Some solutions based on elastic-plastic and creep transition in homogeneous and non-homogeneous shells

Abstract

Elastic-plastic and creep analysis of shell structures have an important role in engineering applications. Engineers have found its increasing application in aerospace, chemical, civil and mechanical industries such as in high-speed centrifugal separators, gas turbines for high-power aircraft engines, spinning satellite structures, certain rotor systems and rotating magnetic shields. To increase the strength of shells, it is therefore very important for engineers to study the behaviour of elastic-plastic and creep transition of shells. A shell is a curved surface, in which the thickness is much smaller than the remaining dimensions. The geometrical properties of shells, i.e. single or double curvature give rise to a tremendous advantage of these light weight structures. The concept of elastic-plastic and creep transition problems is related to the science of forces and motions. The numerical solution of these problems can be achieved through a proper blending of the principles of mechanics with certain postulates and assumptions based on experiments and experience. This study comes under the field of Solid mechanics. The Solid mechanics is the branch of Continuum mechanics that deals with the behaviour of solid materials, especially their motion and deformation under the influence of forces, thermal effects and other different conditions. In order to make a mechanical body safe, a designer must have knowledge of the limiting conditions of stress at which temporary and permanent deformation starts to develop so that danger of yielding or fracture is to be eliminated. Deformations which are recovered after the removal of body forces are called elastic deformations. In this case, a body completely recovers its original configuration. On the other hand, if deformation remains even after body forces have been removed is called plastic deformation. It is one of type of irreversible deformations in which material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress. If the state of deformation in a body remains constant throughout the whole part of the material body is called homogeneous deformation and the tendency of a solid material to move slowly or deform permanently occur as result of long-term exposure to high level of stress is called creep deformation. Creep deformation does not occur suddenly upon the application of stress. Instead, strain accumulates as a result of long-term stress. Therefore, creep is a "time-dependent" deformation. The ‘theory of elasticity’ has been continuously developed by various investigators for anisotropic bodies and isotropic bodies since 1950. This theory is the solid foundation for the scientists in designing of engineering structures because of its increasing application to engineering problems. In the same manner, the designers are interested in the ‘theory of plasticity. It is helpful in understanding the deformation behaviors for avoiding excessive deflection or distortion in machine parts. In 1868, Tresca considered that there exists a ‘mid-zone’ area between the elastic and plastic regions which is against Saint-Venant’s two-zone theory. Although Tresca’s theory was ignored by later research workers for the sake of convenience. Many authors have not recognized mid zone area as a separate part that of elasticity and plasticity. A many attempts have been made in this direction of intermediate state illustrated by Thomas, Green and Seth. Seth has elaborated the concept of this intermediate region. He has named this region as “Transition region”. He has developed a transition theory of elastic-plastic and creep transitions. Seth has defined the concept of generalized strain measure which when applied to the governing differential equation of the medium eliminates ad-hoc assumptions like incompressibility, creep strain law and yield condition but also employs the same constitutive equations to give elastic-plastic and creep results through some transition functions. The most important contribution to be made by generalized measures is that they do not use of semi-empirical laws and jump conditions unnecessary. Therefore, concept of generalized measure in which two parameters are determined experimentally, give a better idea about creep behaviour. The classical macroscopic treatment of solving the problems in (i) plasticity, (ii) creep and (iii) relaxation has to assume semi-empirical yield conditions like those of Tresca and von Mises and Creep strain laws like those of Norton, Odquist and others. This is a direct consequence of using linear strain measures which neglect the non-linear transition region through which the yield occurs and the fact that creep and relaxation strains are never linear. Therefore classical elastic-plastic model does take into consideration the non linear part through which transition takes place. Here in this current literature, the problems of elastic -plastic and creep transitions are solved by considering non liner part through which the transition takes place. The transition theory of B.R. Seth is helpful to deal these types of problems. The motive of study is finding the solution of transition problems of shells with the different conditions. The study is helpful to know the conditions under which life of material parts in machines get longer, reducing wearing on mating parts, better mechanical dampening (less noise), faster operation of equipment, less power needed to run equipment, chemical and corrosion resistance etc. Therefore, the scope of study deals with the providing guidelines to designers for making products. It deals in developing new ideas to meet changing demands of products in machinery around the world so that innovative, collaborative spirit between people can be developed. This can lead to the production of industry’s broadest range of engineering materials. By study the nature of transition in spherical shells, one can determine the applications and mechanical requirements of body containing shell. The whole part of thesis is divided into eight chapters. The first chapter of thesis consists of brief introduction about shells, theory of elasticity, plasticity and creep. The various terms like stress, strain, generalized strain measures, and concept of transition state given by Seth has been discussed. The basic fundamental governing equations are defined in the thesis. The chapter discusses the brief literature review of problems on spherical shells. The chapter also consists of research methodology used in thesis to find the solution of the problems. Further thesis consists of seven more chapters depicting various research problems and their solutions based on elastic-plastic and creep transition. The second chapter deals with problem of “Elastic-plastic stress analysis in a Spherical shell under internal pressure and steady state temperature”. Curves have been drawn between stresses and radii ratio R = r/b for the initial state and fully plastic state. It has been observed that the circumferential stress is the maximum at the external surface. Thermal effect increases the value of circumferential stresses at the outer surface. The spherical shell made of the incompressible material required the higher value of circumferential stresses to start yielding as compared to compressible materials. The third chapter deals with the “Non-homogeneity effect in the spherical shell by using Seth's transition theory”. The effect of non-homogeneity has been discussed numerically and depicted graphically. It is assumed that non-homogeneity varies along the radius of the spherical shell. It has been observed that the spherical shell made of non-homogeneous material requires high pressure to attain fully plastic state from the elastic state as compared to the spherical shell made of homogeneous material. This indicates that spherical shell made of non-homogeneous material is on the safer side of design. The fourth chapter deals with the “Thermal Creep stress and strain analysis in non-homogeneous Spherical shell”. Seth’s transition theory is applied to the problem of creep stresses and strain rates in non-homogeneous spherical shell under steady-state temperature. Neither the yield criterion nor the associated flow rule is assumed here. With the influence of thermal effect, the values of circumferential stress are decrease at the external surface as well as internal surface of the spherical shell for different values of non-homogeneity. The fifth chapter deals with problem of “Elastic-Plastic transition on rotating Spherical shell in dependence of compressibility”. The purpose of paper is to establish the mathematical treatment of elastic-plastic transitions occurring in the rotating spherical shells and to find angular speed required to start yielding in rotating shells for compressible and incompressible materials. The effect of density variation parameter has been discussed numerically and depicted graphically. With the effect of density variation parameter, rotating spherical shells starts yielding at internal surface with the lower values of the angular speed for incompressible/compressible materials. The sixth chapter deals with problem of “Creep transition in the rotating spherical shell under the effect of density variable by Seth’s transition theory". The creep stresses and strain rates have been evaluated with the effect of variable density in a spherical shell. It has been seen that the values of radial/circumferential stresses must be decreased at the internal surface of the spherical shell with the effect of the density variation. This means that the possibility of fracture at the internal surface of the spherical shell decreases with the effect of density variable. The problem in chapter seventh has been discussed in two sections as (i) Elastic-plastic stress analysis of spherical shell under internal and external pressure and (ii) Transition analysis of spherical shell structures under external pressure. The results have been derived numerically and depicted graphically. The effect of the pressure is seen on the radial and circumferential stresses of the spherical shell. The eighth chapter summarizes the salient conclusions of this study. The end chapter also identifies the scope for future work.