Skip to main content
SpringerLink
Log in

An explicit, direct approach to obtaining multiaxial elastic potentials that exactly match data of four benchmark tests for rubbery materials—part 1: incompressible deformations

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

An explicit, direct approach is proposed to obtain multiaxial elastic potentials that exactly match finite strain data of four benchmark tests for incompressible rubberlike materials, including uniaxial, biaxial, plane-strain compression and simple shear tests. This approach is composed of three direct procedures. At first, we utilize a usual interpolating method for single-variable functions to obtain one-dimensional stress–strain relations matching data of uniaxial test and simple shear test, separately, and then obtain two one-dimensional elastic potentials via directly integrating the stress power. After that, we introduce a multiaxial bridging method based on certain invariants of Hencky strain and extend the foregoing two one-dimensional potentials to two potentials for multiaxial cases. Then, we further introduce a multiaxial matching method based also on Hencky invariants and combine the just-mentioned two independent multiaxial potentials to obtain a unified form of multiaxial potential. With two universal relations first disclosed here between the four tests, eventually we demonstrate that each such unified multiaxial potential exactly matches data of four benchmark tests. In particular, we apply the proposed explicit approach with a simple form of rational interpolating function for uniaxial stress–strain relation with two poles and directly from uniaxial stress–strain relation to obtain a new, simple form of multiaxial elastic potential with a limit for strain. It is found that this limit for strain is just a counterpart of the well-known von Mises limit for stress in the theory of elastoplasticity. Also, it is shown that this simple potential accurately matches data of four benchmark tests over the whole range from small to large deformations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  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  Google Scholar 

  3. Aron M.: On certain deformation classes of compressible Hencky materials. Math. Mech. Solids 19, 467–478 (2006)

    MathSciNet  Google Scholar 

  4. 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  Google Scholar 

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

    Article  Google Scholar 

  6. Beatty M.F.: Seven lectures on finite elasticity. In: Hayes, M.A., Saccomandi, G. (eds.) Topics in Finite Elasticity, CISM Courses and Lectures No. 424, Springer, Wien (2001)

    Google Scholar 

  7. Beatty M.F.: An average-stretch full-network model for rubber elasticity. J. Elast. 70, 65–86 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Bechir H., Chevalier L., Idjeri M.: A three-dimensional network model for rubber elasticity: The effect of local entanglement constraints. Int. J. Eng. Sci. 48, 265–274 (2010)

    Article  Google Scholar 

  10. Bergström J.S., Boyce M.C.: Mechanical behaviour of particle filled elastomers. Rubber Chem. Technol. 72, 633–656 (1999)

    Article  Google Scholar 

  11. Blatz P.J., Ko W.L.: Application of finite elasticity theory to the deformation of rubber materials. J. Rheol. 6, 223–252 (1962)

    Article  Google Scholar 

  12. Boyce M.C.: Direct comparison of the Gent and the Arruda-Boyce constitutive models of rubber elasticity. Rubber Chem. Technol. 69, 781–785 (1996)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Bruhns O.T., Xiao H., Meyers A.: Self-consistent Eulerian rate type elastoplasticity models based upon the logarithmic stress rate. Int. J. Plast. 15, 479–520 (1999)

    Article  MATH  Google Scholar 

  15. Bruhns O.T., Xiao H., Meyers A.: The Hencky model of elasticity: a study on Poynting effect and stress response in torsion of tubes and rods. Arch. Mech. 52, 489–509 (2000)

    MathSciNet  MATH  Google Scholar 

  16. Bruhns O.T., Xiao H., Meyers A.: Constitutive inequalities for an isotropic elastic strain energy function based on Hencky’s logarithmic strain tensor. Proc. R. Soc. Lond. A 457, 2207–2226 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bruhns O.T., Xiao H., Meyers A.: Finite bending of a rectangular block of an elastic Hencky material. J. Elast. 66, 237–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  19. 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 hyperelasticity. J. Mech. Phys. Solids 48, 2445–2465 (2000)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  21. Drozdov A.D.: Constitutive equations in finite elasticity of rubbers. Int. J. Solids Struct. 44, 272–297 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  24. Flory J., Rehner J.: Statistical mechanics of cross-linked polymer networks I. Rubberlike elasticity. J. Chem. Phys. 11, 512–520 (1943)

    Article  Google Scholar 

  25. Fried E.: An elementary molecular-statistical basis for the Mooney and Rivlin-Sauders theories of rubber elasticity. J. Mech. Phys. Solids 50, 571–582 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  27. Gao Y.C.: Elastostatic crack tip behaviour for a rubber like material. Theor. Appl. Fract. Mech. 14, 219–231 (1990)

    Article  Google Scholar 

  28. Gao Y.C.: Large deformation field near a crack tip in rubber-like material. Theor. Appl. Fract. Mech. 26, 155–162 (1997)

    Article  Google Scholar 

  29. Gent A.N.: Rubber and rubber elasticity: a review. J. Polym. Sci. 48, 1–17 (1974)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  32. 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 

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

    MATH  Google Scholar 

  34. Haines D.W., Wilson W.D.: Strain-energy density function for rubberlike materials. J. Mech. Phys. Solids 27, 345–360 (1979)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  37. Hayes M.A., Saccomandi G.: Topics in Finite Elasticity. CISM Courses and Lectures No. 424. Springer, Wien (2001)

    Google Scholar 

  38. Hencky, H.: Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Z. Techn. Phys. 9, 215–220; ibid, 457 (1928)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. 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 

  46. Horgan C.O., Saccomandi G.: A molecular-statistical basis for the Gent constitutive model of rubber elasticity. J. Elast. 68, 167–176 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. Horgan C.O., Saccomandi G.: Finite thermoelasticity with limiting chain extensibility. J. Mech. Phys. Solids 51, 1127–1146 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  49. Hutchinson J., Neale K.: Finite strain J 2 deformation theory. In: Carlson, D.E., Shield, R.T. (eds.) Finite Elasticity, pp. 237–247. Martinus Nijhoff, The Hague (1980)

    Google Scholar 

  50. Jones D.F., Treloar L.R.G.: The properties of rubber in pure homogeneous strain. J. Phys. D 8, 1285–1304 (1975)

    Article  Google Scholar 

  51. Knowles J.K., Sternberg E.: An asymptotic finite deformation analysis of the elastostatic field near the tip of a crack. J. Elast. 3, 67–107 (1973)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  53. Lambert-Diani J., Rey Ch.: New phenomenological behavior laws for rubbers and thermoplastic elastomers. Eur. J. Mech. A/Solids 18, 1027–1043 (1999)

    Article  MATH  Google Scholar 

  54. Lurie A.I.: Nonlinear Theory of Elasticity. Elsevier, The Netherlands (1990)

    MATH  Google Scholar 

  55. Mark B., Erman J.E.: Structures and Properties of Rubberlike Networks. Oxford University Press, Oxford (1997)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  57. Mooney M.: A theory of large elastic deformation. J. Appl. Phys. 11, 582–592 (1940)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  59. Murphy J.G.: Some remarks on kinematic modeling of limiting chain extensibility. Math. Mech. Solids 11, 629–641 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  60. Ogden R.W.: Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike materials. Proc. R. Soc. Lond. A 326, 565–584 (1972)

    Article  MATH  Google Scholar 

  61. Ogden R.W.: Large deformation isotropic elasticity-on the correlation of theory and experiment for compressible rubber-like materials. Proc. R. Soc. Lond. A 328, 567–583 (1972)

    Article  MATH  Google Scholar 

  62. Ogden R.W.: Volume changes associated with the deformation of rubber-like solids. J. Mech. Phys. Solids 24, 323–338 (1976)

    Article  Google Scholar 

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

    Google Scholar 

  64. Ogden R.W.: Recent advances in the phenomenological theory of rubber elasticity. Rubber Chem. Technol. 59, 361–383 (1986)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  66. Ogden R.W., Saccomandi G., Sgura I.: On worm-like chain models within the three-dimensional continuum mechanics framework. Proc. R. Soc. Lond. A 462, 749–768 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  67. Rice J.R.: Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanism. In: Argon, A.S. (ed.) Constitutive Equations in Plasticity, pp. 21–79. MIT Press, Cambridge (1975)

    Google Scholar 

  68. Rivlin R.S.: Large elastic deformations of isotropic materials. I. Fundamental concepts. Philos. Trans. R. Soc. Lond. A 240, 459–490 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  69. Rivlin R.S.: Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure, homogeneous deformation. Philos. Trans. R. Soc. Lond. A 240, 491–508 (1948)

    Article  MathSciNet  Google Scholar 

  70. Rivlin R.S.: Large elastic deformations of isotropic materials. III. Some simple problems in cylindrical polar co-ordinates. Philos. Trans. R. Soc. Lond. A 240, 509–528 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  71. Rivlin R.S.: Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philos. Trans. R. Soc. Lond. A 241, 379–397 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  72. Rivlin R.S.: Large elastic deformations of isotropic materials. V. The problem of flexure. Philos. Trans. R. Soc. Lond. A 241, 463–473 (1949)

    MathSciNet  Google Scholar 

  73. Rivlin R.S.: Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Philos. Trans. R. Soc. Lond. A 242, 173–195 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  74. Rivlin R.S., Saunders D.W.: Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Philos. Trans. R. Soc. Lond. A 243, 251–288 (1951)

    Article  MATH  Google Scholar 

  75. Saccomandi G., Ogden R.W.: Mechanics and Thermomechanics of Rubberlike Solids. CISM Courses and Lectures No. 452. Springer, Wien (2004)

    Google Scholar 

  76. Stören S., Rice J.R.: Localized necking in thin sheets. J. Mech. Phys. Solids 23, 421–441 (1975)

    Article  MATH  Google Scholar 

  77. Treloar L.R.G.: The elasticity of a network of long-chain molecules-III. Trans. Faraday Soc. 42, 83–94 (1946)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  79. Treloar L.R.G.: The mechanics of rubber elasticity [and discussions]. Proc. R. Soc. Lond. A. 351, 301–330 (1976)

    Article  MATH  Google Scholar 

  80. 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  MathSciNet  MATH  Google Scholar 

  81. 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 

  82. 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)

    Article  Google Scholar 

  83. Wang M.C., Guth E.: Statistical theory of networks of non-Gaussian flexible chains. J. Chem. Phys. 20, 1144–1157 (1952)

    Article  MathSciNet  Google Scholar 

  84. Wu P.D., Giessen E.: On improved network models for rubber elasticity and their application to orientation hardening in glassy polymers. J. Mech. Phys. Solids 41, 427–456 (1993)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  86. Xiao H., Bruhns O.T., Meyers A.: Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124, 89–105 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  87. Xiao H., Bruhns O.T., Meyers A.: Hypo-elasticity model based upon the logarithmic stress rate. J. Elast. 47, 51–68 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  88. Xiao H., Bruhns O.T., Meyers A.: Existence and uniqueness of the integrable-exactly hypoelastic equation and its significance to finite inelasticity. Acta Mechanica 138, 31–50 (1999)

    Article  MATH  Google Scholar 

  89. Xiao H., Bruhns O.T., Meyers A.: The choice of objective rates in finite elastoplasticity: general results on the uniqueness of the logarithmic rate. Proc. R. Soc. Lond. A 456, 1865–1882 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  90. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  93. Xiao H., Bruhns O.T., Meyers A.: Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects. J. Mech. Phys. Solids 55, 338–365 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  95. Yeoh O.H.: Characterization of elastic properties of carbon-black-filled rubber volcanizates. Rubber Chem. Technol. 63, 792–805 (1990)

    Article  Google Scholar 

  96. Yeoh O.H.: Some forms of the strain energy function for rubber. Rubber Chem. Technol. 66, 754–771 (1993)

    Article  Google Scholar 

  97. Yeoh O.H., Fleming P.D.: A new attempt to reconcile the statistical and phenomenological theories of rubber elasticity. J. Polym. Sci. B 35, 1919–1931 (1997)

    Article  Google Scholar 

  98. Zuniga A.E.: A non-Gaussian network model for rubber elasticity. Polymer 47, 907–914 (2006)

    Article  Google Scholar 

  99. Zuniga A.E., Beatty M.F.: Constitutive equations for amended non-Gaussian network models of rubber elasticity. Int. J. Eng. Sci. 40, 2265–2294 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Yanchang Road 149, Shanghai, 200072, China

    Heng Xiao

Authors
  1. Heng Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Heng Xiao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiao, H. An explicit, direct approach to obtaining multiaxial elastic potentials that exactly match data of four benchmark tests for rubbery materials—part 1: incompressible deformations. Acta Mech 223, 2039–2063 (2012). https://doi.org/10.1007/s00707-012-0684-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-012-0684-2

Keywords

Search

Navigation