Characterization and stability of two-phase piecewise-homogeneous deformations

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 460, Issue 2051, 2004, Pages 3065-3094

  Fu, Y B ^a   Freidin, A B ^b  

^a KEELE UNIVERSITY   (United Kingdom)

^b INSTITUTE OF PROBLEMS OF MECHANICAL ENGINEERING   (Russian Federation)

Author keywords

Interfacial instability; Phase transformation; Stability

Indexed keywords

CORRELATION ANALYSIS; ENERGY; KINETICS; PHASE TRANSITION; REVIEW; SOLID;

EID: 17144371835     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2004.1361     Document Type: Review

References (51)

1. Abeyaratne, R. 1980 Discontinuous deformation gradients in plane finite elastostatics of incompressible materials. J. Elasticity 10, 255-293.
2. Abeyaratne, R. 1981 Discontinuous deformation gradients in the finite twisting of an incompressible elastic tube. J. Elasticity 11, 43-80.
3. Abeyaratne, R. 1983 An admissibility condition for equilibrium shocks in finite elasticity. J. Elasticity 13, 175-184.
4. Abeyaratne, R. & Knowles, J. K. 1989 Equilibrium shocks in plane deformation of incompressible elastic materials. J. Elasticity 22, 63-80.
5. Ball, J. M. 1980 Strict convexity, strong ellipticity, and regularity in the calculus of variations. Math. Proc. Camb. Phil. Soc. 87, 501-513.
6. Barnett, D. M., Lothe, J., Gavazza, S. D. & Musgrave, M. J. 1985 Consideration of the existence of interfacial (Stoneley) waves in bonded anisotropic elastic half-spaces. Proc. R. Soc. Lond. A 402, 153-166.
7. Chadwick, P. 1995 Interfacial and surface-waves in pre-stressed isotropic elastic media. Z. Angew. Math. Phys. 46, S51-S71.
8. Chadwick, P. & Borejko, P. 1994 Existence and uniqueness of Stoneley waves. Geophys. J. Int. 118, 279-284.
9. Cherkaev, A. 1991 Variational methods for structural optimization. Springer.
10. Dowaikh, M. A. & Ogden, R. W. 1991 Interfacial waves and deformations in pre-stressed elastic media. Proc. R. Soc. Lond. A 433, 313-328.
11. Eremeyev, V. A. 1999 On the stability of nonlinear elastic bodies with phase transformations. In Proc. 1st Canadian Conf. on Nonlinear Solids Mechanics (ed. E. M. Croitoro), pp. 519-528. University of Victoria Press.
12. Eremeyev, V. A. & Zubov, L. M. 1991 On the stability of equilibrium of nonlinear elastic bodies with phase transformations. Izv. AN USSR Mekh. Tverdogo Tela 2, 56-65.
13. Eremeyev, V. A., Freidin, A. & Sharipova, L. 2003 Nonuniqueness and stability in problems of equilibrium of elastic two-phase bodies. Dokl. Phys. 48(7), 359-363. (Translated from Dokl. Akad. Nauk SSSR 391(2), 189-193.)
14. Eremeyev, V. A., Freidin, A. & Sharipova, L. 2003 Nonuniqueness and stability in problems of equilibrium of elastic two-phase bodies. Dokl. Phys. 48(7), 359-363. (Translated from Dokl. Akad. Nauk SSSR 391(2), 189-193.)
15. Ericksen, J. L. 1975 Equilibrium of bars. J. Elasticity 5, 191-201.
16. Eshelby, J. D. 1975 The elastic energy-momentum tensor. J. Elasticity 5, 321-335.
17. Fosdick, R. L. & MacSithigh, G. 1983 Helical shear of an elastic, circular tube with a non-convex stored energy. Arch. Ration. Mech. Analysis 84, 31-53.
18. Fosdick, R. L. & Zhang, Y. 1993 The torsion problem for a nonconvex stored energy function. Arch. Ration. Mech. Analysis 122, 291-322.
19. Fosdick, R. L. & Zhang, Y. 1994 Coexistent phase mixtures in the antiplane shear of an elastic tube. Z. Angew. Math. Phys. 45, 202-244.
20. Fosdick, R. L. & Zhang, Y. 1995a A structured phase transition for the antiplane shear of an elastic circular tube. Q. J. Mech. Appl. Math. 48, 189-210.
21. Fosdick, R. L. & Zhang, Y. 1995b Stress and the moment-twist relation in the torsion of a cylinder with a nonconvex stored energy function. Z. Angew. Math. Phys. 46, 146-171.
22. Freidin, A. B. & Chiskis, A. M. 1994a Phase transition zones in nonlinear elastic isotropic materials. Part 1. Basic relations. Izv. AN USSR Mekh. Tverdogo Tela 29, 91-109.
23. Freidin, A. B. & Chiskis, A. M. 1994b Phase transition zones in nonlinear elastic isotropic materials. Part 2. Incompressible materials with a potential depending on one of the deformation invariants. Izv. AN USSR Mekh. Tverdogo Tela 29, 46-58.
24. Freidin, A. B. & Croitoro, E. M. 1999 Phase transition zones and two phase strain fields in elastic solids. In Proc. 1st Canadian Conf. on Nonlinear Solids Mechanics (ed. E. M. Croitoro), pp. 509-518. University of Victoria Press.
25. Freidin, A. B., Viltchevskaya, E. N. & Sharipova, L. 2002 Two-phase deformations within the framework of phase transition zones. Theor. Appl. Mech. 28-29, 149-172.
26. Fu, Y. B. 1999 Buckling of an elastic half-space with surface imperfections. In Proc. 1st Canadian Conf. on Nonlinear Solids Mechanics (ed. E. M. Croitoro), pp. 99-107. University of Victoria Press.
27. Fu, Y. B. & Mielke, A. 2002 A new identity for the surface-impedance matrix and its application to the determination of surface-wave speeds. Proc. R. Soc. Lond. A458, 2523-2543.
28. Grinfeld, M. A. 1980 On conditions of thermodynamic equilibrium of the phases of a nonlinear elastic material. Dokl. Akad. Nauk SSSR 251, 824-827.
29. Grinfeld, M. A. 1991 Thermodynamic methods in the theory of heterogeneous systems. New York: Longman.
30. Gurtin, M. E. 1983 Two-phase deformations of elastic solids. Arch. Ration. Mech. Analysis 84, 1-29.
31. James, R. D. 1981 Finite deformation by mechanical twinning. Arch. Ration. Mech. Analysis 77, 143-177.
32. Knowles, J. K. 1979 On the dissipation associated with equilibrium shocks in finite elasticity. J. Elasticity 9, 131-158.
33. Knowles, J. K. & Sternberg, E. 1978 On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J. Elasticity 8, 329-379.
34. Merodio, J. & Ogden, R. W. 2002 Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation. Arch. Mech. 54, 525-552.
35. Merodio, J. & Ogden, R. W. 2003 Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int. J. Solids Struct. 40, 4707-4727.
36. Merodio, J. & Pence, T. J. 2001a Kink surfaces in a directionally reinforced neo-Hookean material under plane deformation. I. Mechanical equilibrium. J. Elasticity 62, 119-144.
37. Merodio, J. & Pence, T. J. 2001b Kink surfaces in a directionally reinforced neo-Hookean material under plane deformation. II. Kink band stability and maximally dissipative band broadening. J. Elasticity 62, 145-170.
38. Mielke, A. & Fu, Y. B. 2004 Uniqueness of surface-wave speed: a proof that is independent of the Stroh formalism. Math. Mech. Solids 9, 5-15.
39. Morozov, N. F. & Freidin, A. B. 1998 Phase transition zones and phase transformations of elastic solids under different stress states. Proc. Steklov Math. Inst. 223, 220-232.
40. Noether, E. 1918 Invariante Variationsproblems. Kgl. Ges. Wiss. Nachr. Göttingen Math. Physik Kl. 2, 235-257. (English translation by Travel, M. A. 1971 Transport Theory Statist. Phys. 1, 186-207.)
41. Noether, E. 1918 Invariante Variationsproblems. Kgl. Ges. Wiss. Nachr. Göttingen Math. Physik Kl. 2, 235-257. (English translation by Travel, M. A. 1971 Transport Theory Statist. Phys. 1, 186-207.)
42. Ogden, R. W. 1984 Non-linear elastic deformations. New York: Ellis Horwood.
43. Ogden, R. W. & Fu, Y. B. 1996 Nonlinear stability analysis of a pre-stressed elastic half-space. In Contemporary research in the mechanics and mathematics of materials (ed. R. C. Batra & M. F. Beatty), pp. 164-175. Barcelona: CIMNE.
44. Olver, P. J. 1986 Applications of Lie groups to differential equations. Springer.
45. Qiu, G. Y. & Pence, T. J. 1997 Loss of elliptiticy in plane deformations of a simple directionally reinforced incompressible nonlinear elastic solid. J. Elasticity 49, 31-63.
46. Renardy, M. & Rogers, R. C. 1992 An introduction to partial differential equations. Springer.
47. Rosakis, P. 1990 Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Ration. Mech. Analysis 109, 1-37.
48. Rosakis, P. & Jiang, Q. 1993 Deformations with discontinuous gradients in plane elastostatics of compressible solids. J. Elasticity 33, 233-257.
49. Simpson, H. C. & Spector, S. J. 1989 Necessary conditions at the boundary for minimizers in finite elasticity. Arch. Ration. Mech. Analysis 107, 105-125.
50. Simpson, H. C. & Spector, S. J. 1991 Some necessary conditions at an internal boundary for minimizers in finite elasticity. J. Elasticity 26, 203-222.
51. Tommasi, D., Foti, P., Marzano, S. & Piccioni, M. 2001 Incompressible elastic bodies with non-convex energy under dead-load surface tractions. J. Elasticity 65, 149-168.