Finite element implementation of the thermal field dislocation mechanics model: Study of temperature evolution due to dislocation activity

The small deformation formulation of the thermal field dislocation mechanics model (Upadhyay, J. Mech. Phys. Solids, 145 (2020) 104150) is numerically implemented using the finite element method. The implementation consists of solving a first-order div-curl system to obtain an incompatible plastic distortion from a prescribed polar dislocation density along with three governing partial differential equations (PDE): the dislocation transport equation (a first-order hyperbolic PDE), the static equilibrium equation (an elliptic PDE), and the temperature evolution equation (a parabolic PDE). A combination of continuous Galerkin (for the elliptic and parabolic PDEs) and discontinuous Galerkin (for the hyperbolic PDE) space discretizations and Runge-Kutta time discretizations are used to implement these equations in a staggered algorithm and obtain stable solutions at (quasi-)optimal convergence rates. The implementation is verified by comparing the simulation-predicted temperature evolution of a moving edge dislocation with an analytical solution. Next, the contribution of plastic dissipation and thermoelastic effect to the temperature evolution during the motion of an edge and a screw dislocation, annihilation of two edge dislocations and expansion of a dislocation loop are studied in detail. In the case of a moving edge dislocation, contrary to existing literature, the thermoelastic effect is demonstrated to have a more significant contribution to temperature evolution than plastic dissipation for the studied traction boundary condition and dislocation velocity expression. In the dislocation loop expansion case, the role of free surfaces on temperature evolution is highlighted. As the loop approaches the free surfaces, plastic dissipation is found to have an increasing contribution to temperature evolution due to the growing impact of image stresses.