The linear elasticity tensors possess 8 classes of symmetry, determining the number and types of symmetries is a fundamental problem in mechanics theory, these results are useful and even indispensable for erperimentally identifying or theoretically/numerically estimating the tensor components.The tensor of high order, more complex of the symmetry classes analysis. In advanced theories of mechanics, many higher-order tensors are involed, such as the fourth-order flexoelectric tensor and photoelastic tnesor, and the fifth-order elastic stiffness tensor, and their classes of symmetry also need to be clarified. The third-order piezoelectricity tensors seems to be simpler than the fourth-order linear ealsticity one, yet its total number of symmetry classes is still inconclusive. Therefore, it is of great importance to develop an effective method to study the symmetry classes of higher-order tensors.In this paper, by means of the irreducible decomposition of tensor and the multipole representation of a deviator to a scalar and a unit vector set, we develop a new methord to study the symmetry classes of high-order tensors. In this method, we first obtain the deviator symmetry from the unit vector set, then determine the symmetry classes of higher-order tensors by the intersection of point groups of all deviators. This paper has studied the symmetry problem of several important tensors(third- to fifth-order), the main contributions are as follows.(1) A new method is developed to analyze the symmetry properties of tensors. This method has a clear geometric representation so that it is more comprehensible. Furthermore, this method is valid for both even-order and odd-order tensor.(2) We find the maxim of symmetry types of even-order tensors, and prove the conclusion mathematically. Obtain the number and classes of symmetry for all kinds of fourth-order tensors, find that there are only 8 classes or 12 classes possible. We develop a method based on mirror plane to identify the symmetry class of a linear elasticity tensor. Identify the natural coordination and the number of independent material parameters for flexoelectric tensors and photoelastic tnesors. Meanwhile, we prove that the classes of symmetry for fourth-order tensors is not dependent on the numbers of same order deviators on the irreducible decomposition of tensors.(3) We also find the maxim of symmetry types of odd-order tensors, and prove it mathematically, which is a useful foundation for the odd-order tensors symmetry analysis. We study all of the third-order tensors’ symmetry problem, make sure that there are 15 classes of symmetry for piezoelectric tensors. Carry out the study of fifth-order tensors, firstly obtain the 28 classes of symmetry for the fifth-order elastic stiffness tensor M and the number of independent components. For the third- and fifth- order tensors, we prove that the symmetry classes of tensor is not connection with the numbers of same order deviators on the irreducible decomposition of tensor, and there are three possibly numbers of symmetry classes for both of them.