Dislocations and Green's functions in prestressed solids

The present Ph.D. dissertation is divided into two Parts: Green's function and problems of the inclusions and dislocations are addressed in the first Part, while implementation of elastoplastic constitutive laws are treated in the second. These subjects can be seen as different approaches to the investigation of the plastic behaviour of materials. In the first Part, infinite-body two-dimensional Green's functions are derived for the incremental deformation of an incompressible, anisotropic, prestressed body. These functions, given by Bigoni and Capuani, show the response of an infinite body to a concentrated force. The effect of prestress on dislocation (and inclusion) fields in non-linear elastic solids is analyzed by extending previous solutions by Eshelby and Willis. Employing a plane strain constitutive model (for incompressible incremental non-linear elasticity) to describe the behaviour a broad class of (anisotropic) materials, but with a special emphasis on ductile metals (J2-deformation theory of plasticity), it is shown that strongly localized strain patterns emerge, when a dislocation dipole is emitted by a source and the prestress level is high enough. These strain patterns may explain cascade activation of dislocation clustering along slip band directions. Several of the presented results remain valid within a three-dimensional context. Novel infinite-body three-dimensional Green's functions are derived for the incremental deformation of an incompressible, anisotropic, prestressed body. The case of a force dipole is developed within this framework. Results are used to investigate the behaviour of a material deformed near the limit of ellipticity loss and to reveal features related to shear failure cones development in a three-dimensional solid medium. Non-standard elastoplastic constitutive laws are treated in the second Part of the present Ph.D. dissertation, based on pressure-sensitive yield functions, such that proposed by Bigoni and Piccolroaz, which describes the inelastic deformation of ceramic powders and of a broad class of rock-like and granular materials. This yield function is not defined outside the yield locus, so that 'gradient-based' integration algorithms of elastoplasticity cannot be directly employed. Therefore, two ad hoc integration algorithms are proposed: an explicit scheme based on a forward Euler technique with a 'centre-of-mass' return correction and an implicit scheme based on a 'cutoff-substepping' return algorithm.