**Nonlocal Continuum Damage Plasticity: Theory and Computations**

This critical work is concerned with formulating the thermodynamics of nonlocal gradient-dependent plasticity based on the nonlocality energy residual. A thermodynamic based theory for both small strain and finite strain gradient plasticity is developed by introducing gradients for variables associated with kinematic and isotropic hardening. Starting from the classical (local) plasticity theory, thermodynamically consistent equations for the nonlocal plasticity yield criterion and flow rule are derived based on the principle of virtual power and the laws of the thermodynamics. It is shown that the presence of higher-order gradients in the plastic strain enforces the presence of a corresponding history variable brought by the accumulation of the plastic strain gradients. The later is similar to that of the history variable in the classical plasticity theory such that none of the plastic strain or the effective plastic strain exists without the other. Gradients in the plastic strain introduce anisotropy in the form of kinematic hardening and are attributed to the net Burgers vector whereas gradients in the accumulation of the plastic strain introduce isotropic hardening attributed to the additional storage of geometrically necessary dislocations. The equilibrium, or so-called microforce balance, between the internal Cauchy stress and the microstresses that are conjugates to the higher-order gradients, turns out to be the yield criterion which can be simply retrieved from the principle of virtual power. It is also shown that the local Clausius-Duhem inequality does not hold for gradient-dependent material and, therefore, a nonlocal form should be adopted. The nonlocal Clausius-Duhem inequality has an additional term, the nonlocality residual, that results from microstructural long-range energy interchanges between the material points within the body (i.e. the material points influence one another not by contact forces and thermal conduction only, but also by long-distance energy interchanges due to the nonlocality effects). The classical macroscopic boundary conditions are supplemented by non-classical microscopic boundary conditions associated with plastic flow. The developed nonlocal theory preserves the classical assumption of the local plasticity theory such that the plastic flow direction is governed by the deviatoric Cauchy stress. However, it is also argued here that plastic flow direction is the same as if it is governed by the nonlocal microstress. This is not inline with Gurtin (2003) who argued that the plastic flow direction is governed by a microstress and not the deviatoric Cauchy stress.

Some of the contributions of Professor Voyiadjis in this area are:

- finite strain plastic-damage model for high velocity impacts using combined viscosity and gradient localization limiters,
- numerical algorithms for the integration of the thermodynamically consistent formulation of geometrically nonlinear gradient-enhanced visco-inelasticity,
- unified integration algorithms are extensions of the classical rate-independent return mapping algorithms to the rate-dependent problems, namely an operator split structure is used consisting of a trial state followed by the return map by imposing the generalized viscoplastic and viscodamage consistency conditions simultaneously,
- finite deformation scheme based on hypoelastic stress-strain representations and the proposed elastic predictor and coupled viscoplastic-viscodamage corrector algorithm allows for the total uncoupling of geometrical and material nonlinearities,
- a simple and direct computational algorithm is also used for calculation of the higher-order gradients, this algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches for integrating gradient-dependent models,
- the proposed model is implemented in the explicit finite element code ABAQUS via the user subroutine VUMAT,
- model capabilities are illustrated for the dynamic localization of inelastic flow in adiabatic shear bands and the perforation of a thick Weldox 460E steel plates by deformable blunt projectiles at various impact speeds,
- simulated shear band results well illustrated the potential of the proposed model in dealing with the well-known mesh sensitivity problem,
- the introduced implicit and explicit length scale measures are able to predict size effects in localization failures and good agreement is obtained between the numerical simulations and experimental results of the perforation problem.

Professor Voyiadjis was invited to edit/co-edit the following special issues of journals:

**Voyiadjis, G. Z.**, Editor of the special issue “Multi-Scale Modeling of Materials,” in the Mechanics of Materials “MOM” Journal, Vol. 35, No. 8, 2003, pp. 719-844.

**Voyiadjis, G. Z.**, Pijaudier-Cabot, G., and Haj-Ali, R., Co-Editors of the special issue “Multi-Scale Modeling of Damage, and Material Characterization with Microstructure” in the Journal of Engineering Mechanics, ASCE, Vol. 127, No. 7, 2001, pp. 635-746.