Planes and axes of symmetry in an elastic material

Abstract

Physical properties of elastic and piezoelectric materials are studied by using tensors. It is usual to represent a tensor by a matrix. If a tensor is invariant under rotation about a fixed axis, the matrix representing the tensor commutes with rotation matrix. Therefore these two matrices have common eigenvectors, consequently a knowledge of eigenvectors of the rotation matrix provides us with a fair amount of information about eigenvectors of the tensor. This result is utilized to derive familiar representations of a transversely isotropic tensor of rank 2 and the elasticity tensor possessing tetragonal symmetry. Representation of the elasticity tensor belonging to a particular symmetry class can be achieved in an elegant manner. In an arbitrary coordinate system, it is not obvious to identify the symmetry class of the elastic materials under debate. In such circumstances Cowin-Mehrabadi Theorem plays a vital role. Simple proofs are obtained for the Cowin-Mehrabadi Theorem for the identification of a plane of symmetry or an axis of symmetry in an elastic material. Necessary and sufficient Conditions are obtained for the identification of an n-fold axis of symmetry with n ≥ 3. The treatment is then generalized to a Cartesian tensor of arbitrary rank and consequently the necessary and sufficient conditions are also found for the existence of a plane of symmetry or an axis of symmetry for a piezoelectric material. Young’s modulus is a material property that describes the stiffness of an elastic mate rial. It is therefore one of the most important properties in engineering design. The familiar representation derived for a transversely isotropic (or hexagonal) material in this thesis is applied to find an expression for Young’s modulus and consider its optimum values. The expression of Young’s modulus for a hexagonal material is written in terms of one variable only and hence the problem is solved by a straightforward manner.