Multiphysics and Thermodynamic Formulations for Equilibrium and Non-equilibrium Interactions: Non-linear Finite Elements Applied to Multi-coupled Active Materials

Abstract

Combining several theories this paper presents a general multiphysics framework applied to the study of coupled and active materials, considering mechanical, electric, magnetic and thermal fields. The framework is based on thermodynamic equilibrium and nonequilibrium interactions, both linked by a two-temperature model.The multi-coupled governing equations are obtained from energy, momentum and entropy balances; the total energy is the sum of thermal, mechanical and electromagnetic parts. The momentum balance considers mechanical plus electromagnetic balances; for the latter the Abraham representation using the Maxwell stress tensor is formulated. This tensor is manipulated to automatically fulfill the angular momentum balance. The entropy balance is formulated using the classical Gibbs equation for equilibrium interactions and non-equilibrium thermodynamics. For the non-linear finite element formulations, this equation requires the transformation of thermoelectric coupling and conductivities into tensorial form.The two-way thermoelastic Biot term introduces damping: thermomechanical, pyromagnetic and pyroelectric converse electromagnetic dynamic interactions. Ponderomotrix and electromagnetic forces are also considered.The governing equations are converted into a variational formulation with the resulting four-field, multicoupled formalism implemented and validated with two custom-made finite elements in the research code FEAP. Standard first-order isoparametric eight-node elements with seven degrees of freedom (dof) per node (three displacements, voltage and magnetic scalar potentials plus two temperatures) are used. Non-linearities and dynamics are solved with Newton-Raphson and Newmark-β algorithms, respectively.Results of thermoelectric, thermoelastic, thermomagnetic, piezoelectric, piezomagnetic, pyroelectric, pyromagnetic and galvanomagnetic interactions are presented, including non-linear dependency on temperature and some second-order interactions.

Elasto-thermoelectric non-linear, fully coupled, and dynamic finite element analysis of pulsed thermoelectrics

“…[29]). The equations are taken from [17] (with the addition of the thermal inertial term) and the mechanical ones from [21]. These partial differential equations can be discretized following the Galerkin method (viz.…”

“…Five degrees-of-freedom (dof) are required to study elastothermoelectric couplings in three-dimensions (3D): three displacements, temperature and voltage. From [21] (without magnetic field), the assembled "stiffness" matrix is:…”

“…To fill in part this gap, the authors have developed several complete nonlinear and dynamic formulations for thermoelectricity, [17], [18], [19], elastothermoelectricity [20] and fully coupled "four-field" (including magnetic) in [21]. Also, they are working on a one-dimensional (1D) formulation with beam elements for TEs to study their mechanical response minimizing computation time, [22].…”

“…In Section 3, a summary of the coupled and dynamic formulation, both thermodynamic and FE is presented. The formulation follows the steps developed in [21] and includes effects that are not always present in the study of thermoelectric devices, such as Thomson, Biot and the coupling with the mechanical field. The section also incorporates relevant aspects on the FE mesh, computer running and the implementation in the research code FEAP, [25].…”

Peridynamic Formulation for Coupled Thermoelectric Phenomena

“…Furthermore, thermoelectric materials (TEMs) convert temperature differences into electric voltage; hence, it can theoretically convert waste heat to electricity or electricity to heat. We call the first phenomenon the Seebeck effect, where electrical voltage is developed when there is a gradient of temperature between the terminals and was discovered by Seebeck in 1821 while the second is just converse effect of the first one and it was discovered by Peltier in 1834 [1].…”