×

On the effect of dissipation in shape-memory alloys. (English) Zbl 1023.74036

Summary: After reviewing the martensitic phase transformation in shape memory alloys, the conventional models that take into consideration viscosity-like and capillarity-like response are investigated for vanishing ”dissipative effects”. It is shown that they do approach the fully conservative case. Experimental evidence indicates that the response is rate-independent, and thus a model incorporating a phenomenological rate-independent plasticity-type dissipation related with an activated phase-transformation process is investigated.

MSC:

74N05 Crystals in solids
74N20 Dynamics of phase boundaries in solids
74H30 Regularity of solutions of dynamical problems in solid mechanics
35Q72 Other PDE from mechanics (MSC2000)
74C99 Plastic materials, materials of stress-rate and internal-variable type
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abeyaratne, R.; Knowles, J.K., Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids, SIAM J. appl. math., 51, 1205-1221, (1991) · Zbl 0764.73013
[2] Abeyaratne, R.; Knowles, J.K., On the propagation of maximally dissipative phase boundaries in solids, Q. appl. math., 50, 149-172, (1992) · Zbl 0785.73004
[3] Andrews, G., On the existence of solutions to the equation utt=uxxt+σ(ux)x, J. differential equations, 35, 200-231, (1980)
[4] M. Arndt, M. Griebel, T. Roubı́ček, On modelling of cubic to tetragonal martensitic transformation in shape memory alloys. Continuum Mech. Thermodyn., to appear.
[5] Ball, J.M.; Holmes, P.J.; James, R.D.; Pego, R.L.; Swart, P.J., On the dynamics of fine structure, J. nonlinear sci., 1, 17-70, (1991) · Zbl 0791.35030
[6] Bhattacharya, K., Comparison of geometrically nonlinear and linear theories of martensitic transformation, Cont. mech. thermodyn., 5, 205-242, (1993) · Zbl 0780.73005
[7] Brokate, M.; Sprekels, J., Hysteresis and phase transitions, (1996), Springer New York · Zbl 0951.74002
[8] Bubner, N., Landau-Ginzburg model for a deformation-driven experiment on shape memory alloys, Continuum mech. thermodyn., 8, 293-308, (1996) · Zbl 0925.65220
[9] N. Bubner, G. Mackin, R.C. Rogers, Rate dependence of hysteresis in one-dimensional phase transitions, WIAS Preprint no.539, Berlin, 1999.
[10] Chen, Z.; Hoffmann, K.-H., On a one-dimensional nonlinear thermoviscoelastic model for structural phase transitions in shape memory alloys, J. differential equations, 112, 325-350, (1994) · Zbl 0807.73029
[11] Dafermos, C.M., The mixed initial-boundary value problems for equation of nonlinear one-dimensional viscoelasticity, J. differential equations, 6, 71-86, (1969) · Zbl 0218.73054
[12] Demoulini, S., Weak solutions for a class of nonlinear systems of viscoelasticity, Archive rat. mech. anal., 155, 299-334, (2000) · Zbl 0991.74021
[13] Falk, F., Landau theory and martensitic phase transitions, ()
[14] Friesecke, G.; Dolzmann, G., Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy, SIAM J. math. anal., 28, 363-380, (1997) · Zbl 0872.35026
[15] Friesecke, G.; McLeod, J.B., Dynamics as a mechanism preventing the formation of finer and fined microstructure, Archive rat. mech. anal., 133, 199-247, (1996) · Zbl 0920.73345
[16] Garcke, H., Travelling wave solutions as dynamic phase transitions in shape memory alloys, J. differential equations, 121, 203-231, (1995) · Zbl 0829.73007
[17] Hagan, R.; Slemrod, M., The viscosity-capillarity admissibility criterion for shocks and phase transitions, Archive rat. mech. anal., 83, 333-361, (1983) · Zbl 0531.76069
[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. · Zbl 0797.35174
[19] Hoffmann, K.-H.; Zochowski, A., Existence of solutions to some nonlinear thermoelastic systems with viscosity, Math. methods appl. sci., 15, 187-204, (1992) · Zbl 0745.35022
[20] Huo, Y.; Müller, I., Nonequilibrium thermodynamics of pseudoelasticity, Continuum mech. thermodyn., 5, 163-204, (1993) · Zbl 0780.73006
[21] James, R.D., The propagation of phase boundaries in elastic bars, Archive rat. mech. anal., 73, 125-158, (1980) · Zbl 0443.73010
[22] Klouček, P.; Luskin, M., The computation of the dynamics of the martensitic transformation, Continuum mech. thermodyn., 6, 209-240, (1994) · Zbl 0825.73047
[23] Levitas, V.I., Thermomechanical theory of martensitic phase transformations, Int. J. solids & structures, 35, 889-940, (1998) · Zbl 0931.74059
[24] Levitas, V.I.; Idesman, A.V.; Olson, G.B.; Stein, E., Numerical modeling of martensite growth in elastoplastic material, Philosophical magazine, 82, 429-462, (2002)
[25] Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites nonlineáire, (1969), Dunod, Gauthier-Villars Paris
[26] Mielke, A.; Theil, F., A mathematical model for rate-independent phase transformations with hysteresis, (), 117-129
[27] A. Mielke, F. Theil, On rate-independent hysteresis models, Preprint Math.Inst., Uni. Stuttgart, 2001, Nonlin. Diff. Eq. Appl., to appear. · Zbl 1061.35182
[28] Mielke, A.; Theil, F.; Levitas, V.I., A variational formulation of rate-independent phase transformations using extremum principle, Arch. rat. mech. anal., 163, 137-177, (2002) · Zbl 1012.74054
[29] Niezgódka, M.; Sprekels, J., Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys, Math. methods appl. sci., 10, 197-223, (1988) · Zbl 0668.35013
[30] Pawłow, I., Three-dimensional model of thermomechanical evolution of shape memory material, Control cybernet., 29, 341-365, (2000) · Zbl 1205.74140
[31] Pego, R.L., Phase transitions in one-dimensional nonlinear viscoelasticityadmissiblity and stability, Archive rat. mech. anal., 97, 353-394, (1987) · Zbl 0648.73017
[32] Plecháč, P.; Roubı́ček, T., Visco-elasto-plastic model for martensitic phase transformation in shape-memory alloys, Math. methods appl. sci., 25, 1281-1298, (2002) · Zbl 1012.35051
[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] Rajagopal, K.R.; Srinivasa, A.R., Inelastic behavior of materials—part iienergies associated with twinning, Int. J. plasticity, 13, 1-35, (1997) · Zbl 0905.73002
[35] Rajagopal, K.R.; Srinivasa, A.R., On the thermomechanics of shape memory wires, Zeitschrift angew. math. phys., 50, 459-496, (1999) · Zbl 0951.74005
[36] Roubı́ček, T., Dissipative evolution of microstructure in shape memory alloys, (), 45-63 · Zbl 0993.74048
[37] Roubı́ček, T., Relaxation in optimization theory and variational calculus, (1997), W. de Gruyter Berlin · Zbl 0880.49002
[38] Roubı́ček, T., Evolution model for martensitic phase transformation in shape-memory alloys, Interfaces free boundaries, 4, 111-136, (2002), (Preprint MATH-MU-2000/2, MFF UK, Praha) · Zbl 0998.35058
[39] Rybka, P., Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions, Proc. royal soc. Edinburgh, 121A, 101-138, (1992) · Zbl 0758.73004
[40] Rybka, P.; Hoffmann, K.-H., Convergence of solutions to equation of viscoelasticity with capillarity, J. math. anal. appl., 226, 61-81, (1998) · Zbl 0919.35022
[41] A.J.M. Spencer, Theory of invariants, in: A.C. Eringen, (Ed.), Continuum Physics, Vol. 1, Academic Press, New York, 1971.
[42] Sprekels, J., Global existence for thermomechanical processes with nonconvex free energies of Ginzburg-Landau form, J. math. anal. appl., 141, 333-348, (1989) · Zbl 0701.35082
[43] Sprekels, J.; Zheng, S., Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys, Physica D, 39, 39-54, (1989)
[44] Swart, P.J.; Holmes, P.J., Energy minimization and the formation of microstructure in dynamic anti-plane shear, Archive rat. mech. anal., 121, 37-85, (1992) · Zbl 0786.73066
[45] Theil, F., Young-measure solution for a viscoelastically damped wave equation with nonmonotone stress-strain relation, Arch. rat. mech. anal., 144, 47-78, (1998) · Zbl 0936.74040
[46] Truesdell, C.A., A first course in rational continuum mechanics, (1991), Academic Press New York · Zbl 0866.73001
[47] Truskinovsky, L., Equilibrium interface boundaries, Sov. phys. doklady, 27, 551-553, (1982), (Russian orig. Dokl. Akad. Nauk SSSR 265, 306-310)
[48] Truskinovsky, L., Transition to detonation in dynamic phase changes, Archive rat. mech. anal., 125, 375-397, (1994) · Zbl 0813.73009
[49] Vainchtein, A., Dynamics of phase transitions and hysteresis in a viscoelastic Ericksen’s bar on an elastic foundation, J. elasticity, 57, 243-280, (1999) · Zbl 1003.74054
[50] A. Vainchtein, Stick-slip interface motion as a singular limit of the viscosity-capillarity model, Math. Mech. Solids (2002), to appear. · Zbl 1057.74028
[51] Visintin, A., Modified Landau-Lifshitz equation for ferromagnetism, Physica B, 233, 365-369, (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.