Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Abstract

Thermodynamic models for viscoplastic solids are often formulated in the context of continuum thermodynamics and the dissipation principle. The purpose of the current work is to show that models for such material behavior can also be formulated in the form of a General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC), see, e.g., Grmela and Öttinger (Phys Rev E, 56:6620-6632, 1997), Öttinger and Grmela (Phys Rev E, 56:6633-6655, 1997), Grmela (J Non-Newtonian Fluid Mech, 165:980-986, 2010). A GENERIC combines Hamiltonian-dynamics-based modeling of time-reversible processes with Onsager-Casimir-based modeling of time-irreversible processes. The result is a model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to model complex fluids, it has recently been applied to anisotropic inelastic solids in Eulerian (Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:45-52, 2008; Hütter and Tervoort, in J Non-Newtonian Fluid Mech, 152:53-65, 2008; Hütter and Tervoort, in Adv Appl Mech, 42:254-317, 2008) and Lagrangian (Hütter and Svendsen, in J Elast 104:357-368, 2011) settings, as well as to damage mechanics. For simplicity, attention is focused in the current work on the case of thermoelastic viscoplasticity. Central to this formulation is a GENERIC-based form for the viscoplastic flow rule. A detailed comparison with the formulation based on continuum thermodynamics and the dissipation principle is given.

Energy‐momentum‐entropy consistent numerical methods for large‐strain thermoelasticity relying on the GENERIC formalism

“…The last equation governs the time evolution of the thermodynamic variable ∈ { , , u}. Note that the second Piola-Kirchhoff stress tensor S in (7b) assumes the form (5), and the material heat flux vector Q in (8) is given by (6). Choosing = leads to the temperature-based formulation, where the second Piola-Kirchhoff stress tensor follows from (5) and assumes the following form…”

“…GENERIC was originally developed in the context of complex fluids (see the work of Öttinger for a comprehensive account of the GENERIC framework) and later applied to solid mechanics (see the work of Hütter and Svendsen and Mielke). The GENERIC framework was first applied to computational solid mechanics by Romero who coined the notion “thermodynamically consistent (TC) integrator.” Alternatively, Öttinger recently introduced “GENERIC integrators,” which can be regarded as extension of symplectic integrators for Hamiltonian systems to the realm of dissipative systems.…”

A hierarchy of Poisson brackets in non-equilibrium thermodynamics

“…The GENERIC framework contains a large amount of mesoscopic models, e.g. classical hydrodynamics, Boltzmann equation, classical irreversible thermodynamics (CIT), extended irreversible thermodynamics (EIT), see [8], models for polymer flows, visco-elasto-plastic solids [9], etc. The main features of GENERIC are that the reversible part of the evolution equations is constructed from a Poisson bracket while the irreversible from a dissipation potential (or dissipative bracket when thermodynamic forces are small), and that the equations are also automatically compatible with equilibrium thermodynamics as they gradually approach equilibrium.…”

Thermodynamic formulation of flowing soft matter with transient forces

“…The General Equation for the Nonequilibrium Reversible-Irreversible Coupling (GENERIC) framework provides a convenient and rigorous tool to assess the mechanical and thermodynamic validity of transport equations while highlighting the separation of reversible and irreversible dynamics contributions (Öttinger and Grmela 1997;Grmela and Öttinger 1997;Öttinger 2005). It has been successful in both proving the consistency, as well as extending a wide variety of models, including rheological models of colloids (Wagner 2001;Ellero et al 2003) and polymer solutions (Öttinger 2001), elasticity (Mielke 2011) and viscoplasticity (Hütter and Svendsen 2012), and even relativistic hydrodynamics (Öttinger 1998a, b, 1999Ilg and Öttinger 1999) or dissipative quantum systems (Öttinger 2011).…”