We report a continuum theory for 2D strain gradient materials accounting for a class of dissipation phenomena. The continuum description is constructed by means of a (reversible) placement function and by (irreversible) damage and plastic functions. Besides, expressions of elastic and dissipation energies have been assumed as well as the postulation of a hemi-variational principle. No flow rules have been assumed and plastic deformation is also compatible, that means it can be derived by a placement function. Strain gradient Partial Differential Equations (PDEs), boundary conditions (BCs) and Karush-Kuhn-Tucker (KKT) type conditions are derived by a hemi variational principle. PDEs and BCs govern the evolution of the placement descriptor and KKT conditions that of damage and plastic variables. Numerical experiments for the investigated homogeneous cases do not need the use of Finite Element simulations and have been performed to show the applicability of the model. In particular, the induced anisotropy of the response has been investigated and the coupling between damage and plasticity evolution has been shown.

This paper presented a physico-mathematical model for dynamic fracture propagation in brittle materials with a purely continuum mechanics hemi-variational-based strain gradient theory. As for the quasi-static case, the simulation results, obtained by means of finite elements, revealed that strain gradient effects significantly affected the fracture propagation, leading to finite fracture thickness that was independent of the mesh size. It was also observed that nonsymmetric loading rate lead to a deviation from standard mode-Ⅰ crack propagation that cannot be revealed in the quasi-static case. The model results were compared against experimental data from fracture tests on notched specimens taken from the literature. The comparison showed good agreement between the model predictions and the experimental measurements. The presented model and simulation results can be useful in the design and optimization of structural components subjected to dynamic loading conditions.