Investigations on some models in gradient elastoplasticity

Gradient elastoplasticity theories include the dependence on the gradient of the plastic distortion field within a solid body. This is motivated by the need to incorporate an internal characteristic length scale and describe size effects. Adding this dependence also brings mathematical benefits for existence theorems, ensuring sufficient compactness to perform limit passages when nonlinearities—arising from complex constitutive models or large deformations—are involved. In this thesis, we develop and analyze two models of gradient plasticity, mainly addressing modeling aspects and existence theory. To develop these models, we systematically adopt a framework based on the principle of virtual powers, inspired by Gurtin and Anand's approach to gradient plasticity, where a microscopic balance of microforces—describing the plastic response—is added alongside the usual macroscopic balance of forces. The first developed model is a simplified one-dimensional model of a metallic cylindrical bar. Configurations are described by solely three fields along its length: the cross section's axial displacement, its transversal dilation, and the axial plastic distortion. We build a linearized model for small deformations, and a geometrically nonlinear model for large deformations. In both cases, we develop an existence theory, resorting to Han and Reddy's evolutionary variational inequalities for the small deformation model, and to Mielke and Theil's energetic solutions of rate-independent systems for the large deformation model. We then analyze the boundary value problem describing the tensile test—where an assigned tensile force or elongation is prescribed at the ends—to understand whether the model describes the necking phenomenon. We show the linearized model cannot predict necking, and perform a formal stability analysis at large deformations, gaining qualitative insights about its onset. The second gradient plasticity model describes plastic materials whose yield strength depends on hydrostatic pressure. Instead of adopting non-associated plasticity theory—commonly used in geotechnical modeling—we propose an alternative approach. It formally reproduces the classical one under particular constitutive relations, despite differing foundations. This avoids non-associated flow rules (which are harder to handle) and directly provides a clearer variational structure. The key idea is to include a kinematic constraint on the volumetric part of the plastic distortion (linking it to a damage field describing microcracks), impose it through a Lagrange multiplier representing internal pressure, and make the constitutive relation of the deviatoric plastic microforce dependent on it. To prove that the model admits solutions, we develop a general existence theory for constrained mechanical systems with a dissipation potential possibly depending on the Lagrange multiplier. The proof is based on a time discretization approach: the existence of a time-discrete approximate solution is established; then, the continuous-in-time solution is recovered in the limit of vanishing time step. The incremental problem has a nonlinear saddle-point structure, solved as follows: the constraint is exploited to explicitate one unknown in terms of the rest; these are found for an arbitrary Lagrange multiplier by solving a static nonlinear problem (using a non-smooth version of Minty-Browder theorem). Finally, the Lagrange multiplier is recovered as the solution of a fixed-point problem via the Schauder theorem. Having established existence, we propose a finite element numerical scheme to approximate solutions, validate it against an explicit analytical solution, and provide the numerical solution of a particular boundary value problem.