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Formulation of thermoelastic dissipative material behavior using GENERIC. (English) Zbl 1272.74137
Summary: We show that the coupled balance equations for a large class of dissipative materials can be cast in the form of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). In dissipative solids (generalized standard materials), the state of a material point is described by dissipative internal variables in addition to the elastic deformation and the temperature. The framework GENERIC allows for an efficient derivation of thermodynamically consistent coupled field equations, while revealing additional underlying physical structures, like the role of the free energy as the driving potential for reversible effects and the role of the free entropy (Massieu potential) as the driving potential for dissipative effects. Applications to large and small-strain thermoplasticity are given. Moreover, for the quasistatic case, where the deformation can be statically eliminated, we derive a generalized gradient structure for the internal variable and the temperature with a reduced entropy as driving functional.

MSC:
74F05 Thermal effects in solid mechanics
74A15 Thermodynamics in solid mechanics
82C35 Irreversible thermodynamics, including Onsager-Machlup theory
Software:
GENERIC
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[1] Alber, H.-D.: Materials with Memory. Lecture Notes in Mathematics, vol. 1682. Springer, Berlin (1998)
[2] Bartels S., Roubíček T.: Thermoviscoplasticity at small strains. Z. Angew. Math. Mech. (ZAMM) 88, 735–754 (2008) · Zbl 1153.74011
[3] Bartels, S., Roubíček, T.: Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion. Math. Model. Numer. Anal. (M2AN) (2010). Submitted (Preprint INS, Univ. Bonn) · Zbl 1267.74037
[4] Berdichevsky V.L.: Structure of equations of macrophysics. Phys. Rev. E 68, 066126 26 (2003)
[5] Dzyaloshinskii I.E., Volovick G.E.: Poisson brackets in condensed matter physics. Ann. Phys. 125, 67–97 (1980)
[6] Edwards B.J.: An analysis of single and double generator thermodynamics formalisms for the macroscopic description of complex fluids. J. Non-Equilib. Thermodyn. 23(4), 301–333 (1998) · Zbl 0932.76007
[7] Grmela, M.: Particle and bracket formulations of kinetic equations. In: Fluids and Plasmas: Geometry and Dynamics (Boulder, Colo., 1983), vol. 28. Contemp. Math. Am. Math. Soc. Providence, RI, pp. 125–132 (1984) · Zbl 0558.58012
[8] Grmela M.: Bracket formulation of dissipative time evolution equations. Phys. Lett. A 111(1–2), 36–40 (1985)
[9] Grmela M.: Reciprocity relations in thermodynamics. Phys. A 309(3–4), 304–328 (2002) · Zbl 0995.82046
[10] Grmela M.: Why GENERIC?. J. Non-Newtonian Fluid Mech. 165, 980–986 (2010) · Zbl 1425.80004
[11] Grmela M., Öttinger H.C.: Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E (3) 56(6), 6620–6632 (1997)
[12] Hackl K.: Generalized standard media and variational principles in classical and finite strain elastoplasticity. J. Mech. Phys. Solids 45(5), 667–688 (1997) · Zbl 0974.74512
[13] Hütter M., Tervoort T.A.: Finite anisotropic elasticity and material frame indifference from a nonequilibrium thermodynamics perspective. J. Non-Newtonian Fluid Mech. 152, 45–52 (2008) · Zbl 1138.74007
[14] Hütter M., Tervoort T.A.: Thermodynamic considerations on non-isothermal finite anisotropic elasto-viscoplasticity. J. Non-Newtonian Fluid Mech. 152, 53–65 (2008) · Zbl 1210.74033
[15] Kaufman A.: Dissipative Hamiltonian systems. Phys. Lett. A 100(8), 419–422 (1984)
[16] Muschik W., Gümbel S., Kröger M., Öttinger H.: A simple example for comparing generic with rational non-equilibrium thermodynamics. Physica A 285, 448–466 (2000) · Zbl 1060.82522
[17] Mielke, A.: Hamiltonian and Lagrangian Flows on Center Manifolds. With Applications to Elliptic Variational Problems. Lecture Notes in Mathematics, vol. 1489. Springer, Berlin (1991) · Zbl 0747.58001
[18] Mielke A.: A mathematical framework for generalized standard materials in the rate-independent case. In: Helmig, R., Mielke, A., Wohlmuth, B.I. (eds) Multifield Problems in Solid and Fluid Mechanics, pp. 351–379. Springer, Berlin (2006) · Zbl 1298.74006
[19] Mielke, A.: A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity (2010). Submitted. WIAS preprint 1485 · Zbl 1227.35161
[20] Mielke, A.: On thermodynamically consistent models and gradient structures for thermoplasticity. GAMM Mitt. (2010) Submitted · Zbl 1279.74008
[21] Morrison P.J.: A paradigm for joined Hamiltonian and dissipative systems. Phys. D 18(1–3), 410–419 (1986) · Zbl 0661.70025
[22] Öttinger H.C., Grmela M.: Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys. Rev. E (3) 56(6), 6633–6655 (1997)
[23] Öttinger H.C.: Beyond Equilibrium Thermodynamics. Wiley, New Jersey (2005)
[24] Öttinger H.C.: Nonequilibrium thermodynamics for open systems. Phys. Rev. E (3) 73(3), 036126, 10 (2006)
[25] Penrose O., Fife P.C.: Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43(1), 44–62 (1990) · Zbl 0709.76001
[26] Penrose O., Fife P.C.: On the relation between the standard phase-field model and a ”thermodynamically consistent” phase-field model. Physica D 69(1–2), 107–113 (1993) · Zbl 0799.76084
[27] Roubíček T.: Thermodynamics of rate independent processes in viscous solids at small strains. SIAM J. Math. Anal. 42, 256–297 (2010) · Zbl 1213.35279
[28] Sprekels J., Zheng S.M.: Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions. J. Math. Anal. Appl. 176(1), 200–223 (1993) · Zbl 0804.35063
[29] Ziegler, H., Wehrli, C.: The derivation of constitutive relations from the free energy and the dissipation function. In: Advances in Applied Mechanics, vol. 25, pp. 183–237. Academic Press, Orlando, FL (1987) · Zbl 0719.73001
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