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A nonlinear viscoelastic constitutive model taking into account of physical aging

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Abstract

Polymers exhibit viscoelastic behavior: their mechanical response depends on the loading time, or on the loading frequency. In addition, if a polymer structure has a long service life, the mechanical behavior can also depend on physical aging and chemical degradation. This paper describes a thermodynamically consistent constitutive law taking into account the viscoelastic phenomena and the physical aging. First, an original nonlinear viscoelastic law, depending on the physical aging time, is developed. Then, considering experimental values of dynamic modulus from the literature, the model parameters are identified, using a new method based on the discrete form of the spectrum of relaxation time. The obtained model is numerically implemented and compared to experimental results.

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References

  • Amin, A., Lion, A., Sekita, S., Okui, Y.: Nonlinear dependence of viscosity in modeling the rate-dependent response of natural and high damping rubbers in compression and shear: experimental identification and numerical verification. Int. J. Plast. 22, 1610–1657 (2006)

    MATH  Google Scholar 

  • Arruda, E., Boyce, M.: A three dimensional model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41(2), 389–412 (1993)

    MATH  Google Scholar 

  • Avazmohammadi, R., Castaneda, P.: Tangent second order estimates for the large strain, macroscopic response of particle reinforced elastomers. J. Elast. 112, 139–183 (2012)

    MathSciNet  MATH  Google Scholar 

  • Baumgaertel, M., Winter, H.: Interrelation between continuous and discrete relaxation time spectra. J. Non-Newton. Fluid Mech. 44, 15–36 (1992)

    Google Scholar 

  • Bradshaw, R.D., Brinson, L.C.: Physical aging in polymers and polymer composites: an analysis and method for time-aging time superposition. Polym. Eng. Sci. 37, 31–44 (1997)

    Google Scholar 

  • Brinson, L.C.: Effects of physical aging on long term creep of polymers and polymer matrix composites. Int. J. Solids Struct. 32, 827–846 (1995)

    MATH  Google Scholar 

  • Brinson, L.C.: Polymer Engineering Science and Viscoelasticity. An Introduction. Springer, Berlin (2008)

    Google Scholar 

  • Cerrada, M.L., McKenna, G.B.: Physical aging of amorphous PEN: isothermal, isochronal and isostructural results. Macromolecules 33, 3065–3076 (2000)

    Google Scholar 

  • Christensen, R.: A nonlinear theory of viscoelasticity for application to elastomers. J. Appl. Mech. 47, 762–768 (1980)

    MathSciNet  MATH  Google Scholar 

  • Coleman, B., Gurtin, M.: Thermodynamics with internal state variables. J. Chem. Phys. 47, 597–613 (1967)

    Google Scholar 

  • Diani, J., Brieu, M., Gilormini, P.: Observation and modeling of the anisotropic visco hyperelastic behavior of a rubber like material. Int. J. Solids Struct. 43, 3044–3056 (2006)

    MATH  Google Scholar 

  • Drozdov, A.D., Dorfmann, A.: Physical aging and the viscoelastic response of glassy polymers: comparison of observations in mechanical and dilatometric tests. Math. Comput. Model. 37, 665–681 (2003)

    MathSciNet  MATH  Google Scholar 

  • Findley, W., Lai, J., Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials (1976)

    MATH  Google Scholar 

  • Ghoreishy, M., Firouzbakht, M., Naderi, G.: Parameter determination and experimental verification of bergström boyce hysteresis model for rubber compounds reinforced by carbon black blends. Mater. Des. 53, 457–465 (2014)

    Google Scholar 

  • Goudarzi, T., Lopez-Pamies, O.: Numerical modeling of the nonlinear elastic response of filled elastomers via composite-sphere assemblages. J. Appl. Mech. 47, 1028–1036 (2013)

    Google Scholar 

  • Govindjee, S., Mihalic, P.: Viscoelastic constitutive relations for the steady spinning of a cylinder. Report, University of California, Berkeley (1998)

  • Guo, J., Lic, Z., Longb, J., Xiao, R.: Modeling the effect of physical aging on the stress response of amorphous polymers based on a two-temperature continuum theory. Mech. Mater. 143, 103335 (2020)

    Google Scholar 

  • Haidar, B., Vidal, A.: Time and temperature dependence of amorphous polymer dynamic properties after a small static deformation. J. Phys. IV 6, C8 (1996)

    Google Scholar 

  • Halphen, B., Nguyen, Q.: Sur les matériaux standards generalisés. J. Méc. 14, 39–63 (1975)

    MATH  Google Scholar 

  • Holzapfel, G.A.: Nonlinear Solid Mechanics, a Continuum Approach for Engineering. Wiley, New York (2006)

    Google Scholar 

  • Huber, N., Tsakmakis, C.: Finite deformation viscoelasticity laws. Mech. Mater. 32, 1–18 (2000)

    MATH  Google Scholar 

  • Jalocha, D.: Payne effect: a constitutive model based on a dynamic strain amplitude dependent spectrum of relaxation time. Mech. Mater. 148, 103526 (2020)

    Google Scholar 

  • Jalocha, D., Constantinescu, A., Neviere, R.: Prestrain dependent viscosity of a highly filled elastomer: experiments and modeling. Mech. Time-Depend. Mater. 19, 78–98 (2015a)

    Google Scholar 

  • Jalocha, D., Constantinescu, A., Neviere, R.: Prestrained biaxial DMA investigation of viscoelastic nonlinearities in highly filled elastomers. Polym. Test. 42, 37–44 (2015b)

    Google Scholar 

  • Jalocha, D., Constantinescu, A., Neviere, R.: Revisiting the identification of generalized Maxwell models from experimental results. Int. J. Solids Struct. 64, 169–181 (2015c)

    Google Scholar 

  • Kaminski, M.: Homogenization with uncertainty in Poisson ratio for polymers with rubber particles. Composites, Part B 69, 267–277 (2015)

    Google Scholar 

  • Knauss, W., Emri, L., Lu, H.: Handbook of Experimental Solid Mechanics. Springer, Berlin (2006)

    Google Scholar 

  • LeTallec, P., Rahier, C.: Numerical models of steady rolling for non linear viscoelastic structures in finite deformations. Int. J. Numer. Methods Eng. 37, 1159–1186 (1994)

    MATH  Google Scholar 

  • LeTallec, P., Rahier, C., Kaiss, A.: Three dimensional incompressible viscoelasticity in large strains: formulation and numerical approximation. Comput. Methods Appl. Mech. Eng. 109, 223–258 (1993)

    MathSciNet  MATH  Google Scholar 

  • Lion, A.: On the large deformation behavior of reinforced rubber at different temperatures. J. Mech. Phys. Solids 45, 1805–1834 (1997)

    Google Scholar 

  • Lion, A., Retka, J., Rendek, M.: On the calculation of predeformation dependent dynamic modulus tensors in finite nonlinear viscoelasticity. Mech. Res. Commun. 36, 500–520 (2009)

    MATH  Google Scholar 

  • Lopez Jimenez, F.: Modeling of soft composites under three-dimensional loading. Composites, Part B 59, 173–180 (2014)

    Google Scholar 

  • Lopez-Pamies, O.: A new i1 based hyperelastic model for rubber elastic materials. C. R., Méc. 338, 3–11 (2010)

    MATH  Google Scholar 

  • Markovitz, A., Hershel, J.: Boltzmann and the beginnings of rheology. Trans. Soc. Rheol. 21(3), 381–398 (1977)

    MathSciNet  MATH  Google Scholar 

  • McKenna, G.B.: On the physics required for prediction of long term performance of polymers and their composites. J. Res. Natl. Inst. Stand. Technol. 99, 169–189 (1994)

    Google Scholar 

  • Ogden, R., Roxburgh, D.: A pseudo elastic model for the Mullins effect in filled rubber. Proc. R. Soc. Lond. 455, 2861–2877 (1999)

    MathSciNet  MATH  Google Scholar 

  • Ozupek, S.: Constitutive Modeling of Hight Elongation Solid Propellants. PhD thesis, University of Texas (1989)

  • Ozupek, S., Becker, E.: Constitutive modeling of high elongation solid propellants. J. Eng. Mater. Technol. 114, 111–115 (1992)

    Google Scholar 

  • Park, S., Schapery, R.: Methods of interconversion between linear viscoelastic material functions. Part I—a numerical method based on Prony series. Int. J. Solids Struct. 36, 1653–1675 (1999)

    MATH  Google Scholar 

  • Schwarzl, F.: On the interconversion between viscoelastic material functions microsymposia on macromolecules. In: Microsymposium on Macromolecules, Prague, PMM, Macromolecules. Prague, IV, V and VI, Prague, Czechoslovakia, 1969-09-01–1969-09-11 (1969)

    Google Scholar 

  • Simo, J.: On a fully three dimensional finite strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60, 153–173 (1987)

    MATH  Google Scholar 

  • Simo, J., Hughes, T.: Computational Inelasticity. Interdisciplinary Applied Mathematics (1998)

    MATH  Google Scholar 

  • Smith, T.: Empirical equations for representing viscoelastic functions and for deriving spectra. J. Polym. Sci. 35, 39–50 (1971)

    Google Scholar 

  • Steinmann, P., Hossain, M., Possart, G.: Hyperelastic models for rubber like materials: consistent tangent operators and suitability for treloar s data. Arch. Appl. Mech. 82, 1183–1217 (2012)

    MATH  Google Scholar 

  • Swanson, S., Christensent, L.: A constitutive formulation for high elongation propellants. J. Spacecr. 20, 559–566 (1983)

    Google Scholar 

  • Widder, D.: The Laplace Transform. Princeton University Press, Princeton (1946)

    Google Scholar 

  • Williams, M., Landel, R., Ferry, J.: The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. J. Am. Chem. Soc. 77, 3701–3707 (1955)

    Google Scholar 

  • Wollscheid, D.: Predeformation and frequency dependence of filler reinforced rubber under vibration. PhD thesis, Universitat der Bundeswehr Munchen (2014)

  • Wollscheid, D., Lion, A.: Predeformation- and frequency-dependent material behaviour of filler-reinforced rubber: experiments, constitutive modelling and parameter identification. Int. J. Solids Struct. 50, 1217–1225 (2013)

    Google Scholar 

  • Xu, F., Aravas, N., Sofronis, P.: Constitutive modeling of solid propellant materials with evolving microstructural damage. J. Mech. Phys. Solids 56, 2050–2073 (2008)

    MATH  Google Scholar 

  • Zhang, X., Andrieux, F., Sun, D.: Pseudo elastic description of polymeric foams at finite deformation with stress softening and residual strain effects. Mater. Des. 32, 877–884 (2011)

    Google Scholar 

  • Zheng, Y., Priestley, R.D., McKenna, G.B.: Physical Aging of an Epoxy Subsequent to Relative Humidity Jumps Through the Glass Concentration. Wiley-Interscience, New York (2003). https://doi.org/10.1002/20084

    Book  Google Scholar 

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  1. CEA CESTA, 15 Avenue des Sablieres, 33114, Le Barp, France

    D. Jalocha

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Jalocha, D. A nonlinear viscoelastic constitutive model taking into account of physical aging. Mech Time-Depend Mater 26, 21–31 (2022). https://doi.org/10.1007/s11043-020-09473-x

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