Introduction 

The purpose of this example is to demonstrate a system identification process by using Microsoft Excel and xlOptimizer. A specific Shape Memory Alloy (SMA) model, developed by Charalampakis and Tsiatas [1] is used in this case. Estimation of the unknown parameters is performed for experimental data obtained from the work by Zhang and Zhu [2] using graph digitization.

Two words about Shape Memory Alloys 

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 martensitic transformation between a parent phase called austenite (A), stable at high temperature and low stress, and a product phase called martensite (M), metastable at low temperature and high stress. The martensitic transformation is at the origin of the two main effects observed in SMAs, namely the shape-memory effect and superelasticity.

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. For applications in structural engineering, however, uniaxial phenomenological models are of special interest as integration with existing reliable FEM codes.

A simple uniaxial SMA model has been recently proposed by Charalampakis and Tsiatas [1]. This model is used in this study. You can refer to the published paper for more information.

Step 1 and only

We have already created the spreadsheet for you. You can modify the values in the yellow cells and press the Calculate button to evaluate the model for these specific values. You can go to the xlOptimizer menu and press the Single Objective button in the Action pane to optimize the parameter values. All the variables, constraints, and algorithms are already set-up. The value in cell G1 is the error between the experimental values and the predicted values. Note that the optimization may take some time because the digitization was very dense in order to minimize the error.

The file for this example can be downloaded here. Note that it is saved a macro enabled book with the extension .xlsm. You need to enable macros to run it. 

References

[1] Charalampakis, A. E., Tsiatas, G. C., “A simple rate-independent uniaxial Shape Memory Alloy (SMA) model”, Frontiers in Built Environment, 4 (2018): 46, doi: 10.3389/fbuil.2018.00046.
[2] Zhang, Y., Zhu, S. "A shape memory alloy-based reusable hysteretic damper for seismic hazard mitigation", Smart Mater. Struct. 16, (2007) 1603–1613. doi: 10.1088/0964-1726/16/5/014.