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Variational Problems for Crystals with Defects

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Microstructure and Phase Transition

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 54))

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Abstract

Equilibrium configurations for crystals with defects are analyzed. Using the theory of compensated compactness the state functions are characterized in the case where minimizing sequences develop oscillations. A new class of variational problems involving bulk and surface energy terms is studied.

The research of the first author was partially supported by the National Science Foundation under Grant No. DMS-8803315. This collaboration took place during a visit of I. Fonseca to the University of Bath (U.K.) in May-July 1990, supported by a visiting fellowship of the Science and Engineering Research Council of the U.K.

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References

  1. Ball, J.M., Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh 88A (1981), pp. 315–328.

    MATH  Google Scholar 

  2. Chipot, M. And D. Kinderlehrer, Equilibrium configurations of crystals, Arch. Rat. Mech. Anal. 103 (1988), pp. 237–277.

    Article  MathSciNet  MATH  Google Scholar 

  3. Davini, C., A proposal for a continuum theory of defective crystals, Arch. Rat. Mech. Anal. 96 (1986), pp. 295–317.

    Article  MathSciNet  MATH  Google Scholar 

  4. Davini, C. And G. Parry, On defect-preserving deformations in crystals, Int. J. of Plasticity 5 (1989), pp. 337–369.

    Article  MATH  Google Scholar 

  5. Davini, C. And G. Parry, A complete list of invariants for defective crystals, to appear.

    Google Scholar 

  6. Ericksen, J.L., Loading devices and stability of equilibrium, in Nonlinear Elasticity, Academic Press, N.Y. (1973), pp. 161–173.

    Google Scholar 

  7. Ericksen, J.L., Twinning of crystals I, in Metastability and Incompletely Posed Problems, IMA Vol. in Appl. Math 3 (1987), pp. 77–96.

    MathSciNet  Google Scholar 

  8. Fonseca, I., Variational methods for elastic crystals,Arch. Rat. Mech. Anal. 97 (1987), pp. 189–220.

    Article  MathSciNet  MATH  Google Scholar 

  9. Fonseca, I., The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. Pures et Appl. 67 (1988), pp. 175–195.

    MathSciNet  MATH  Google Scholar 

  10. Fonseca, I., The Wulff Theorem revisited, Proc. R. Soc. Lond. A 432 (1991), pp. 125–145.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Fonseca, I. And S. MÜLLER, An uniqueness proof for the Wulff Problem, Proc. Roy. Soc. Edinburgh 119 A (1991), pp. 125–136.

    MATH  Google Scholar 

  12. Fonseca, I. And G. Parry, Equilibrium configurations of defective crystals,to appear.

    Google Scholar 

  13. Fonseca, I. And G. Parry, On a class of invariant integrals,to appear.

    Google Scholar 

  14. Herring, C., Some theorems on the free energies of crystal surfaces, Phys. Rev. 82 (1951), pp. 87–93.

    Article  ADS  MATH  Google Scholar 

  15. Kinderlehrer, D., Twinning of crystals II, in Metastability and Incompletely Posed Problems, IMA Vol. in Appl. Math 3 (1987), pp. 185–211.

    MathSciNet  Google Scholar 

  16. Tartar, L., Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, vol. IV, Res. Notes in Math. 39, Pitman, 1979, pp. 136–212.

    Google Scholar 

  17. Taylor, G.I., The mechanism of plastic deformation of crystals, I,II, Proc. Roy. Soc. A 145 (1934), pp. 362–387, 388–404.

    ADS  Google Scholar 

  18. Taylor, J., Existence and structure of solutions to a class of nonelliptic variational problems, Symposia Mathematica 14 (1974), pp. 499–508.

    Google Scholar 

  19. Taylor, J., Unique structure of solutions to a class of nonelliptic variational problems, Proc. Symp. Pure Math., A.M.S., 27 (1975), pp. 419–427.

    Google Scholar 

  20. Taylor, J., Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), pp. 568–588.

    Article  MathSciNet  MATH  Google Scholar 

  21. Wulff, G., Zur Frage der Geschwindigkeit des Wachstums and der Auflösung der Kristallflachen, Zeitschrift für Kristallographie 34 (1901), pp. 449–530.

    Google Scholar 

  22. Young, L.C., Generalized surfaces in the calculus of variations MI, Ann. of Math. 43 (1942), pp. 84–103, 530–544.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Carnegie Mellon University, Pittsburgh, 15213, PA, USA

    Irene Fonseca

  2. School of Mathematical Sciences, University of Bath, Claverton Down, Bath, Avon, BA2 7AY, UK

    Gareth Parry

Authors
  1. Irene Fonseca
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  2. Gareth Parry
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    David Kinderlehrer

  2. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455, USA

    Richard James

  3. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Mitchell Luskin

  4. 5378 Buckskin Bob Drive, Florence, OR, 97439, USA

    Jerry L. Ericksen

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© 1993 Springer-Verlag New York, Inc.

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Fonseca, I., Parry, G. (1993). Variational Problems for Crystals with Defects. In: Kinderlehrer, D., James, R., Luskin, M., Ericksen, J.L. (eds) Microstructure and Phase Transition. The IMA Volumes in Mathematics and its Applications, vol 54. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8360-4_7

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  • DOI: https://doi.org/10.1007/978-1-4613-8360-4_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8362-8

  • Online ISBN: 978-1-4613-8360-4

  • eBook Packages: Springer Book Archive

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