SMA Constitutive Model Research Articles

The Stiffness and Damping Characteristics of a Rubber-Based SMA Composite Shock Absorber with a Hyper-Elastic SMA-Constitutive Model Considering the Loading Rate.

Shock absorbers are essential in enhancing vehicle ride comfort by mitigating vibrations. However, traditional rubber shock absorbers are constrained by their fixed stiffness and damping properties, limiting their adaptability to varying loads and thus affecting the ride comfort, especially under extreme road conditions. Shape Memory Alloys (SMAs), known for their intelligent material properties, offer a unique solution by adjusting stiffness and damping in response to temperature changes or strain rates, making them ideal for advanced vibration control applications. This study builds upon the Auricchio constitutive model to propose an enhanced SMA hyper-elastic constitutive model that accounts for different loading rates. This new model elucidates the impact of loading rates on the stiffness and damping characteristics of SMAs. Additionally, we introduce an innovative circular rubber-based SMA composite vibration reduction structure. Through a parameterized model and finite element simulation, we comprehensively analyze the stiffness and damping properties of the composite damper under various loading rates and harmonic excitations. Our findings suggest a novel approach to improving the vehicle ride comfort, offering significant potential for engineering applications and practical value.

Post-buckling and vibration analysis of double-curved sandwich panels with SMA embedded faces

Shape memory alloys, especially in the form of wire, can provide structures with magnificent improvements in buckling and frequency responses. The SMA's ability to recover its original shape can produce a sizable amount of recovery stress, resulting in enhancing natural frequency and critical temperature and lowering the maximum deflection of structures. In this study, in addition to considering the effects of SMA presence in double-curved sandwich plate on the aforementioned characteristics, the influences of heat treatment of curing SMAs on their properties, calculated by differential scanning calorimetry (DSC) and tensile testing, and sandwich structures will be considered. Furthermore, the effects of recovery stress, the volume fraction, and the position of SMA wires on thermal and mechanical stability will be studied. The Boyd & Lagoudas SMA constitutive model is utilized in the ABAQUS UMAT subroutine.

On the underlying material response of pseudoelastic NiTi

In some temperature regimes NiTi can be deformed to strain of 5–7% that recovers fully on unloading. Under tensile loading it traces a closed hysteresis with two stress plateaus associated with the reversible solid-state phase transformation between the austenitic (A) and the martensitic (M) phases, during which the deformation localizes and propagates through the specimen. Hallai and Kyriakides (2013) extracted the underlying softening response associated with A-M transformation from a tensile test on a NiTi strip laminated between two hardening steel face-strips that ensured that it deformed uniformly. The test finishes with the laminate permanently deformed to a strain of 6%. Extraction of the softening response of the M-A transformation requires compressing the thin laminate back to zero strain. The present paper presents a new experimental set-up that enables completion of the load/reverse load test on such a laminate by laterally supporting the specimen during compression to prevent buckling. The extracted response of NiTi exhibits softening branches with up-down-up trajectories for both the A-M and M-A transformations, and Maxwell stresses that match those of the stress plateaus of the tensile hysteresis of a NiTi strip. The extracted response is used to calibrate a custom SMA constitutive model, which when incorporated in a finite element analysis reproduces the measured hysteresis of NiTi nearly perfectly for the first time. Extraction of the complete underlying partially unstable response of SMAs enables modeling of the unstable behavior of SMA structures with the confidence provided by measured data.

Evolution of multiple Martensite variants in a SMA thick-walled cylinder loaded by internal pressure

The stress and deformation fields in a SMA ring or a thick-walled cylinder loaded by internal pressure at constant temperature (over the start temperature of the martensitic transformation) are determined in closed form under plane stress loading conditions. The phenomenological SMA constitutive model incorporates the volume fractions of multi-variants Martensite, which are assumed to evolve linearly with the Tresca effective stress, according to the associative flow rule and the corner flow rule. Initially, the cylinder is everywhere in a state of Austenite. The application of an internal pressure then triggers the martensitic transformation starting from the inner radius of the cylinder wall and extending towards the outer radius. If the wall thickness is large enough, the tangential stress may vanish at the inner radius and correspondingly the stress state may reach a corner of the Tresca transformation condition, thus originating two different Martensite variants according to the corner transformation rule. The admissible phase partitions within the wall thickness originating during the loading process have been systematically investigated according to the ratio between the outer and inner radii. The results obtained here suggest that the loading process should be interrupted soon after the complete martensitic transformation is achieved at the inner radius of the cylinder to avoid permanent plastic deformations.

An enhanced Brinson model with modified kinetics for martensite transformation

We propose an enhanced Brinson model with modified kinetics for martensite transformation. Two additional material constants are considered to follow the stress-temperature diagram above austenite start temperature (As) along with treatment to keep the continuity of the martensite volume fraction and the path dependency of the phase transformation. To demonstrate the performance of the proposed model, we implement this algorithm into ABAQUS user subroutine, then conduct several numerical simulations and compare their results with SMA wire experiments as well as those of three-dimensional SMA constitutive models. From the results, it turns out that the proposed model is as accurate as the three-dimensional models and shows better accuracy over original Brinson model in terms of recovery stress.

Modeling of hydrogen effect on the superelastic behavior of Ni-Ti shape memory alloy wires

Superelastic NiTi wires are widely used in orthodontic treatments, but sometimes fracture can be observed after few months of use in buccal cavity and attributed to the degradation of NiTi mechanical properties due to hydrogen absorption. In this paper, a modeling approach is proposed in order to describe the effect of hydrogen diffusion on the transformation properties of NiTi SMAs. In order to experimentally predict such effects, cathodic hydrogen charging was performed at a current density of 10 for 6h, 24h, 48h and 72h in 0.9% NaCl aqueous solution at room temperature. Tensile tests were carried out shortly after hydrogen charging. The obtained stress-strain curves showed an increase of yield transformation stresses for forward and reverse martensitic transformations and a decrease of maximum transformation strain. Using Fick’s second law, the transformation temperatures variation can be expressed as a function of the mean concentration of absorbed hydrogen and then taked into account in the SMA constitutive model developed by Chemisky et al (2011). The numerical results are compared to the experimental ones to calibrate the proposed method. Simulations showed that hydrogen diffusion induces a shifting of transfomation temperatures, a decreasing of maximum transformation strain and an increasing of yield transfomation stresses.

On Behaviors of Functionally Graded SMAs under Thermo-Mechanical Coupling

An analytical solution is obtained for the Functionally Graded Shape Memory Alloy (FG-SMA) composites subjected to thermo-mechanical coupling. Young's modulus and thermal expansion coefficient of the material are assumed to vary in different forms of power function through the thickness, with the Poisson's ratio being constant. An SMA constitutive model is combined with the averaging techniques of composite to determine the mechanical properties of the FG-SMA composites. Different phase transformation steps and the corresponding stress distributions through the thickness direction are given. The results show that the average stresses decrease as the transformations proceed.

The Research of the Difference between Small-Strain and Large-Strain Formulations for Shape Memory Alloys

Based on thermodynamics and phase transformation driving force, we apply a SMA constitutive model to analyze the large and small deformation of SMA materials. Simulations under different loading, uniaxial tension and shear conditions, illustrate the characteristics of the model in large strain deformation and small strain deformation. The results indicate that the difference between the two methods is small under the uniaxial tension case, while the large deformation and the small deformation results are very different under shear deformation case. It lays a foundation for the further studies of the constitutive model of SMA, especially in the multiaxial non-proportional loading aspects.

Transformation characteristics of shape memory alloy composites

Shape memory alloy (SMA) composites are being used in an ever-expanding set ofapplications. For new applications, SMA composites are being developed incorporating awide variety of matrices. The effect of these new compositions on the transformationbehavior of the SMA inhomogeneities and on the effective composite behavior is exploredhere. An analytic methodology combining micromechanical methods with an SMAconstitutive model is developed to determine the overall transformation properties of thecomposite. Specifically, the effective phase diagram, the effective transformation strains andthe composite stress state before and after transformation are determined. The resultsobtained from the analyses of an SMA–ceramic composite show that after transformationthe stress distribution between the two phases is modified such that the stress in thedirection of applied loading in the SMA phase is reduced while the stress in the samedirection in the ceramic phase increases. This stress redistribution decreases the localtransformation strain in the direction of loading and results in an increase of the appliedstress necessary to initiate and complete the forward and reverse transformation at aspecific temperature. The effects of the elastic modulus of the matrix and volumefraction of the SMA inhomogeneities on the transformation behavior are exploredthrough a parametric study to understand their influence on SMA composite design.

Advanced methods for the analysis, design, and optimization of SMA-based aerostructures

Engineers continue to apply shape memory alloys to aerospace actuation applications dueto their high energy density, robust solid-state actuation, and silent and shock-freeoperation. Past design and development of such actuators relied on experimental trial anderror and empirically derived graphical methods. Over the last two decades, however, it hasbeen repeatedly demonstrated that existing SMA constitutive models can capturestabilized SMA transformation behaviors with sufficient accuracy. This work buildsupon past successes and suggests a general framework by which predictive toolscan be used to assess the responses of many possible design configurations in anautomated fashion. By applying methods of design optimization, it is shown thatthe integrated implementation of appropriate analysis tools can guide engineersand designers to the best design configurations. A general design optimizationframework is proposed for the consideration of any SMA component or assembly ofsuch components that applies when the set of design variables includes manymembers. This is accomplished by relying on commercially available software andutilizing tools already well established in the design optimization community. Suchtools are combined with finite element analysis (FEA) packages that consider amultitude of structural effects. The foundation of this work is a three-dimensionalthermomechanical constitutive model for SMAs applicable for arbitrarily shaped bodies. Areduced-order implementation also allows computationally efficient analysis of structuralcomponents such as wires, rods, beams and shells. The use of multiple optimizationschemes, the consideration of assembled components, and the accuracy of theimplemented constitutive model in full and reduced-order forms are all demonstrated.

Theoretical and Experimental Study of the Shape Memory Effect of Beams in Bending Conditions

In this study, the shape memory effect of SMA beams under complex stress conditions is studied by means of a finite element model. The 1D version of a well-established SMA constitutive model is utilized in the numerical computations and the required parameters are obtained experimentally starting from thermal cycling tests in tension under different constant loads. After being calibrated, the model is used to compute the deformation of beams loaded in bending and undergoing thermal cycling; three-point bending and cantilever configurations are considered in this stage. Finally, the response predicted by the model is compared to experimental results and model capabilities are discussed. In particular, insight of the stress and strain evolution in bending is provided.

On the robustness and efficiency of integration algorithms for a 3D finite strain phenomenological SMA constitutive model

AbstractMost devices based on shape memory alloys experience large rotations and moderate or finite strains. This motivates the development of finite‐strain constitutive models together with the appropriate computational counterparts. To this end, in the present paper a three‐dimensional finite‐strain phenomenological constitutive model is investigated and a robust and efficient integration algorithm is proposed. Properly defining the variables, extensively used regularization schemes are avoided and a nucleation–completion criterion is defined. Moreover, introducing a logarithmic mapping, a new form of time‐discrete equations is proposed. The solution algorithm as well as a suitable initial guess for the resultant nonlinear equations are also deeply discussed. Extensive numerical tests are performed to show robustness as well as efficiency of the proposed integration algorithm. Implementation of the integration algorithm within a user‐defined subroutine UMAT in the commercial nonlinear finite element software ABAQUS/Standard makes also possible the solution of a variety of boundary value problems. The obtained results show the efficiency and robustness of the proposed approach and confirm the improved efficiency (in terms of solution CPU time) when a nucleation–completion criterion is used instead of regularization schemes, as well as when a logarithmic mapping is used for the time‐discrete evolution equation instead of an exponential mapping. Copyright © 2010 John Wiley & Sons, Ltd.

A semi-analytic analysis of shape memory alloy thick-walled cylinders under internal pressure

In this paper, a new method for analysis of the pseudoelastic response of shape memory alloy thick-walled cylinders subjected to internal pressure is proposed. Two cases of short and long cylinders are considered by assuming the plane stress and plane strain conditions. In each case, a three-dimensional phenomenological SMA constitutive model is simplified to obtain the corresponding two-dimensional constitutive relations. The cylinder is partitioned into a finite number of narrow annular regions, and appropriate assumptions are made in order to find a closed-form solution for the equilibrium equations in each annular region. The global solution is obtained by enforcing the stress continuity condition at the interface of the annular regions and imposing the boundary conditions. Several numerical examples are presented to demonstrate the efficiency of the proposed method, and the results are compared with three-dimensional finite element simulations.

Nonlinear Coupled Field Finite Element Procedure for Modeling of Shape Memory Alloy Components in Multi-physics Applications

In this paper, we introduce a one-dimensional finite element procedure for modeling shape memory alloy components used in multi-physics based applications including MEMS and biomedical applications. The procedure addressed in this paper is based on a sequential solution of two main finite-element procedures which are then combined after each iteration. The procedure was applied to model an SMA wire and the results were compared with corresponding numerical and experimental results from the literature for different loading cases and boundary conditions. The results showed very good agreement with both experimental and numerical results. The procedure was found to be capable of capturing system characteristics such as structural and thermal non-linearity, variable SMA material properties and the ability to model coupled electrical, thermal and structural loading and boundary conditions and was general enough to be extensible to various geometrical shapes and different SMA constitutive models.

Development of a Quasi-static Model of NiMnGa Magnetic Shape Memory Alloy

A quasi-static model for NiMnGa magnetic shape memory alloy is formulated along the lines of the Brinson and Tanaka SMA constitutive models. Since the shape memory effect (SME) and pseudoelasticity exist in both NiTi and NiMnGa, constitutive models for SMAs offer a basis for ferromagnetic shape memory alloys (FSMA) modeling. Two types of quasi-static tests involving constant external magnetic field and constant stress are conducted to identify nine model parameters. These model parameters include free strain, Young's moduli, fundamental critical stresses, fundamental critical threshold fields, and stress-influence coefficients. The Young's moduli of the material in its field and stress preferred states are determined to be 450 and 820 MPa respectively, while the free strain is measured to be 6.5%. These test data are used to assemble a critical stress—magnetic field intensity profile that is useful for determining the model parameters and for predicting the various states of the material for a wide range of magnetic or mechanical loading conditions. Although all of the parameters can be obtained from constant magnetic field testing, useful insight into NiMnGa actuator behavior can be gained from constant axial stress tests. Once implemented, the analytical model shows good correlation with the test data, capturing both the magnetic shape memory effect and pseudoelasticity. Because the model is piecewise linear, it does not capture material behavior resulting from nonlinear effects such as magnetic saturation. Despite its inherent limitations, this model shows encouraging results, providing a solid basis for future modeling efforts.

A quasi-static model for NiMnGa magnetic shape memory alloy

A quasi-static model for NiMnGa magnetic shape memory alloy (MSMA) is formulated inparallel to the Brinson and Tanaka thermal SMA constitutive models. Since the shapememory effect (SME) and pseudoelasticity exist in both NiTi and NiMnGa, constitutivemodels for SMAs can serve as a basis for MSMA behavioral modeling. The simplified,linear, quasi-static model for NiMnGa was characterized by nine material parametersidentified by conducting a series of uniaxial compression tests in a constant fieldenvironment. These model parameters include free strain, Young’s modulus, fundamentalcritical stresses, fundamental threshold fields, and stress-influence coefficients. The Young’smoduli of the material in both its field and stress preferred configurations were determinedto be 450 MPa and 820 MPa respectively, while the free strain was measured to be 5.8%.These test data were used to assemble a critical stress profile that is useful for determiningmodel parameters and for understanding the dependence of critical stresses onmagnetic fields. Once implemented, the analytical model shows good correlationwith test data for all modes of NiMnGa quasi-static behavior, capturing both themagnetic shape memory effect and pseudoelasticity. Furthermore, the model isalso capable of predicting partial pseudoelasticity, minor hysteretic loops andstress–strain behaviors. To correct for the effects of magnetic saturation, a series ofstress influence functions were developed from the critical stress profile. Althoughrequiring further refinement, the model’s results are encouraging, indicating thatthe model is a useful analytical tool for predicting NiMnGa actuator behavior.

Implementation strategy for the dual transformation region in the Brinson SMAconstitutive model

The transformation kinetics formulation is the principal factor underlying the constitutivemodel of shape memory alloys (SMAs). Therefore, the transformation kinetics formulation,which is applicable to any status of stress and temperature, is essential for predicting thematerial behavior of SMAs. In this work we show that the transformation kinetics of theoriginal Brinson model, which is the most widely used one-dimensional model,has shortcomings in the case where temperature decreases at low temperature(T<Ms). In addition, we propose a modified transformation kinetics formula that can be used fordual transformation conditions. The martensite transformation kinetics is modified so thatthe transformation from austenite into temperature-induced martensite, due tothe decrease in temperature, is coupled with a transformation from austeniteor temperature-induced martensite into stress-induced martensite, due to theincrease in stress. Through this modification, the suggested formulation can properlydescribe the behavior of martensite fractions in the dual transformation region.

Development of a Strain-Rate Dependent Model for Uniaxial Loading of SMA Wires

This paper describes a modeling approach to incorporate the effects of nonquasistatic loading on the extensional behavior of an SMA wire. Experimental results under a variety of loading conditions indicate that the instantaneous temperature of the material is closely related to the applied strainrate in the material. In order to predict this behavior, a coupled thermo-mechanical analysis with the rate form of SMA constitutive models is formulated. The temperatures and temperature rates are not prescribed, but are derived from energy conservation of the material. The stress rate is simultaneously derived using the rate form of SMA constitutive models. Important parameters governing heat transfer such as specific heat, heat transfer and latent heat are modeled and validated with experimental data. For the material tested (0.38 mm diameter SMA wire), the quasistatic strain rate below which no significant deviation in material characteristics are observed was empirically determined to be about 0.0005/s. At a strain rate of 0.01/s, the transformation stresses in the material are increased by approximately 0.5 × 10 8 Pa, accompanied by an increase in temperature of about 2°C over the quasistatic values. The predictions for the stress-strain curves are compared with experimental data at different strain rates over a range of environmental temperatures, and are found to be in good qualitative agreement. The model is also shown to be in good qualitative agreement with the experimental behavior under more complex loading conditions involving two different strain rates of loading. Parametric variation of the model coefficients is discussed. Although the qualitative aspects of the model are in good agreement with experimental data, it is argued that more comprehensive estimation of the model parameters is required to assess the quantitative aspects of the model. The model described in the paper uses the rate forms of the Brinson model. However, it is equally applicable with a modified rate-dependant form of any other quasistatic model describing SMA behavior.

Impact induced propagation of phase transformation in a shape memory alloy rod

Abstract In this work we study the propagation of a phase-transformation front induced by an impact loading. Our main goals in this paper are to demonstrate the wave structure of the solution to the impact loading and to estimate the velocity of phase front propagation using a thermodynamically based constitutive model for SMAs with internal variables. First, we study the impact problem in an isothermal setting; this study provides an important insight into the wave structure of the solution. The non-uniqueness issue inherent to such problems due to nonconvex SMA constitutive models is thoroughly discussed. Then we develop the solution for an adiabatic approximation for a simplified SMA model. Several numerical examples based on a first order Lax-Friedrichs finite difference scheme are presented.

Phenomenological Multiaxial Constitutive Model for Shape Memory Alloys and Its Application. 2nd Report, Micromechanical Analysis for Hysteretic Behavior of TiNi-SMA Embedded Unidirectional Composite Laminae.

Hysteretic behavior of unidirectional TiNi SMA/Epoxy composites subjected to a single isothemal loading and unloading cycle has been analyzed using a three-dimensional micromechanical unitcell model. The multiaxial constitutive equations developed in the previous study were applied to the description of the stress-induced rhombohedral and martensitic transformations of the SMA fibers embedded in the epoxy matrix. It was clearly observed that the global behavior of the SMA composites was governed by the pseudoelastic property of the SMA fibers. The hysteretic responses of SMA composites were significantly influenced by the SMA fiber volume fraction. For a high volume fraction of the SMA fibers, the rhombohedral transformation of the embedded SMA fibers characterized a pseudoelasticity-like behavior of SMA composites. This predication indicates a feature unique to the multiaxial SMA constitutive model developed in the previous report.