Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
T. Alazard, Low Mach number limit of the full Navier-Stokes equations. Arch. Ration. Mech. Anal. 180 (1), 1–73 (2006). https://doi.org/10.1007/s00205-005-0393-2. ISSN 1432-0673
R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics (Dover, New York, 1990)
R.C. Batra, Universal relations for transversely isotropic elastic materials. Math. Mech. Solids 7, 421–437 (2002)
J. Carlson, A. Jaffe, A. Wiles, The Millennium Prize Problems (American Mathematical Society, Providence, 2006). ISBN 9780821836798
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) https://doi.org/10.1103/PhysRevE.96.023101. https://link.aps.org/doi/10.1103/PhysRevE.96.023101
M.S. Cramer, Numerical estimates for the bulk viscosity of ideal gases. Phys Fluids 24 (6), 066102 (2012). https://doi.org/10.1063/1.4729611
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). https://doi.org/10.1016/S0022-5096(01)00121-1. http://www.sciencedirect.com/science/article/pii/S0022509601001211. ISSN 0022-5096
A.S. Dukhin, P.J. Goetz, Bulk viscosity and compressibility measurement using acoustic spectroscopy. J. Chem. Phys. 130 (12), 124519 (2009). https://doi.org/10.1063/1.3095471
A.C. Eringen, Microcontinuum Field Theories II. Fluent Media (Springer, New York, 2001)
M. Frewer, More clarity on the concept of material frame-indifference in classical continuum mechanics. Acta Mech. 202, 213–246 (2009)
M.E. Gurtin, L.C. Martins, Cauchy’s theorem in classical physics. Arch. Ration. Mech. Anal. 60 (4), 305–324 (1976). https://doi.org/10.1007/BF00248882. ISSN 1432-0673
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). https://doi.org/10.1016/0022-247X(68)90181-9. http://www.sciencedirect.com/science/article/pii/0022247X68901819. ISSN 0022-247X
K. Hutter, K. Johnk, Continuum Methods of Physical Modeling (Springer, New York, 2004)
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)
P. Moon, D.E. Spencer, Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions (Springer, New York, 1988)
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)
J. Nordström, A roadmap to well posed and stable problems in computational physics. J. Sci. Comput. 71 (1), 365–385 (2017). https://doi.org/10.1007/s10915-016-0303-9
R.S. Rivlin, J.L. Ericksen, Stress-deformation relations for isotropic materials. J. Ration. Mech. Anal. 4, 323–425 (1955). http://www.jstor.org/stable/24900365. ISSN 19435282, 19435290
S. Schochet, The incompressible limit in nonlinear elasticity. Commun. Math. Phys. 102 (2), 207–215 (1985). https://doi.org/10.1007/BF01229377. ISSN 1432-0916
C.G. Speziale, Comments on the material frame-indifference controversy. Phys. Rev. A At. Mol. Opt. Phys. 36, 4522–4525 (1987)
C.G. Speziale, A review of material frame-indifference in mechanics. Appl. Mech. Rev. 51, 489–504 (1998)
B. Svendsen, A. Bertram, On frame-indifference and form-invariance in constitutive theory. Acta Mech. 132, 195–207 (1999)
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (American Mathematical Society, Providence, 2001)
C.C. Wang, A new representation theorem for isotropic functions. Arch. Ration. Mech. Anal. 36 (3), 198–223 (1970). https://doi.org/10.1007/BF00272242. ISSN 1432-0673
H. Xiao, O.T. Bruhns, A. Meyers, On isotropic extension of anisotropic constitutive functions via structural tensors. ZAMM 86 (2), 151–161 (2006). https://doi.org/10.1002/zamm.200410226. https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.200410226
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Holmes, M.H. (2019). Continuum Mechanics: Three Spatial Dimensions. In: Introduction to the Foundations of Applied Mathematics. Texts in Applied Mathematics, vol 56. Springer, Cham. https://doi.org/10.1007/978-3-030-24261-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-24261-9_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-24260-2
Online ISBN: 978-3-030-24261-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)