Abstract
The aim of this paper is to propose an approximate homogenization method for deriving the effective strain energy density function of small volume fraction particle-reinforced hyperelastic matrix composites. This method is inspired by the finite element (FE) simulation of a single-particle representative volume element (RVE) model, in which an incompressible Yeoh constitutive model is used to describe the mechanical behavior of the matrix and the particle is regarded as a rigid material. According to the FE results, the RVE can be partitioned into different regions with uniform strain distributions, and then, the equivalent strain energy density function of the whole composite can be calculated based on a volume averaging treatment. With the new method, the stress–strain response of a mica particle-filled rubber matrix composite under uniaxial tension is theoretically predicted, and it shows a good agreement with both experimental measurements and numerical calculations, especially when the particle volume fraction is relatively small. Due to the simple derivation and concise formulation of the strain energy density function, the present method should be of practical value for engineering use in predicting the large deformation behavior of hyperelastic composites.
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References
Mooney, M.: A theory of large elastic deformation. J. Appl. Phys. 11, 582–592 (1940)
Ogden, R.W.: Large deformation isotropic elasticity–correlation of theory and experiment for incompressible rubber like solids. Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 326, 565–584 (1972)
Valanis, K.C., Landel, R.F.: The strain-energy function of a hyperelastic material in terms of the extension ratios. J. Appl. Phys. 38, 2997–3002 (1967)
Arruda, E.M., Boyce, M.C.: A 3-dimensional constitutive model for the large stretch behavior of rubber elastic-materials. J. Mech. Phys. Solids 41, 389–412 (1993)
Debotton, G., Shmuel, G.: A new variational estimate for the effective response of hyperelastic composites. J. Mech. Phys. Solids 58, 466–483 (2010)
Duan, S., Wen, W., Fang, D.: A predictive micropolar continuum model for a novel three-dimensional chiral lattice with size effect and tension-twist coupling behavior. J. Mech. Phys. Solids 121, 23–46 (2018)
Bergstrom, J.S., Boyce, M.C.: Mechanical behavior of particle filled elastomers. Rubber Chem. Technol. 72, 633–656 (1999)
Treloar, L.R.G.: Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc. 40, 0059–0069 (1944)
Treloar, L.R.G., Montgomery, D.J.: The Physics of Rubber Elasticity. Clarendon Press, Oxford (1958)
Mullins, L., Tobin, N.R.: Stress softening in rubber vulcanizates. Part I. Use of a strain amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber. J. Appl. Polym. Sci. 9, 2993–3009 (1965)
Hill, R.: Constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 326, 131–147 (1972)
Hill, R., Rice, J.R.: Elastic potentials and the structure of inelastic constitutive laws. SIAM J. Appl. Math. 25, 448–461 (1973)
Lopez-Pamies, O., Castañeda, P.P.: On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—Theory. J. Mech. Phys. Solids 54, 831–863 (2006)
Hashin, Z.: Large isotropic elastic deformation of composites and porous media. Int. J. Solids Struct. 21, 711–720 (1985)
Ogden, R.W.: Extremum principles in non-linear elasticity and their application to composites—I: theory. Int. J. Solids Struct. 14, 265–282 (1978)
Castaneda, P.P.: The overall constitutive behaviour of nonlinearly elastic composites. Proc. R. Soc. Lond. 422, 147–171 (1989)
Lopez-Pamies, O., Castaneda, P.P.: Second-order estimates for the large-deformation response of particle-reinforced rubbers. c. r. Mec. 331, 1–8 (2003)
Lopez-Pamies, O., Ponte Castaneda, P.: Second-order estimates for the macroscopic response and loss of ellipticity in porous rubbers at large deformations. J. Elast. 76, 247–287 (2004)
Dorfmann, A., Ogden, R.W.: A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber. Int. J. Solids Struct. 41, 1855–1878 (2004)
Drozdov, A.D.: Constitutive equations in finite elasticity of rubbers. Int. J. Solids Struct. 44, 272–297 (2007)
Guo, Z.Y., Chen, Y., Wan, Q., et al.: A hyperelastic constitutive model for chain-structured particle reinforced neo-Hookean composites. Mater. Des. 95, 580–590 (2016)
Lopez-Pamies, O.: An exact result for the macroscopic response of particle-reinforced Neo-Hookean solids. J. Mech. 77, 021016 (2010)
Lopez-Pamies, O., Idiart, M.: An exact result for the macroscopic response of porous Neo-Hookean solids. J. Elast. 95, 99–105 (2009)
Qi, H.J., Boyce, M.C.: Constitutive model for stretch-induced softening of the stress–stretch behavior of elastomeric materials. J. Mech. Phys. Solids 52, 2187–2205 (2004)
Bouchart, V., Brieu, M., Bhatnagar, N., et al.: A multiscale approach of nonlinear composites under finite deformation: experimental characterization and numerical modeling. Int. J. Solids Struct. 47, 1737–1750 (2010)
Goudarzi, T., Lopez-Pamies, O.: Numerical modeling of the nonlinear elastic response of filled elastomers via composite-sphere assemblages. J. Appl. Mech. 80, 050906 (2013)
Guo, Z.Y., Shi, X.H., Chen, Y., et al.: Mechanical modeling of incompressible particle-reinforced Neo-Hookean composites based on numerical homogenization. Mech. Mater. 70, 1–17 (2014)
Li, X., Xia, Y., Li, Z.R., et al.: Three-dimensional numerical simulations on the hyperelastic behavior of carbon-black particle filled rubbers under moderate finite deformation. Comput. Mater. Sci. 55, 157–165 (2012)
Ilseng, A., Skallerud, B.H., Clausen, A.H.: An experimental and numerical study on the volume change of particle-filled elastomers in various loading modes. Mech. Mater. 106, 44–57 (2017)
Segurado, J., Llorca, J.: A numerical approximation to the elastic properties of sphere-reinforced composites. J. Mech. Phys. Solids 50, 2107–2121 (2002)
Gilormini, P., Toulemonde, P.A., Diani, J., et al.: Stress-strain response and volume change of a highly filled rubbery composite: experimental measurements and numerical simulations. Mech. Mater. 111, 57–65 (2017)
Yeoh, O.H.: Some forms of the strain-energy function for rubber. Rubber Chem. Technol. 66, 754–771 (1993)
Yeoh, O.H., Fleming, P.D.: A new attempt to reconcile the statistical and phenomenological theories of rubber elasticity. J. Polym. Sci. Part B Polym. Phys. 35, 1919–1931 (2015)
Acknowledgements
The work reported here is supported by NSFC through Grants (Nos. 11532013, 11872114, 11772333) and the Project of State Key Laboratory of Explosion Science and Technology (No. ZDKT17-02). The authors thank Jian Liu and Prof. Xiaohong Li (Engineering Research Center for Nanomaterials, Henan University) for their assistance in providing natural rubber-based particulate composite samples.
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Li, D., Yao, Y. An approximate method to predict the mechanical properties of small volume fraction particle-reinforced composites with large deformation matrix. Acta Mech 230, 3307–3315 (2019). https://doi.org/10.1007/s00707-019-02444-5
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DOI: https://doi.org/10.1007/s00707-019-02444-5