Axial Rotation Symmetry Of Third-Order Nonlinear Elasticity Of Crystals

In this thesis, the axis rotation symmetry of third-order nonlinear elasticity was researched with the help of group theoretical methods. Using the properties of irreducible representation of group SO(2), The program for calculating the basis functions of the irreducible representation and independent components of the third-order nonlinear elastic coefficient was compiled with MATLAB. At the same time, the calculation result was checked.In this program, the higher-order basis functions of group SO(2) were constructed with the lower-order basis functions. The facultative order basis function can be derived with the help of the program. Then, a set of linearly independent basis functions of identity representation was picked out.With the programs, 141 sixth-order basis functions of irreducible representation of group SO(2) which are linearly independent were given out according to group representation theory. And the relationships among components of sixth-order tensor having the symmetry of the group SO(2) were obtained on the basis of that. It is indicated that, in the six-order tensor having the symmetry of the group SO(2), 364 tensor components are zero. Among the 365 non-zero ones, 141 components are independent and 224 components are dependent.The general form of the third-order nonlinear elastic coefficient tensor having the symmetry of the group SO(2) was obtained by delivering the symmetry of third-order nonlinear elastic coefficient tensor to the sixth-order tensor obtained. It is indicated that, there are 279 non-zero tensor components, 10 of those are independent.Contrasting the form of the known third-order nonlinear elastic coefficient tensors for variety of crystals and quasi-crystals with the general form obtained, it is indicated that,the third-order nonlinear elastic coefficient tensors for 32 point group crystal classes and pentagonal do not meet the requirements of general form obtained. So they do not have the arbitrary rotation symmetry with respect to the X3 axis in physical coordinate system (superposition with the higher-fold axis in crystal or quasi-crystal). While the phonon third-order nonlinear elastic coefficient tensors for decagonal quasi-crystal classes meet the requirements. It can be predicted that they are invariant under the rotation operation with respect to the X3 axis (superposition with the ten-fold symmetry axis in quasi-crystal classes).