A continuum formulation for elasto-plastic–damage strain gradient solids with granular microstructure

This presentation is devoted to the development of a continuum theory accounting for elastic [A] and some dissipative phenomena like damage and plasticity [B]. The physical object is defined as those materials having granular microstructure but can be generalized for any continuum model in the same way Navier [C] and Cauchy [D] did in their works. In other words, the continuum description is constructed assuming expressions of elastic and dissipation energies as well as postulating a hemi-variational principle, without incorporating any additional postulate like flow rules. Granular micromechanics is connected kinematically to the continuum scale through Piola’s ansatz (also called Cauchy–Born approximation). In elasticity, expressions for geometrically non-linear second gradient coefficients, both in the 2D and in the 3D cases, have been identified in terms of Young’s modulus, of Poisson’s ratio and of a microstructural length. Besides, Karush–Kuhn–Tucker (KKT)-type conditions, providing evolution equations for damage and plastic variables associated with grain–grain interactions, are derived solely from the fundamental postulates. Numerical experiments have been performed to investigate the applicability of the model. Cyclic loading–unloading histories have been considered to elucidate the material hysteretic features of the continuum, which emerge from simple grain–grain interactions. We also assess the competition between damage and plasticity, each having an effect on the other. Further, the evolution of the loadfree shape is shown not only to assess the plastic behavior, but also to make tangible the point that, in the proposed approach, plastic strain is found to be intrinsically compatible with the existence of a placement function. References [A] Emilio Barchiesi, Anil Misra, Placidi L, Emilio Turco (2021). Granular micromechanics-based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations. Zeitschrift für Angewandte Mathematik und Mechanik, vol. 101, e202100059. [B] Placidi L, Emilio Barchiesi, Anil Misra, Dmitry Timofeev (2021). Micromechanics-based elasto plastic– damage energy formulation for strain gradient solids with granular microstructure. Continuum Mechanics and Thermodynamics, vol. 33, p. 2213-2241. [C] Navier, CL. Sur les lois de l’equilibre et du mouvement des corps solides elastiques. Memoire de l’Academie Royale de Sciences 1827; 7: 375–393. [D] Cauchy, A-L. Sur l’equilibre et le mouvement d’un systeme de points materiels sollicites par des forces d’attraction ou de repulsion mutuelle. Excercises de Mathematiques 1826–1830; 3: 188–212.