# Thermodynamically consistent coupled thermo-mechanical gradient-enhanced continuum model of plasticity

The classical continuum theory of plasticity and heat transfer fails in predicting the microscale responses of metals in both space and time as they usually ignore the effect of the microstructure and its evolution in the course of plastic deformation and heat conduction. The evidence of such a behavior can be found in the size dependency of materials over a scale that ranges from a fraction of a micron to a hundred microns

under inhomogeneous plastic flow along with the heat transport responses under both short time and spatial scales. Moreover, for microstructural optimization of material properties, the plastic deformation mechanisms on the grain level play a significant role. A similar strengthening effect is also associated with decreasing the grain size in polycrystalline materials due to the increase in yield stress which is referred to as the Hall–Petch effect.

Professor Voyiadjis has worked on the gradient-enhanced continuum theory to assess the thermal and mechanical responses of small-scale metallic compounds under fast transient processes. Such framework is based on a system of microscopic force balances, derived from the principle of virtual power, a thermomechanical version of the second law that includes, work performed during plastic flow, and the heat generation and partial dissipation due to the plastic work and fast transient time via the energy balance relation. When combined with thermodynamically consistent constitutive relations, the microscopic force balances become nonlocal flow rules in the form of partial differential equations (PDEs) requiring boundary conditions. A rate and temperature dependent grain boundary flow rule is also presented which accounts for the energetic state of a plastically strained boundary along with boundary resistance against the slip transfer.

In this research, the free energy and dissipation potentials for the bulk and interface are postulated based on dominant microstructural phenomena involved in the problem of interest (i.e., thin film substrate system). These potentials are then accepted as the primary function and proposed to derive the stresses following Ziegler’s notion, for the thermodynamic gradient theory based on the same concept. Consequently, the mechanical and thermal conjugate forces of bulk and interface are decomposed into energetic and dissipative counterparts, which endowed the governing equations to have both energetic and dissipative gradient length scales.

The contributions in this work include:

- the coupled thermo-mechanical strain gradient plasticity theory is developed to assess the thermal and mechanical responses of small-scale metallic compounds under fast transient processes,
- the formulation incorporates both energetic hardening and dissipative strengthening,
- the classical heat equation is enhanced by involving the terms representing heating due to thermomechanical coupling and inelastic dissipation,
- the rate and temperature dependent flow rules for grain interior as well as grain boundary are proposed, correspondingly the enhanced model possesses four material length scales, i.e. energetic and dissipative length scales for grain interior and grain boundary,
- the proposed theory is applied to both small and large deformation framework,
- the 1D- and 2D finite element method is applied to the simple shear plate problem to investigate the size dependent behavior of small-scale metallic compounds under fast transient processes,
- the size dependent material behavior arising from the boundary effects is also addressed,
- the thermal responses are also explored through the finite element simulations considering the gradient-enhanced heat equation,
- the mesh dependency tests are carried out to examine the proposed gradient-enhanced theory on the regularization of localization.

Some results of Voyiadjis’s work appeared in more than 320 journal articles and the following book chapters:

Book Chapters in Handbooks: