A thermodynamical constitutive model for shape memory materials. Part II. The SMA composite material

The phenomenological SMA equations developed in Part I are used in this second paper to derive the free energy and dissipation of a SMA composite material. The derivation consists of solving a boundary value problem formulated over a mesoscale representative volume element, followed by an averaging procedure to obtain the macroscopic composite constitutive equations. Explicit equations are derived for the transformation tensors that relate the composite transformation strain rate to the phase transformation rate in the fiber and matrix. Some key findings for the two-way SME in a SMA fiber/elastomer matrix composite are that processing-induced residual stresses alter the composite austenite start and martensite start temperatures, as well as the amount of composite strain recovered during a complete cycle of temperature and fiber martensite volume fraction. Relative to the two-way SME response of stiff-matrix composites, it was found that compliant-matrix composites: (1) complete the phase transformation over a narrower temperature range; (2) exhibit greater transformation strain during the reverse transformation; and (3) undergo an incomplete strain cycle during a complete cycle of temperature and fiber martensite volume fraction. Due to the interaction of the fiber and matrix during transformation, macroscopic proportional stressing of the composite results in non-proportional fiber stressing, which in turn causes a small amount of martensitic reorientation to occur simultaneously with the transformation.     

A three-dimensional constitutive model for shape memory alloys

Shape memory alloys (SMAs) are materials that, among other characteristics, have the ability to present high deformation levels when subjected to mechanical loading, returning to their original form after a temperature change. Literature presents numerous constitutive models that describe the phenomenological features of the thermomechanical behavior of SMAs. The present paper introduces a novel three-dimensional constitutive model that describes the martensitic phase transformations within the scope of standard generalized materials. The model is capable of describing the main features of the thermomechanical behavior of SMAs by considering four macroscopic phases associated with austenitic phase and three variants of martensite. A numerical procedure is proposed to deal with the nonlinearities of the model. Numerical simulations are carried out dealing with uniaxial and multiaxial single-point tests showing the capability of the introduced model to describe the general behavior of SMAs. Specifically, uniaxial tests show pseudoelasticity, shape memory effect, phase transformation due to temperature change and internal subloops due to incomplete phase transformations. Concerning multiaxial tests, the pure shear stress and hydrostatic tests are discussed showing qualitatively coherent results. Moreover, other tensile–shear tests are conducted modeling the general three-dimensional behavior of SMAs. It is shown that the multiaxial results are qualitative coherent with the related data presented in the literature.     

A macroscopic multi-mechanism based constitutive model for the thermo-mechanical cyclic degeneration of shape memory effect of NiTi shape memory alloy

A macroscopic based multi-mechanism constitutive model is constructed in the framework of irreversible thermodynamics to describe the degeneration of shape memory effect occurring in the thermo-mechanical cyclic deformation of NiTi shape memory alloys(SMAs). Three phases,austenite A, twinned martensite Mtand detwinned martensite M~d, as well as the phase transitions occurring between each pair of phases( A → M~t, M~t→ A, A → M~d,M~d→ A, and M~t→ M~d) are considered in the proposed model. Meanwhile, two kinds of inelastic deformation mechanisms, martensite transformation-induced plasticity and reorientation-induced plasticity, are used to explain the degeneration of shape memory effects of NiTi SMAs. The evolution equations of internal variables are proposed by attributing the degeneration of shape memory effect to the interaction between the three phases(A, M~t, and M~d) and plastic deformation. Finally, the capability of the proposed model is verified by comparing the predictions with the experimental results of NiTi SMAs. It is shown that the degeneration of shape memory effect and its dependence on the loading level can be reasonably described by the proposed model.     

A constitutive model for shape memory alloys accounting for thermomechanical coupling

This paper presents a generalized Zaki-Moumni (ZM) model for shape memory alloys (SMAs) [cf. Zaki, W., Moumni, Z., 2007a. A three-dimensional model of the thermomechanical behavior of shape memory alloys. J. Mech. Phys. Solids 55, 2455-2490 accounting for thermomechanical coupling. To this end, the expression of the Helmholtz free energy is modified in order to derive the heat equation in accordance with the principles of thermodynamics. An algorithm is proposed to implement the coupled ZM model into a finite element code, which is then used to solve a thermomechanical boundary value problem involving a superelastic SMA structure. The model is validated against experimental data available in the literature. Strain rate dependence of the mechanical pseudoelastic response is taken into account with good qualitative as well as quantitative accuracy in the case of moderate strain rates and for mechanical results in the case of high strain rates. However, only qualitative agreement is achieved for thermal results at high strain rates. It is shown that this discrepancy is mainly due to localization effects which are note taken into account in our model. Analyzing the influence of the heat sources on the material response shows that the mechanical hysteresis is mainly due to intrinsic dissipation, whereas the thermal response is governed by latent heat. In addition, the variation of the area of the hysteresis loop with respect to the strain rate is discussed. It is found that this variation is not monotonic and reaches a maximum value for a certain value of strain rate.     

Superelastic shape memory alloy cables: Part I – Isothermal tension experiments

Cables (or wire ropes) made from NiTi shape memory alloy (SMA) wires are relatively new and unexplored structural elements that combine many of the advantages of conventional cables with the adaptive properties of SMAs (shape memory and superelasticity) and have a broad range of potential applications. In this two part series, an extensive set of uniaxial tension experiments was performed on two Nitinol cable constructions, a 7 × 7 right regular lay and a 1 × 27 alternating lay, to characterize their superelastic behavior in room temperature air. Details of the evolution of strain and temperature fields were captured by simultaneous stereo digital image correlation and infrared imaging, respectively. Here in Part I, the nearly isothermal, superelastic responses of the two cable designs are presented and compared. Overall, the 7 × 7 construction has a mechanical response similar to that of straight wires with propagating transformation fronts and distinct stress plateaus during stress-induced transformations. The 1 × 27 construction, however, exhibits a more compliant and stable mechanical response, trading a decreased force for additional elongation, and does not exhibit transformation fronts due to the deeper helix angles of the layers. In Part II that follows, selected subcomponents are dissected from the two cable’s hierarchical constructions to experimentally break down the cable’s responses.     

Three-dimensional constitutive model for shape memory alloys based on microplane model

A new model for the behavior of polycrystalline shape memory alloys (SMA), based on a statically constrained microplane theory, is proposed. The new model can predict three-dimensional response by superposing the effects of inelastic deformations computed on several planes of different orientation, thus reproducing closely the actual physical behavior of the material. Due to the structure of the microplane algorithm, only a one-dimensional constitutive law is necessary on each plane. In this paper, a simple constitutive law and a robust kinetic expression are used as the local constitutive law on the microplane level. The results for SMA response on the macroscale are promising: simple one-dimensional response is easily reproduced, as are more complex features such as stress-strain subloops and tension-compression asymmetry. A key feature of the new model is its ability to accurately represent the deviation from normality exhibited by SMAs under nonproportional loading paths.     

A constitutive theory for shape memory polymers. Part II: A linearized model for small deformations

A constitutive theory is developed for shape memory polymers. It is to describe the thermomechanical properties of such materials under large deformations. The theory is based on the idea, which is developed in the work of Liu et al. [2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modelling. Int. J. Plasticity 22, 279-313], that the coexisting active and frozen phases of the polymer and the transitions between them provide the underlying mechanisms for strain storage and recovery during a shape memory cycle. General constitutive functions for nonlinear thermoelastic materials are used for the active and frozen phases. Also used is an internal state variable which describes the volume fraction of the frozen phase. The material behavior of history dependence in the frozen phase is captured by using the concept of frozen reference configuration. The relation between the overall deformation and the stress is derived by integration of the constitutive equations of the coexisting phases. As a special case of the nonlinear constitutive model, a neo-Hookean type constitutive function for each phase is considered. The material behaviors in a shape memory cycle under uniaxial loading are examined. A linear constitutive model is derived from the nonlinear theory by considering small deformations. The predictions of this model are compared with experimental measurements.     

A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings

This paper presents a new phenomenological constitutive model for shape memory alloys, developed within the framework of irreversible thermodynamics and based on a scalar and a tensorial internal variable. In particular, the model uses a measure of the amount of stress-induced martensite as scalar internal variable and the preferred direction of variants as independent tensorial internal variable. Using this approach, it is possible to account for variant reorientation and for the effects of multiaxial non-proportional loadings in a more accurate form than previously done. In particular, we propose a model that has the property of completely decoupling the pure reorientation mechanism from the pure transformation mechanism. Numerical tests show the ability to reproduce main features of shape memory alloys in proportional loadings and also to improve prediction capabilities under non-proportional loadings, as proven by the comparison with several experimental results available in the literature.     

A Preisach type model for temperature driven hysteresis memory erasure in shape memory materials

We establish the well-posedness and thermodynamic consistency of a variational inequality modeling temperature-induced memory erasure in shape memory materials. It is shown that the input–output operator is continuous with respect to uniform convergence.     

A constitutive relation for pseudo-elastic behaviour in shape memory alloys

A constitutive relation to describe pseudo-elastic deformation in shape memory alloys is presented in this paper. It is capable of describing deformation behaviour of polycrystalline materials under triaxial stress state as well as of monocrystalline materials under one-dimensional condition. Total strain rate is supposed to be composed of elastic strain rate and transformation strain rate. Deformation behaviour of Cu−Zn−Sn alloy and Ti−ni alloy is simulated by use of the proposed constitutive relation. it is shown that simulated results are in a good agreement with experimental data. The project supported by National Natural Science Foundation of China.     

On the Fremond’s constitutive model for shape memory alloys

The remarkable properties of shape memory alloys have increasing the interest in applications in different areas varying from biomedical to aerospace hardware. Despite the large number of applications, the modeling of SMA is the objective of many researches developed in order to describe all details of its thermomechanical behavior. The present contribution revisits a constitutive model presented by Savi et al. (2002), which is built up on the classical Fremond’s model, in order to contemplate the horizontal enlargement of the stress–strain hysteresis loop. Numerical simulations present qualitative agreement with experimental data, showing pseudoelastic, one-way and two-way shape memory effects.     

A 3D finite strain phenomenological constitutive model for shape memory alloys considering martensite reorientation

Most devices based on shape memory alloys experience both finite deformations and non-proportional loading conditions in engineering applications. This motivates the development of constitutive models considering finite strain as well as martensite variant reorientation. To this end, in the present article, based on the principles of continuum thermodynamics with internal variables, a three-dimensional finite strain phenomenological constitutive model is proposed taking its basis from the recent model in the small strain regime proposed by Panico and Brinson (J Mech Phys Solids 55:2491–2511, 2007). In the finite strain constitutive model derivation, a multiplicative decomposition of the deformation gradient into elastic and inelastic parts, together with an additive decomposition of the inelastic strain rate tensor into transformation and reorientation parts is adopted. Moreover, it is shown that, when linearized, the proposed model reduces exactly to the original small strain model.     

A shape memory alloy model by a second order phase transition

Motivated by the experimental results of the paper [1] and unlike the general theories of shape memory alloys (SMAs), in this paper we suggest for such materials a phase field model by a second order phase transition. So that, with this new system we obtain a simulation of phase dynamics very convenient to describe the natural behavior of these materials. The differential system is governed by the motion equation, the heat equation and the Ginzburg–Landau (GL) equation and by a constitutive law between the phase field, the temperature, the strain and the stress. The use of this new model is characterized by new potentials of the GL equation and by a new dependence on the temperature in the constitutive equation. Using this new model, we obtain simulations in better agreement with experimental data and respect to previous work [2].     

A nonlinear constitutive model for magnetostrictive materials

A general nonlinear constitutive model is proposed for magnetostrictive materials, based on the important physical fact that a nonlinear part of the elastic strain produced by a pre-stress is related to the magnetic domain rotation or movement and is responsible for the change of the maximum magnetostrictive strain with the pre-stress. To avoid the complicity of determining the tensor function describing the nonlinear elastic strain part, this paper proposes a simplified model by means of linearizing the nonlinear function. For the convenience of engineering applications, the expressions of the 3-D (bulk), 2-D (film) and 1-D (rod) models are, respectively, given for an isotropic material and their applicable ranges are also discussed. By comparison with the experimental data of a Terfenol-D rod, it is found that the proposed model can accurately predict the magnetostrictive strain curves in low, moderate and high magnetic field regions for various compressive pre-stress levels. The numerical simulation further illustrates that, for either magnetostrictive rods or thin films, the proposed model can effectively describe the effects of the pre-stress or residual stress on the magnetization and magnetostrictive strain curves, while none of the known models can capture all of them. Therefore, the proposed model enjoys higher precision and wider applicability than the previous models, especially in the region of the high field.The project supported by the National Natural Science Foundation of China (10132010 and 90405005)     

A constitutive theory for shape memory polymers. Part I: Large deformations

A constitutive theory is developed for shape memory polymers. It is to describe the thermomechanical properties of such materials under large deformations. The theory is based on the idea, which is developed in the work of Liu et al. [2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling. Int. J. Plasticity 22, 279-313], that the coexisting active and frozen phases of the polymer and the transitions between them provide the underlying mechanisms for strain storage and recovery during a shape memory cycle. General constitutive functions for nonlinear thermoelastic materials are used for the active and frozen phases. Also used is an internal state variable which describes the volume fraction of the frozen phase. The material behavior of history dependence in the frozen phase is captured by using the concept of frozen reference configuration. The relation between the overall deformation and the stress is derived by integration of the constitutive equations of the coexisting phases. As a special case of the nonlinear constitutive model, a neo-Hookean type constitutive function for each phase is considered. The material behaviors in a shape memory cycle under uniaxial loading are examined. A linear constitutive model is derived from the nonlinear theory by considering small deformations. The predictions of this model are compared with experimental measurements.     

Parameter determination for thermodynamical models of the pseudoelastic behaviour of shape memory alloy by resistivity measurements and infrared thermography

Two thermodynamical models of pseudoelastic behaviour of shape memory alloys have been formulated. The first corresponds to the ideal reversible case. The second takes into account the hysteresis loop characteristic of this shape memory alloys.Two totally independent techniques are used during a loading-unloading tensile test to determine the whole set of model parameters, namely resistivity and infrared thermography measurements. In the ideal case, there is no difficulty in identifying parameters.Infrared thermography measurements are well adapted for observing the phase transformation thermal effects.Notations 1 austenite 2 martensite - ₍₎ Macroscopic infinitesimal strain tensor of phase - ₍₂₎ ^f Traceless strain tensor associated with the formation of martensite phase - Macroscopic infiniesimal strain tensor - Macroscopic infinitesimal strain tensor deviator - _f Trace - Equivalent strain - ^pe Macroscopic pseudoelastic strain tensor - x Distortion due to parent (austenite =1)product (martensite =2) phase transformation (traceless symmetric second order tensor) - M Total mass of a system - M() Total mass of phase - V Total volume of a system - V() Total volume of phase - z=M(2)/M Weight fraction of martensite - 1-z=M(1)/M Weight fraction of austenite - u ₀ ^* () Specific internal energy of phase (=1,2) - s ₀ ^* () Specific internal entropy of phase - Specific configurational energy - Specific configurational entropy - ₀ ^f (T) Driving force for temperature-induced martensitic transformation at stress free state ( ₀ ^f T) = T ^*–Ts ^*) - Kirchhoff stress tensor - Kirchhoff stress tensor deviator - Equivalent stress - Cauchy stress tensor - Mass density - K Bulk moduli (K ₀=K) - L Elastic moduli tensor (order 4) - E Young modulus - Energetic shear (₀ = ) - Poisson coefficient - M _s ^o (M _F ^o ) Martensite start (finish) temperature at stress free state - A _s ^o (A _F ^o ) Austenite start (finish) temperature at stress free state - C _v Specific heat at constant volume - k Conductivity - Pseudoelastic strain obtained in tensile test after complete phase transformation (AM) (unidimensional test) - ₀ Thermal expansion tensor - r Resistivity - 1MPa 10⁶ N/m ² - ⁽⁾ Specific free energy of phase - _n Specific free energy at non equilibrium (R model) - _n ^eq Specific free energy at equilibrium (R model) - _n ^v Volumic part of ^eq - Specific free energy at non equilibrium (R _L model) - _conf Specific coherency energy (R _L model) - _c Specific free energy at constrained equilibria (R _L model) - _it (T) Coherency term (R _L model)