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Edited by: Nikos D. Lagaros, National Technical University of Athens, Greece

Reviewed by: Dimitrios Giagopoulos, University of Western Macedonia, Greece; Savvas Triantafyllou, University of Nottingham, United Kingdom

This article was submitted to Computational Methods in Structural Engineering, a section of the journal Frontiers in Built Environment

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this paper, three specific uniaxial phenomenological models commonly used for the description of a Shape Memory Alloy (SMA) behavior are examined in detail. In particular, the models examined are the Graesser-Cozzarelli model, the Wilde-Gardoni-Fujino model, and the Zhang-Zhu model. The pertinent model parameters are examined with respect to their physical representation, if any. Based on this analysis, a new simple rate-independent model is proposed which addresses all issues in a unified manner. Finally, powerful metaheuristics are employed for system identification, producing excellent fit with experimental data while revealing valuable information regarding the relative sensitivity of the proposed model parameters.

Throughout history, humanity has sought shelter from natural phenomena. Natural shelters, such as caves, were abandoned for artificial structures made of primitive materials, i.e., wood, stone, and brick. The building blocks were held in place by either sole gravity or by primitive paste. In the classical era, the Romans used

Seeking the next revolution in the construction industry, researchers drew their attention to the so-called smart materials. These materials exhibit extraordinary properties, ranging from piezoelectricity and pH-sensitivity to magnetostriction and self-healing. A popular class of these materials, commonly known as Shape Memory Alloys (SMAs), exhibit physical and mechanical characteristics that allow their integration into structures. SMAs are capable of sustaining large inelastic strains that can be recovered by heating or unloading, depending on prior loading history. The origin of this unusual behavior is the ability of SMAs to undergo a first-order solid-solid diffusionless, and reversible phase change called

Various innovative systems and devices, mainly using NiTi and Cu-based SMAs, have been developed for seismic energy absorption, damping control, structural retrofit. Several prototypes of SMA braces for the seismic protection of structures have been designed, numerically assessed and experimentally tested (Clark et al.,

Obviously, proper modeling of the extraordinary behavior of SMAs is important and, thus, several models have been proposed in the literature. These can be broadly categorized into microscopic thermodynamic models, based on the Ginzburg-Landau theory or molecular dynamics; micro-macro models, based on micromechanics, micro-planes or micro-spheres; and macroscopic models, based on the theory of plasticity, thermodynamic potentials, finite strains or statistical physics (Cisse et al.,

The primary forms of SMA behavior which are pertinent to the applications examined in this study are shown in Figure _{f}, where _{f} is the temperature at which the microstructure of the material is fully martensitic. Although the loop strongly resembles the one observed in most conventional steels, the hysteretic mechanism is quite different. In steels, hysteresis in cyclic loading is due to dislocation glide but in SMAs it is due to twinning deformation of martensite that occurs by rotation, growth, and shrinkage of individual variants of martensite (Graesser and Cozzarelli,

Schematic representation SMA behavior _{f}), _{f}).

Figure _{f}, where _{f} is the temperature at which the microstructure of the material is fully austenitic. This behavior exhibits two very important properties, i.e., energy dissipation and zero residual strain upon unloading, and thus it is termed

By modifying Ozdemir's model (Ozdemir,

where, (˙) = ordinary time derivative, σ = stress, ε = strain, β = backstress, _{y}/(_{y}) = parameter controlling the post-elastic slope of the curve (_{y} = post-elastic modulus), _{in} = _{T},

and

Note that parameter

The Graesser and Cozzarelli model is rate-independent, as is the Bouc-Wen model (Charalampakis and Koumousis, _{max} → −_{max} → _{max} → …, will produce the same result.

Twinning hysteresis in the Graesser and Cozzarelli model is observed by setting _{T} = 0 in Equation (2). The base model shown in Figure _{max} = 0.016.

Twinning hysteresis (_{T} = 0) with the Graesser-Cozzarelli model,

In Figure _{y} is also affected since _{y} = α

In Figure

In Figure _{y} without affecting the initial stiffness

In Figure

Superelasticity in the Graesser and Cozzarelli model is observed by setting _{T} > 0 in Equation (2). The base model shown in Figure _{T} = 0.07; _{max} = 0.016. In Figure _{T} is shown. It is observed that high values of _{T} reduce the size of the energy-dissipating loops exhibited during the material transformations. Apparently, the sensitivity of the model with respect to _{T} is high. In Figure ^{c} during the experiment. The responses for ^{c} ≅ 1 for ^{c} takes a form not radically different from the error function _{T}, _{T}, ^{c} in Equation (2) is deemed redundant and will be removed in the proposed model which will be presented later.

Superelasticity (_{T} > 0) with the Graesser-Cozzarelli model, _{T} = 0.07, _{T} ^{c} for various values of _{T},

The final note on the Graesser-Cozzarelli model refers to the error function itself. As the evaluation of this function is cumbersome [see Equation (3)], its substitution with the hyperbolic tangent is proposed herein. The hyperbolic tangent is remarkably similar to the error function but more comfortable to evaluate, e.g., by using:

In Figure

Comparison of the error and hyperbolic tangent functions.

The Wilde-Gardoni-Fujino model (Wilde et al.,

where the functions _{I} (_{II} (_{III} (

In this model, the response is divided into separate regions which are activated or deactivated by the flags described by Equations (8)–(10). The strain _{m} defines the point at which the transformation from austenite to martensite is completed. Beyond this strain, the response is linear elastic with modulus equal to _{m}, due to the term _{III}(

Although the model is capable of simulating the martensitic phase of the SMA, its usage is cumbersome. For instance, the coefficients _{1}, _{2}, and _{3} do not have physical representation and plausible value ranges are not easy to establish. As is the case with the Graesser-Cozzarelli model, the term (

The Zhang-Zhu model (Zhang and Zhu,

where the functions _{I} (_{II} (_{III} (

and the signum function is defined as

Note that parameters

The problem with the potentially negative term (

The coefficients _{1}, _{2}, and _{3} have been replaced.

The error function has been replaced.

The following, however, can be listed as disadvantages of the model:

The number of parameters is high.

Plausible value ranges are not provided for the parameters not having a physical representation.

The response is still divided into many phases controlled with on/off flags.

Based on the observations above, a simple rate-independent uniaxial phenomenological model is proposed herein which is described by the following terms:

where, _{t} and 1 for |_{t}. Note that _{t}) = 0.5, i.e., at this level of strain the weights of the Graesser-Cozzarelli term and the elastic martensitic term in Equation (19) are equal. The coefficient

In total, the model contains only nine parameters (_{T}, _{m}, _{t}, _{m}, _{T}, _{m}, _{1}, _{1}, _{2}, _{3}) and the eleven to fourteen parameters of the Zhang-Zhu model (_{l}, _{u}, _{m}, _{Y}, _{l}, _{u}, _{l}, _{u}, _{T}, _{m}, _{1},

Brief description of proposed model parameters and their effect on the overall response.

Initial modulus during the austenitic phase | |

“Yield” stress | |

Control of post-elastic stiffness | |

Control of abruptness of transition between initial elastic and post-elastic phases | |

_{T} |
Control between twinning hysteresis and superelasticity |

Smoothness around the origin during cyclic loading | |

_{m} |
Modulus during the fully martensitic phase |

_{t} |
Strain of middle point of transition between Graesser-Cozzarelli and martensitic terms |

Control of abruptness of transition between Graesser-Cozzarelli and martensitic terms |

Despite its simplicity, the proposed model can accurately capture all the pertinent characteristics of the response curve. System identification based on metaheuristics produces excellent fit with experimental data obtained from the literature, as will be demonstrated. The optimum (best) parameter values were evaluated by Differential Evolution, a powerful metaheuristic algorithm (Storn and Price, _{r} = 0.9. Ten independent runs were conducted with different random seeds. Each run was terminated after 20000 function evaluations. The sum of squares of the difference between the measured and the predicted time history of stress is used as the objective function to be minimized. Cast in discrete form, this function can be written as

where

In Figure _{t}, _{m}, _{T}, and

Statistical analysis of identified parameters of the proposed model for the experimental data obtained from Zhang and Zhu (

10,000 | 60,000 | 46,823.455 | 34,519.290 | 46,823.455 | 42,422.809 | 3,534.132 | 0.083 | |

200 | 500 | 314.844 | 311.916 | 317.408 | 315.654 | 1.929 | 0.006 | |

0 | 1 | 0.148 | 0.148 | 0.218 | 0.168 | 0.020 | 0.118 | |

0.5 | 3 | 1.327 | 1.327 | 2.926 | 1.726 | 0.452 | 0.262 | |

_{T} |
0.1 | 1 | 0.064 | 0.061 | 0.066 | 0.064 | 0.001 | 0.021 |

1 | 1,000 | 194.224 | 194.224 | 225.739 | 202.093 | 8.711 | 0.043 | |

_{m} [MPa] |
10,000 | 60,000 | 19,291.590 | 18,991.328 | 19,906.820 | 19,230.288 | 263.735 | 0.014 |

_{t} |
0.01 | 0.08 | 0.052 | 0.050 | 0.052 | 0.051 | 0.000 | 0.006 |

1 | 1,000 | 99.135 | 96.101 | 107.987 | 98.857 | 3.439 | 0.035 |

Figure _{m}, _{t}, _{t} > 0.037, any _{m}, and large

Statistical analysis of identified parameters of the proposed model for the experimental data obtained from Zhuang et al. (

100 | 2,000 | 584.159 | 584.159 | 587.764 | 586.236 | 1.530 | 0.003 | |

1 | 10 | 4.255 | 4.253 | 4.260 | 4.256 | 0.002 | 0.000 | |

0 | 1 | 0.395 | 0.390 | 0.395 | 0.392 | 0.002 | 0.006 | |

0.5 | 3 | 1.399 | 1.382 | 1.399 | 1.389 | 0.007 | 0.005 | |

_{T} |
0.1 | 1 | 0.017 | 0.017 | 0.017 | 0.017 | 0.000 | 0.004 |

1 | 1,000 | 665.339 | 663.530 | 671.228 | 666.635 | 3.191 | 0.005 |

Figure

Statistical analysis of identified parameters of the proposed model for the experimental data obtained from Ozbulut et al. (

10,000 | 60,000 | 39,286.352 | 32,783.866 | 40,068.672 | 36,609.716 | 2,971.842 | 0.081 | |

200 | 500 | 284.271 | 282.443 | 290.882 | 285.069 | 2.691 | 0.009 | |

0 | 1 | 0.130 | 0.124 | 0.165 | 0.144 | 0.015 | 0.108 | |

0.5 | 3 | 1.112 | 1.018 | 1.678 | 1.315 | 0.254 | 0.193 | |

_{T} |
0.1 | 1 | 0.081 | 0.078 | 0.085 | 0.080 | 0.002 | 0.028 |

1 | 1,000 | 135.944 | 130.274 | 150.642 | 140.804 | 7.059 | 0.050 | |

_{m} [MPa] |
10,000 | 60,000 | 12,084.968 | 10,988.091 | 12,690.244 | 12,122.933 | 508.070 | 0.042 |

_{t} |
0.01 | 0.08 | 0.054 | 0.054 | 0.055 | 0.055 | 0.000 | 0.007 |

1 | 1,000 | 181.706 | 172.829 | 217.115 | 190.377 | 12.437 | 0.065 |

In Figure

Identified parameters of the proposed model for the experimental data shown in Figures

^{°}C−1 Hz |
^{°}C−1.5 Hz |
^{°}C−2 Hz |
^{°}C−0.5 Hz |
|||
---|---|---|---|---|---|---|

72,478.332 | 56,469.931 | 38,944.849 | 60,000.000 | 40,754.313 | 60,000.000 | |

619.072 | 385.812 | 204.114 | 243.092 | 338.771 | 444.085 | |

0.023 | 0.075 | 0.164 | 0.088 | 0.174 | 0.120 | |

1.682 | 2.249 | 1.076 | 1.141 | 2.083 | 2.550 | |

_{T} |
0.530 | 0.114 | 0.031 | 0.061 | 0.071 | 0.100 |

179.385 | 254.366 | 1,000.000 | 382.420 | 178.101 | 208.790 | |

_{m} [MPa] |
– | 14,899.926 | 14,932.520 | 16,418.358 | 22,441.737 | 10,000.000 |

_{t} |
– | 0.059 | 0.036 | 0.042 | 0.048 | 0.032 |

– | 58.753 | 150.688 | 144.455 | 142.488 | 108.515 |

In order to demonstrate the advantages of the proposed model, the same identification procedure is applied to the Wilde-Gardoni-Fujino model with data obtained from Zhang and Zhu (

Statistical analysis of identification results for the Wilde-Gardoni-Fujino model with data obtained from Zhang and Zhu (

10,000 | 100,000 | 35,995.179 | 35,048.908 | 53,552.683 | 44,130.931 | 7,752.798 | 0.176 | |

_{m} [MPa] |
10,000 | 100,000 | 16,072.799 | 10,000.000 | 16,072.799 | 11,360.786 | 2,483.071 | 0.219 |

100 | 800 | 359.596 | 172.570 | 417.298 | 308.840 | 89.187 | 0.289 | |

_{m} |
0.01 | 0.08 | 0.064 | 0.010 | 0.065 | 0.037 | 0.020 | 0.544 |

_{1} |
0.01 | 0.08 | 0.043 | 0.010 | 0.078 | 0.058 | 0.023 | 0.393 |

_{T} |
0.1 | 1 | 0.180 | 0.103 | 0.997 | 0.558 | 0.341 | 0.612 |

0.01 | 1 | 0.139 | 0.033 | 0.152 | 0.081 | 0.046 | 0.565 | |

1 | 2 | 1.939 | 1.160 | 2.000 | 1.737 | 0.302 | 0.174 | |

1 | 5,000 | 283.772 | 271.270 | 4,982.904 | 3,102.826 | 2,097.170 | 0.676 | |

0.0001 | 1 | 0.143 | 0.102 | 0.368 | 0.255 | 0.096 | 0.376 | |

_{1} |
0 | 100,000 | 26,367.269 | 825.089 | 99,018.466 | 36,926.151 | 33,488.155 | 0.907 |

_{2} |
0 | 100,000 | 27,821.814 | 0.000 | 94,387.750 | 37,703.847 | 34,095.015 | 0.904 |

_{3} |
0 | 100,000 | 9,386.547 | 0.000 | 97,085.336 | 28,741.415 | 35,912.720 | 1.250 |

Objective function | 5.211E+05 | 5.211E+05 | 1.055E+07 | 4.834E+06 | 3.825E+06 | 0.791 |

In this paper, three specific uniaxial phenomenological models commonly used for the description of a Shape Memory Alloy (SMA) behavior were examined in detail, and a new simple rate-independent model was proposed which addresses all issues in a unified manner. From the presented analysis and the numerical results, the following main conclusions can be drawn:

In total, the proposed model contains only nine parameters as opposed to the thirteen parameters of the Wilde-Gardoni-Fujino model and the eleven to fourteen parameters of the Zhang-Zhu model.

Despite its simplicity, the proposed model can accurately capture all the pertinent characteristics of the response curve.

For the proposed model, system identification based on metaheuristics produced excellent fit with experimental data obtained from the literature.

Apart from the best result, all runs produced quality solutions in the same region of the design space, and the identified parameters had small coefficients of variation. This is a strong indication of a well-behaved model, with uniquely defined parameters, each one with a distinct role in the response curve.

On the contrast, the application of the same identification process to the Wilde-Gardoni-Fujino model yielded significantly inferior results, i.e., great variability in parameter values and quality of solutions.

A significant advantage of the proposed uniaxial model is that it can be incorporated within a Finite Element code such as OpenSees straightforwardly.

AC had the research idea, drafted the article, and contributed to the derivation of the numerical examples. GT contributed to the conception of the work and interpretation of the results. Both authors contributed to the writing of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The handling editor declared a shared affiliation, though no other collaboration, with one of the authors AC at time of review.