Defects and metric anomalies in Foppl-von Karman surfaces

A general framework is developed to study the deformation and stress response in Foppl-von Karman shallow shells for a given distribution of defects, such as dislocations, disclinations and interstitials, and metric anomalies, such as thermal and growth strains. The theory includes dislocations and disclinations whose defect lines can both pierce the two-dimensional surface and lie within the surface. An essential aspect of the theory is the derivation of strain incompatibility relations for stretching and bending strains with incompatibility sources in terms of the various defect and metric anomaly densities. The incompatibility relations are combined with balance laws and constitutive assumptions to obtain the inhomogeneous Foppl-von Karman equations for shallow shells. Several boundary value problems are posed, and solved numerically, by first considering only dislocations and then disclinations coupled with growth strains.