Theory of Cosserat-type Elastic Shells with Distributed Dislocations and Disclinations

Abstract

AbstractA mathematical model is constructed for Cosserat elastic shells containing continuously distributed dislocations and disclinations. Displacements, rotations, and strains are assumed to be small, i.e., the geometrically linear 6-parameter theory of shells is used. The system of equations describing the static deformations of the shells is derived by passing to the limit from a discrete set of isolated dislocations and disclinations to their continuous distribution. Deformational boundary conditions are derived, and a boundary value problem is formulated for the equilibrium of an elastic shell with given dislocation and disclination densities. With the help of stress functions, a variational statement of the boundary value problem of the statics of shells with distributed defects is given. A static-geometry analogy of the Cosserat shells theory considering distributed dislocations and disclinations is established. The dual boundary value problems of shell statics are formulated. The dual problems are mathematically equivalent, but completely different in their physical formulation. The problem of the stress state of a spherical shell with uniformly distributed dislocations and disclinations is solved.KeywordsDislocation and disclination densitiesDeformational boundary conditionsStress functionsStatic-geometry analogyDual boundary value problems