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The stress driven instability in elastic crystals: Mathematical models and physical manifestations

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Summary

Equilibrium equations and stability conditions for the simple deformable elastic body are derived by means of considering a minimum of the static energy principle. The energy is supposed to be sum of the volume (elastic) and the surface terms. The ability to change relative positions of different material particles is taken into account, and appropriate natural definitions of the first and second variations of the energy are introduced and calculated explicitly. Considering the case of negligible magnitude of the surface tension, we establish that an equilibrium state of a nonhydrostatically stressed simple elastic body (of any physically reasonable elastic energy potential and of any symmetry) possessing any small smooth part of free surface is always unstable with respect to relative transfer of the material particles along the surface. Surface tension suppresses the mentioned instability with respect to sufficiently short disturbances of the boundary surface and thus can probably provide local smoothness of the equilibrium shape of the crystal. We derive explicit formulas for critical wavelength for the simplest models of the internal and surface energies and for the simplest equilibrium configurations. We also formulate the simplest problem of mathematical physics, revealing peculiarities and difficulties of the problem of equilibrium shape of elastic crystals, and discuss possible manifestations of the above-mentioned instability in the problems of crystal growth, materials science, fracture, physical chemistry, and low-temperature physics.

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References

  1. Woodruff, D. P.,The Solid-Liquid Interface. Cambridge University Press, Cambridge (1973).

    Google Scholar 

  2. Martin, J. W., Doherty, R. D.,Stability of Microstructure in Metallic Systems. Cambridge University Press, Cambridge (1975).

    Google Scholar 

  3. Langer, J. S., “Instabilities and Pattern Formation in Crystal Growth.”Rev. Mod. Phys., 52, (1980), pp. 1–27.

    Article  Google Scholar 

  4. Pelce, P.,Dynamics of Curved Fronts. Academic Press, New York (1988).

    MATH  Google Scholar 

  5. Paterson, M. S., “Nonhydrostatical Thermodynamics and its Geological Applications.”Rev. Geophys. Space Phys., 11, (1974), pp. 355–389.

    Google Scholar 

  6. Roitburd, A. L., “Martensitic Transformation as a Typical Phase Transformation in Solids.”Sol. State Phys. 33, (1978), pp. 317–390.

    Google Scholar 

  7. Larch, F., Cahn, J. W., “A Nonlinear Theory of Thermochemical Equilibrium of Solids under Stress.”Acta Metal., 26, (1978), pp. 53–60.

    Article  Google Scholar 

  8. Khachaturyan, A. G.,Theory of Structural Transformation in Solids. John Wiley and Sons, New York (1983).

    Google Scholar 

  9. Gurtin, M. E.,Phase Transformations and Material Instabilities in Solids. Academic Press, New York (1984).

    MATH  Google Scholar 

  10. Antman, S. S., Ericksen, J. L., Kinderlehrer, D., Muller, I., Eds.,Metastability and Incompletely Posed Problems. Springer-Verlag, New York (1987).

    MATH  Google Scholar 

  11. Rascle, M., Serre, D., Slemrod, M., Eds.,PDEs and Continuum Models of Phase Transitions. Springer-Verlag, New York (1989).

    MATH  Google Scholar 

  12. Grinfeld, M. A.,Thermodynamic Methods in the Theory of Heterogeneous Systems. Longman, New York (1991).

    Google Scholar 

  13. Gibbs, J. W., “On the Equilibrium of Heterogeneous Substances.” In:Collected Papers of J.W. Gibbs, (1876, 1878).

  14. Grinfeld, M. A., “The Influence of Surface Tension on Heterogeneous Equilibria,”Doklady, Earth Sci. Sect., 283, (1985), pp. 27–30. (English translation, Russian origin:Dokl. AN SSSR, 283, (1985), pp. 1139–1143).

    Google Scholar 

  15. Grinfeld, M. A., “Instability of the Separation Boundary between a Non-hydrostatically Stressed Elastic Body and a Melt.”Soviet Physics Doklady, 31, (1986), pp. 831–834. (English translation, Russian origin:Dokl. AN SSSR, 290, (1986), pp. 1358–1364).

    Google Scholar 

  16. Grinfeld, M. A., “Instability of the Equilibrium of a Non-hydrostatically Stressed Body and a Melt.”Fluid Dynam., 22, (1987). (English translation, Russian origin:Izv. AN SSSR, Mekhan, Zhidk. Gaza, No. 2, (1987), pp. 3–7).

  17. Alexander, J. I. D., Johnson, W. C., “Thermomechemical Equilibrium in Solid-Fluid Systems with Curved Interfaces.”J. Appl. Phys., 58, (1985), pp. 816–824.

    Article  Google Scholar 

  18. Ball, J. M., James, R. D., “Fine Phase Mixtures as Minimizers of Energy.”Arch. Rat. Mech. Anal., 97, (1987), pp. 13–52.

    Article  MathSciNet  Google Scholar 

  19. Dacorogna, B.,Direct Methods in the Calculus of Variations. Springer-Verlag, New York (1989).

    MATH  Google Scholar 

  20. Fonseca, I., “Interfacial Energy and the Maxwell Rule.”Arch. Rat. Mech. Anal., 101, (1988), pp. 63–93.

    MathSciNet  Google Scholar 

  21. Sternberg, P., “The Effect of Singular Perturbation on Nonconvex Variational Problems.”Arch. Rat. Mech. Anal., 101, (1988), pp. 209–260.

    Article  MATH  MathSciNet  Google Scholar 

  22. Luckhaus, S., Modica, L., “The Gibbs-Thomson Relation within the Gradient Theory of Phase Transitions.”Arch. Rat. Mech. Anal., 107, (1989), pp. 71–83.

    Article  MATH  MathSciNet  Google Scholar 

  23. Kohn, R., “Surface Energy and Microstructure in Coherent Phase Transitions.” IMA workshopMicrostructure and Phase Transitions, (1990), (to be published).

  24. Gurtin, R.,Thermomechanics of Evolving Phase Boundaries. (manuscript), (1990).

  25. Caroli, B., Caroli, C., Roulet, B., Voorhees, P. W., “Effect of Elastic Stresses on the Morphological Stability of a Solid Sphere Growing from a Supersaturated Melt.”Acta Metal., 37, (1989), pp. 257–268.

    Article  Google Scholar 

  26. Srolovitz, D. J., “On the Instability of Surfaces of Stressed Solids.”Acta Metal., 37, (1989), pp. 621–625.

    Article  Google Scholar 

  27. Leo, P. H., Sekerta, R. F., “The Effect of Surface Stress on Crystal-Melt and Crystal-Crystal Equilibrium.”Acta Metal., 37, (1989), pp. 3119–3138.

    Article  Google Scholar 

  28. Leo, P. H., Sekerka, R. F., “The Effect of Elastic Fields on the Morphological Stability of a Precipitate Grown from Solid Solution.”Acta Metal., 37, (1989), pp. 3139–3149.

    Article  Google Scholar 

  29. Nozières, P.,Growth and Shape of Crystals. Lectures given at Beg-Rohu, Brittany, Summer School, (1989), (mimeographed—will be published soon).

  30. Landau, L. D., Lifshitz, E. M.,Statistical Physics. Nauka (in Russian), (1964).

  31. Taylor, J. E., “Crystalline Variational Problems.”Bull. Amer. Math. Soc., 84, (1978), pp. 568–588.

    Article  MATH  MathSciNet  Google Scholar 

  32. Finn, R.,Equilibrium Capillary Surfaces. Springer-Verlag, New York (1986).

    MATH  Google Scholar 

  33. Virga, E., “Drops of Nematic Liquid Crystals.”Arch. Rat. Mech. Anal., 107, (1989), pp. 371–390.

    Article  MATH  MathSciNet  Google Scholar 

  34. Thomas, T. Y.,Plastic Flow and Fracture in Solids. Academic Press, New York (1961).

    MATH  Google Scholar 

  35. Grinfeld, M. A., “The Construction of a Continuum Theory of Recrystallization in Heterogeneous Systems.”Soviet Physics Doklady, 31, (1986), pp. 866–868. (English translation, Russian origin:Dokl. AN SSSR, 291, (1986), pp. 63–67).

    Google Scholar 

  36. Davini, C., “A Proposal for a Continuum Theory of Defective Crystals.”Arch. Rat. Mech. Anal., 96, (1986), pp. 295–317.

    Article  MATH  MathSciNet  Google Scholar 

  37. Grinfeld, M. A., “Stability of Heterogeneous Equilibrium in Systems Containing Solid Elastic Phases.”Doklady, Earth Sci. Sect., 265, (1982), pp. 21–24. (English translation, Russian origin:Dokl. AN SSSR, 265, (1982), pp. 836–840.

    MathSciNet  Google Scholar 

  38. Maddocks, J. H. “On the Second-Order Conditions in Constrained Variational Principles.” (To appear onJ.O.T.A.).

  39. Rhebinder, P. A., Shchukin, E. D., “Surface Phenomena in Solids During the Course of Their Deformation and Failure.”Soviet Physics, Uspekhi, 15, (1973).

  40. Bridgmen, P. W.,The Physics of High Pressure. Bell and Sons, Ltd., (1931).

  41. Kapustyan, V. M., “Polymerization of Monomers in Solid State under High Pressures and Shear Stresses Reactions in Solids.”Dokl. AN SSSR, 179, (1968), pp. 627–628 (in Russian).

    Google Scholar 

  42. Enikolopyan, N. S., “Explosive Chemical Reactions in Solids.”Dokl. AN SSSR, 292, (1987), pp. 1165–1169 (in Russian).

    Google Scholar 

  43. Enikolopyan, N. S., “The Influence of Elastic and Thermoelastic Stresses in Ignition of Explosion of Solids under High Pressures.”Dokl. AN SSSR, 293, (1987), pp. 389–392 (in Russian).

    Google Scholar 

  44. Enikolopyan, N. S., “Detonation—A Solid-Phase Chemical Reaction.”Dokl. AN SSSR, 302, (1988), pp. 630–634, (in Russian).

    Google Scholar 

  45. Matthews, J. W., “Coherent Interfaces and Misfit Dislocations.” InEpitaxial Growth, Part B. Academic Press, New York (1975).

    Google Scholar 

  46. van der Merwe, J. H., “The Role of Lattice Misfit in Epitaxy.” InChemistry and Physics of Solid Surfaces, Vol. II, CRC Press, Inc., Boca Raton, FL, (1979).

    Google Scholar 

  47. Gilmer, G. H., Grabow, M. H., “Models of Thin Film Growth Modes.”Journal of Metals, June, (1987).

  48. Antman, S. S., Carbone, E. R., “Shear and Necking Instabilities in Nonlinear Elasticity.”J. of Elasticity, 7, (1977), pp. 125–151.

    Article  MATH  MathSciNet  Google Scholar 

  49. Miyazaki, T., “The Formation of γ Precipitate Doublets in Ni-Al Alloys and their Energetic Stability.”Matert. Sci. Engng., 54, (1982), pp. 9–15.

    Article  Google Scholar 

  50. Won-Hyuk, Rhee, Yoon, Duk, N., “The Instability of Solid-Liquid Interface in Mo-Ni Alloy Induced by Diffusional Coherency Strain.”Acta Metall., 35, (1987), pp. 1447–1451.

    Article  Google Scholar 

  51. Rittel, D., “The Influence of Microstructure on the Macroscopic Pattern of Surface Instabilities in Metals.”Scripta Meta. Mater., 24, (1990), pp. 1759–1764.

    Article  Google Scholar 

  52. Balibar, S., Edwards, D. D., Saam, W. F., “The Effect of Heat Flow and Non-Hydrostatic Strain on the Surface of Helium Crystals.”J. Low Temp. Phys., 82, (1991), pp. 119–143.

    Article  Google Scholar 

  53. Bodensohn, J., Nicolai, K., Leiderer, P., “The Growth of Atomically Rough4He Crystals.”Z. Phys., B, Condensed Matter, 64, (1986), pp. 55–64.

    Article  Google Scholar 

  54. Grinfeld, M., “The Stress Driven Instabilities in Crystals: Mathematical Models and Physical Manifestations.” IMA preprint series 819, June (1991).

  55. Gao, H., “Stress Concentration at Slightly Undulating Surfaces.”J. Mech. Phys. Solids, 39, (1991), pp. 443–458.

    Article  MATH  Google Scholar 

  56. Spencer, B. J., Voorhees, P. W., Davis, S. H., “Morphological Instability in Epitaxially Strained Dislocation-Free Solid Films.”Phys. Rev. Lett., 67, (1991), pp. 3696–3699.

    Article  Google Scholar 

  57. Bowley, R. M., Nozières, P., “The Effect of Heat Current on the Stability of the Liquid Solid Interface.”J. Phys., 2, (1992), pp. 433–441.

    Google Scholar 

  58. Torii, R. H., Balibar, S., “Helium Crystals Under Stress: the Grinfeld Instability.”J. Low Temp. Phys., 89, (1992), pp. 391–400.

    Article  Google Scholar 

  59. Nozières, P., “Amplitude Expansion for the Grinfeld Instability due to Uniaxial Stress at a Solid Surface.”J. Phys. (to appear).

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  1. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    M. A. Grinfeld

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Communicated by Robert Kohn

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Grinfeld, M.A. The stress driven instability in elastic crystals: Mathematical models and physical manifestations. J Nonlinear Sci 3, 35–83 (1993). https://doi.org/10.1007/BF02429859

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