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Journal of Advances in Applied Mathematics
JAAM > Volume 6, Number 3, July 2021

Predictive Continuum Constitutive Modeling of Unfilled and Filled Rubbers

Download PDF  (4300.4 KB)PP. 146-161,  Pub. Date:August 13, 2021
DOI: 10.22606/jaam.2021.63002

Author(s)
Fuzhang Zhao
Affiliation(s)
APD Optima Study, Lake Forest, CA 92630, USA
Abstract
The general CSE model fits Treloar’s uniaxial extension test and predicts unfitted
uniaxial compression, equibiaxial extension, biaxial extension, pure shear, and simple shear tests. As
a newly proposed method, the general CSE model, along with the stress-softening ratio, the residualstretch
ratio, and the weighted piecewise two-point interpolation function, fits the Cheng–Chen’s
test and the Diani–Fayolle–Gilormini’s test in cyclic uniaxial extension at different pre-stretches
and predicts corresponding responses at untested pre-stretches. Physical mechanisms of the Mullins
effect have also been predicted based on the evolution of constitutive parameters.
Keywords
Filled rubber, general CSE functional, Mullins effect, predictive constitutive modeling, unfilled rubber.
References
  • [1]  L. Mullins, “Effect of stretching on the properties of rubber,” Rubber Chemistry and Technology, vol. 21, no. 2, pp. 281–300, 1948.
  • [2]  A. Dorfmann and R. W. Ogden, “A constitutive model for the Mullins effect with permanent set in particlereinforced rubber,” International Journal of Solids and Structures, vol. 41, no. 7, pp. 1855–1878, 2004.
  • [3]  L. Mullins and N. R. Tobin, “Theoretical model for the elastic behavior of filler-reinforced vulcanized rubbers,” Rubber Chemistry and Technology, vol. 30, no. 2, pp. 555–571, 1957.
  • [4]  A. S. Wineman and K. R. Rajagopal, “On a constitutive theory for materials undergoing microstructural changes,” Archives of Mechanics, vol. 42, no. 1, pp. 53–75, 1990.
  • [5]  A. S. Wineman and H. E. Huntley, “On a constitutive theory for materials undergoing microstructural changes,” International Journal of Solids and Structures, vol. 31, no. 23, pp. 3295–3313, 1994.
  • [6]  M. A. Johnson and M. F. Beatty, “A constitutive equation for the Mullins effect in stress controlled extension experiments,” Continuum Mechanics and Thermodynamics, vol. 5, no. 4, pp. 301–318, 1993.
  • [7]  M. F. Beatty and S. Krishnaswamy, “A theory of stress-softening in incompressible isotropic materials,” Journal of the Mechanics and Physics of Solids, vol. 48, no. 9, pp. 1931–1965, 2000.
  • [8]  A. E. Zú?iga and M. F. Beatty, “A new phenomenological model for stress-softening in elastomers,” Zeitschrift f¨ur angewandte Mathematik und Physik, vol. 53, no. 5, pp. 794–814, 2002.
  • [9]  H. J. Qi and M. C. Boyce, “Constitutive model for stretch-induced softening of the stress–stretch behavior of elastomeric materials,” Journal of the Mechanics and Physics of Solids, vol. 52, no. 4, pp. 2187–2205, 2004.
  • [10]  M. E. Gurtin and E. C. Francis, “Simple rate-independent model for damage,” Journal of Spacecraft, vol. 18, no. 3, pp. 285–286, 1981.
  • [11]  J. C. Simo, “On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects,” Computer Methods in Applied Mechanics and Engineering, vol. 60, no. 2, pp. 153–173, 1987.
  • [12]  S. Govindjee and J. C. Simo, “A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating Mullins’ effect,” Journal of the Mechanics and Physics of Solids, vol. 39, no. 1, pp. 87–112, 1991.
  • [13]  ——, “Mullins effect and the strain amplitude dependence of the storage modulus,” International Journal of Solids and Structures, vol. 29, no. 14-15, pp. 1737–1751, 1992.
  • [14]  E. A. De Souza Neto, D. Peri′c, and D. R. J. Owen, “A phenomenological three-dimensional rate-independent continuum damage model for highly filled polymers: formulation and computational aspects,” Journal of the Mechanics and Physics of Solids, vol. 42, no. 10, pp. 1533–1550, 1994.
  • [15]  C. Miehe, “Discontinuous and continuous damage evolution in Ogden-type large-strain elastic materials,” European Journal of Mechanics A/Solids, vol. 14, no. 5, pp. 697–720, 1995.
  • [16]  C. Miehe and J. Keck, “Superimposed finite elastic-viscoelastic-plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation,” Journal of the Mechanics and Physics of Solids, vol. 48, no. 2, pp. 323–365, 2000.
  • [17]  M. Klüppel and J. Schramm, “A generalized tube model of rubber elasticity and stress softening of filler reinforced elastomer systems,” Macromolecular Theory and Simulations, vol. 9, no. 9, pp. 742–754, 2000.
  • [18]  H. Lorenz and M. Klüppel, “Microstructure-based modelling of arbitrary deformation histories of fillerreinforced elastomers,” Journal of the Mechanics and Physics of Solids, vol. 60, no. 11, pp. 1842–1861, 2012.
  • [19]  R. Raghunath, D. Juhre, and M. Klüppel, “A physically motivated model for filled elastomers including strain rate and amplitude dependency in finite viscoelasticity,” International Journal of Plasticity, vol. 78, pp. 223–241, 2016.
  • [20]  J. Plagge and M. Klüppel, “A physically based model of stress softening and hysteresis of filled rubber including rate- and temperature dependency,” International Journal of Plasticity, vol. 89, pp. 173–196, 2017.
  • [21]  R. Dargazany and M. Itskov, “A network evolution model for the anisotropic mullins effect in carbon black filled rubbers,” International Journal of Solids and Structures, vol. 46, no. 3, pp. 2967–2977, 2009.
  • [22]  ——, “Constitutive modeling of Mullins effect and cyclic stress softening in filled elastomers,” Physical Review E, vol. 88, no. 1, pp. 012 602(1–24), 2013.
  • [23]  R. Dargazany, V. N. Khiêm, and M. Itskov, “A generalized network decomposition model for the quasi-static inelastic behavior of filled elastomers,” International Journal of Plasticity, vol. 63, no. 12, pp. 94–109, 2014.
  • [24]  V. N. Khiêm and M. Itskov, “An averaging based tube model for deformation induced anisotropic stress softening of filled elastomers,” International Journal of Plasticity, vol. 90, no. 12, pp. 96–115, 2017.
  • [25]  ——, “Analytical network-averaging of the tube model: Mechanically induced chemiluminescence in elastomers,” International Journal of Plasticity, vol. 102, no. 11, pp. 1–15, 2018. 26.
  • [26]  Mullins effect in filled rubber,” Proceedings of Royal Society London A, vol. 455, no. 1988, pp. 2861–2877, 1999.
  • [27]  A. Dorfmann and R. W. Ogden, “A pseudo-elastic model for loading, partial unloading and reloading of particle-reinforced rubber,” International Journal of Solids and Structures, vol. 40, no. 11, pp. 2699–2714, 2003.
  • [28]  S. R. Rickaby and N. H. Scott, “A cyclic stress softening model for the Mullins effect,” International Journal of Solids and Structures, vol. 50, no. 1, pp. 111–120, 2013.
  • [29]  C. Naumann and J. Ihlemann, “On the thermodynamics of pseudo-elastic material models which reproduce the Mullins effect,” International Journal of Solids and Structures, vol. 69-70, pp. 360–369, 2015.
  • [30]  E. Septanika and L. Ernst, “Application of the network alteration theory for modeling the time-dependent behavior of rubber. Part I. General theory,” Mechanics of Materials, vol. 30, no. 4, pp. 253–263, 1998.
  • [31]  G. Marckmann, E. Verron, L. Gornet, G. Chagnon, P. Charrier, and P. Fort, “A theory of network alteration for the mullins effect,” Journal of the Mechanics and Physics of Solids, vol. 50, no. 9, pp. 2011–2028, 2002.
  • [32]  J. Diani, M. Brieu, and J. M. Vacherand, “A damage directional constitutive model for Mullins effect with permanent set and induced anisotropy,” European Journal of Mechanics A/Solids, vol. 25, no. 3, pp. 483–496, 2006.
  • [33]  X. Zhao, “A theory for large deformation and damage of interpenetrating polymer networks,” Journal of the Mechanics and Physics of Solids, vol. 60, no. 2, pp. 319–332, 2012.
  • [34]  Q. Wang and Z. Gao, “A constitutive model of nanocomposite hydrogels with nanoparticle crosslinkers,” Journal of the Mechanics and Physics of Solids, vol. 94, pp. 127–147, 2016.
  • [35]  P. Zhu and Z. Zhong, “Modelling the mechanical behaviors of double-network hydrogels,” International Journal of Solids and Structures, vol. 193-194, pp. 492–501, 2020.
  • [36]  ——, “Development of the network alteration theory for the Mullins softening of double-network hydrogels,” Mechanics of Materials, vol. 152, pp. 103 658(1–7), 2021.
  • [37]  S. Cantournet, R. Desmorat, and J. Besson, “Mullins effect and cyclic stress softening of filled elastomers by internal sliding and friction thermodynamics model,” International Journal of Solids and Structures, vol. 46, no. 11-12, pp. 2255–2264, 2009.
  • [38]  L. Mullins, “Softening of rubber by deformation,” Rubber Chemistry and Technology, vol. 42, no. 1, pp. 339–362, 1969.
  • [39]  J. Diani, B. Fayolle, and P. Gilormini, “A review on the Mullins effect,” European Polymer Journal, vol. 45, no. 3, pp. 601–612, 2009.
  • [40]  R. Diaz, J. Diani, and P. Gilormini, “Physical interpretation of the Mullins softening in a carbon-black filled SBR,” European Polymer Journal, vol. 55, no. 19, pp. 4942–4947, 2014.
  • [41]  T. A. Vilgis, G. Heinrich, and M. Klüppel, Reinforcement of polymer nano-composites: theory, experiments and applications, 1st ed. Cambridge: Cambridge University Press, 2009.
  • [42]  R. S. Rivlin, “Large elastic deformations of isotropic materials IV. Further developments of the general theory,” Philosophical Transactions of the Royal Society A, vol. 241, pp. 379–397, 1948.
  • [43]  F. Zhao, “Continuum constitutive modeling for isotropic hyperelastic materials,” Advances in Pure Mathematics, vol. 6, no. 9, pp. 571–582, 2016.
  • [44]  ——, “Modeling and implementing compressible isotropic finite deformation without the isochoric–volumetric split,” Journal of Advances in Applied Mathematics, vol. 5, no. 2, pp. 57–70, 2020.
  • [45]  L. R. G. Treloar, “Stress-strain data for vulcanised rubber under various types of deformation,” Transactions of the Faraday Society, vol. 40, pp. 59–70, 1944.
  • [46]  M. C. Boyce and E. M. Arruda, “Constitutive models of rubber elasticity: A review,” Rubber Chemistry and Technology, vol. 73, no. 3, pp. 504–523, 2000.
  • [47]  G. Chagnon, G. Marckmann, and E. Verron, “A comparison of the Hart-Smith model with Arruda-Boyce and Gent formulations for rubber elasticity,” Rubber Chemistry and Technology, vol. 77, no. 4, pp. 724–735, 2004.
  • [48]  P. Steinmann, M. Hossain, and G. Possart, “Hyperelastic models for rubber-like materials: Consistent tangent operators and suitability for Treloar’s data,” Archive for Applied Mechanics, vol. 82, no. 9, pp. 1183–1217, 2012.
  • [49]  F. Zhao, “On constitutive modeling of arteries,” Journal of Advances in Applied Mathematics, vol. 4, no. 2, pp. 54–68, 2019.
  • [50]  M. Cheng and W. Chen, “Experimental investigation of the stress-stretch behavior of EPDM rubber with loading rate effects,” International Journal of Solids and Structures, vol. 40, no. 18, pp. 4749–4768, 2003.
  • [51]  G. Marckmann and E. Verron, “Comparison of hyperelastic models for rubberlike materials,” Rubber Chemistry and Technology, vol. 79, no. 5, pp. 835–858, 2006.
  • [52]  L. R. G. Treloar, The Physics of Rubber Elasticity, 3rd ed. New York: Oxford University Press, 2005.
  • [53]  G. Chagnon, E. Verron, L. Gornet, G. Marckmann, and P. Charrier, “On the relevance of continuum damage mechanics as applied to the Mullins effect in elastomers,” Journal of the Mechanics and Physics of Solids, vol. 52, no. 7, pp. 1627–1650, 2004.
  • [54]  F. Bueche, “Molecular basis for the Mullins effect,” Journal of Applied Polymer Science, vol. 4, no. 10, pp. 107–114, 1960.
  • [55]  R. Houwink, “Slipping of molecules during the deformation of reinforced rubber,” Rubber Chemistry and Technology, vol. 29, no. 3, pp. 888–893, 1956.
  • [56]  F. Cl′ement, L. Bokobza, and L. Monnerie, “On the Mullins effect in silica filled polydimethylsiloxane networks,” Rubber Chemistry and Technology, vol. 74, no. 5, pp. 846–870, 2001.
  • [57]  T. Sui, E. Salvati, S. Ying, G. Sun, I. P. Dolbnya, K. Dragnevski, C. Prisacariu, and A. M. Korsunsky, “Strain softening of nano-scale fuzzy interfaces causes Mullins effect in thermoplastic polyurethane,” Scientific Reports, vol. 7, pp. 916 (1–9), 2017.
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