A non-local dissipative Lagrangian modelling for generalized thermoelasticity in solids

A generalized thermoelasticity theory emanating from the discretized Euler-Lagrange equation with dissipation in the non-local realm is formulated and proposed. The main idea of the present work is to interpolate the coupled thermal and deformation fields in solids via the discrete Lagrangian and Dissipation associated with the interaction of particles in a discrete system. The resulting governing equations for different fields are described with a full Lagrangian description such that the updating of the weight functions is eliminated in the numerical procedure. Numerical examples include (a) a one-dimensional bar with various thermal flux models, such as Fourier-type, Jeffrey-type, and Cattaneo-type thermal fluxes and (b) a two-dimensional plate with insulted cracks. In the one-dimensional case, the numerical results with a small particle influence domain show excellent agreement with the classical local model in each case while the non-locality leads to a stiffer thermal and deformation profiles of the system. In the two-dimensional case, the thermodynamic responses are easily predicted using the proposed Lagrangian-based approach. Moreover, a unified time integration is exploited to achieve the time marching of the coupled first/second order in time system.