Continuum Mechanics and Thermodynamics

, Volume 4, Issue 1, pp 59–79 | Cite as

Stability of a two-phase process involving a planar phase boundary in a thermoelastic solid

  • E. Fried


This investigation is directed toward understanding the role of coupled mechanical and thermal effects in the linear stability of an isothermal antiplane shear motion which involves a steadily propagatingnormal planar phase boundary in anon-elliptic thermoelastic material. When the relevant process is static — so that the phase boundary does not move prior to the imposition of the disturbance —it is shown to be linearly stable. However, when the process involves a moving phase boundary it may be linearly unstable. Various conditions sufficient to guarantee the linear instability of the process are obtained. These depend on the monotonicity of thekinetic response function — a constitutively supplied entity which relates thedriving traction acting on a phase boundary to the local absolute temperature and the normal velocity of the phase boundary-and, in certain cases, on the spectrum of wave-numbers associated with the perturbation to which the process is subjected. Inertia is found to play an insignificant role in the qualitative features of the aforementioned sufficient conditions. It is shown, in particular, that instability can arise even when the normal velocity of the phase boundary is an increasing function of the driving traction if the temperature dependence in the kinetic response function is of a suitable nature. The instability which is present in this setting occurs only in thelong waves of the Fourier decomposition of the moving phase boundary, implying that the interface prefers to be highly wrinkled.


Response Function Thermal Effect Phase Boundary Absolute Temperature Linear Stability 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fried, E.: Stability of a two-phase process involving a planar phase boundary in an elastic solid. To appear in J. Elast.Google Scholar
  2. 2.
    Owen, R. F.; Schoen, F. J.; Srinivasan, G. R.: The growth of a plate of martensite. In Phase Transformations, 157–180, American Society for Metals, Metals Park, Ohio, 1968Google Scholar
  3. 3.
    Clapp, P. C.; Yu, Z.-Z.: Growth dynamics study of the martensitic transformation in Fe-30 pct Ni alloys: part 1, quantitative measurements of growth velocity. Metal. Trans. 20A (1989) 1601–1615Google Scholar
  4. 4.
    Grujicic, M.; Olsen, G. B.; Owen, W. S.: Mobility of the β1−γ1 martensitic interface in Cu−Ni−Al: part 1, experimental measurements. Metal. Trans. 16A (1985) 1723–1734Google Scholar
  5. 5.
    Cong Dahn, N.; Morphy, D.; Rajan, K.: Kinetics of the martensitic F.C.C.→B.C.C. transformation in Co−Cr−Mo alloy powders. Acta Metal. 9 (1984) 1317–1322Google Scholar
  6. 6.
    Langer, J. S.: Dendrites, viscous fingers, and the theory of pattern formation. Science 243 (1989) 1150–1156Google Scholar
  7. 7.
    Mullins, W. W.; Sekerka, R. F.: Stability of a planar interface during solidification of a binary alloy. J. App. Phys. 35 (1964) 444–451Google Scholar
  8. 8.
    Strain, J.: Linear stability of planar solidification fronts. Phys. D30 (1988) 277–320Google Scholar
  9. 9.
    Fried, E.: Linear stability of a two-phase process involving a steadily propagating planar phase boundary in a solid: part 2. thermomechanical case. Technical Report No. 5, ONR Grant N00014-90-J-1871 1991Google Scholar
  10. 10.
    Jiang, Q.: On modeling of thermo-mechanical phase transformations in solids. To appear in J. Elast.Google Scholar
  11. 11.
    Knowles, J. K.: On finite anti-plane shear for incompressible elastic materials. J. Aust. Math. Soc. 19 Series B (1976), 400–415Google Scholar
  12. 12.
    Rosakis, P.: Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Rat. Mech. Anal. 109 (1990) 1–37Google Scholar
  13. 13.
    Jiang, Q.; Knowles, J. K.: A class of compressible elastic materials capable of sustaining finite anti-plane shear. To appear in J. Elast.Google Scholar
  14. 14.
    Fosdick, R.; Kao, B. G.: Transverse deformations associated with rectilinear shear in elastic solids. J. Elast. 8 (1978) 117–142Google Scholar
  15. 15.
    Fosdick, R.; Serrin, J.: Rectilinear steady flow of simple fluids. Proceedings of the Royal Society of London SERIES A 332 (1973) 311–333Google Scholar
  16. 16.
    Abeyaratne, R.; Knowles, J. K.: On the driving traction acting on a surface of strain discontinuity in a continuum. J. Mech. Phys. Solids 38 (1990) 345–360Google Scholar
  17. 17.
    Abeyaratne, R.; Knowles, J. K.: Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids. To appear in SIAM J. Appl. Math.Google Scholar
  18. 18.
    Abeyaratne, R.; Knowles, J. K.: Kinetic relations and the propagation of phase boundaries in elastic solids. Arch. Rat. Mech. Anal. 114 (1991) 119–154Google Scholar
  19. 19.
    Abeyaratne, R.; Knowles, J. K.: On the propagation of maximally dissipative phase. boundaries in solids. To appear in Quart. Appl. Math.Google Scholar
  20. 20.
    Gurtin, M. E.; Struthers, A.: Multiphase thermomechanics with interfacial structure. 3. Evolving phase boundaries in the presence of bulk deformation. Arch. Rat. Mech. Anal. 112 (1990) 97–160Google Scholar
  21. 21.
    Gurtin, M. E.; Pego, R. L.: Forthcoming.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • E. Fried
    • 1
  1. 1.Division of Engineering and Applied ScienceCalifornia Institute of TechnologyPasadena

Personalised recommendations