Advertisement

Continuum Mechanics: One Spatial Dimension

  • Mark H. Holmes
Chapter
  • 1.5k Downloads
Part of the Texts in Applied Mathematics book series (TAM, volume 56)

Abstract

In the previous chapter we investigated how to model the spatial motion of objects (cars, molecules, etc.) but omitted the possibility that the objects exert forces on each other. The objective now is to introduce this into the modeling. The situations where this is needed are quite varied and include the deformation of an elastic bar, the stretching of a string, or the flow of air or water.

References

  1. E. Abraham, O. Penrose, Physics of negative absolute temperatures. Phys. Rev. E 95, 012125 (2017).  https://doi.org/10.1103/PhysRevE.95.012125. https://link.aps.org/doi/10.1103/PhysRevE.95.012125
  2. R. Aris, Review of rational thermodynamics. Am. Math. Mon. 94, 562–564 (1987)Google Scholar
  3. C. Atkinson, G.E.H. Reuter, C.J. Ridler-Rowe, Traveling wave solution for some nonlinear diffusion equations. SIAM J. Math. Anal. 12 (6), 880–892 (1981). https://doi.org/10.1137/0512074 MathSciNetCrossRefGoogle Scholar
  4. T.A. Blackledge, C.Y. Hayashi, Silken toolkits: biomechanics of silk fibers spun by the orb web spider Argiope argentata (Fabricius 1775). J Exp Biology 209, 2452–2461 (2006)CrossRefGoogle Scholar
  5. D. Chiron, Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one. Nonlinearity 25 (3), 813 (2012). http://stacks.iop.org/0951-7715/25/i=3/a=813
  6. M. Finnis, Interatomic Forces in Condensed Matter (Oxford University Press, Oxford, 2010)Google Scholar
  7. F. Giustino, Materials Modelling Using Density Functional Theory (Oxford University Press, Oxford, 2014)Google Scholar
  8. G.W. Griffiths, W.E. Schiesser, Linear and nonlinear waves. Scholarpedia 4 (7), 4308 (2009).  https://doi.org/10.4249/scholarpedia.4308. revision #154041
  9. E.-Y. Guo, H.-X. Xie, S.S. Singh, A. Kirubanandham, T. Jing, N. Chawla, Mechanical characterization of microconstituents in a cast duplex stainless steel by micropillar compression. Mater. Sci. Eng. A Struct. Mater. 598 (Supplement C), 98–105 (2014). https://doi.org/10.1016/j.msea.2014.01.002. http://www.sciencedirect.com/science/article/pii/S0921509314000100. ISSN 0921-5093
  10. M.H. Holmes, Finite deformation of soft tissue: Analysis of a mixture model in uni-axial compression. J. Biomech. Eng. 108 (4), 372–381 (1986)CrossRefGoogle Scholar
  11. D. Kiener, W. Grosinger, G. Dehm, R. Pippan, A further step towards an understanding of size-dependent crystal plasticity: in situ tension experiments of miniaturized single-crystal copper samples. Acta Mater. 56 (3), 580–592 (2008). https://doi.org/10.1016/j.actamat.2007.10.015. http://www.sciencedirect.com/science/article/pii/S1359645407006969. ISSN 1359-6454
  12. M. Kwan, A Finite Deformation Theory for Nonlinearly Permeable Cartilage and Other Soft Hydrated Connective Tissues and Rheological Study of Cartilage Proteoglycans, PhD Thesis, RPI (1985)Google Scholar
  13. K.M. Liew, B.J. Chen, Z.M. Xiao, Analysis of fracture nucleation in carbon nanotubes through atomistic-based continuum theory. Phys Rev B: Condens. Matter 71, 235424 (2005)CrossRefGoogle Scholar
  14. J.E. Mark, B. Erman, Rubberlike Elasticity: A Molecular Primer, 2nd edn. (Cambridge University Press, Cambridge, 2007)CrossRefGoogle Scholar
  15. J.E. Marsden, T.J.R. Hughes, Mathematical Foundations of Elasticity (Dover, New York, 1994)zbMATHGoogle Scholar
  16. Y. Qi, T. Cagin, Y. Kimura, W.A. Goddard, Molecular-dynamics simulations of glass formation and crystallization in binary liquid metals: Cu-Ag and Cu-Ni. Phys. Rev. B 59, 3527–3533 (1999).  https://doi.org/10.1103/PhysRevB.59.3527. https://link.aps.org/doi/10.1103/PhysRevB.59.3527
  17. P. Raos, Modelling of elastic behaviour of rubber and its application in FEA. Plast. Rubber Compos. Process. Appl. 19, 293–303 (1993)Google Scholar
  18. R.H. Swendsen, How physicists disagree on the meaning of entropy. Am. J. Phys. 79 (4), 342–348 (2011). https://doi.org/10.1119/1.3536633 CrossRefGoogle Scholar
  19. R.H. Swendsen, Thermodynamics of finite systems: a key issues review. Rep. Prog. Phys. 81 (7), 072001 (2018). http://stacks.iop.org/0034-4885/81/i=7/a=072001
  20. C. Truesdell, Rational Thermodynamics, 2nd edn. (Springer, New York, 1984)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

Personalised recommendations