Advertisement

Elasticity with Disarrangements

  • David R. Owen
Part of the International Centre for Mechanical Sciences book series (CISM, volume 447)

Abstract

The subject of elasticity provides, among many other things, a continuum field theory for the dynamical evolution of bodies that undergo large deformations, that respond to changes in geometry through a stored energy function, and that experience internal dissipation in isothermal situations only during non-smooth processes. The central position of this field theory within mechanics has provided a starting point for many approaches to obtaining field theories that capture the effects of submacroscopic material structure or of submacroscopic geometrical changes on the macroscopic evolution of a body. The developments described here employ structured deformations and structured motions in order to formulate field theories for the dynamical evolution of bodies undergoing smooth geometrical changes at the macrolevel, while undergoing only piecewise smooth geometrical changes at submacroscopic levels. In keeping with elasticity, the new field equations should describe bodies that undergo large deformations and that store energy, while, in transcending elasticity in its standard form, the field equations should permit the body to experience internal dissipation during smooth dynamical processes and should provide a connection between the internal dissipation and the non-smooth geometrical changes (“disarrangements”) experienced by the body.

Keywords

Structure Deformation Linear Momentum Rigid Motion Helmholtz Free Energy Consistency Relation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Del Piero, G., Foundations of the theory of structured deformations; in this volume.Google Scholar
  2. [2]
    Paroni, R., Second-order structured deformations: approximation theorems and energetics; in this volume.Google Scholar
  3. [3]
    Deseri, L., Crystalline plasticity and structured deformations; in this volume.Google Scholar
  4. [4]
    Deseri, L., and Owen, D. R., Toward a field theory for elastic bodies undergoing disarrangements, Journal of Elasticity 70, 197–236, 2003.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Del Piero, G., and Owen, D.R., Structured deformations of continua, Archive for Rational Mechanics and Analysis 124, 99–155, 1993.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Del Piero, G., and Owen, D.R., Integral-gradient formulae for structured deformations, Archive for Rational Mechanics and Analysis 131, 121–138, 1995.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Deseri, L., and Owen, D.R., Invertible Structured Deformations and the Geometry of Multiple Slip in Single Crystals, International Journal of Plasticity 18, 833–849, 2002.CrossRefMATHGoogle Scholar
  8. [8]
    Del Piero, G., and Owen, D.R., Structured Deformations, Quaderni dell’ Istituto Nazionale di Alta Matematica, Gruppo Nazionale di Fisica Matematica. No. 58, 2000.Google Scholar
  9. [9]
    Owen, D.R., Structured deformations and the refinements of balance laws induced by microslip, International Journal of Plasticity 14, 289–299, 1998.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    Haupt, P., Continuum Mechanics and Theory of Materials, Springer Verlag, Berlin, etc., 2000.CrossRefMATHGoogle Scholar
  11. [11]
    Noll, W., La mécanique classique, basée sur un axiome d’objectivité, pp. 47–56 of La Méthode Axiomatique dans les Mécaniques Classiques and Nouvelles (Colloque International, Paris, 1959 ), Paris, Gauthier-Villars, 1963.Google Scholar
  12. [12]
    Green, A.E., and Rivlin, R.S., On Cauchy’s equations of motion, Journal of Applied Mathematics and Physics 15, 290–292, 1964.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Curtin, M.E., An Introduction to Continuum Mechanics, Academic Press, New York, etc., 1981.Google Scholar
  14. [14]
    Coleman, B.D., and Noll, W., The thermodynamics of elastic materials with heat conduction and viscosity, Archive for Rational Mechanics and Analysis 13, 167–178, 1963.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Dafermos, C.M., Quasilinear hyperbolic systems with involutions, Archive for Rational Mechanics and Analysis 94, 373–389, 1986.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Choksi, R., Del Piero, G., Fonseca, I., and Owen, D.R., Structured deformations as energy minimizers in models of fracture and hysteresis, Mathematics and Mechanics of Solids 4, 321–356, 1999.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Deseri, L., and Owen, D.R., Energetics of two-level shears and hardening of single crystals, Mathematics and Mechanics of Solids 7, 113–147, 2002.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Noll, W., On the continuity of solid and fluid states, Journal of Rational Mechanics and Analysis, 4, 3–91, 1955.MathSciNetMATHGoogle Scholar
  19. [19]
    Silhavÿ, M., and Kratochvíl, J., A theory of inelastic behavior of materials, Part I, Archive for Rational Mechanics and Analysis, 65, 97–129, 1977MathSciNetCrossRefMATHGoogle Scholar
  20. Silhavÿ, M., and Kratochvíl, J., Part II, Archive for Rational Mechanics and Analysis, 65, 131–152, 1977.MathSciNetCrossRefMATHGoogle Scholar
  21. [20]
    Bertram, A., An alternative approach to finite plasticity based on material isomorphisms, International Journal of Plasticity, 14, 353–374, 1999.CrossRefGoogle Scholar
  22. [21]
    Owen, D.R., and Paroni, R., Second-order structured deformations, Archive for Rational Mechanics and Analysis, 155, 215–235, 2000.MathSciNetCrossRefMATHGoogle Scholar
  23. [22]
    Owen, D.R., Twin balance laws for bodies undergoing structured motions, in Rational Continua: Classical and New, P. Podio-Guidugli and M. Brocato, eds., Springer Verlag, New York, etc., 2002.Google Scholar

Copyright information

© Springer-Verlag Wien 2004

Authors and Affiliations

  • David R. Owen
    • 1
  1. 1.Department of Mathematical SciencesCarnegie MellonPittsburghUSA

Personalised recommendations