Crystalline solids with a uniform distribution of dislocations

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In this thesis, we investigate the problem of a crystalline solid containing a uniform distribution of elementary dislocations. In the first part, we introduce the basic concepts of a material manifold describing the intrinsic properties of a solid and a crystalline structure similar to that of a Lie algebra. We define the dislocation density as the continuum limit of a distribution of elementary dislocations in a crystal lattice. It is represented by a Burger’s vector. In the general case, the dynamics are then described by mappings from space-time to the material manifold. In the static case, the Euler-Lagrange equations – a system of elliptic partial differential equations expressing the divergence freeness of the stress tensor – are derived in the most general form. We complete the boundary value problem by adding the proper boundary conditions for a mapping from the material manifold into Euclidean space (static spacetime). They reflect the vanishing of the external forces on the boundary of the solid. The interchange of the roles of the domain and target space prevents us from solving a free boundary problem. Moreover, we state the Legendre-Hadamard conditions to justify the choice of the energy density. These conditions are given in terms of the derivatives of the energy with respect to the thermodynamic configuration, a symmetric bilinear form on the crystalline structure. The second part ends with a discussion of the equivalences for both the crystalline structures and the mechanics of a solid. In the third part, which constitutes the core part of the thesis, we study the special cases of uniform distributions of the two elementary types of dislocations, edge and screw dislocations in two and three space dimensions, respectively. It turns out that the material manifold can then be given the structure of a (non-Abelian) Lie group, the corresponding Lie algebra representing the crystalline structure. In the first case, upon choosing an isotropic energy density, the problem reduces to the study of energy minimizing mappings from a domain in the hyperbolic plane to Euclidean space. We show that the associated boundary value problem has a unique solution up to rotations and translations. This result is achieved by an iteration scheme, in which we choose a parameter reflecting the curvature sufficiently small, followed by a scaling argument. In the second case, however, the isotropy is broken by the dislocation lines, and we are left with mappings from the Heisenberg group manifold to Euclidean space. Using an anisotropic type of energy, we give the strategy to solve the problem, which, in contrast to the two-dimensional case, is not purely Riemannian.