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Thermodynamic considerations on non-isothermal finite anisotropic elasto-viscoplasticity. (English) Zbl 1210.74033
Summary: A class of dynamic models to describe finite anisotropic elasto-viscoplastic behavior in an Eulerian setting are examined from the perspective of non-equilibrium thermodynamics. A proper description of anisotropic elastic behavior, using the deformation gradient as an internal variable, serves as a starting point. A general model for elasto-viscoplastic deformation is then formulated by allowing isochoric relaxation of the total deformation gradient, giving rise to an elastic deformation gradient and a plastic strain rate. We discuss thermodynamic restrictions on specific forms of the plastic strain rate for anisotropic materials, using the representation theorem for tensor functions. In addition, the thermodynamic admissibility of a multi-mode formulation of the elasto-viscoplastic model is demonstrated. Finally, the model is applied to describe rate-dependent inelastic deformation of an amorphous polymer glass, and anisotropic strain-rate dependent crystalline slip as found in metal plasticity.

##### MSC:
 74C20 Large-strain, rate-dependent theories of plasticity 74A15 Thermodynamics in solid mechanics 74E10 Anisotropy in solid mechanics
Mathematica
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##### References:
 [1] Lee, E. H.: Elastic – plastic deformation at finite strains, J. appl. Mech. 36, 1 (1969) · Zbl 0179.55603 [2] Green, A. E.; Naghdi, P. M.: A general theory of an elastic – plastic continuum, Arch. rat. Mech. anal. 18, 251 (1965) · Zbl 0133.17701 [3] Green, A. E.; Naghdi, P. M.: A thermodynamic development of elastic-plastic continua, , 117-131 (1966) · Zbl 0179.55601 [4] Green, A. E.; Naghdi, P. M.: Some remarks on elastic-plastic deformation at finite strain, Int. J. Eng. sci. 9, 1219 (1971) · Zbl 0226.73022 [5] Casey, J.; Naghdi, P. M.: A remark on the use of the decomposition F=Fe$$\cdot$$Fp in plasticity, Appl. mech. 47, 672 (1980) · Zbl 0472.73035 [6] Naghdi, P. M.: A critical review of the state of finite plasticity, J. appl. Math. phys. (Z. Angew. math. Phys.) 41, 315 (1990) · Zbl 0712.73032 [7] Besseling, J. F.; Van Der Giessen, E.: Mathematical modelling of inelastic deformation, Applied mathematics and mathematical computation 5 (1994) · Zbl 0809.73003 [8] Rice, J. R.: Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity, J. mech. Phys. solids 19, 433 (1971) · Zbl 0235.73002 [9] Boyce, M. C.; Parks, D. M.; Argon, A. S.: Large inelastic deformation of glassy polymers. Part I. Rate dependent constitutive model, Mech. mater. 7, 15 (1988) [10] Tervoort, T. A.; Smit, R. J. M.; Brekelmans, W. A. M.; Govaert, L. E.: A constitutive equation for the elasto-viscoplastic deformation of glassy polymers, Mech. time-dep. Mater. 1, 269 (1998) [11] Bird, R. B.; Curtiss, C. F.; Armstrong, R. C.; Hassager, O.: Dynamics of polymeric liquids. Vol. 2: kinetic theory, (1987) [12] Green, M. S.; Tobolsky, A. V.: A new approach to the theory of relaxing polymeric media, J. chem. Phys. 14, No. 2, 80-92 (1946) [13] Doi, M.; Edwards, S. F.: Dynamics of concentrated polymer systems. 1. Brownian motion in equilibrium state, J. chem. Soc., Faraday trans. II 74, 1789 (1978) [14] Doi, M.; Edwards, S. F.: Dynamics of concentrated polymer systems. 2. Molecular motion under flow, J. chem. Soc., Faraday trans. II 74, 1802 (1978) [15] Leonov, A. I.: Non-equilibrium thermodynamics and rheology of viscoelastic polymer media, Rheol. acta 15, 85 (1976) · Zbl 0351.73001 [16] Beris, A. N.; Edwards, B. J.: Thermodynamics of flowing systems, (1994) [17] Öttinger, H. C.: Beyond equilibrium thermodynamics, (2005) [18] M. Hütter, T.A. Tervoort, Finite anisotropic elasticity and material frame indifference from a non-equilibrium thermodynamics perspective, J. Non-Newtonian Fluid Mech. 152 (2008) 45 – 52. · Zbl 1138.74007 [19] Grmela, M.; Öttinger, H. C.: Dynamics and thermodynamics of complex fluids. I. development of a general formalism, Phys. rev. E, 6620 (1997) [20] Öttinger, H. C.; Grmela, M.: Dynamics and thermodynamics of complex fluids. II. illustrations of a general formalism, Phys. rev. E 56, 6633 (1997) [21] Hunter, S. C.: Mechanics of continuous media, (1976) · Zbl 0385.73002 [22] Truesdell, C.; Noll, W.: The non-linear field theories of mechanics, (1992) · Zbl 0779.73004 [23] Boyce, M. C.; Weber, G. G.; Parks, D. M.: On the kinematics of finite strain plasticity, J. mech. Phys. solids 37, 647 (1989) · Zbl 0692.73043 [24] Rubin, M. B.: Plasticity theory formulated in terms of physically based microstructural variables. Part 1. Theory, Int. J. Solids struct. 31, 2615 (1994) · Zbl 0943.74507 [25] Rubin, M. B.: On the treatment of elastic deformation in finite elastic-viscoplastic theory, Int. J. Plasticity 12, 951 (1996) · Zbl 1002.74509 [26] Beris, A. N.; Edwards, B. J.: Poisson bracket formulation of viscoelastic flow equations of differential type: a unified approach, J. rheol. 34, 503 (1990) · Zbl 0697.76015 [27] Edwards, B. J.; Beris, A. N.: Non-canonical Poisson bracket for nonlinear elasticity with extensions to viscoelasticity, J. phys. A.: math. Gen. 24, 2461 (1991) · Zbl 0735.73011 [28] Dafalias, Y. F.: The plastic spin, J. appl. Mech. -T. ASME 52, 865 (1985) · Zbl 0587.73052 [29] Dafalias, Y. F.: Plastic spin: necessity or redundancy?, Int. J. Plasticity 14, 909 (1998) · Zbl 0947.74008 [30] Loret, B.: On the effects of plastic rotation in the finite deformation of anisotropic elastoplastic materials, Mech. mater. 2, 287 (1983) [31] Van Der Giessen, E.: Micromechanical and thermodynamic aspects of the plastic spin, Int. J. Plasticity 7, 365 (1991) · Zbl 0759.73017 [32] Wang, C. C.: A new representation theorem for isotropic functions: an answer to Professor G.F. Smith’s criticism of my paper on ’representations for isotropic functions’, Arch. rat. Mech. anal. 36, 198 (1970) · Zbl 0327.15030 [33] Spencer, A. J. M.: A.c.eringenpart III. Theory of invariants in continuum physics mathematics, Part III. Theory of invariants in continuum physics mathematics 1, 239-353 (1971) [34] Dafalias, Y. F.: Corotational rates for kinematic hardening at large plastic deformations, J. appl. Mech. 50, 561 (1983) · Zbl 0524.73047 [35] M. Hütter, T.A. Tervoort, Plastic strain rate and plastic spin derived from microscopic fluctuations, submitted for publication. [36] Tervoort, T. A.; Klompen, E. T. J.; Govaert, L. E.: A multi-mode approach to finite, three-dimensional, nonlinear viscoelastic behavior of polymer glasses, J. rheol. 40, 779 (1996) [37] Tervoort, T. A.; Govaert, L. E.: Strain-hardening behavior of polycarbonate in the glassy state, J. rheol. 44, 1263 (2000) [38] Govaert, L. E.; Timmermans, P. H. M.; Brekelmans, W. A. M.: The influence of intrinsic strain softening on strain localization in polycarbonate: modeling and experimental validation, J. eng. Mater. -T. ASME 122, 177 (2000) [39] Klompen, E. T. J.; Engels, T. A. P.; Govaert, L. E.; Meijer, H. E. H.: Modeling of the postyield response of glassy polymers: influence of thermomechanical history, Macromolecules 38, 6997 (2005) [40] Mathematica \textregistered  software package, Wolfram Research Inc., http://www.wolfram.com/. [41] W. Thompson, On the dynamical theory of heat, Trans R. Soc. Ed. 20 (1853) 261. · ERAM 045.1239cj [42] Asaro, R. J.: J.w.hutchinsont.y.wuadvances in applied mechanics, Advances in applied mechanics 23, 1-115 (1983) [43] Peirce, D.; Asaro, R. J.; Needleman, A.: Material rate depenence and localized deformation in crystalline solids, Acta metall. 31, 1951 (1983) [44] Hutchinson, J. W.: Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. R. Soc. lond. A 348, No. 1652, 101 (1976) · Zbl 0319.73059 [45] Pan, J.; Rice, J. R.: Rate sensitivity of plastic-flow and implications for yield-surface vertices, Int. J. Solids struct. 19, 973 (1983) · Zbl 0543.73033
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