On the effect of dissipation in shape-memory alloys

Nonlinear Analysis: Real World Applications

Volumn 4, Issue 4, 2003, Pages 581-597

  Rajagopal, K R ^a   Roubíček, Tomáš ^b,c  

^a Texas A M University   (United States)

^b CHARLES UNIVERSITY   (Czech Republic)

^c INSTITUTE OF INFORMATION THEORY AND AUTOMATION   (Czech Republic)

Author keywords

Capillarity; Martensitic transformation; Plasticity; Rate independent dissipation; Regularity; Viscosity

Indexed keywords

EID: 0037412208     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/S1468-1218(02)00079-2     Document Type: Article

References (51)

1. R. Abeyaratne, J.K. Knowles, Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids, SIAM J. Appl. Math. 51 (1991) 1205-1221.
2. R. Abeyaratne, J.K. Knowles, On the propagation of maximally dissipative phase boundaries in solids, Q. Appl. Math. 50 (1992) 149-172.
3. x , J. Differential Equations 35 (1980) 200-231.
4. M. Arndt, M. Griebel, T. Roubéček, On modelling of cubic to tetragonal martensitic transformation in shape memory alloys. Continuum Mech. Thermodyn., to appear.
5. J.M. Ball, P.J. Holmes, R.D. James, R.L. Pego, P.J. Swart, On the dynamics of fine structure, J. Nonlinear Sci. 1 (1991) 17-70.
6. K. Bhattacharya, Comparison of geometrically nonlinear and linear theories of martensitic transformation, Cont. Mech. Thermodyn. 5 (1993) 205-242.
7. M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.
8. N. Bubner, Landau-Ginzburg model for a deformation-driven experiment on shape memory alloys, Continuum Mech. Thermodyn. 8 (1996) 293-308.
9. N. Bubner, G. Mackin, R.C. Rogers, Rate dependence of hysteresis in one-dimensional phase transitions, WIAS Preprint no.539, Berlin, 1999.
10. Z. Chen, K.-H. Hoffmann, On a one-dimensional nonlinear thermoviscoelastic model for structural phase transitions in shape memory alloys, J. Differential Equations 112 (1994) 325-350.
11. C.M. Dafermos, The mixed initial-boundary value problems for equation of nonlinear one-dimensional viscoelasticity, J. Differential Equations 6 (1969) 71-86.
12. S. Demoulini, Weak solutions for a class of nonlinear systems of viscoelasticity, Archive Rat. Mech. Anal. 155 (2000) 299-334.
13. F. Falk, Landau theory and martensitic phase transitions, in: L. Delaey, M. Chandrasekaran (Eds.), Proceedings of the Conference on Martensitic Transformations, Les Editions de Physique, Les Ulis, 1982.
14. G. Friesecke, G. Dolzmann, Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy, SIAM J. Math. Anal. 28 (1997) 363-380.
15. G. Friesecke, J.B. McLeod, Dynamics as a mechanism preventing the formation of finer and fined microstructure, Archive Rat. Mech. Anal. 133 (1996) 199-247.
16. H. Garcke, Travelling wave solutions as dynamic phase transitions in shape memory alloys, J. Differential Equations 121 (1995) 203-231.
17. R. Hagan, M. Slemrod, The viscosity-capillarity admissibility criterion for shocks and phase transitions, Archive Rat. Mech. Anal. 83 (1983) 333-361.
18. H. Hattori, K. Mischaikow, On a global dynamics of a phase transition problem in: J.M. Chadam, H. Rasmussen (Eds.), Proceedings of the Free Boundary Problems: Theory and Applications, Vol.5, Pitman Research Notes in Mathematics, Vol. 281, Longmann, Harlow, Essex, (1993) 226-231.
19. K.-H. Hoffmann, A. Zochowski, Existence of solutions to some nonlinear thermoelastic systems with viscosity, Math. Methods Appl. Sci. 15 (1992) 187-204.
20. Y. Huo, I. Müller, Nonequilibrium thermodynamics of pseudoelasticity, Continuum Mech. Thermodyn. 5 (1993) 163-204.
21. R.D. James, The propagation of phase boundaries in elastic bars, Archive Rat. Mech. Anal. 73 (1980) 125-158.
22. P. Klouček, M. Luskin, The computation of the dynamics of the martensitic transformation, Continuum Mech. Thermodyn. 6 (1994) 209-240.
23. V.I. Levitas, Thermomechanical theory of martensitic phase transformations, Int. J. Solids & Structures, 35 (1998) 889-940.
24. V.I. Levitas, A.V. Idesman, G.B. Olson, E. Stein, Numerical modeling of martensite growth in elastoplastic material, Philosophical magazine, 82 (2002) 429-462.
25. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlineáire, Dunod, Gauthier-Villars, Paris, 1969.
26. A. Mielke, F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in: H.-D. Alder, R. Balean, R. Farwig (Eds.), Models of Continuum Mechanics in Analysis and Engineering, Shaker Ver, Aachen, 1999, pp.117-129.
27. A. Mielke, F. Theil, On rate-independent hysteresis models, Preprint Math. Inst., Uni. Stuttgart, 2001, Nonlin. Diff. Eq. Appl., to appear.
28. A. Mielke, F. Theil, V.I. Levitas, A variational formulation of rate-independent phase transformations using extremum principle, Arch. Rat. Mech. Anal. 163 (2002) 137-177.
29. M. Niezgódka, J. Sprekels, Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys, Math. Methods Appl. Sci. 10 (1988) 197-223.
30. I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory material, Control Cybernet. 29 (2000) 341-365.
31. R.L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: admissiblity and stability, Archive Rat. Mech. Anal. 97 (1987) 353-394.
32. P. Plecháč, T. Roubíček, Visco-elasto-plastic model for martensitic phase transformation in shape-memory alloys, Math. Methods Appl. Sci. 25 (2002) 1281-1298.
33. K.R. Rajagopal, Multiple configurations in continuum mechanics, Report of the Institute for Computational and Applied Mechanics, No. 6, University of Pittsburgh, 1995.
34. K.R. Rajagopal, A.R. Srinivasa, Inelastic behavior of materials-Part II: Energies associated with twinning, Int. J. Plasticity 13 (1997) 1-35.
35. K.R. Rajagopal, A.R. Srinivasa, On the thermomechanics of shape memory wires, Zeitschrift angew. Math. Phys.50 (1999) 459-496.
36. T. Roubíček, Dissipative evolution of microstructure in shape memory alloys, in: H.-J. Bungartz, R.H.W. Hoppe, C. Zenger (Eds.), Lectures on Applied Mathematics, Springer, Berlin, 2000, pp. 45-63.
37. T. Roubíček, Relaxation in Optimization Theory and Variational Calculus, W. de Gruyter, Berlin, 1997.
38. T. Roubíček, Evolution model for martensitic phase transformation in shape-memory alloys (Preprint MATH-MU-2000/2, MFF UK, Praha), Interfaces Free Boundaries 4 (2002) 111-136.
39. P. Rybka, Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions, Proc. Royal Soc. Edinburgh 121A (1992) 101-138.
40. P. Rybka, K.-H. Hoffmann, Convergence of solutions to equation of viscoelasticity with capillarity, J. Math. Anal. Appl. 226 (1998) 61-81.
41. A.J.M. Spencer, Theory of invariants, in: A.C. Eringen, (Ed.), Continuum Physics, Vol. 1, Academic Press, New York, 1971.
42. J. Sprekels, Global existence for thermomechanical processes with nonconvex free energies of Ginzburg-Landau form, J. Math. Anal. Appl. 141 (1989) 333-348.
43. J. Sprekels, S. Zheng, Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys, Physica D 39 (1989) 39-54.
44. P.J. Swart, P.J. Holmes, Energy minimization and the formation of microstructure in dynamic anti-plane shear, Archive Rat. Mech. Anal. 121 (1992) 37-85.
45. F. Theil, Young-measure solution for a viscoelastically damped wave equation with nonmonotone stress-strain relation, Arch. Rat. Mech. Anal. 144 (1998) 47-78.
46. C.A. Truesdell, A First Course in Rational Continuum Mechanics, Academic Press, New York, 1991.
47. L. Truskinovsky, Equilibrium interface boundaries, Sov. Phys. Doklady 27 (1982) 551-553 (Russian orig. Dokl. Akad. Nauk SSSR 265, 306-310).
48. L. Truskinovsky, Transition to detonation in dynamic phase changes, Archive Rat. Mech. Anal. 125 (1994) 375-397.
49. A. Vainchtein, Dynamics of phase transitions and hysteresis in a viscoelastic Ericksen's bar on an elastic foundation, J. Elasticity 57 (1999) 243-280.
50. A. Vainchtein, Stick-slip interface motion as a singular limit of the viscosity-capillarity model, Math. Mech. Solids (2002), to appear.
51. A. Visintin, Modified Landau-Lifshitz equation for ferromagnetism, Physica B 233 (1997) 365-369.