69. Khan AS, Lopez-Pamies O, Kazmi R (2006) Thermo-mechanical large deformation response
and constitutive modeling of viscoelastic polymers over a wide range of strain rates and
temperatures. Int J Plast 22:581–601
70. Boiko AV, Kulik VM, Seoudi BM, Chun H, Lee I (2010) Measurement method of complex
viscoelastic material properties. Int J Solids Struct 47:374–382
71. Lee JH, Kim KJ (2001) Characterization of complex modulus of viscoelastic materials
subject to static compression. Mech Time-Depend Mater 5:255–271
72. Gottenberg W, Christensen R (1972) Prediction of the transient response of a linear viscoelastic solid. J Appl Mech 6:448–450
73. Osanaiye GJ (1996) Effects of temperature and strain amplitude on dynamic mechanical
properties of EPDM gum and its carbon black compounds. J Appl Polym Sci 59:567–575
74. Luo W, Hu X, Wang C, Li Q (2010) Frequency- and strain-amplitude-dependent dynamical
mechanical properties and hysteresis loss of CB-filled vulcanized natural rubber. Int J Mech
Sci 52:168–174
75. Pipkin A (1986) Lectures on viscoelasticity theory. Springer, Berlin
76. Williams M, Landel R, Ferry J (1955) The temperature dependence of relaxation mechanisms
in amorphous polymers and other glass-forming liquids. J Am Chem Soc 77:3701–3707
77. Christensen R (2003) Theory of viscoelasticity, 2nd edn. Dover Publications, New York
78. Singh A, Lakes R, Gunasekaran S (2006) Viscoelastic characterization of selected foods over
an extended frequency range. Rheol Acta 46:131–142
79. Mullins L (1947) Effect of stretching on the properties of rubber. J Rubber Res 16:275–289
80. Dorfmann A, Ogden RW (2003) A pseudo-elastic model for loading, partial unloading and
reloading of particle-reinforced rubber. Int J Solids Struct 40:2699–2714
81. Dorfmann A, Ogden RW (2004) A constitutive model for the Mullins effect with permanent
set in particle-reinforced rubber. Int J Solids Struct 41:1855–78
82. 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
39:814–22
83. Harwood JAC, Payne AR (1966) Stress softening in natural rubber vulcanizates III. Carbon
black filled vulcanizates. J Appl Polym Sci 10:315–23
84. Mullins L, Tobin NR (1957) Theoretical model for the elastic behavior of filler reinforced
vulcanized rubbers. Rubber Chem Technol 30:555–71
85. Klu ¨ppel M, Schramm M (2000) A generalized tube model of rubber elasticity and stress
softening of filler reinforced elastomer systems. Macromol Theory Simul 9:742–54
86. Diani J, Brieu M, Vacherand JM (2006) A damage directional constitutive model for Mullins
effect with permanent set and induced anisotropy. Eur J Mech Solids/A 25:483–96
87. Kakavas PA (1996) Mechanical properties of bonded elastomer discs subjected to triaxial
stress. J Appl Polym Sci 59:251–61
88. Flamm M, Steinweger T, Spreckels J, Bru ¨ger T (2008) In mechanical properties of EPDM.
In: Boukamel A, Laiarinandrasana L, Me ´o S, Verron E (eds) In constitutive models for
rubber. V. Balkema, Netherlands, pp 233–242
89. Cle ´ment F, Bokobza L, Monnerie L (2001) On the Mullins effect in silica filled polydimethylsiloxane networks. Rubber Chem Technol 74:846–70
90. Mullins L (1948) Effect of stretching on the properties of rubber. J Rubber Res 16:275–82
91. Stevenson I, David L, Gauthier C, Arambourg L, Davenas J, Vigier G (2001) Influence of
SiO2 fillers on the radiation ageing of silicone rubbers. Polymer 42:9287–92
92. Hanson DE, Hawley M, Houlton R, Chitanvis K, Rae P, Orler EB et al (2005) Stress softening
experiments in silica-filled polydimethylsiloxane provide insight into a mechanism for the
Mullins effect. Polymer 46:10989–95
93. Blanchard AF, Parkinson D (1952) Breakage of carbon-rubber networks by applied stress.
J Ind Eng Chem 44:799–812
94. Mullins L, Tobin N (1957) Theoretical model for the elastic behavior of fillerreinforced
vulcanized rubbers. Rubber Chem Technol 30:551–571
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
267
and constitutive modeling of viscoelastic polymers over a wide range of strain rates and
temperatures. Int J Plast 22:581–601
70. Boiko AV, Kulik VM, Seoudi BM, Chun H, Lee I (2010) Measurement method of complex
viscoelastic material properties. Int J Solids Struct 47:374–382
71. Lee JH, Kim KJ (2001) Characterization of complex modulus of viscoelastic materials
subject to static compression. Mech Time-Depend Mater 5:255–271
72. Gottenberg W, Christensen R (1972) Prediction of the transient response of a linear viscoelastic solid. J Appl Mech 6:448–450
73. Osanaiye GJ (1996) Effects of temperature and strain amplitude on dynamic mechanical
properties of EPDM gum and its carbon black compounds. J Appl Polym Sci 59:567–575
74. Luo W, Hu X, Wang C, Li Q (2010) Frequency- and strain-amplitude-dependent dynamical
mechanical properties and hysteresis loss of CB-filled vulcanized natural rubber. Int J Mech
Sci 52:168–174
75. Pipkin A (1986) Lectures on viscoelasticity theory. Springer, Berlin
76. Williams M, Landel R, Ferry J (1955) The temperature dependence of relaxation mechanisms
in amorphous polymers and other glass-forming liquids. J Am Chem Soc 77:3701–3707
77. Christensen R (2003) Theory of viscoelasticity, 2nd edn. Dover Publications, New York
78. Singh A, Lakes R, Gunasekaran S (2006) Viscoelastic characterization of selected foods over
an extended frequency range. Rheol Acta 46:131–142
79. Mullins L (1947) Effect of stretching on the properties of rubber. J Rubber Res 16:275–289
80. Dorfmann A, Ogden RW (2003) A pseudo-elastic model for loading, partial unloading and
reloading of particle-reinforced rubber. Int J Solids Struct 40:2699–2714
81. Dorfmann A, Ogden RW (2004) A constitutive model for the Mullins effect with permanent
set in particle-reinforced rubber. Int J Solids Struct 41:1855–78
82. 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
39:814–22
83. Harwood JAC, Payne AR (1966) Stress softening in natural rubber vulcanizates III. Carbon
black filled vulcanizates. J Appl Polym Sci 10:315–23
84. Mullins L, Tobin NR (1957) Theoretical model for the elastic behavior of filler reinforced
vulcanized rubbers. Rubber Chem Technol 30:555–71
85. Klu ¨ppel M, Schramm M (2000) A generalized tube model of rubber elasticity and stress
softening of filler reinforced elastomer systems. Macromol Theory Simul 9:742–54
86. Diani J, Brieu M, Vacherand JM (2006) A damage directional constitutive model for Mullins
effect with permanent set and induced anisotropy. Eur J Mech Solids/A 25:483–96
87. Kakavas PA (1996) Mechanical properties of bonded elastomer discs subjected to triaxial
stress. J Appl Polym Sci 59:251–61
88. Flamm M, Steinweger T, Spreckels J, Bru ¨ger T (2008) In mechanical properties of EPDM.
In: Boukamel A, Laiarinandrasana L, Me ´o S, Verron E (eds) In constitutive models for
rubber. V. Balkema, Netherlands, pp 233–242
89. Cle ´ment F, Bokobza L, Monnerie L (2001) On the Mullins effect in silica filled polydimethylsiloxane networks. Rubber Chem Technol 74:846–70
90. Mullins L (1948) Effect of stretching on the properties of rubber. J Rubber Res 16:275–82
91. Stevenson I, David L, Gauthier C, Arambourg L, Davenas J, Vigier G (2001) Influence of
SiO2 fillers on the radiation ageing of silicone rubbers. Polymer 42:9287–92
92. Hanson DE, Hawley M, Houlton R, Chitanvis K, Rae P, Orler EB et al (2005) Stress softening
experiments in silica-filled polydimethylsiloxane provide insight into a mechanism for the
Mullins effect. Polymer 46:10989–95
93. Blanchard AF, Parkinson D (1952) Breakage of carbon-rubber networks by applied stress.
J Ind Eng Chem 44:799–812
94. Mullins L, Tobin N (1957) Theoretical model for the elastic behavior of fillerreinforced
vulcanized rubbers. Rubber Chem Technol 30:551–571
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
267