This work deals with the macroscopic modeling of solid continua undergoing incompatible deformations due to the presence of microscopic defects like dislocations in single crystals. We propose a novel approach relying on a geometrical description of the medium by the strain tensor and the representation of internal efforts using zero-th and second-order strain gradients in an infinitesimal framework. At the same time, energetic arguments allow to monitor the corresponding tangent moduli. We provide mathematical and numerical results to support these ideas in the framework of isotropic constitutive laws. Discretization is made by means of Finite elements. The numerical method is innovative, based on concepts of exterior calculus and the notion of "Complexes", here applied to the so-called elasticity complex. New elements are implemented to address the specific tensor and 4th-order nature of the equations.  We will present the model and a series of numerical simulations on several benchmarks examples in plasticity. Hencky perfect plasticity as well as time-incremental plasticity with hardening will be presented. Comparison with classical models will be provided, in order to assess the validity of our model and highlight its main features. Comparison of the classical and new FE results will be presented as well.