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Journal of Elasticity

, Volume 13, Issue 2, pp 125–147 | Cite as

Linear elastic materials with voids

  • Stephen C. Cowin
  • Jace W. Nunziato
Article

Abstract

A linear theory of elastic materials with voids is presented. This theory differs significantly from classical linear elasticity in that the volume fraction corresponding to the void volume is taken as an independent kinematical variable. Following a discussion of the basic equations, boundary-value problems are formulated, and uniqueness and weak stability are established for the mixed problem. Then, several applications of the theory are considered, including the response to homogeneous deformations, pure bending of a beam, and small-amplitude acoustic waves. In each of these applications, the change in void volume induced by the deformation is determined. In the final section of the paper, the relationship between the theory presented and the effective moduli approach for porous materials is discussed.

In the two year period between the submission of this manuscript and the receipt of the page proof, there have been some extensions of the results reported here. In the context of the theory described, the classical pressure vessel problems and the problem of the stress distribution around a circular hole in a field have uniaxial tension have been solved [19,22]. The solution given in the present paper for the pure bending of a beam when the rate effect of the theory is absent is extended to case when the rate effect is present in [21]. The various implications of the rate effect in the void volume deformation are pursued all the subsequent works [19,20,21,22].

Keywords

Rate Effect Elastic Material Uniaxial Tension Void Volume Linear Elasticity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Nunziato, J.W., and S.C. Cowin, “A Nonlinear Theory of Elastic Materials with Voids,” Arch. Rational Mech. Anal. 72, 175 (1979).CrossRefGoogle Scholar
  2. [2]
    Goodman, M.A., and S.C. Cowin, “A Continuum Theory for Granular Materials”, Arch. Rational Mech. Anal. 44, 249 (1972).CrossRefGoogle Scholar
  3. [3]
    Cowin, S.C., and M.A. Goodman, “A Variational Principle for Granular Materials,” ZAMP 56, 281 (1976).Google Scholar
  4. [4]
    Passman, S.L., “Mixtures of Granular Materials,” Int. J. Engrg. Sci. 15, 117 (1977).CrossRefGoogle Scholar
  5. [5]
    Nunziato, J.W., and E.K. Walsh, “On Ideal Multiphase Mixtures with Chemical Reactions and Diffusion,” Arch. Rational Mech. Anal. 73, 285 (1980).CrossRefGoogle Scholar
  6. [6]
    Jenkins, J.T., “Static Equilibrium of Granular Materials,” J. Appl. Mech. 42, 603 (1975).Google Scholar
  7. [7]
    Mindlin, R.D., “Microstructure in Linear Elasticity,” Arch. Rational Mech. Anal. 16, 51 (1964).CrossRefGoogle Scholar
  8. [8]
    Toupin, R.A., “Theories of Elasticity with Couple Stress,” Arch. Rational Mech. Anal. 17, 85 (1964).CrossRefGoogle Scholar
  9. [9]
    Cowin, S.C., and F.M. Leslie, “On Kinetic Energy and Momenta in Cosserat Continua,” ZAMP 31, 247 (1980).Google Scholar
  10. [10]
    Capriz, G., and P. Podio-Guidugli, “Materials with Spherical Structure,” Arch. Rational Mech. Anal., 75, 269 (1981).CrossRefGoogle Scholar
  11. [11]
    Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Cambridge, 1927.Google Scholar
  12. [12]
    Eshelby, J.D., “The Force on an Elastic Singularity,” Phil. Trans. Roy. Soc. A 244, 87 (1951).Google Scholar
  13. [13]
    Atkin, R.J., S.C. Cowin, and N. Fox, “On Boundary Conditions for Polar Materials,” ZAMP 28, 1017 (1977).Google Scholar
  14. [14]
    Nunziato, J.W., and E.K. Walsh, “Small-Amplitude Wave Behavior in One-Dimensional Granular Materials,” J. Appl. Mech. 44, 559 (1977).Google Scholar
  15. [15]
    Nunziato, J.W., and E.K. Walsh, “On the Influence of Void Compaction and Material Non-Uniformity on the Propagation of One-Dimensional Acceleration Waves in Granular Materials,” Arch. Rational Mech. Anal. 64, 299 (1977); Addendum, ibid, 67, 395 (1978).CrossRefGoogle Scholar
  16. [16]
    MacKenzie, J.K., “The Elastic Constants of a Solid Containing Spherical Holes,” Proc. Phys. Soc. B 63, 2 (1950).CrossRefGoogle Scholar
  17. [17]
    Christensen, R.M., Mechanics of Composite Materials, Wiley, 1979.Google Scholar
  18. [18]
    Truesdell, C., and R.A. Toupin, “The Classical Field Theories,” Handbuch der Physik, Vol. III/1 (ed. by S. Flügge), Springer, 1960.Google Scholar
  19. [19]
    Cowin, S.C. and Puri, P., The Classical Pressure Vessel Problems for Linear Elastic Materials with Voids, J. Elasticity, 13 (1983) 157.Google Scholar
  20. [20]
    Passman, S.L., Stress Relaxation, Creep, Failure and Hysteresis in a Linear Elastic Material with Voids, J. Elasticity, in press (accepted May 12, 1982).Google Scholar
  21. [21]
    Cowin, S.C., A Note on the Problem of Pure Bending for a linear Elastic Material with Voids, J. Elasticity, in press (accepted July 26, 1982).Google Scholar
  22. [22]
    Cowin, S.C., The Stresses Around a Hole in a Linear Elastic Material with Voids, Quart. J. Mech. Appl. Math., in press (accepted November 5, 1982), to appear in vol. 36, no. 4, 1983.Google Scholar

Copyright information

© Martinus Nijhoff Publishers 1983

Authors and Affiliations

  • Stephen C. Cowin
    • 1
  • Jace W. Nunziato
    • 2
  1. 1.Department of Biomedical EngineeringTulane UniversityNew OrleansUSA
  2. 2.Fluid Mechanics and Heat Transfer Division ISandia National LaboratoriesAlbuquerqueUSA

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