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.
