Generalized Elastic-Plastic Decomposition in Defective Crystals

  • G. P. Parry
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


I outline ideas which allow one to prove a rigorous type of elasticplastic decomposition between related crystal states. The context of the work is a theory of defective crystals where the microstructure is represented by fields of lattice vectors, and the construction of the “generalized elastic-plastic decomposition” is a geometrical procedure which does not involve any notion of stress. The decomposition of the change of state has the form F 1 e F p F 2 e , where F 1 e and F 2 e are piecewise elastic deformations and F p is a rearrangement of “unit cells” of a related crystal structure. The relevant unit cells do not derive from any supposed perfect lattice-like properties (e.g., translational symmetry) of the crystal states (no such presumption is appropriate since the crystals, and corresponding unit cells, support continuously distributed defects) but are constructed by means of the piecewise elastic deformations using a self-similarity property of the crystal states. Further, each individual unit cell, with its attendant distribution of lattice vector fields and defect densities, is translated to its new position by the rearrangement F p , so that the corresponding mapping of points of the body represents a “change of shape” of the crystal domain


Elastic Deformation Lattice Vector Directional Derivative Crystal State Elastic Scalar 
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  1. [1]
    C. Davini, A proposal for a continuum theory of defective crystals, Arch. Rational Mech. Anal., 96 (1986), 295–317.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    C. Davini and G. P. Parry, On defect-preserving deformations in crystals, Internat. J. Plasticity, 5 (1989), 337–369.MATHCrossRefGoogle Scholar
  3. [3]
    C. Davini and G. P. Parry, A complete list of invariants for defective crystals, Proc. Roy. Soc. London Sect. A, 432 (1991), 341–365.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    I. Fonseca and G. P. Parry, Equilibrium configurations of defective crystals, Arch. Rational Mech. Anal., 120 (1992), 245–283.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    M. Glanville and G. P. Parry, Elastic invariants in complex materials, in P. M. Mariano, V. Sepe, and M. Lacagnina, eds., AIMETA’01: XV Congressa AIMETA di Meccanica Teorica e Applicata, Tipografia dell’ Universita’, Catania, Italy, 2001.Google Scholar
  6. [6]
    E. H. Lee, Elastic-plastic deformation at finite strains, J. Appl. Mech., 36 (1969), 1–6.MATHCrossRefGoogle Scholar
  7. [7]
    E. H. Lee and D. T. Liu, Finite-strain elastic-plastic theory with application to plane-wave analysis, J. Appl. Phys., 38 (1967), 19–27.CrossRefGoogle Scholar
  8. [8]
    P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, UK, 1996.Google Scholar
  9. [9]
    G. P. Parry, Defects and rearrangements in crystals, in L. M. Brock, ed., Defects and Anelasticity in the Characterization of Crystalline Solids, ASME, New York, 1992, 117–132.Google Scholar
  10. [10]
    G. P. Parry, Elastic-plastic decompositions of defect-preserving changes of state in solid crystals, in M. D. P. Monteiro Marques and J. F. Rodrigues, eds., Trends in Applications of Mathematics to Mechanics, Longman, Harlow, UK, 1994, 154–163.Google Scholar
  11. [11]
    G. P. Parry, The moving frame, and defects in crystals, Internat. J. Solids Structures, 38 (2001), 1071–1087.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    G. P. Parry and M. Šilhavý, Elastic invariants in the theory of defective crystals, Proc. Roy. Soc. London Sect. A, 455 (1999), 4333–4346.MATHCrossRefGoogle Scholar
  13. [13]
    G. P. Parry and M. Šilhavý, Invariant line integrals in the theory of defective crystals, Rend. Mat. Acc. Lincei. 9, 11 (2000), 111–140.MATHGoogle Scholar
  14. [14]
    C. Truesdell and R. A. Toupin, The Classical Field Theories, Handbuch der Physik Band III/1, Springer-Verlag, Berlin, Heidelberg, New York, 1960.Google Scholar

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© Springer Science+Business Media New York 2004

Authors and Affiliations

  • G. P. Parry
    • 1
  1. 1.Division of Theoretical Mechanics, School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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