Quasistatic elastoplasticity via Peridynamics: existence and localization

Abstract

Peridynamics is a nonlocal continuum-mechanical theory based on minimal regularity on the deformations. Its key trait is that of replacing local constitutive relations featuring spacial differential operators with integrals over differences of displacement fields over a suitable positive interaction range. The advantage of such perspective is that of directly including nonregular situations, in which discontinuities in the displacement field may occur. In the linearized elastic setting, the mechanical foundation of the theory and its mathematical amenability have been thoroughly analyzed in the last years.We present here the extension of Peridynamics to linearized elastoplasticity. This calls for considering the time evolution of elastic and plastic variables, as the effect of a combination of elastic energy storage and plastic energy dissipation mechanisms. The quasistatic evolution problem is variationally reformulated and solved by time discretization. In addition, by a rigorous evolutive Γ-convergence argument we prove that the nonlocal peridynamic model converges to classic local elastoplasticity as the interaction range goes to zero. 2

A peridynamic-based machine learning model for one-dimensional and two-dimensional structures

“…Peridynamics can be applicable for both elastic and plastic materials [6][7][8][9][10], composite and polycrystalline materials [11][12][13][14][15][16], multiphysics [17][18][19], large deformation problems [20], topology optimization [21] and multiscale modeling [22,23]. Peridynamics can also be suitable for structural idealization to analyze slender structures by using PD beam models [24][25][26][27] or thin wall structures by using PD plate and shell models [27][28][29][30][31][32].…”

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

“…Statebased material models are distinguished as ordinary and non-ordinary models. For ordinary state-based PD, the following material models are available: Elastic brittle [14,[33][34][35], Plasticity [36][37][38], Composite [39], Eulerian fluid [40], position-aware linear solid (PALS) [41], and Viscoelastic [42][43][44][45]. For non-ordinary state-based PD the correspondence model [14,46], the beam\plate model [47], and a model for cementitious composites [48] are available.…”

Review of Peridynamics: Theory, Applications, and Future Perspectives

“…Stefanelli et al [67], in the framework of nonlocal vector calculus, analysed the solution of the peridynamic elasto-plastic problem. The existence and convergence to the CCM solution were proved.…”