×

Bifurcation and stability of martensitic transformation dynamics. (English) Zbl 1165.34361

Summary: We have studied the mathematical properties and their physical implications of a system of nonlinear ordinary differential equations with two variables and six parameters, which was proposed to model the martensitic transformation of shape memory alloys. While the system could not have limit cycles, bifurcation to hysteresis was found in load and/or temperature controlled processes with large enough interfacial energies. This agrees qualitatively quite well with the experimental observation and theoretical understandings of the stress-strain-temperature hysteresis of the alloys. A three-dimensional bifurcation diagram was identified. Nonhyperbolic equilibrium points were found as saddle nodes and high order node points. The local behavior was studied and the phase portrait of the system was obtained for the load and temperature parameters.
Accordingly, a stable node represents the stable martensitic or austenitic phase, a saddle stands for the unstable phase mixture, and a saddle node corresponds to the beginning or the end of the transformation. Therefore, for a thermal-mechanical loading path, the martensitic transformation process accords with the qualitative changes of the equilibrium points.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
74N30 Problems involving hysteresis in solids
34C55 Hysteresis for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Achenbach M., J. de Phys. 12 pp 163–
[2] Achenbach M., Arch. Mech. 37 pp 573–
[3] DOI: 10.1016/0020-7683(86)90006-5 · Zbl 0576.73033 · doi:10.1016/0020-7683(86)90006-5
[4] DOI: 10.1016/0749-6419(89)90023-5 · doi:10.1016/0749-6419(89)90023-5
[5] Andronov A. A., Qualitative Theory of Second-Order Dynamical Systems (1973)
[6] Arnold V. I., Bifurcation Theory and Catastrophe Theory (1999)
[7] DOI: 10.1007/s00419-006-0101-1 · Zbl 1179.74095 · doi:10.1007/s00419-006-0101-1
[8] Awrejcewicz J., Int. Rev. Mech. Eng. 1 pp 1–
[9] DOI: 10.1103/PhysRevLett.84.3077 · doi:10.1103/PhysRevLett.84.3077
[10] DOI: 10.1090/S0002-9939-99-04719-X · Zbl 0915.76095 · doi:10.1090/S0002-9939-99-04719-X
[11] DOI: 10.1080/13873950500076404 · Zbl 1097.74528 · doi:10.1080/13873950500076404
[12] DOI: 10.1016/j.jmps.2005.07.009 · Zbl 1120.74392 · doi:10.1016/j.jmps.2005.07.009
[13] DOI: 10.1007/BF00042524 · Zbl 0765.73011 · doi:10.1007/BF00042524
[14] DOI: 10.1007/BF01177231 · doi:10.1007/BF01177231
[15] DOI: 10.1063/1.353672 · doi:10.1063/1.353672
[16] DOI: 10.1063/1.530789 · Zbl 0797.58048 · doi:10.1063/1.530789
[17] DOI: 10.1007/BF01126524 · Zbl 0780.73006 · doi:10.1007/BF01126524
[18] DOI: 10.1016/j.actamat.2004.02.016 · doi:10.1016/j.actamat.2004.02.016
[19] DOI: 10.1063/1.2408484 · Zbl 1146.76427 · doi:10.1063/1.2408484
[20] DOI: 10.1023/A:1024423626386 · Zbl 1062.70599 · doi:10.1023/A:1024423626386
[21] DOI: 10.1016/j.ijsolstr.2003.10.015 · Zbl 1045.74530 · doi:10.1016/j.ijsolstr.2003.10.015
[22] Lee J. H., Discr. Contin. Dyn. Syst.-Ser. B 6 pp 339–
[23] DOI: 10.1155/2004/717986 · doi:10.1155/2004/717986
[24] DOI: 10.1007/BF02729036 · doi:10.1007/BF02729036
[25] DOI: 10.1007/BF01141998 · doi:10.1007/BF01141998
[26] DOI: 10.1016/S0895-7177(01)00134-0 · Zbl 1066.74043 · doi:10.1016/S0895-7177(01)00134-0
[27] DOI: 10.1088/0964-1726/16/1/020 · doi:10.1088/0964-1726/16/1/020
[28] Otsuka K., Shape Memory Alloys (1998)
[29] DOI: 10.1007/978-1-4684-0392-3 · doi:10.1007/978-1-4684-0392-3
[30] DOI: 10.1103/PhysRevB.48.864 · doi:10.1103/PhysRevB.48.864
[31] DOI: 10.1103/PhysRevA.43.669 · doi:10.1103/PhysRevA.43.669
[32] Seelecke S., Appl. Mech. Rev. 57 pp 27–
[33] DOI: 10.1016/j.amc.2004.04.048 · Zbl 1177.65163 · doi:10.1016/j.amc.2004.04.048
[34] DOI: 10.1016/S0045-7825(98)00205-9 · Zbl 0949.74049 · doi:10.1016/S0045-7825(98)00205-9
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.