Filled polymers which are used in various areas of engineering usually show a significant stress-softening, commonly known as Mullins effect which is pretty well investigated in the time-regime (in form of stress–strain relations). As these materials are often exposed to dynamic loads (e.g. tires, engine-mounts, etc.), it is very common to test them by using the dynamic mechanical analysis (DMA) and describe the material behaviour in the frequency domain (in terms of dynamic moduli). In this contribution, it is experimentally investigated how the Mullins effect influences the dynamic properties (storage and loss moduli) of the material and it is shown that the stress-softening is not only dependent on the maximum strain in history, but is also influenced by the rate of deformation during the preconditioning process. In the second part, a modelling approach for this phenomenon is presented while the anisotropic behaviour of the Mullins effect is considered by use of the concept of representative directions. Lastly, the contribution is closed with simulations of the experiments performed to show the capability of the model.
MullinsL. Softening of rubber by deformation. Rubber Chem Technol1969; 42: 339–362.
4.
OgdenRWRoxburghDG. A pseudoelastic model for the Mullins effect in filled rubber. Proc R Soc London, Ser A1999; 455: 2861–2877.
5.
MarckmannGVerronEGornetL. A theory of network alteration for the Mullins effect. J Mech Phys Solids2002; 50: 2011–2028.
6.
QiHJBoyceMC. Constitutive model for stretch-induced softening of the stress-stretch behavior of elastomeric materials. J Mech Phys Solids2004; 52: 2187–2205.
7.
ChagnonGVerronEMarckmannG. Development of new constitutive equations for the Mullins effect in rubber using the network alteration theory. Int J Solids Struct2006; 43: 6817–6831.
8.
KlüppelMMeierJDämgenM. Modelling of stress softening and filler induced hysteresis of elastomer materials. Constitutive models for rubber IV, London: Taylor & Francis, 2005, pp. 171–171.
9.
LorenzHKlüppelMHeinrichG. Microstructure-based modelling and FE implementation of filler-induced stress softening and hysteresis of reinforced rubbers. J Appl Math Mech2012; 92: 608–631.
10.
HorganCOOgdenRWSaccomandiG. A theory of stress softening of elastomers based on finite chain extensibility. Proc R Soc London, Ser A2004; 460: 1737–1754.
11.
DianiJBrieuMVacherandJM. A damage directional constitutive model for Mullins effect with permanent set and induced anisotropy. Eur J Mech A Solids2006; 25: 483–496.
12.
DargazanyRItskovM. A network evolution model for the anisotropic Mullins effect in carbon black filled rubbers. Int J Solids Struct2009; 46: 2967–2977.
13.
Merckel Y. Experimental characterization and modeling of the mechanical behavior of filled rubbers under cyclic loading conditions. PhD thesis, Ecole Centrale de Lille, 2012.
14.
Diercks N and Lion A. Modelling deformation-induced anisotropy using 1D-laws for Mullins-effect. In: Constitutive models for rubber VIII, 2013. CRC Press/Balkema: The Netherlands, Leiden, pp.419–424.
15.
MieheCGöktepeSLuleiF. A micro-macro approach to rubber-like materials. J Mech Phys Solids2004; 52: 2617–2660.
16.
GöktepeSMieheC. A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage. J Mech Phys Solids2005; 53: 2259–2283.
17.
FreundMIhlemannJ. Generalization of one-dimensional material models for the finite element method. J Appl Math Mech2010; 90: 399–417.
18.
Freund M. Verallgemeinerung eindimensionaler Materialmodelle für die Finite-Elemente-Methode. PhD thesis. Technische Universität Chemnitz, 2012.
19.
TobolskyAV. Mechanische Eigenschaften und Struktur von Polymeren, Berliner Union, 1967.
20.
Haupt P. Continuum mechanics and theory of materials. Advanced texts in physics. Berlin: Springer, 2002.
21.
HauptPLionA. On finite linear viscoelasticity of incompressible isotropic materials. Acta Mech2002; 159: 87–124.
22.
KariL. On the dynamic stiffness of preloaded vibration isolators in the audible frequency range: Modeling and experiments. J Acoust Soc Am2003; 113: 1909–1921.
23.
HöferPLionA. Modelling of frequency- and amplitude-dependent material properties of filler-reinforced rubber. J Mech Phys Solids2009; 57: 500–520.
24.
GovindjeeSSimoJC. Mullins effect and the strain amplitude dependence of the storage modulus. Int J Solids Struct1992; 29: 1737–1751.
25.
FletcherWPGentAN. Non-linearity in the dynamic properties of vulcanised rubber compounds. Trans Inst Rubber Ind1953; 29: 266–280.
26.
PayneAR. The dynamic properties of carbon black-loaded natural rubber vulcanizates. Part I. J Appl Polym Sci1962; 6: 57–63.
LionAKardelkyC. The Payne effect in finite viscoelasticity: constitutive modelling based on fractional derivatives and intrinsic time scales. Int J Plast2004; 20: 1313–1345.
29.
RendekMLionA. Amplitude dependence of filler-reinforced rubber: Experiments, constitutive modelling and FEM-Implementation. Int J Solids Struct2010; 47: 2918–2936.
30.
Rendek M. Transient effects of filler-reinforced rubber with respect to the payne effect: experiments, constitutive modelling and FEM implementation. IRIS, 2011.
31.
BažantPOhBH. Efficient numerical integration on the surface of a sphere. J Appl Math Mech1986; 66: 37–49.
32.
LionADiercksNCaillardJ. On the directional approach in constitutive modelling: A general thermomechanical framework and exact solutions for Mooney-Rivlin type elasticity in each direction. Int J Solids Struct2013; 50: 2518–2526.
33.
HeinrichGStraubeEHelmisG. Rubber elasticity of polymer networks: theories. In: Polymer physics. vol. 85 of Advances in polymer science, Berlin: Springer, 1988, pp. 33–87.
34.
FreundMIhlemannJRabkinM. Modelling of the Payne effect using a 3-d generalization technique for the finite element method. In: Constitutive models for rubber VII, CRC Press, 2011, pp. 235–240.
