Applying notions from Eshelby and Maugin's mechanics to the non-linear continuum theory of displocation

The purpose of the present thesis is the study of continuum bodies with a continuous distribution of dislocations using notions from the theory of material mechanics. The whole context is within the geometrically nonlinear theory. We are trying to investigate the benefits one can obtain by studying the static and the dynamic theory of dislocations from the point of view of the material space. In Chapter 2 we give some historical clues together with the basic references on the subject. The first ideas concerning the crystalline dislocations until their discovery are mentioned. Their interaction with the differential geometry and the main investigators behind this project are also indicated together with the founders of the contemporary Continuum Mechanics. The chapter concludes with those who laid the foundations of the theory of material mechanics. In Chapter 3 we present the basic notions from the non-linear theory of elasticity. The balance laws of mass, momentum and angular momentum are formulated. Also, the first and the second law of thermodynamics are set out. Chapter 4 deals with the theory of dislocations. Starting from their discrete picture in monocrystalline bodies we pass to their continuum analog and, a step further, to bodies with a continuous distribution of dislocations. The ways the material momentum equation can be derived is the object of Chapter 5. This equation can be produced either by pulling back momentum” s equation to the material manifold, either by using notions from the calculus of variations and the Noether“s theorem for the direct and the inverse description of elasticity. The second derivation is feasible when we are in the context of the classical Lagrangian field theory. The duality in the derivation of the material momentum”s equation is highlighted also with the use of the balance of energy by requiring to be invariant under a suitable group of transformations as we present in the same chapter and constitutes a novel approach. The whole framework of this chapter is within the bounds of elasticity. Chapter 6 starts with the basic notions for materially uniform but inhomogeneous materials. For this kind of bodies we use Noether“s theorem in order to produce momentum”s and material“s momentum equation. Although only the second equation is affected from the inhomogeneities presence, it is an identity for the equations of momentum. Thus, we need new equations to describe phenomena where the field of defects can change as time elapses. These equations are derived in Chapter 8. In Chapter 7 we give three paradigms of the theory presented in the previous chapter. We study three materially uniform but inhomogeneous bodies subjected to anti-plane shear. It’s about three bodies with a continuous distribution of dislocations-two of the edge and one of the screw type. We investigate the form that the deformation takes since dislocations are present. For the cases under consideration the dislocations are not allowed to move. The arithmetic considerations are performed with the help of Femlab. The equations necessary for the case when the field of defects changes as time elapses are presented in Chapter 8. Starting from a generalized balance of energy we derive this equations by requiring invariance under a suitable group of transformations in the field of the defects. Also, we lay out the energy theorem and the Clausius-Duhem inequality that correspond to this generalised balance of energy. The conclusions of the thesis is the object of Chapter 9