Abstract
The effect of physical aging on the mechanics of amorphous solids as well as mechanical rejuvenation is modeled with nonequilibrium thermodynamics, using the concept of two thermal subsystems, namely a kinetic one and a configurational one. Earlier work (Semkiv and Hütter in J Non-Equilib Thermodyn 41(2):79–88, 2016) is extended to account for a fully general coupling of the two thermal subsystems. This coupling gives rise to hypoelastic-type contributions in the expression for the Cauchy stress tensor, that reduces to the more common hyperelastic case for sufficiently long aging. The general model, particularly the reversible and irreversible couplings between the thermal subsystems, is compared in detail with models in the literature (Boyce et al. in Mech Mater 7:15–33, 1988; Buckley et al. in J Mech Phys Solids 52:2355–2377, 2004; Klompen et al. in Macromolecules 38:6997–7008, 2005; Kamrin and Bouchbinder in J Mech Phys Solids 73:269–288 2014; Xiao and Nguyen in J Mech Phys Solids 82:62–81, 2015). It is found that only for the case of Kamrin and Bouchbinder (J Mech Phys Solids 73:269–288, 2014) there is a nontrivial coupling between the thermal subsystems in the reversible dynamics, for which the Jacobi identity is automatically satisfied. Moreover, in their work as well as in Boyce et al. (Mech Mater 7:15–33, 1988), viscoplastic deformation is driven by the deviatoric part of the Cauchy stress tensor, while for Buckley et al. (J Mech Phys Solids 52:2355–2377, 2004) and Xiao and Nguyen (J Mech Phys Solids 82:62–81, 2015) this is not the case.
Article PDF
Similar content being viewed by others
References
Struik, L.C.E.: Physical aging in amorphous glassy polymers. Ann. NY Acad. Sci. 279, 78–85 (1976)
Hutchinson, J.M.: Physical aging of polymers. Prog. Polym. Sci. 20, 703–760 (1995)
Boyce, M.C., Parks, D.M., Argon, A.S.: Large inelastic deformation of glassy polymers. Part I: rate dependent constitutive model. Mech. Mater. 7, 15–33 (1988)
G’Sell, C., McKenna, G.B.: Influence of physical ageing on the yield response of model DGEBA/poly(propylene oxide) epoxy glasses. Polymer 33, 2103–2113 (1992)
Meijer, H.E.H., Govaert, L.E.: Mechanical performance of polymer systems: the relation between structure and properties. Prog. Polym. Sci. 30, 915–938 (2005)
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, 6997–7008 (2005)
Engels, T.A.P., Govaert, L.E., Peters, G.W.M., Meijer, H.E.H.: Processing-induced properties in glassy polymers: application of structural relaxation to yield stress development. J. Polym. Sci. Polym. Phys. 44, 1212–1225 (2006)
Struik, L.C.E.: On the rejuvenation of physically aged polymers by mechanical deformation. Polymer 38, 4053–4057 (1997)
Tool, A.Q.: Relaxation of stresses in annealing glass. J. Res. Natl. Bur. Stand. (US) 34, 199–211 (1945)
Buckley, C.P., Dooling, P.J., Harding, J., Ruiz, C.: Deformation of thermosetting resins at impact rates of strain. Part 2: constitutive model with rejuvenation. J. Mech. Phys. Solids 52, 2355–2377 (2004)
Nguyen, T.D., Qi, H.J., Castro, F., Long, K.N.: A thermoviscoelastic model for amorphous shape memory polymers: incorporating structural and stress relaxation. J. Mech. Phys. Solids 56, 2792–2814 (2008)
Bouchbinder, E., Langer, J.S.: Nonequilibrium thermodynamics of driven amorphous materials II. Effective-temperature theory. Phys. Rev. E 80, 031132 (2009)
Nguyen, T.D., Yakacki, C.M., Brahmbhatt, P.D., Chambers, M.L.: Modeling the relaxation mechanisms of amorphous shape memory polymers. Adv. Mater. 22, 3411–3423 (2010)
Kamrin, K., Bouchbinder, E.: Two-temperature continuum thermomechanics of deforming amorphous solids. J. Mech. Phys. Solids 73, 269–288 (2014)
Xiao, R., Nguyen, T.D.: An effective temperature theory for the nonequilibrium behavior of amorphous polymers. J. Mech. Phys. Solids 82, 62–81 (2015)
Semkiv, M., Hütter, M.: Modeling aging and mechanical rejuvenation of amorphous solids. J. Non-Equilib. Thermodyn. 41, 79–88 (2016)
Cugliandolo, L.F., Kurchan, J., Peliti, L.: Energy flow, partial equilibration, and effective temperatures in systems with slow dynamics. Phys. Rev. E 55, 3898–3914 (1997)
Nieuwenhuizen, T.M.: Thermodynamic description of a dynamical glassy transition. J. Phys. A Math. General 31, L201–L207 (1998)
Nieuwenhuizen, T.M.: Thermodynamics of the glassy state: effective temperature as an additional system parameter. Phys. Rev. Lett. 80, 5580–5583 (1998)
Sciortino, F., Kob, W., Tartaglia, P.: Inherent structure entropy of supercooled liquids. Phys. Rev. Lett. 83, 3214–3217 (1999)
Berthier, L., Barrat, J.L., Kurchan, J.: A two-time-scale, two-temperature scenario for nonlinear rheology. Phys. Rev. E 61, 5464–5472 (2000)
Nieuwenhuizen, T.M.: Formulation of thermodynamics for the glassy state: configurational energy as a modest source of energy. J. Chem. Phys. 115, 8083–8088 (2001)
Öttinger, H.C.: Nonequilibrium thermodynamics of glasses. Phys. Rev. E 74, 011113 (2006)
Leuzzi, L.: A stroll among effective temperatures in aging systems: limits and perspectives. J. Non-Cryst. Solids 355, 686–693 (2009)
Grmela, M., Öttinger, H.C.: Dynamics and thermodynamics of complex fluids, I, development of a general formalism. Phys. Rev. E 56, 6620–6632 (1997)
Öttinger, H.C., Grmela, M.: Dynamics and thermodynamics of complex fluids, II, Illustrations of a general formalism. Phys. Rev. E. 56, 6633–6655 (1997)
Öttinger, H.C.: Beyond Equilibrium Thermodynamics. Wiley, Hobroken (2005)
Hütter, M., Tervoort, T.A.: Finite anisotropic elasticity and material frame indifference from a nonequilibrium thermodynamics perspective. J. Non-Newton. Fluid Mech. 152, 45–52 (2008)
Hütter, M., Tervoort, T.A.: Thermodynamic considerations on non-isothermal finite anisotropic elasto-viscoplasticity. J. Non-Newton. Fluid Mech. 152, 53–65 (2008)
Ngan, A.H.W.: Canonical ensemble for static elastic structures with random microstructures. J. Mech. Phys. Solids 57, 803–811 (2009)
Stillinger, F.H., Weber, T.A.: Hidden structure in liquids. Phys. Rev. A 25, 978–989 (1982)
Doghri, I.: Mechanics of Deformable Solids. Springer, Berlin (2000)
Öttinger, H.C.: Modeling complex fluids with a tensor and a scalar as structural variables. Rev. Mex. Fís. 48(Suppl. 1), 220–229 (2002)
Kolvin, I., Bouchbinder, E.: Simple nonlinear equation for structural relaxation in glasses. Phys. Rev. E 86, 010501(R) (2012)
Beris, A.N., Edwards, B.J.: Thermodynamics of Flowing Systems with Internal Microstructure. Oxford University Press, New York (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Öchsner.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Semkiv, M., Anderson, P.D. & Hütter, M. Two-subsystem thermodynamics for the mechanics of aging amorphous solids. Continuum Mech. Thermodyn. 29, 647–663 (2017). https://doi.org/10.1007/s00161-016-0550-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-016-0550-x