Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A constitutive model for compressible elastomeric solids

Abstract

A non-linear thermo-elastic constitutive model for the large deformations of isotropic materials is formulated. This model is specialized to account for the physics and thermodynamics of the elastic deformation of rubber-like materials, and based on these molecular considerations a constitutive model for compressible elastomeric solids is proposed. The new constitutive model generalizes the incompressible and isothermal model of Arruda and Boyce (1993) to include the compressibility and thermal expansion of these materials. The model is fit to existing experimental data on vulcanized natural rubbers to determine the material parameters for the rubbers examined. The fit between the simple model and the data is found to be very good for large stretches and moderate volume changes.

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

Abbreviations

x\s=f(p):

Deformation function

p:

Material point of a body in a reference configuration

x:

Place occupied by material point p in the current configuration

F(p)\eq(\t6/\t6p) f(p):

Deformation gradient

J\s=det F\s>0:

Determinant of F

F\s=RU\s=VR:

Polar decompositions of F

U, V:

Right and left stretch tensors; positive definite and symmetric

R:

Rotation tensor; proper orthogonal

U=Σ 1−1 3 λ 1 2 r1⊗r1 :

Spectral representation of U

V=Σ 1=1 3 λ t 2 1t⊗11 :

Spectral representation of V

λt > 0:

Principal stretches

{ri}:

Right principal basis

{li}:

Left principal basis

C\s=FTF, B\s=FFT :

Right and left Cauchy-Green strain tensors

\gq\s>0:

Absolute temperature

\ge:

Internal energy density/unit reference volume

\gh:

Entropy density/unit reference volume

\gy\s=\ge\t-\gq\gh:

Helmholtz free energy/unit reference volume

References

  1. Adams, L. H.; Gibson, R. E. 1930: The Compressibility of Rubber. Journal of the Washington Academy of Sciences 20: 213–223

  2. Arruda, E.; Boyce, M. C. 1993: A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials. J. Mech. Phys. Solids 41: 389–412

  3. Ericksen, J. L. 1991: Introduction to the Thermodynamics of Solids. London: Chapman & Hall

  4. Flory, P. J. 1961: Thermodynamic Relations for High Elastic Materials. Trans. Faraday Soc. 57: 829–838

  5. Fong, J. T.; Penn, R. W. 1975: Construction of a Strain-Energy Function for Isotropic Elastic Material. Transactions of the Society of Rheology 19: 99–113

  6. Gurtin, M. 1981: An Introduction to Continuum Mechanics. pp. 175–177. New York: Academic Press

  7. James, H. M.; Guth, E. 1943: Theory of the Elastic Properties of Rubber. Journal of Chemical Physics 11: 455–481

  8. Jones, D. F.; Treloar, L. R. G. 1975: The Properties of Rubber in Pure Homogeneous Strain. J. Phys. D: Appl. Phys. 8: 1285–1304

  9. Kuhn, W.; Grün, F. 1942: Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung höchelastischer Stoffe. Kolloidzeitschrift 101: 248–271

  10. Meyer, K. H.; Ferri, C. 1935: Sur l'élasticité du caoutchouc. Helv. Chim. Acta. 18: 570–589

  11. Ogden, R. W. 1982: Elastic Deformations of Rubber-like Solids. In: Hopkins H. G.; Sewell, M. J. (eds.) Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, pp. 499–537 Oxford: Pergamon Press

  12. Ogden, R. W. 1984: Non-Linear Elastic Deformations. New York: John Wiley & Sons

  13. Peng, S. T. J.; Landel, R. F. 1975: Stored Energy Function and Compressibility of Compressible Rubber-like Materials under Large Strain. Journal of Applied Physics 46: 2599–2604

  14. Penn, R. W. 1970: Volume Changes Accompanying the Extension of Rubber. Transactions of the Society of Rheology 14: 509–517

  15. Treloar, L. R. G. 1944: Stress-strain Data for Vulcanized Rubber under Various Types of Deformation. Trans. Faraday Soc. 40: 59–70

  16. Treloar, L. R. G. 1975: The Physics of Rubber Elasticity Oxford: Clarendon Press

  17. Truesdell, C.; Noll, W. 1965: The Non-Linear Field Theories of Mechanics, Handbuch Der Physik, Vol III/3: 294–304

  18. Weiner, J. H. 1983: Statistical Mechanics of Elasticity. Chap. 5. New York: John Wiley & Sons

  19. Wood, L. A.; Martin, G. M. 1964: Compressibility of Natural Rubber at Pressures below 500 kg/cm2. Journal of Research of the National Bureau of Standards—A. Physics and Chemistry 68 A: 259–268

Download references

Author information

Additional information

Communicated by S. N. Atluri, 27 March 1996

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Anand, L. A constitutive model for compressible elastomeric solids. Computational Mechanics 18, 339–355 (1996). https://doi.org/10.1007/BF00376130

Download citation

Keywords

  • Experimental Data
  • Rubber
  • Thermal Expansion
  • Simple Model
  • Information Theory