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Research Papers

# Low Dimensional Approximations to Ferroelastic Dynamics and Hysteretic Behavior Due to Phase Transformations

[+] Author and Article Information
Linxiang X. Wang

Institute of Mechatronic Engineering, Hangzhou Dianzi University, Hangzhou 310037, China; MCI, Faculty of Science and Engineering, University of Southern Denmark, Sonderborg DK-6400, Denmark

Roderick V. N. Melnik1

M2 NeT Lab, Wilfrid Laurier University, 75 University Avenue West, Waterloo, ON, N2L 3C5, Canadarmelnik@wlu.ca

1

Corresponding author.

J. Appl. Mech 77(3), 031015 (Feb 24, 2010) (12 pages) doi:10.1115/1.4000381 History: Received January 18, 2006; Revised August 03, 2009; Published February 24, 2010; Online February 24, 2010

## Abstract

In this paper, a low dimensional model is constructed to approximate the nonlinear ferroelastic dynamics involving mechanically and thermally-induced martensite transformations. The dynamics of the first order martensite transformation is first modeled by a set of nonlinear coupled partial differential equations (PDEs), which is obtained by using the modified Ginzburg–Landau theory. The Chebyshev collocation method is employed for the numerical analysis of the PDE model. An extended proper orthogonal decomposition is then carried out to construct a set of empirical orthogonal eigenmodes of the dynamics, with which system characteristics can be optimally approximated (in a specified sense) within a range of different temperatures and under various mechanical and thermal loadings. The performance of the low dimensional model is analyzed numerically. Results on the dynamics involving mechanically and thermally-induced phase transformations and the hysteresis effects induced by such transformations are presented.

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## Figures

Figure 1

The first three empirical eigenmodes for the strain distributions ((a), (b), and (c)) and temperature distributions ((d), (e), and (f))

Figure 2

The first empirical eigenmode for the strain distributions (a) and temperature distributions (b) in the second block of snapshots using the extended POD

Figure 3

The absolute approximation error to the second block of snapshots using different numbers of empirical eigenmodes: (a) seven modes for the strain distribution; (b) ten modes for the strain distribution; (c) seven modes for the temperature distribution; (d) ten modes for the temperature distribution

Figure 4

Numerical results for thermally-induced phase transformations in a SMA rod obtained with the developed low dimensional model. (a) Strain distribution, (b) thermal hysteresis at x=0.1 cm, (c) temperature distribution, and (d) thermal hysteresis at x=0.9 cm.

Figure 5

Numerical results for a SMA rod obtained with the developed low dimensional model. (a) Strain distribution, (b) temperature distribution, and (c) mechanical hysteresis with θ=255 deg.

Figure 6

Comparison of hysteretic behavior due to mechanically induced phase transformations: theoretical analysis ((a), (b), and (c)) and numerical simulation ((d), (e), and (f)); θ=220 deg ((a) and (d)), θ=240 deg ((b) and (e)), θ=320 deg ((c) and (f))

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