Three-dimensional model of thermomechanical evolution of shape memory materials

Control and Cybernetics

Volumn 29, Issue 1, 2000, Pages 340-365

  Pawlow, Irena ^a  

^a SYSTEMS RESEARCH INSTITUTE   (Poland)

Author keywords

Martensitic phase transformation; Shape memory; Thermodynamically consistent model; Thermomechanical evolution

Indexed keywords

EID: 0347573291     PISSN: 03248569     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (42)

1. ALT, H.W., PAWLOW, I. (1995) Thermodynamical models of phase transitions with multicomponent order parameter, in: Trends in Applications of Mathematics to Mechanics, M.D.P.M. Marques, J. F. Rodrigues (Eds), Pitman Monographs and Surveys in Pure and Applied Mathematics 77, Longman, 87-98.
2. ALT, H.W., PAWLOW, I. (1996) On the entropy principle of phase transition models with a conserved order parameter. Advances in Mathematical Sciences and Applications, 6, 291-376.
3. ABEYARATNE, R., KNOWLES, J.K. (1991) Implications of viscosity and straingradient effects for the kinetics of propagating phase boundaries in solids. SIAM J. Appl. Math., 51, 5, 1205-1221.
4. BARSCH, G.R., KRUMHANSL, J.A. (1984) Twin boundaries in ferroelastic media without interface dislocations. Phys. Rev. Letters, 53, 11, 1069-1072.
5. BALL, J.M., HOLMES, P.J., JAMES, R.D., PEGO, R.L., SWART, P.J. (1991) On the dynamics of fine structure. J. Nonlinear Science, 1, 17-70.
6. COLLI, P. (1995) Global existence for the three-dimensional Frémond model of shape memory alloys. Nonlinear Analysis, T.M.A., 24, 1565-1579.
7. COLLI, P., FRÉMOND, M., VISINTIN, A. (1990) Thermo-mechanical evolution of shape memory alloys. Q. Appl. Math., 48, 31-47.
8. COLLI, P., SPREKELS, J. (1992) Global existence for the three-dimensional model for the thermo-mechanical evolution of shape memory alloys. Nonlinear Anal., 18, 873-888.
9. COLLI, P., SPREKELS, J. (1993) Positivity of temperature in the general Frémond model for shape memory alloys. Continuum Mech. Thermodyn., 5, 255-264.
10. EDELEN, D.G.B. (1973) On the existence of symmetry relations and dissipation potentials. Arch. Ration. Mech. Analysis, 51, 218-227.
11. EDELEN, D.G.B. (1974) Primitive thermodynamics: A new look at the Clausius-Duhem inequality. Int. J. Engng Sci., 12, 121-141.
12. ERICKSEN, J.L. (1986) Constitutive theory for some constrained elastic crystals. Int. J. Solids and Structures, 22, 951-964.
13. FALK, F. (1982) Landau theory and martensitic phase transitions. Journal de Physique C4, 43, 3-15.
14. FALK, F. (1990) Elastic phase transitions and nonconvex energy functions. In: Free Boundary Problems: Theory and Applications, K. -H. Hoffmann and J. Sprekels, Eds, vol. I, Longman, New York, 45-55.
15. FALK, F., KONOPKA, P. (1990) Three-dimensional Landau theory describing the martensitic phase transformation of shape memory alloys. J. Phys.: Condens. Matter, 2, 61-77.
16. FRÉMOND, M. (1987) Matériaux à mémoire de forme, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 304, 7, 239-244.
17. FELDERHOF, B.U. (1970) Dynamics of the diffuse gas-liquid interface near the critical point. Physica, 48, 541-560.
18. FRIED, E., GURTIN, M.E. (1994) Dynamic solid-solid transitions with phase characterized by an order parameter. Physica D, 72, 287-308.
19. FOSDICK, R.L., SERRIN, J. (1979) On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids. Journal of Elasticity, 9, 1, 83-89.
20. GURTIN, M.E. (1981) An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, 158, Academic Press.
21. HOFFMANN, K.-H., NIEZGÓDKA, M., ZHENG, S. (1990) Existence and uniqueness of global solutions to an extended model of the dynamical developments in shape memory alloys. Nonlinear Analysis, TMA, 15, 977-990.
22. HOFFMANN, K.-H., ŻOCHOWSKI, A. (1992) Existence of solutions to some non-linear thermoelastic systems with viscosity. Math. Meth. Appl. Sci., 15, 187-204.
23. KLOUČEK, P., LUSKIN, M. (1994) The computation of the dynamics of the martensitic transformation. Continuum Mech. Thermodyn., 6, 209-240.
24. LANDAU, L.D., LIFSHYTZ, E.M. (1987) Elasticity Theory. Nauka, Moscow (in Russian).
25. LEVITAS, V.I. (1998) Thermomechanical theory of martensitic phase transformations in inelastic materials. Int. J. Solids Structures, 35, 9-10, 889-940.
26. MÜLLER, I. (1985) Thermodynamics, Interaction of Mechanics and Mathematics Series, Pitmann, Boston.
27. NIEZGÓDKA, M., SPREKELS, J. (1988) Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys. Math. Meth. Appl. Sci., 10, 197-223.
28. PAWLOW, I. (2000) Mathematical models for media with microstructures. Advances in Mathematical Sciences and Applications, 10, 1.
29. PAWLOW, I., ŻOCHOWSKI, A. (2000) Nonlinear thermoelastic system with viscosity and nonlocality. In: Proceedings Free Boundary Problems. Theory and Applications I, N. Kenmochi, ed., Gakuto International Series. Mathematical Sciences and Applications, 13, 251-265.
30. PAWLOW, I., ŻOCHOWSKI, A. (2000) Existence and uniqueness of solutions for a three-dimensional thermoelastic system, to appear.
31. ROUBÍČEK, T. (1999) Dissipative evolution of microstructure in shape memory alloys. In: Lectures on Applied Mathematics, II.-J. Bungartz, R.W. Hoppe, C. Zenger, Eds., Springer, Berlin, 45-63.
32. RYBKA, P. (1992) Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions. Proc. Roy. Soc. Edinburgh, 121 A, 101-138.
33. RYBKA, P. (1994) A priori estimates for the gradient of the solution to the system of viscoelasticity in several dimensions. Topol. Methods Nonlinear Anal., 3, 235-256.
34. RYBKA, P. (1997) The viscous damping prevents propagation of singularities in the system of viscoelasticity. Proc. Roy. Soc. Edinburgh, 127 A, 1067-1074.
35. SLEMROD, M. (1983) Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Archieve for Rational Mechanics and Analysis, 81, 301-315.
36. SLEMROD, M. (1984) Dynamics of first order phase transitions. In: Phase Transformations and Material Instabilities in Solids, Academic Press.
37. SPREKELS, J. (1990) Shape memory alloys: mathematical models for a class of first order solid-solid phase transitions in metals. Control and Cybernetics, 19, 287-308.
38. SPREKELS, J., ZHENG, S. (1989) Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys. Physica D, 39, 59-76.
39. TOUPIN, R.A. (1964) Theories of elasticity with couple-stress. Arch. Rat. Mech. Anal., 17, 2, 85-112.
40. TRUSKINOVSKY, L. (1994) About the "normal growth" approximation in the dynamical theory of phase transitions. Continuum Mech. Thermodyn., 6, 185-208.
41. ŻOCHOWSKI, A. (1992) Mathematical Problems in Shape Optimization and Shape Memory Materials, Peter Lang Verlag, in the series Methoden und Verfahren der mathematischen Physik, 38.
42. ŻOCHOWSKI, A. (1993) Regularity of solutions to certain viscosity systems, presentation at the 10th Conference on Problems and Methods in Mathematical Physics, September 13-17, 1993, Chemnitz, Germany.