In this work, a new constitutive model of the behavior of shape-memory alloys is presented, based on earlier models, showing a very good agreement with the existing experimental results. A simple approximate application concerning the use of these alloys modelled as dissipation devices in a special truss-moment frame is demonstrated. The results obtained are considered sufficiently encouraging as a motivation for the ongoing work.

Shape-memory alloys (SMAs) are unique materials having the ability to recover their original shape after mechanical distortion via the application of heat, which indicates the shape-memory effect, or by unloading, which indicates the superelastic effect, through the phase change (austenitic-martensitic). This ability has made SMAs quite popular for a wide variety of applications, such as aerospace, automotive, dental, biomedical, and, of course, structural. An interesting review of the use of SMAs in civil engineering is the one by Song et al. [

In the literature, a limited number of SMA constitutive (behavioral) models have been reported [

Finally, a preliminary investigation on the potential use of this model as an energy dissipation device in the special segments of steel special truss moment frames [

In this section, an overview of the features and characteristics of SMAs is presented, as well as a short description of the most relevant constitutive models. This is considered needed to allow the reader to comprehensively follow the subject of the present work.

Shape-memory alloys (SMAs) are unique materials having the ability to recover their original shape after mechanical distortion via the application of heat, which indicates the shape-memory effect, or by unloading, which indicates the superelastic effect, through the phase change (austenitic-martensitic).

SMAs have the ability to form two crystal structures through the rearrangement of atoms within the crystal lattice, the austenitic and the martensitic. Their features are the consequences of shape changes due to martensitic phase transformations. This kind of transformations is reversible diffusionless shear transformations which occur by some form of cooperative movement of a relatively large number of atoms, each being displaced by a small distance relative to its neighbours and that results in a change in the crystal structure. The diffusionless character makes the martensitic transformation almost instantaneous.

The austenitic crystal structure, which SMAs form, is characterized by a cubic high-symmetry structure stable at higher temperatures, whereas the martensitic one by a monoclinic low-symmetry structure is stable at lower temperatures. The austenitic phase is called the “parent” phase, and it presents only one crystal-orientation direction, which is called a variant, whereas the martensitic phase presents twenty-four variants, and their structure depends on the type of the transformation the material has undergone. This is because, when the austenite shears to form the martensite, there are different directions to do this.

In a stress-free state, an SMA is characterized by four transformation temperatures: _{s}, _{f}, _{s}, and _{f}. The first two, with _{s} smaller than _{f}, indicate the temperatures at which the transformation of austenite to martensite starts and finishes during cooling. The latter, with _{s} smaller than _{f}, is the temperatures at which the transformation of martensite into austenite starts and finishes during heating.

The loading and unloading of an SMA in the austenitic state results in a hysteresis loop with zero or insignificant residual strain. This is due to the fact that the austenitic phase is loaded elastically up to a “yield” stress where a stress-induced transformation from austenite to martensite takes place. The martensite that is formed due to the application of stress is called detwinned martensite or stress-induced martensite, and its formation process consists of the spatial re-orientation of the original martensite variants. The product phase is then called a single-variant martensite and is characterized by a detwinned structure [

After the complete transformation to martensite, further straining causes its elastic loading at a modulus lower than that of elastic austenite but much higher than that of the phase transition portion of the loading curve. Upon unloading, since martensite is stable due to the presence of the applied stress, the reverse transformation takes place but at a lower stress plateau. After full unloading, the material ideally returns to its undeformed geometry [

There are limits to stress ranges among which superelastic deformation can occur. Excessive deformation beyond that, which can be accommodated by transformation to martensite, will lead to plastic deformation by slip, which is an irreversible process.

A typical schematic picture of a superelastic loading path is presented in Figure

A superelastic loading path (from Lagoudas [

Schematic stress-strain curve of superelastic effect of SMAs.

Buehler et al. [

Since then, the first serious attempt to develop a rational force-displacement relationship for a metallic damper was made by Özdemir [

In this particular model of interest, the equation of the stress was identical to the one by Özdemir given as follows:

Parameters

The unit-step function will activate the added term of the expression of the back stress only during unloading. The branch of the hysteresis loop during loading is unaffected by it. The motivation for selecting this particular form of the back stress arises from the requirement of zero residual strain, which is necessary when describing superelastic material behavior. For a superelastic type of response, parameters

If the parameter

Typical cyclic response of Graesser– Cozzarelli model for (a)

A feasible SMA constitutive model must be able to represent (a) the elastic loading of martensite that follows the complete transformation of austenite to martensite in a superelastic type of response and (b) the rate-dependent nature.

All the models referenced above, more or less, have some serious drawbacks. More specifically, the ones presented by Özdemir [

In the sequel, a new constitutive model is presented herein, in order to capture the aforementioned characteristics of SMAs; it is based on Graesser-Cozzarelli model, as previously stated.

The stress exerted on the SMA is the stress

In order to determine the form of

In Figure

Comparison of the experimental responses at frequencies of 0.025, 0.5, and 1 Hz.

As depicted in Figure

Figure _{t} = 0.268,

Comparison of Graesser–Cozzarelli model prediction to experimental response at

Moreover, comparing the Graesser–Cozzarelli model prediction to the experimental stress-strain curve at frequencies of 0.5 and 1 Hz, a significant difference is observed due to the rate-independent character of the model. The behavior predicted is the same for any frequency as shown in Figure

Comparison of Graesser – Cozzarelli model prediction to experimental response at (a)

According to the above comparisons, we may then proceed to the definitions of the additional stresses of Equation (

Comparison of the proposed model prediction to the experimental response at a frequency of 0.025 Hz, which corresponds to quasi-static conditions, shows that the elastic loading of martensite is well captured and that the overall behavior approximates the test data in an adequate manner. As far as the dynamic conditions are concerned, comparing the proposed model prediction to the experimental response at frequencies of 0.05 and 1 Hz, it is readily perceived that the influence of strain rate is also captured. These findings are presented in Figures

Comparison of the proposed model prediction to experimental response at frequencies (a) of 0.025 Hz and (b) of 0.5 Hz.

Comparison of the proposed model prediction to experimental response at a frequency of 1 Hz.

In order to obtain the above results, a MATLAB code was utilised (not presented herein for brevity), while the experimental evidence was digitized from [

Special truss moment frames (STMFs) are a rather new type of steel structural system that was developed for resisting forces and deformations induced by severe earthquake ground motions [

Another type of truss moment frame, namely, the DTMF (Ductile TMF), was proposed by Longo et al. [

Traditional or dual-moment resisting frames are also a successful alternative in steel buildings [

Herein, we consider an STMF with a Vierendeel middle segment [

The undeformed and the deformed shape of the STMF configuration.

The columns and the members outside the special segment are designed to respond elastically, and plastic hinges will form only in the chord members and more specifically at the end of the middle segment, i.e., at points J, C, D, and L.

In what follows, the STMF is simulated via a 2-DOF mechanical model, based on certain simplifying assumptions. First, the mass of the beams is considered to be concentrated at the top of the column (lumped-mass model) neglecting the mass of the column. Second, a linear elastic response for all the members of the frame is considered. Column masses have finite rotational inertia, while all the masses of the other members of the frame have infinite rotational inertia. The axial deformation of the columns is insignificant, thus neglected. The columns and the beams attached to them preserve their verticality and their initial length. The energy-dissipating device undergoes axial deformations. The lumped mass can move horizontally and can also rotate; hence, the generalized coordinates (degrees of freedom) are the horizontal translation and the rotation. In Figure

The 2-DOF lumped-mass mechanical model (a) and its free body diagram (b).

The generalized forces exerted on the model are: the inertial moment _{I}, the elastic moment of the column _{c,φ} (due to the rotation _{c,ux} (due to the horizontal translation _{x}), the spring force of the column _{C,ux} (due to _{x}), and the vertical and horizontal components of the damping forces

Thereafter, the dynamic response of the model is studied under various types of simple time-dependent loading, as reported below.

Under the action of a lateral step load of infinite duration (of 10 kN), the damper dissipates some of the input energy and, finally, the system equilibrates in a deformed (point attractor) configuration as shown throughout Figures

Damper response.

Displacement-velocity phase plane (a) and angle-rotation phase plane (b).

A load with decreasing amplitude of finite duration, starting from 9 kN and decaying linearly (followed by the model’s free vibration) is studied here. Figure

Decreasing load followed by free vibration. External load (a) and damper response (b).

Decreasing load followed by free vibration. (a) The displacement-velocity phase plane and (b) the angle-rotation phase plane.

A load derived by superposing random sinusoidal loads is applied to the model for 10 seconds. Afterwards, the model vibrates freely for 15 seconds, according to the contents of Figure

Random load followed by free vibration. (a) The external force applied and (b) the damper response.

Random load followed by free vibration. (a) The displacement-velocity phase plane and (b) the angle-rotation phase plane.

From the above results, a rather expected and well-captured response is reported. Future work (ongoing by the authors) involves (a) full-scale tests, (b) more sophisticated models, and (c) analyses via FE software, in order to fully validate the proposed model.

In this paper, after developing a new rate-dependent constitutive model of superelastic SMA bars, found in a very good agreement with existing experimental results, a 2-degree-of-freedom simulation is proposed, in order to capture the dynamics of a special truss moment frame simulation with an SMA bar incorporated, in an adequate manner. The model, giving satisfactory results for the loads under which it is acted upon, may be used to further examine the behavior of the STMF under loads that are more complicated and on multistory buildings, constituting the basis for ongoing research.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.