Numerical studies of stresses in isotropic and transversely isotropic materials

Abstract

Mechanics, the science of forces and motions, has numerous engineering applications. These applications have been achieved through a proper blending of the principles of mechanics with certain postulates and assumptions based on experiments and experience. Solid mechanics deals with the mechanical behavior of the deformable bodies subjected to various types of external forces. The research in solid mechanics basically deals with the understanding of mechanical phenomenon such as mechanical and structural technology. This in turn enhances the advance development of aerospace industry, surface transportation vehicles, earthquake resistant design, offshore structures, orthopedic devices, material processing and manufacturing technologies. Advances in the subject are central to assure the safety, reliability and economy in the design of the devices required for these diverse areas. The theory of elasticity, plasticity and creep for anisotropic bodies has been continually developed and enriched with new investigations for the last several decades. These materials have been widely used in many areas because of their excellent static and dynamic behavior. Unidirectional laminated materials (i.e. single response in one direction and different in other two directions) belong to the group of transversely isotropic materials. Five independent elastic constants are used to characterize these materials. Two independent elastic constants known as Lame’s parameter are used to characterize the isotropic materials. Investigators have worked out problems of stress analysis under elastic-plastic and creep theory by assuming certain ad-hoc and semi empirical laws, such as yield conditions, creep strain laws etc. In classical treatment, different constitutive equations are used for each state, which is based on some hypotheses that simplify the problem. First, the deformations are assumed to be small to make infinitesimal strain theory applicable. Secondly, the constitutive equation of the material are simplified by assuming incompressibility of the material and in some cases without this assumption, it is not even possible to find the solution of the problem in closed form. Seth (1962-1964) has developed a new transition theory of elastic-plastic and creep deformation. Many authors in the literature have explained that the transition from one state into another is an asymptotic phenomenon. Seth has argued that at transition the differential system should attain some criticality. Once the critical points are recognized, the asymptotic solutions at these transition points give the solution corresponding to the transition state which is neglected in the classical theory. This theory is applied to a large number of problems related to cylinders, disks, shells, etc. under different loading conditions. In this thesis, an attempt has been made to study elastic-plastic and creep problems in isotropic and transversely isotropic materials by using the transition theory as well as the classical theory.