Applications of fully conservative schemes in nonlinear thermoelasticity: modelling shape memory materials

Матус, П. П.; Melnik, Roderick V. N.; Wang, L.; Rybak, Iryna

doi:10.1016/j.matcom.2004.01.012

Cited by 27 publications

(33 citation statements)

References 29 publications

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“…Computational results presented in [2,13] confirmed robustness of our modelling approach for a number of practically important cases. The main results of the present paper are pertinent to the general threedimensional case.…”

Section: Introductionsupporting

confidence: 58%

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A Dynamic Model for Phase Transformations in 3D Samples of Shape Memory Alloys

Mahapatra

Melnik

2005

Lecture Notes in Computer Science

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Abstract. Despite recent progress in modelling the shape memory alloy (SMA) behaviour, many difficulties remain due to various limitations of the existing free energy models and strong nonlinearity of these nonlinear materials. Phase kinetics of SMA coupled with thermoelastodynamics is still not fully tractable, and one needs to deal with complicated multiscale character of SMA materials requiring a linkage between their microstructure and macroscopic properties. In this paper we develop a new dynamic model of 3D SMA which employs an improved version of the microscopic Landau theory. Essential properties of the single and multivariant martensitic phase transformations are recovered using consistent steps, which eliminates the problem of non-uniqueness of energy partitioning and relaxes the over-sensitivity of the free energy due to many unknown material constants in previously reported models. We exemplify our results on a model for cubic to tetragonal transformations in a rectangular SMA bar by showing key algorithmic steps which can be extended to more complex cases.

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Section: Introductionsupporting

confidence: 58%

“…Note that earlier, a new low-dimensional model for the time-dependent dynamics of SMA was derived in [2] and a class of such reduced models was rigorously analyzed in [13]. Computational results presented in [2,13] confirmed robustness of our modelling approach for a number of practically important cases.…”

Section: Introductionmentioning

confidence: 58%

A Dynamic Model for Phase Transformations in 3D Samples of Shape Memory Alloys

Mahapatra

Melnik

2005

Lecture Notes in Computer Science

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“…For the full system (Eq. (1)), the well-posedness results are found in, for example, [11][12][13] and so on, and numerical results are found in Hoffmann-Zou [14], Niezgodka-Sprekels [15] by finite element method and in Matus-Melnik-Wang-Rybak [16] and Yoshikawa [17,18] by the finite difference method. In particular, in Ref.…”

Section: Introductionmentioning

confidence: 92%

Shape Memory Wires in R3

Okabe¹,

Suzuki²,

Yoshikawa³

2017

Shape Memory Alloys - Fundamentals and Applications

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We propose a new model describing the dynamics of wire made of shape memory alloys, by combining an elastic curve theory and the Ginzburg-Landau theory. The wire is assumed to be a closed curve and is not to be stretched with deformation. The derived system of nonlinear partial differential equations consists of a thermoelastic system and a geometric evolution equation under the inextensible condition. We also show that the system has dual variation structure as well as a straight material case. The structure implies stability of infinitesimally stable stationary state in the Lyapunov sense.

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“…We start our consideration from a mathematical model for the SMA dynamics based on a coupled system of the three fundamental laws, conservation of mass, linear momentum, and energy balance, in a way we described previously in [21,17,26]. Using these laws, the system that describes coupled thermo-mechanical wave interactions for the first order martensitic phase transformations in a three dimensional SMA structure can be written as follows [18,21,24] …”

Section: Mathematical Model For Sma Dynamicsmentioning

confidence: 99%

“…Mathematical modelling tools play an important role in studying such transformations and computational experiments, based on mathematical models, can be carried out to predict the response of the material under various loadings, different types of phase transformations, and reorientations. The development of such tools is far from straightforward even in the one-dimensional case where the analysis of the dynamics is quite involved due to a strongly nonlinear pattern of interactions between mechanical and thermal fields (e.g., [3,17] and references therein). For a number of practical applications a better understanding of the dynamics of SMA structures with dimensions higher than one becomes critical.…”

Section: Introductionmentioning

confidence: 99%

Finite volume analysis of nonlinear thermo-mechanical dynamics of shape memory alloys

Wang

Melnik

2006

Heat Mass Transfer

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In this paper, the finite volume method is developed to analyze coupled dynamic problems of nonlinear thermoelasticity. The major focus is given to the description of martensitic phase transformations essential in the modelling of shape memory alloys. Computational experiments are carried out to study the thermo-mechanical wave interactions in a shape memory alloy rod, and a patch. Both mechanically and thermally induced phase transformations, as well as hysteresis effects, in a one-dimensional structure are successfully simulated with the developed methodology. In the two-dimensional case, the main focus is given to square-to-rectangular transformations and examples of martensitic combinations under different mechanical loadings are provided.

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Applications of fully conservative schemes in nonlinear thermoelasticity: modelling shape memory materials

Cited by 27 publications

References 29 publications

A Dynamic Model for Phase Transformations in 3D Samples of Shape Memory Alloys

A Dynamic Model for Phase Transformations in 3D Samples of Shape Memory Alloys

Shape Memory Wires in R3

Finite volume analysis of nonlinear thermo-mechanical dynamics of shape memory alloys

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