We propose and analyze elasto-plastic models for crystalline materials as extension of Teodosiu's models in two directions: non-local crystal plasticity and crystalline elasto-plastic materials with microstructural defects. Kroner and Teodosiu (1972) argue that plasticity and viscoplasticity are typical properties of crystalline materials, which are generated by existing inside defects.
 
The local stress free (relaxed) isoclinic configurations where introduced by Teodosiu (1970) in order to define the multiplicative decomposition of the (anholonomic) deformation gradient into elastic and plastic parts. Material response is elastic with respect to the isoclinic configurations, while the plastic distortion and internal variables are defined by the appropriate evolution equations, see also Teodosiu and Sidoroff (1976), as well as Cleja-Tigoiu and Soos (1990) for the material symmetry concept. As a natural generalization of the models developed by Teodosiu et al. (1993), Cleja-Tigoiu and Pascan (2013) proposed a crystal plasticity model based on a multi-slip flow rule, coupled with diffusion like evolution equations for scalar dislocation densities, with hardening influenced by dislocations and the back stress involved in the activation condition dependent on the gradients of dislocation densities.

Teodosiu formulated and solved boundary value problems for "Elastic models of crystalline defects" (1982) within linear elasticity. Second grade hyperelastic materials with strain energy dependent on the non-compatible elastic distortion and its covariant derivative have been considered by Teodosiu in "Contributions to continuum theory of dislocations and initial stresses, I-III," (1967), Rev. Roum. Sci. Tech. The quasi-dislocations (described by the torsion tensor associated with a metric connection) are the only sources of the initial stresses and hyperstresses.

Our elasto-plastic models published in Cleja-Tigoiu (2010), Cleja-Tigoiu et al. (2016), (2019), aim

- to describe the behaviour of crystalline materials containing defects by non-local fields that are smooth over an interatomic length scale and at the time of micro-seconds;

- to elaborate a strategy to solve the initial boundary value problems for elasto-plastic materials with defects, such as dislocations, dislinations and grain boundaries;

- to propose the algorithms to simultaneously solve the incremental equilibrium equation, coupled with partial differential equations which describe the defects evolution.

At the initial moment we consider that certain heterogeneous distribution of the defects exists inside the material, which is characterized by a disclination density tensor, or a dislocation density tensor, respectively. We use arrays of disclination dipoles for modeling the grain boundaries, as they represent the misfit between the lattice orientation of the two single crystalline materials in contact. These defects become active if and only if the elastic stresses, which are determined by solving the elastic problem, reached the appropriate critical values.

Using the Riemann-Cartan differential geometry concepts the dislocation density measures the incompatibility of the plastic distortion and the disclination tensor characterizes the non-metric property of the so-called plastic connection with respect to the plastic metric tensor. The basic concepts as balance equations for macro and micro stresses and stress momenta, the principle of free energy imbalance were introduced and developed within continuum mechanics framework. The free energy density is function of elastic strain and defects, expressed by the torsion tensor associated with the plastic connection, the disclination tensor and its gradient.

The non-local evolution equations for plastic distortion and disclination tensor were defined to be compatible with the principle of the free energy imbalance. The initial and boundary value problems were formulated also for models with edge dislocation and wedge disclination. In the presented here examples we restricted to small distortions, a framework which is sufficient for sustaining the principal ideas from our paper concerning the evolution of dislocations, disclinations as well as their densities.