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Direct determination of multi-axial elastic potentials for incompressible elastomeric solids: an accurate, explicit approach based on rational interpolation

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Abstract

With a novel approach based on certain logarithmic invariants, we demonstrate that a multi-axial elastic potential for incompressible, isotropic rubber-like materials may be obtained directly from two one-dimensional elastic potentials for uniaxial case and simple shear case, in a sense of exactly matching finite strain data for four benchmark tests, including uniaxial extension, simple shear, bi-axial extension, and plane-strain extension. As such, determination of multi-axial elastic potentials may be reduced to that of two one-dimensional elastic potentials. We further demonstrate that the latter two may be obtained by means of rational interpolating procedures for uniaxial data and shear data displaying strain-stiffening effects. Numerical examples are presented in fitting Treloar’s data and other data.

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References

  1. Anand L.: On H. Hencky’s approximate strain-energy function for moderate deformations. J. Appl. Mech. 46, 78–82 (1979)

    Article  ADS  MATH  Google Scholar 

  2. Anand L.: Moderate deformations in extension-torsion of incompressible isotropic elastic materials. J. Mech. Phys. Solids 34, 293–304 (1986)

    Article  ADS  Google Scholar 

  3. Arruda E.M., Boyce M.C.: A three-dimensional constitutive model for the large stretch behaviour of rubber elastic materials. J. Mech. Phys. Solids 41, 389–412 (1993)

    Article  ADS  Google Scholar 

  4. Beatty M.F.: Topic in finite elasticity: Hyper-elasticity of rubber, elastomers, and biological tissues-with examples. Appl. Mech. Rev. 40, 1699–1733 (1987)

    Article  ADS  Google Scholar 

  5. Beatty M.F.: On constitutive models for limited elastic, molecular based materials. Math. Mech. Solids 13, 375–387 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Boyce M.C., Arruda E.M.: Constitutive models of rubber elasticity: a review. Rubber Chem. Technol. 73, 504–523 (2000)

    Article  Google Scholar 

  7. Charlton D.J., Yang J.: A review of methods to characterize rubber elastic behavior for use in finite element analysis. Rubber Chem. Technol. 67, 481–503 (1994)

    Article  Google Scholar 

  8. Criscione S.C.: Direct tensor expression for natural strain and fast, accurate approximation. Comput. Struct. 80, 1895–1905 (2002)

    Article  MathSciNet  Google Scholar 

  9. Criscione S.C.: Rivlin’s representation formula is ill-conceived for the determination of response functions via biaxial testing. J. Elast. 70, 129–147 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Criscione J.C., Humphrey J.D., Douglas A.S., Hunter W.C.: An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyper-elasticity. J. Mech. Phys. Solids 48, 2445–2465 (2000)

    Article  ADS  MATH  Google Scholar 

  11. Diani J., Gilormini P.: Combining the logarithmic strain and the full-network model for a better understanding of the hyper-elastic behaviour of rubber-like materials. J. Mech. Phys. Solids 53, 2579–2596 (2005)

    Article  ADS  MATH  Google Scholar 

  12. Drozdov A.D., Gottlieb M.: Ogden-type constitutive equations in finite elasticity of elastomers. Acta Mech. 183, 231–252 (2006)

    Article  MATH  Google Scholar 

  13. Drozdov A.D.: Polymer networks with slip-links: 1. Constitutive equations for an uncross-linked network. Continuum Mech. Therm. 18, 157–170 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Drozdov A.D.: Polymer networks with slip-links: 2. Constitutive equations for a cross-linked network. Continuum Mech. Therm. 18, 171–193 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Eterovic A.L., Bathe K.J.: A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures. Int. J. Numer. Methods Eng. 30, 1099–1115 (1990)

    Article  MATH  Google Scholar 

  16. Fitzjerald S.: A tensorial Hencky measure of strain and strain rate for finite deformation. J. Appl. Phys. 51, 5111–5115 (1980)

    Article  ADS  Google Scholar 

  17. Fu Y.B., Ogden R.W.: Nonlinear Elasticity. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  18. Gendy A.S., Saleeb A.F.: Nonlinear material parameter estimation for characterizing hyperelastic large strain models. Comput. Mech. 25, 66–77 (2001)

    Article  Google Scholar 

  19. Gent A.N.: A new constitutive relation for rubber. Rubber Chem. Technol. 69, 59–61 (1996)

    Article  MathSciNet  Google Scholar 

  20. Gent A.N.: Extensibility of rubber under different types of deformation. J. Rheol. 49, 271–275 (2005)

    Article  ADS  Google Scholar 

  21. Green A.E., Zerna W.: Theoretical Elasticity. Clarendon Press, Oxford (1968)

    MATH  Google Scholar 

  22. Hartmann S.: Parameter estimation of hyper-elasticity relations of generalized polynomial-type with constraint conditions. Int. J. Solids Struct. 38, 7999–8018 (2001)

    Article  MATH  Google Scholar 

  23. Haupt P.: Continuum Mechanics and Theory of Materials. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  24. Hencky H.: Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Z. Techn. Phys. 9, 215–220 (1928)

    Google Scholar 

  25. Hencky H.: The law of elasticity for isotropic and quasi-isotropic substances by finite deformations. J. Rheol. 2, 169–176 (1931)

    Article  ADS  Google Scholar 

  26. Hencky H.: The elastic behavior of vulcanized rubber. Rubber Chem. Technol. 6, 217–224 (1933)

    Article  Google Scholar 

  27. Hill R.: Constitutive inequalities for simple materials. J. Mech. Phys. Solids 16, 229–242 (1968)

    Article  ADS  MATH  Google Scholar 

  28. Hill R.: Constitutive inequalities for isotropic elastic solids under finite strain. Proc. R. Soc. Lond. A 326, 131–147 (1970)

    Article  ADS  Google Scholar 

  29. Hill R.: Aspects of invariance in solid mechanics. Adv. Appl. Mech. 18, 1–75 (1978)

    Article  MATH  Google Scholar 

  30. Horgan C.O., Murphy J.G.: Limiting chain extensibility constitutive models of Valanis-Landel type. J. Elast. 86, 101–111 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Horgan C.O., Murphy J.G.: A generalization of Hencky’s strain-energy density to model the large deformation of slightly compressible solid rubber. Mech. Mater. 41, 943–950 (2009)

    Article  Google Scholar 

  32. Horgan C.O., Saccomandi G.: Phenomenological hyper-elastic strain-stiffening constitutive models for rubber. Rubber Chem. Technol. 79, 1–18 (2006)

    Article  Google Scholar 

  33. Kakavas P.A.: A new development of the strain energy function for hyper-elastic materials using a logarithmic strain approach. J. Appl. Polym. Sci. 77, 660–672 (2000)

    Article  Google Scholar 

  34. Lion A.: A constitutive model for carbon black filled rubber: experimental investigations and mathematical representation. Continuum Mech. Therm. 8, 153–169 (1994)

    Article  ADS  Google Scholar 

  35. Miehe C., Apel N., Lambrecht M.: Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials. Comput. Meth. Appl. Mech. Eng. 191, 5383–5425 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Miehe C., Göktepe S., Lulei F.: A micro-macro approach to rubber-like materials-Part 1: the non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids 52, 2617–2660 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Muhr A.H.: Modeling the stress-strain behaviour of rubber. Rubber Chem. Technol. 78, 391–425 (2005)

    Article  Google Scholar 

  38. Ogden R.W.: Non-Linear Elastic Deformations. Ellis Horwood, Chichester (1984)

    Google Scholar 

  39. Ogden R.W., Saccomandi G., Sgura I.: Fitting hyperelastic models to experimental data. Comput. Mech. 34, 484–502 (2004)

    Article  MATH  Google Scholar 

  40. Perić D., Owen D.R.J., Honnor M.E.: A model for finite strain elastoplasticity based on logarithmic strains: Computational issues. Comput. Methods Appl. Mech. Eng. 94, 35–61 (1992)

    Article  ADS  MATH  Google Scholar 

  41. Treloar L.R.G.: The Physics of Rubber Elasticity. Oxford University Press, Oxford (1975)

    Google Scholar 

  42. Twizell B.H., Ogden R.W.: Non-linear optimization of the material constants in Ogden’s stress-deformation function for incompressible isotropic elastic materials. J. Aust. Math. Soc. B 24, 424–434 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  43. Vahapoglu V., Karadenitz S.: Constitutive equations for isotropic rubber-like materials using phenomenological approach: a bibliography (1930-2003). Rubber Chem. Technol. 79, 489–499 (2005)

    Article  Google Scholar 

  44. Xiao H.: Hencky strain and Hencky model: extending history and ongoing tradition. Multidiscip. Model. Mater. Struct. 1, 1–52 (2005)

    Google Scholar 

  45. Xiao, H.: An explicit, direct approach to obtaining multi-axial elastic potentials that exactly match data of four benchmark tests for rubbery materials-part 1: incompressible deformations. Acta Mech. (2012). doi:10.1007/s00707-012-0684-2

  46. Xiao H., Bruhns O.T., Meyers A.: Basic issues concerning finite strain measures and isotropic stress-deformation relations. J. Elast. 67, 1–23 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  47. Xiao H., Bruhns O.T., Meyers A.: Explicit dual stress-strain and strain-stress relations of incompressible isotropic hyper-elastic solids via deviatoric Hencky strain and Cauchy stress. Acta Mech. 168, 21–33 (2004)

    Article  MATH  Google Scholar 

  48. Xiao H., Bruhns O.T., Meyers A.: Elastoplasticity beyond small deformations. Acta Mech. 182, 31–111 (2006)

    Article  MATH  Google Scholar 

  49. Xiao, H., Chen, L.S.: Hencky’s logarithmic strain measure and dual stress-strain and strain-stress relations in isotropic finite hyper-elasticity. Int. J. Solids Struct. 40, 1455–1463 (2003)

    Google Scholar 

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Authors and Affiliations

  1. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Yanchang Road 149, Shanghai, 200072, China

    Yu-Yu Zhang, Hao Li, Xiao-Ming Wang, Zheng-Nan Yin & Heng Xiao

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  1. Yu-Yu Zhang
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  2. Hao Li
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  3. Xiao-Ming Wang
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  4. Zheng-Nan Yin
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  5. Heng Xiao
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Correspondence to Heng Xiao.

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Communicated by Andreas Öchsner.

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Zhang, YY., Li, H., Wang, XM. et al. Direct determination of multi-axial elastic potentials for incompressible elastomeric solids: an accurate, explicit approach based on rational interpolation. Continuum Mech. Thermodyn. 26, 207–220 (2014). https://doi.org/10.1007/s00161-013-0297-6

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