Continuum Mechanics: Three Spatial Dimensions

  • Mark H. Holmes
Part of the Texts in Applied Mathematics book series (TAM, volume 56)


The water in the ocean, the air in the room, and a rubber ball have a common characteristic, they appear to completely occupy their respective domains. What this means is that the material occupies every point in the domain.


  1. T. Alazard, Low Mach number limit of the full Navier-Stokes equations. Arch. Ration. Mech. Anal. 180 (1), 1–73 (2006). ISSN 1432-0673MathSciNetCrossRefGoogle Scholar
  2. R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics (Dover, New York, 1990)zbMATHGoogle Scholar
  3. R.C. Batra, Universal relations for transversely isotropic elastic materials. Math. Mech. Solids 7, 421–437 (2002)MathSciNetCrossRefGoogle Scholar
  4. J. Carlson, A. Jaffe, A. Wiles, The Millennium Prize Problems (American Mathematical Society, Providence, 2006). ISBN 9780821836798Google Scholar
  5. A. Colagrossi, D. Durante, J.B. Bonet, A. Souto-Iglesias, Discussion of Stokes’ hypothesis through the smoothed particle hydrodynamics model. Phys. Rev. E 96, 023101 (2017)
  6. M.S. Cramer, Numerical estimates for the bulk viscosity of ideal gases. Phys Fluids 24 (6), 066102 (2012).
  7. M. Destrade, P.A. Martin, T.C.T. Ting, The incompressible limit in linear anisotropic elasticity, with applications to surface waves and elastostatics. J. Mech. Phys. Solids 50 (7), 1453–1468 (2002). ISSN 0022-5096
  8. A.S. Dukhin, P.J. Goetz, Bulk viscosity and compressibility measurement using acoustic spectroscopy. J. Chem. Phys. 130 (12), 124519 (2009).
  9. A.C. Eringen, Microcontinuum Field Theories II. Fluent Media (Springer, New York, 2001)Google Scholar
  10. M. Frewer, More clarity on the concept of material frame-indifference in classical continuum mechanics. Acta Mech. 202, 213–246 (2009)CrossRefGoogle Scholar
  11. M.E. Gurtin, L.C. Martins, Cauchy’s theorem in classical physics. Arch. Ration. Mech. Anal. 60 (4), 305–324 (1976). ISSN 1432-0673MathSciNetCrossRefGoogle Scholar
  12. M.E. Gurtin, V.J. Mizel, W.O. Williams, A note on Cauchy’s stress theorem. J. Math. Anal. Appl. 22 (2), 398–401 (1968). ISSN 0022-247X
  13. K. Hutter, K. Johnk, Continuum Methods of Physical Modeling (Springer, New York, 2004)CrossRefGoogle Scholar
  14. E. Lauga, M.P. Brenner, H.A. Stone, Microfluidics: the no-slip boundary condition, in Handbook of Experimental Fluid Dynamics, ed. by C. Tropea, A.L. Yarin, J.F. Foss (Springer, New York, 2007)Google Scholar
  15. P. Moon, D.E. Spencer, Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions (Springer, New York, 1988)zbMATHGoogle Scholar
  16. A. Murdoch, Some primitive concepts in continuum mechanics regarded in terms of objective space-time molecular averaging: the key role played by inertial observers. J. Elast. 84, 69–97 (2006)CrossRefGoogle Scholar
  17. J. Nordström, A roadmap to well posed and stable problems in computational physics. J. Sci. Comput. 71 (1), 365–385 (2017). MathSciNetCrossRefGoogle Scholar
  18. R.S. Rivlin, J.L. Ericksen, Stress-deformation relations for isotropic materials. J. Ration. Mech. Anal. 4, 323–425 (1955). ISSN 19435282, 19435290
  19. S. Schochet, The incompressible limit in nonlinear elasticity. Commun. Math. Phys. 102 (2), 207–215 (1985). ISSN 1432-0916MathSciNetCrossRefGoogle Scholar
  20. C.G. Speziale, Comments on the material frame-indifference controversy. Phys. Rev. A At. Mol. Opt. Phys. 36, 4522–4525 (1987)CrossRefGoogle Scholar
  21. C.G. Speziale, A review of material frame-indifference in mechanics. Appl. Mech. Rev. 51, 489–504 (1998)CrossRefGoogle Scholar
  22. B. Svendsen, A. Bertram, On frame-indifference and form-invariance in constitutive theory. Acta Mech. 132, 195–207 (1999)MathSciNetCrossRefGoogle Scholar
  23. R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (American Mathematical Society, Providence, 2001)CrossRefGoogle Scholar
  24. C.C. Wang, A new representation theorem for isotropic functions. Arch. Ration. Mech. Anal. 36 (3), 198–223 (1970). ISSN 1432-0673CrossRefGoogle Scholar
  25. H. Xiao, O.T. Bruhns, A. Meyers, On isotropic extension of anisotropic constitutive functions via structural tensors. ZAMM 86 (2), 151–161 (2006).

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Authors and Affiliations

  • Mark H. Holmes
    • 1
  1. 1.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA

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