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Computational Mechanics

, Volume 57, Issue 4, pp 653–677 | Cite as

Mesoscale constitutive modeling of non-crystallizing filled elastomers

  • Ajay B. HarishEmail author
  • Peter Wriggers
  • Juliane Jungk
  • Nils Hojdis
  • Carla Recker
Original Paper

Abstract

Elastomers are exceptional materials owing to their ability to undergo large deformations before failure. However, due to their very low stiffness, they are not always suitable for industrial applications. Addition of filler particles provides reinforcing effects and thus enhances the material properties that render them more versatile for applications like tyres etc. However, deformation behavior of filled polymers is accompanied by several nonlinear effects like Mullins and Payne effect. To this day, the physical and chemical changes resulting in such nonlinear effect remain an active area of research. In this work, we develop a heterogeneous (or multiphase) constitutive model at the mesoscale explicitly considering filler particle aggregates, elastomeric matrix and their mechanical interaction through an approximate interface layer. The developed constitutive model is used to demonstrate cluster breakage, also, as one of the possible sources for Mullins effect observed in non-crystallizing filled elastomers.

Keywords

Filled elastomers Mesoscale constitutive modeling  Carbon black Finite element method Mullins damage  Cluster breakage 

Notes

Acknowledgments

We would like to thank Continental Aktiengesellschaft, Hannover (Germany) for the generous funding.

References

  1. 1.
    Alexander AP, Lazurkin JS (1946) Strength of amorphous and crystallizing rubberlike polymers. Rubber Chem Technol 19:42–45CrossRefGoogle Scholar
  2. 2.
    Arruda EM, Boyce MC (1993) A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 41:389–412CrossRefGoogle Scholar
  3. 3.
    Bergström JS, Boyce MC (1998) Constitutive modeling of the large strain time-dependent behavior of elastomers. J Mech Phys Solids 46:931–954CrossRefzbMATHGoogle Scholar
  4. 4.
    Bergström JS, Boyce MC (1999) Mechanical behavior of particle filled elastomers. Rubber Chem Technol 72:633–656CrossRefGoogle Scholar
  5. 5.
    Bergström JS, Boyce MC (2000) Large strain time-dependent behavior of elastomers. Mech Mater 32:627–644CrossRefGoogle Scholar
  6. 6.
    Bernstein B, Kearsley EA, Zapas LJ (1963) A study of stress relaxation with finite strain. Trans Soc Rheol 7:391–410CrossRefzbMATHGoogle Scholar
  7. 7.
    Besdo D, Ihlemann J (2003) A phenomenological constitutive model for rubberlike materials and its numerical applications. Int J Plast 19:1019–1036CrossRefzbMATHGoogle Scholar
  8. 8.
    Besdo D, Ihlemann J (2003) Properties of rubberlike materials under large deformations explained by self-organizing linkage patterns. Int J Plast 19:1001–1018CrossRefzbMATHGoogle Scholar
  9. 9.
    Bhattacharya A, Medvedev GA, Caruthers JM (2011) Time dependent mechanical behavior of carbon black filled elastomers. Rubber Chem Technol 84:296–324CrossRefGoogle Scholar
  10. 10.
    Blanchard AF, Parkinson D (1952) Breakage of carbon-rubber networks by applied stress. Ind Eng Chem 44:799–812CrossRefGoogle Scholar
  11. 11.
    Bouasse H, Carriere Z (1903) Sur les courbes de traction du caoutchouc vulcanise. Annales de la facult des sciences de Toulouse 5:257–283MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Boyce MC, Kear K, Socrate S, Shaw K (2001) Deformation of thermoplastic vulcanizates. J Mech Phys Solids 49:1073–1098CrossRefzbMATHGoogle Scholar
  13. 13.
    Browning R, Gurtin ME, Williams WO (1984) A one-dimensional viscoplastic constitutive theory for filled polymers. Int J Solids Struct 20:921–934CrossRefzbMATHGoogle Scholar
  14. 14.
    Bueche F (1960) Molecular basis for the mullins effect. J Appl Polym Sci 4:107–114CrossRefGoogle Scholar
  15. 15.
    Bueche F (1961) Mullins effect and rubber-filler interaction. J Appl Polym Sci 5:271–281CrossRefGoogle Scholar
  16. 16.
    Cantournet S, Desmorat R, Besson J (2009) Mullins effect and cyclic stress softening of filled elastomers by internal sliding and friction thermodynamics model. Int J Solids Struct 46:2255–2264CrossRefzbMATHGoogle Scholar
  17. 17.
    Chagnon G, Verron E, Marckmann G, Gornet L (2006) Development of new constitutive equations for the mullins effect in rubber using the network alteration theory. Int J Solids Struct 43:6817–6831CrossRefzbMATHGoogle Scholar
  18. 18.
    Christensen RM (1980) A nonlinear theory of viscoelasticity for application to elastomers. J Appl Mech 47:762–768MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cotten GR (1984) Mixing of carbon black with rubber: IV. Measurement of dispersion rate by changes in mixing torque. Rubber Chem Technol 57:118–133CrossRefGoogle Scholar
  20. 20.
    Cotten GR (1987) Mixing of carbon black with rubber: IV. Effect of carbon black characteristics. Plast Rubber Process Appl 7:173–178Google Scholar
  21. 21.
    Dal H, Kaliske M (2009) Bergström-boyce model for nonlinear finite rubber viscoelasticity: theoretical aspects and algorithmic treatment for the FE method. Comput Mech 44:809–823MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dannenberg EM, Brennan JJ (1965) Strain energy as a criterion for stress softening in carbon-black filled vulcanizates. Rubber Chem Technol 39:597–608CrossRefGoogle Scholar
  23. 23.
    Dargazany R, Itskov M (2009) A network evolution model for the anisotropic mullins effect in carbon black filled rubbers. Int J Solids Struct 46:2967–2977CrossRefzbMATHGoogle Scholar
  24. 24.
    Dargazany R, Itskov M (2013) Constitutive modeling of the mullins effect and cyclic stress softening in filled elastomers. Phys Rev E 88:012,602-1–012,602-13CrossRefGoogle Scholar
  25. 25.
    Diani J, Gilormini P (2005) Combining the logarithmic strain and the full network for a better understanding of the hyperelastic behavior of rubberlike materials. J Mech Phys Solids 53:2579–2596CrossRefzbMATHGoogle Scholar
  26. 26.
    Diani J, Brieu M, Vacherand JM, Rezgui A (2004) Directional model for isotropic and anisotropic hyperelastic rubberlike materials. Mech Mater 36:313–321CrossRefGoogle Scholar
  27. 27.
    Diani J, Brieu M, Gilormini P (2006) Observation and modeling of anisotropic visco-hyperelastic behavior of a rubberlike material. Int J Solids Struct 43:3044–3056CrossRefzbMATHGoogle Scholar
  28. 28.
    Diani J, Brieu M, Vacherand JM (2006) A damage directional constitutive model for mullins effect with permanent set and induced anisotropy. Eur J Mech A Solids 25:483–496CrossRefzbMATHGoogle Scholar
  29. 29.
    Diani J, Fayolle B, Gilormini P (2009) A review on the mullins effect. Eur Polymer J 45:601–612CrossRefGoogle Scholar
  30. 30.
    Diani J, Gilormini P, Merckel Y, Vion-Loisel F (2013) Micromechanical modeling of the linear viscoelasticity of carbon black filled styrene butadiene rubbers: The role of filler - rubber interphase. Mech Mater 59:65–72CrossRefGoogle Scholar
  31. 31.
    Diaz R, Diani J, Gilormini P (2014) Physical interpretation of mullins effect in a carbon-black filled SBR. Polymer 55:4942–4947CrossRefGoogle Scholar
  32. 32.
    Donnet J (1993) Carbon black: science and technology. CRC Press, New YorkGoogle Scholar
  33. 33.
    Dorfmann A, Ogden RW (2004) A constitutive model for the mullins effect with permanent set in particle reinforced rubber. Int J Solids Struct 41:1855–1878CrossRefzbMATHGoogle Scholar
  34. 34.
    Dorfmann A, Pancheri FQ (2012) A constitutive model for the mullins effect with changes in material symmetry. Int J Non-Linear Mech 47:874–887CrossRefGoogle Scholar
  35. 35.
    Drozdov AD, Dorfmann A (2001) Stress-strain relations in finite viscoelastoplasticity of rigid-rod networks: applications to the mullins effect. Continuum Mech Thermodyn 13:183–205MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Fletcher WP, Gent AN (1953) Nonlinearity in the dynamic properties of vulcanised rubber compounds. Rubber Chem Technol 27:266–280Google Scholar
  37. 37.
    Freund M, Ihlemann J (2010) Generalization of one-dimensional material models for the finite element method. J Appl Math Mech 90:399–417MathSciNetzbMATHGoogle Scholar
  38. 38.
    Freund M, Lorenz H, Juhre D, Ihlemann J, Klüppel M (2011) Finite element implementation of a microstructure based model for filled elastomers. Int J Plast 27:902–919CrossRefzbMATHGoogle Scholar
  39. 39.
    Fritzsche J, Klüppel M (2011) Structural dynamics and interfacial properties of filled-reinforced elastomers. J Condens Matter Phys 23:035,104-01–035,104-11Google Scholar
  40. 40.
    Fukahori Y (2005) New progress in the theory and model of carbon black reinforcement of elastomers. J Appl Polym Sci 95:60–67CrossRefGoogle Scholar
  41. 41.
    Fukahori Y (2007) Generalized concept of the reinforcement of elastomers. Part I: carbon black reinforcement of rubbers. Rubber Chem Technol 80:701–725CrossRefGoogle Scholar
  42. 42.
    Gent AN (1954) Crystallization and the relaxation of stress in stretched natural rubber vulcanizates. Trans Faraday Soc 50:521–533CrossRefGoogle Scholar
  43. 43.
    Gent AN (1962) Relaxation processes in vulcanized rubber. I. Relation among stress relaxation, creep, recovery and hysteresis. J Appl Polym Sci 6:433–441CrossRefGoogle Scholar
  44. 44.
    Göktepe S, Miehe C (2005) A micro-macro approach to rubberlike materials—part III: the microsphere model of anisotropic mullins-type damage. J Mech Phys Solids 52:2259–2283CrossRefzbMATHGoogle Scholar
  45. 45.
    Göktepe S, Miehe C (2008) Efficient two-scale modeling of finite rubber viscoelasticity. Technische Mechanik 28:22–31Google Scholar
  46. 46.
    Govindjee S, Simo J (1991) A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating mullins effect. J Mech Phys Solids 39:87–112MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Govindjee S, Simo J (1992) Transition from micro-mechanics to computationally efficient phenomenology: carbon black filled rubers incorporating mullins effect. J Mech Phys Solids 40:213–233CrossRefzbMATHGoogle Scholar
  48. 48.
    Greene MS, Li Y, Chen W, Liu WK (2014) The archetype-genome exemplar in molecular dynamics and continuum mechanics. Comput Mech 53:687–737MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Guth E (1945) Theory of filler reinforcement. J Appl Phys 16:20–25CrossRefGoogle Scholar
  50. 50.
    Häfner S, Eckardt S, Luther T, Könke C (2006) Mesoscale modeling of concrete: geometry and numerics. Comput Struct 84:450–461CrossRefGoogle Scholar
  51. 51.
    Hamed GR, Hatfield S (1989) On the role of bound rubber in carbon black reinforcement. Rubber Chem Technol 62:143–156CrossRefGoogle Scholar
  52. 52.
    Hanson DE, Hawley M, Houlton R, Chitanvis K, Rae P, Orler EB (2005) Stress softening experiments in silica-filled polydimethylsiloxane provide insight into a mechanism for the mullins effect. Polymer 46:10989–10995CrossRefGoogle Scholar
  53. 53.
    Harish A, Wriggers P (2013) Micromechanical model for mullins effect induced anisotropic damage in filled elastomers. In: Proceedings of 12th U.S. national congress on computational mechanics (USNCCM12), USACM, Raleigh, USAGoogle Scholar
  54. 54.
    Harwood JAC, Payne AR (1966) Stress softening in natural rubber vulcanizates. Part III. Carbon black filled vulcanizates. J Appl Polym Sci 10:315–323CrossRefGoogle Scholar
  55. 55.
    Harwood JAC, Payne AR (1966) Stress softening in natural rubber vulcanizates. Part IV. Unfilled vulcanizates. J Appl Polym Sci 10:1203–1211CrossRefGoogle Scholar
  56. 56.
    Harwood JAC, Mullins L, Payne AR (1966) Stress softening in natural rubber vulcanizates. Part II. Stress softening effects in pure gum and filler loaded rubbers. Rubber Chem Technol 39:814–822CrossRefGoogle Scholar
  57. 57.
    Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behavior of multiphase materials. J Mech Phys Solids 11:127–140MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Hausler K, Sayir MB (1995) Nonlinear viscoelastic response of carbon black reinforced rubber derived from moderately large deformation in torsion. J Mech Phys Solids 43:295–318CrossRefGoogle Scholar
  59. 59.
    Heinrich G, Vilgis TA (1995) Physical adsorption of polymers on disordered filler surfaces. Rubber Chem Technol 68:26–36CrossRefGoogle Scholar
  60. 60.
    Horgan CO, Ogden RW, Saccomandi G (2004) A theory of stress softening of elastomers based on finite chain extensibility. Proc R Soc Lond A 460:1737–1754MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Houwink R (1956) Slippage of molecules during the deformation of reinforced rubber. Rubber Chem Technol 29:888–893CrossRefGoogle Scholar
  62. 62.
    Huber G, Vilgis TA (1999) Universal properties of filled rubbers: mechanisms for reinforcement on different length scales. Kautsch Gummi Kunstst 52:102–107Google Scholar
  63. 63.
    Huber G, Vilgis TA (2002) On the mechanism of hydrodynamic reinforcement in elastic composites. Macromolecules 35:9204–9210CrossRefGoogle Scholar
  64. 64.
    Itskov M, Ehret AE, Makovska RK, Weinhold GW (2010) A thermodynamically consistent phenomenological model of the anisotropic mullins effect. J Appl Math Mech 90:370–396zbMATHGoogle Scholar
  65. 65.
    James AG, Green A (1975) Strain energy functions of rubber II. Characterization of filled vulcanizates. J Appl Polym Sci 19:2033–2058CrossRefGoogle Scholar
  66. 66.
    Kaliske M, Heinrich G (1999) An extended tube-model for rubber elasticity: statistical-mechanical theory and finite element implementation. Rubber Chem Technol 72:602–632CrossRefGoogle Scholar
  67. 67.
    Kaliske M, Nasdala L, Rothert H (2001) On damage modelling for elastic and viscoelastic materials at large strain. Comput Struct 79:2133–2141CrossRefGoogle Scholar
  68. 68.
    Klüppel M (2003) The role of disorder in filler reinforcement of elastomers on various length scales. Adv Polym Sci 164:1–86CrossRefGoogle Scholar
  69. 69.
    Klüppel M, Schramm M (2000) A generalized tube model of rubber elasticity and stress softening of filler reinforced elastomer systems. Macromol Theory Simul 9:742–754CrossRefGoogle Scholar
  70. 70.
    Kraus G (1978) Reinforcement of elastomers by carbon black. Rubber Chem Technol 51:297–321CrossRefGoogle Scholar
  71. 71.
    Kraus G, Childers CW, Rollman KW (1966) Stress softening in carbon black reinforced vulcanizates: strain rate and temperature effects. J Appl Polym Sci 10:229–240CrossRefGoogle Scholar
  72. 72.
    Leblanc JL (2002) Rubber-filler interactions and rheological properties in filled compounds. Prog Polym Sci 27:627–687CrossRefGoogle Scholar
  73. 73.
    Leblanc JL, Evo C, Lionnet R (1994) Composite design experiments to study the relationships between the mixing behavior and the rheological properties of SBR compounds. Kautsch Gummi Kunstst 47:401–407Google Scholar
  74. 74.
    Li Y, Tang S, Abberton BC, Kröger M, Burkhart C, Jiang B, Papkonstantopoulos GJ, Poldneff M, Liu WK (2012) A predictive multiscale computational framework for viscoelastic properties of linear polymers. Polymer 53:5935–5952CrossRefGoogle Scholar
  75. 75.
    Li Y, Kröger M, Liu WK (2014) Dynamic structure of unentangled polymer chains in the vicinity of non-attractive nanoparticles. Soft Matter 10:1723–1737CrossRefGoogle Scholar
  76. 76.
    Lin RC, Schomburg U (2003) A finite elastic-viscoelastic-elastoplastic material law with damage: theoretical and numerical aspects. Comput Methods Appl Mech Eng 192:1591–1627CrossRefzbMATHGoogle Scholar
  77. 77.
    Lion A (1996) A constitutive model for carbon black filled rubber: experimental investigations and mathematical representation. Continuum Mech Thermodyn 8:153–169CrossRefGoogle Scholar
  78. 78.
    Lion A (1997) A physically based method to represent the thermo-mechanical behavior of elastomers. Acta Mech 123:1–25CrossRefzbMATHGoogle Scholar
  79. 79.
    Liu Y, Greene MS, Chen W, Dikin DA, Liu KW (2013) Computational microstructure characterization and reconstruction for stochastic multiscale material design. Comput Aided Des 45:65–76CrossRefGoogle Scholar
  80. 80.
    Liu Z, Moore JA, Aldousari SM, Hedia HS, Asiri SA, Liu WK (2015) A statistical descriptor based volume-integral micromechanics model of heterogeneous material with arbitrary inclusion shape. Comput Mech 55:963–981MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Lorenz H, Freund M, Juhre D, Ihlemann J, Klüppel M (2011) Constitutive generalization of a microstructure based model for filled elastomers. Macromol Theory Simul 20:110–123CrossRefzbMATHGoogle Scholar
  82. 82.
    Lorenz H, Klüppel M, Heinrich G (2012) Microstructure based modelling and FE implementation of filler induced stress softening and hysteresis of reinforced rubbers. J Appl Math Mech 92:608–631MathSciNetGoogle Scholar
  83. 83.
    Lubliner J (1985) A model of rubber viscoelasticity. Mech Res Commun 12:93–99CrossRefGoogle Scholar
  84. 84.
    Lüchow H, Breier E, Grownski W (1997) Characterisation of polymer adsorption on disordered filler surfaces by transversal H-NMR relaxation. Rubber Chem Technol 70:747–758CrossRefGoogle Scholar
  85. 85.
    Machado G, Chagnon G, Favier D (2014) Theory and identification of a constitutive model of induced anisotropy by the mullins effect. J Mech Phys Solids 63:29–39MathSciNetCrossRefGoogle Scholar
  86. 86.
    Maiti A, Small W, Gee RH, Weisgraber TH, Chinn SC, Wilson TS, Maxwell RS (2014) Mullins effect in a filled elastomer under uniaxial tension. Phys Rev E 89(012):602–6Google Scholar
  87. 87.
    Marckmann G, Verron E, Gornet L, Chagnon G, Charrier P, Fort P (2002) A theory of network alteration the mullins effect. J Mech Phys Solids 50:2011–2028CrossRefzbMATHGoogle Scholar
  88. 88.
    Martinez JRS, Cam JBL, Balandraud X, Toussaint E, Caillard J (2014) New elements concerning the mullins effect: a thermomechanical analysis. Eur Polymer J 55:98–107CrossRefGoogle Scholar
  89. 89.
    Medalia AI (1967) Morphology of aggregates I. calculation of shape and bulkiness factors; application to computer-simulated random flocs. J Colloid Interface Sci 24:393–404CrossRefGoogle Scholar
  90. 90.
    Medalia AI, Kraus G (1994) Reinforcement of elastomers by particulate fillers. In: Mark JE, Erman B, Eirich FR (eds) Science and technology of rubber. Academic Press Inc., Cambridge, pp 387–418Google Scholar
  91. 91.
    Meinecke EA, Taftaf MI (1988) Effect of carbon black on the mechanical properties of elastomers. Rubber Chem Technol 61:534–547CrossRefGoogle Scholar
  92. 92.
    Merckel Y, Brieu M, Diani J, Caillard J (2012) A mullins softening criterion for general loading conditions. J Mech Phys Solids 60:1257–1264CrossRefGoogle Scholar
  93. 93.
    Merckel Y, Diani J, Brieu M, Caillard J (2013) Constitutive modeling of the anisotropic behavior of mullins softened filled rubbers. Mech Mater 57:30–41CrossRefGoogle Scholar
  94. 94.
    Miehe C (1995) Discontinuous and continuous damage evolution in ogden type large strain elastic materials. Eur J Mech A Solids 14:697–720zbMATHGoogle Scholar
  95. 95.
    Miehe C, Göktepe S (2005) A micro-macro approach to rubberlike materials—part II: the microsphere model of finite rubber viscoelasticity. J Mech Phys Solids 52:2231–2258CrossRefzbMATHGoogle Scholar
  96. 96.
    Miehe C, Keck J (2000) Superimposed finite elastic–viscoelastic–plastoelastic stress response with damage in filled rubbery polymers. experiments, modelling and algorithmic implementation. J Mech Phys Solids 48:323–365CrossRefzbMATHGoogle Scholar
  97. 97.
    Miehe C, Göktepe S, Lulei F (2004) A micro-macro approach to rubberlike materials—Part I: the non-affine microsphere model of rubber elasticity. J Mech Phys Solids 52:2617–2660MathSciNetCrossRefzbMATHGoogle Scholar
  98. 98.
    Mooney M (1940) A theory of large elastic deformation. J Appl Phys 11:582–592CrossRefzbMATHGoogle Scholar
  99. 99.
    Mooney M (1951) The viscosity of a concentrated suspension of spherical particles. J Colloid Sci 6:162–170CrossRefGoogle Scholar
  100. 100.
    Moore JA, Ma R, Domel AG, Liu WK (2014) An efficient multiscale model of damping properties for filled elastomers with complex microstructures. Compos Part B Eng 62:262–270CrossRefGoogle Scholar
  101. 101.
    Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Mater 21:571–574CrossRefGoogle Scholar
  102. 102.
    Mullins L (1948) Effect of stretching on the properties of rubber. Rubber Chem Technol 21:281–300CrossRefGoogle Scholar
  103. 103.
    Mullins L (1949) Permanent set in vulcanized rubber. India Rubber World 120:63–66Google Scholar
  104. 104.
    Mullins L (1950) Thixotropic behavior of carbon black in rubber. Rubber Chem Technol 23:239–251CrossRefGoogle Scholar
  105. 105.
    Mullins L (1969) Softening of rubber by deformation. Rubber Chem Technol 42:339–362CrossRefGoogle Scholar
  106. 106.
    Mullins L, Tobin NR (1957) Theoretical model for the elastic behavior of filler reinforced vulcanized rubber. Rubber Chem Technol 30:555–571CrossRefGoogle Scholar
  107. 107.
    Mullins L, Tobin NR (1965) Stress softening in rubber vulcanizates. Part I. use of strain amplification factor to describe elastic behavior of filled-reinforced vulcanized rubber. J Appl Polym Sci 9:2993–3007CrossRefGoogle Scholar
  108. 108.
    Neto EADS, Peric D, Owen DRJ (1994) A phenomenological three-dimensional rate-independent continuum damage model for highly filled polymers: formulation and computational aspects. J Mech Phys Solids 42:1533–1550CrossRefzbMATHGoogle Scholar
  109. 109.
    Netzker C, Dal H, Kaliske M (2010) An endochronic plasticity formulation for filled rubber. Int J Solids Struct 47:2371–2379CrossRefzbMATHGoogle Scholar
  110. 110.
    Ogden RW (1972) Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc Lond Ser A 324:565–584CrossRefzbMATHGoogle Scholar
  111. 111.
    Ogden RW, Roxburgh DG (1999) A pseudo-elastic model for the mullins effect in filled rubber. Proc R Soc Lond A 455:2861–2877MathSciNetCrossRefzbMATHGoogle Scholar
  112. 112.
    Payne AR (1962) The dynamic properties of carbon black loaded natural rubber vulcanizates. Part I. J Appl Polym Sci 6:57–63CrossRefGoogle Scholar
  113. 113.
    Pliskin I, Tokita N (1972) Bound rubber in elastomers: analysis of elastomer, filler interaction and its effects on viscosity and modulus of composite systems. J Appl Polym Sci 16:473–492CrossRefGoogle Scholar
  114. 114.
    Qi HJ, Boyce MC (2004) Constitutive model for stretch-induced softening of the stress–stretch behavior of elastomeric materials. J Mech Phys Solids 52:2187–2205CrossRefzbMATHGoogle Scholar
  115. 115.
    Rajagopal KR, Wineman AS (1992) A constitutive equation for the nonlinear solids which undergo deformation induced microstructural changes. Int J Plast 8:385–395CrossRefzbMATHGoogle Scholar
  116. 116.
    Reuss A (1929) Berechnung der fliessgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle. J Appl Math Mech 9:49–58zbMATHGoogle Scholar
  117. 117.
    Rickaby SR, Scott NH (2013) A cyclic stress-softening model for the mullins effect. Int J Solids Struct 50:111–120CrossRefzbMATHGoogle Scholar
  118. 118.
    Rigbi Z (1980) Reinforcement of rubber by carbon black. Adv Polym Sci 36:21–68CrossRefGoogle Scholar
  119. 119.
    Rivlin RS (1948) Large elastic deformations of isotropic materials. I. Fundamental concepts. Philos Trans R Soc Lond A 240:459–490MathSciNetCrossRefzbMATHGoogle Scholar
  120. 120.
    Schmoller KM, Bausch AR (2013) Similar nonlinear mechanical responses in hard and soft materials. Nat Mater 12:278–281CrossRefGoogle Scholar
  121. 121.
    Schröder J, Neff P, Balzani D (2005) A variational approach for materially stable anisotropic hyperelasticity. Int J Solids Struct 42:4352–4371MathSciNetCrossRefzbMATHGoogle Scholar
  122. 122.
    Sichel EK, Gittleman JI, Sheng P (1982) Electrical properties of carbon-polymer composites. J Electron Mater 11:699–747CrossRefGoogle Scholar
  123. 123.
    Smallwood HM (1944) Limiting law of the reinforcement of rubber. J Appl Phys 15:758–766CrossRefGoogle Scholar
  124. 124.
    So H, Chen UD (1991) A nonlinear mechanical model for solid-filled polymers. Polym Eng Sci 6:410–416CrossRefGoogle Scholar
  125. 125.
    Sodhani D, Reese S (2014) Finite element based micromechanical modeling of microstructure morphology in filler reinforced elastomer. Macromolecules 47:3161–3169CrossRefGoogle Scholar
  126. 126.
    Suzuki N, Ito M, Yatsuyanagi F (2005) Effects of rubber/filler interactions on deformation behavior of silica filled SBR systems. Polymer 46:193–201CrossRefGoogle Scholar
  127. 127.
    Tang S, Greene MS, Liu WK (2012) Two-scale mechanism-based theory of nonlinear viscoelasticity. J Mech Phys Solids 60:199–226MathSciNetCrossRefzbMATHGoogle Scholar
  128. 128.
    Tommasi DD, Puglisi G, Saccomandi G (2006) A micromechanics-based model for the mullins effect. J Rheol 50:495–512CrossRefGoogle Scholar
  129. 129.
    Vilgis TA, Heinrich G, Klüppel M (2009) Reinforcement of polymer nano-composites. Cambridge university press, CambridgeCrossRefGoogle Scholar
  130. 130.
    Voigt W (1889) Ueber die beziehung zwischen den beiden elasticitätsconstanten isotroper körper. Annalen der physik 274:573–587Google Scholar
  131. 131.
    Wang ZM, Kwan AKH, Chan HC (1999) Mesoscopic study of concrete i: generation of random aggregate structure and finite element mesh. Comput Struct 70:533–544CrossRefzbMATHGoogle Scholar
  132. 132.
    Wriggers P, Moftah SO (2006) Mesoscale models for concrete: homogenization and damage behaviour. Finite Elem Anal Des 42:623–636CrossRefGoogle Scholar
  133. 133.
    Xu H, Greene MS, Deng H, Dikin D, Brinson C, Liu KW, Burkhart C, Papakonstantopoulos G, Poldneff M, Chen W (2013) Stochastic reassembly strategy for managing information complexity in heterogeneous materials analysis and design. J Mech Des 135(101):010Google Scholar
  134. 134.
    Xu H, Dikin D, Burkhart C, Chen W (2014) Descriptor-based design methodology for statistical characterization and 3d reconstruction of microstructural materials. Comput Mater Sci 85:206–216CrossRefGoogle Scholar
  135. 135.
    Xu H, Li Y, Brinson C, Chen W (2014) Descriptor-based design methodology for developing heterogeneous microstructural materials system. J Mech Des 136:05.2007Google Scholar
  136. 136.
    Zohdi TI, Wriggers P (2005) An introduction to computational micromechanics. Springer, BerlinCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Ajay B. Harish
    • 1
    Email author
  • Peter Wriggers
    • 1
  • Juliane Jungk
    • 2
  • Nils Hojdis
    • 2
  • Carla Recker
    • 2
  1. 1.Institute of Continuum MechanicsHannoverGermany
  2. 2.Continental Reifen Deutschland GmbHHannoverGermany

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