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Cowin, Stephen C.; Nunziato, Jace W.

Linear elastic materials with voids. (English) Zbl 0523.73008

J. Elasticity 13, 125-147 (1983).
The authors treat the title subject by taking volume fraction of voids as an independent kinematic variable. Boundary value problems are formulated; uniqueness and weak stability are established for the mixed problem. Applications, including pure bending of a beam and small-amplitude acoustic waves, are discussed. The paper is concluded with a discussion of the relationship between the theory and effective moduli for porous materials. This is a straightforward and clearly presented work of general interest.
Reviewer: R. H. Lance (Ithaca)
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Cited in 15 Reviews
Cited in 345 Documents

MSC:

74B99 Elastic materials
74H99 Dynamical problems in solid mechanics
74E05 Inhomogeneity in solid mechanics
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics

Keywords:

elastic materials with voids; weak stability; response to homogeneous deformations; pure bending of beam; small-amplitude acoustic waves; change in void volume induced by deformation; effective moduli approach; porous materials
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References:

[1] Nunziato, J.W., and S.C. Cowin, ”A Nonlinear Theory of Elastic Materials with Voids,” Arch. Rational Mech. Anal. 72, 175 (1979). · Zbl 0444.73018 · doi:10.1007/BF00249363
[2] Goodman, M.A., and S.C. Cowin, ”A Continuum Theory for Granular Materials”, Arch. Rational Mech. Anal. 44, 249 (1972). · Zbl 0243.76005 · doi:10.1007/BF00284326
[3] Cowin, S.C., and M.A. Goodman, ”A Variational Principle for Granular Materials,” ZAMP 56, 281 (1976). · Zbl 0336.73006
[4] Passman, S.L., ”Mixtures of Granular Materials,” Int. J. Engrg. Sci. 15, 117 (1977). · Zbl 0352.73006 · doi:10.1016/0020-7225(77)90027-1
[5] Nunziato, J.W., and E.K. Walsh, ”On Ideal Multiphase Mixtures with Chemical Reactions and Diffusion,” Arch. Rational Mech. Anal. 73, 285 (1980). · Zbl 0453.76092 · doi:10.1007/BF00247672
[6] Jenkins, J.T., ”Static Equilibrium of Granular Materials,” J. Appl. Mech. 42, 603 (1975). · doi:10.1115/1.3423649
[7] Mindlin, R.D., ”Microstructure in Linear Elasticity,” Arch. Rational Mech. Anal. 16, 51 (1964). · Zbl 0119.40302 · doi:10.1007/BF00248490
[8] Toupin, R.A., ”Theories of Elasticity with Couple Stress,” Arch. Rational Mech. Anal. 17, 85 (1964). · Zbl 0131.22001 · doi:10.1007/BF00253050
[9] Cowin, S.C., and F.M. Leslie, ”On Kinetic Energy and Momenta in Cosserat Continua,” ZAMP 31, 247 (1980). · Zbl 0454.73001 · doi:10.1007/BF01590750
[10] Capriz, G., and P. Podio-Guidugli, ”Materials with Spherical Structure,” Arch. Rational Mech. Anal., 75, 269 (1981). · Zbl 0484.73002 · doi:10.1007/BF00250786
[11] Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Cambridge, 1927.
[12] Eshelby, J.D., ”The Force on an Elastic Singularity,” Phil. Trans. Roy. Soc. A 244, 87 (1951). · Zbl 0043.44102 · doi:10.1098/rsta.1951.0016
[13] Atkin, R.J., S.C. Cowin, and N. Fox, ”On Boundary Conditions for Polar Materials,” ZAMP 28, 1017 (1977). · Zbl 0385.76003 · doi:10.1007/BF01601669
[14] Nunziato, J.W., and E.K. Walsh, ”Small-Amplitude Wave Behavior in One-Dimensional Granular Materials,” J. Appl. Mech. 44, 559 (1977). · Zbl 0374.73092 · doi:10.1115/1.3424135
[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). · Zbl 0366.73028 · doi:10.1007/BF00282342
[16] MacKenzie, J.K., ”The Elastic Constants of a Solid Containing Spherical Holes,” Proc. Phys. Soc. B 63, 2 (1950). · Zbl 0034.40801
[17] Christensen, R.M., Mechanics of Composite Materials, Wiley, 1979. · Zbl 0393.46049
[18] Truesdell, C., and R.A. Toupin, ”The Classical Field Theories,” Handbuch der Physik, Vol. III/1 (ed. by S. Flügge), Springer, 1960.
[19] Cowin, S.C. and Puri, P., The Classical Pressure Vessel Problems for Linear Elastic Materials with Voids, J. Elasticity, 13 (1983) 157. · Zbl 0547.73077 · doi:10.1007/BF00041232
[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). · Zbl 0535.73080
[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).
[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. · Zbl 0543.73016
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