We present the theory of novel time-stepping algorithms for general nonlinear, non-smooth, coupled, and thermomechanical problems. The proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics: for isolated systems, they preserve the total energy and the entropy never decreases. Extending previous works on the subject, the newly proposed integrators are applicable to coupled mechanical systems with non-smooth kinetics and can be formulated in arbitrary variables. The ideas are illustrated with a simple non-smooth problem: a rheological model for a thermo-elasto-plastic material with hardening. Numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability, which stems precisely from their ability to preserve the structure of the evolution equations they discretize.
ENERGY-ENTROPY-MOMENTUM INTEGRATION SCHEMES
777As a result, symplectic, variational, and energy-momentum integrators, among others, have been developed and widely employed in the last decades. As these works show, preserving (part of) the Hamiltonian structure has proven very effective in the numerical solution of stiff models arising in nonlinear mechanics of solids [6,7], flexible and rigid multibody analysis [8,9], contact mechanics [10,11], elastic beams and shells [12][13][14], and so on. For practical applications in solid mechanics, however, all these models fall short because many (if not most) of the interesting problems are not elastic, but rather elastoplastic, or visco-elastic, or exhibit damage, or are coupled with temperature and so on.The class of solids that exhibit some kind of dissipative behavior is much larger than that of elastic solids, and a common mathematical structure for the former is not as apparent as in the Hamiltonian case. This lack of unifying formalism has hindered the formulation of structure preserving methods to nonlinear dissipative phenomena in solids. A few remarkable methods have been proposed in the past for single, specific problems (for example, [15][16][17][18]) but cannot be employed beyond the boundaries of the problem they were designed for.The only class of structure preserving methods for general (smooth) dissipative solids that have been proposed so far is, to the authors' knowledge, the energy-entropy-momentum (EEM) integrators initially proposed in [19] for finite dimensional thermomechanical problems and later extended to the infinite dimensional case [20,21]. In addition to thermoelastic problems, the EEM method has been applied to phase field modeling [22], discrete thermo-visco-plasticity [23], and thermo-visco-elasticity [24][25][26].Energy-entropy-momentum methods could be applied to a broad class of thermomechanical systems after realizing that many of the latter can be formulated as metriplectic models [27]. This mathematical and geometrical formalism generalizes the ideas of Hamiltonian problems and is able to encompass several dissipative phenomen...
