Simulation of the effect of elastic precipitates in SMA materials based on a micromechanical model
Introduction
Shape memory alloys (SMA) became well known for their adaptive and functional behavior owing to the reversible austenite to martensite phase transformation. They are finding increasing wider applications due to their specific thermomechanical behavior characteristics such as superelasticity, shape memory effect and rubbery effect. A wide range of SMA that exhibit such particular properties are studied in literature. They can be classified in three families: the NiTi-based, the Cu-based and the Fe-based SMA. Each of them possesses some advantages and disadavantages. NiTi-based SMAs exhibit a good Shape Memory Effect (SME), biocompatibility and good corrosion resistance even though they have a relative high cost. Cu-based SMA also have a good SME but they are brittle due to their intragranular anisotropy leading to high stress levels at grain boundaries. Despite their low cost, the Fe-based SMA have an allowable SME and a weak corrosion resistance.
In order to improve performances of SMA belonging to the three above mentioned families, two main approaches have been investigated in literature during elaboration. The first one corresponds to the application of various kinds of heat treatments in order to control the values of forward and reverse transformation temperatures as well as the maximum transformation strain. For instance, Cuniberti et al. [8], [9], Khalil-Allafi et al. [15] and Zheng et al. [33] demonstrated the influence of aging on transformation properties for CuZnAl and NiTi SMA. The second approach considers the introduction of additional elements in the SMA composition that could modify the SMA transformation properties. As examples, Ti3Ni4 precipitates for NiTi Buchheit and Wert [2], Shakeri et al. [26], Zhou et al. [34], TiCu precipitates for TiNiCu Tong and Liu [29] and Ti2Ni and Ti2Pd precipitates for TiNiPb Zarnetta et al. [32] increase the austenite yield strength, enhance fatigue properties, shape memory effect and superelasticity. However the precipitates reduce the maximum transformation strains and may restrict the detwinning process. The Cu9Al4 precipitates for CuAlBe SMA Cuniberti et al. [8] induce a higher value of transformation yield stress and, for a volume fraction exceeding 14%, increase the transformation temperatures. Therefore, these elements could also have a negative or positive influence on the conventional properties of these SMA such as ductility, corrosion resistance and strength: Ying et al. [31] and Efstathiou and Sehitoglu [10]. These additional elements may be dissolved in the SMA and/or form precipitates Liu et al. [19]; Khalil-Allafi and Amin-Ahmadi [14], Ying et al. [31]. These precipitates have an elastic or elastic plastic behavior. They induce deformation incompatibilities inside grains creating internal stresses that modify the SMA thermomechanical properties. As an example, ductile niobium precipitates induce a larger hysteresis size in NiTi due to their plastic behavior, Ying et al. [31]. Hard zirconium precipitates with elastic behavior shift the transformation temperature values in CuZnAl SMA, Chung et al. [5].
To better describe and optimize the effects of hard or ductile precipitates on the thermomechanical behavior, constitutive models have been proposed in literature. They are based either on phenomenological approaches coupled to scale transition techniques Piotrowski et al. [23], [24]; Chemisky et al. [3], or on finite element Lester et al. [17] or micromechanical approaches Collard et al. [6]. Atomistic modeling based on Molecules Dynamics Simulation Sato et al. [25] or phase field approach Zhou et al. [34] also allow to predict such effects. The phenomenological approach considers a biphasic material composed of elastic or elastic plastic inclusions embedded in an SMA matrix. A phenomenological constitutive model is assigned to each phase. The effective behavior is then deduced by the Mori–Tanaka scale transition technique. The finite element approach adopts a geometrical model composed of a matrix containing generally spherical inclusions. It is meshed with solid continuum finite elements and the effective behavior is deduced by a structural analysis for various boundary conditions and loadings. The micromechanical modeling considers two scale transition steps leading to the effective behavior. At the first step, the homogenized thermomechanical behavior at the grain scale is derived by using the Mori–Tanaka scale transition technique. The corresponding Representative Volume Element (RVE) is assumed to be composed of an SMA matrix with embedded elastic or elastic plastic spherical inclusions. Micromechanical crystalline plasticity-like constitutive models are generally adopted for each phase. A second scale transtion technique such as the self-consistent scheme is then used to derive the effective behavior from the homogenized behavior of each grain. This last scale transition is applied to a second RVE made of polycrystalline agregates.
This paper deals with the development of a micromechanical model that describes the effect of elastic precipitates on the SMA macroscopic behavior following the work presented in Collard et al. [6]. In the new version of the model, the numerical integration scheme is substantially modified leading to a behavior simulation indepedent of the increment size. Under the assumption of a uniform temperature in the material, it allows to simulate the evolution of maximum transformation strain and tranformation temperatures following precipates volume fraction. In addition, it predicts the effect of the precipitates hardness versus the SMA matrix on the macroscopic behavior. Finally, additional loading cases, such as shear and rolling, are considered and more detailed results at grain scale and new ones at the macroscopic scale are given.
The present article contains three main sections in addition to introduction and conclusion. Section 2 details the constitutive equations of the proposed model. Section 3 deals with the numerical aspects related to the resolution of the obtained non-linear system of constitutive equations. Section 4 gives and comments the results of the numerical simulations at the grain and at the polycrystal levels.
Section snippets
Homogenization strategy
Let’s consider a RVE described in Fig. 1 and composed of an SMA polycrystal. This polycrystal is an agregate of composite grains composed of an SMA matrix with embedded elastic precipitates. Its effective thermomechanical behavior is built on two successive scale transitions. They allow to derive the macroscopic constitutive equations from the granular ones. At the grain scale, the composite is assumed to be a continuum medium composed of two homogeneous phases, corresponding to elastic
Numerical resolution technique
The previous equations are now computed in Simula+ [27]. For the SMA single crystal problem, if the control variables are stress and temperature, this problem will not usually be solved since the matrix Hnm cannot be inverted. For a strain increment of and a temperature increment of ΔT, an Euler explicit integration scheme gives (see Appendix A.1):The stress rate increment is then given by:where the thermo-mechanical modulii
Results at the grain scale
In order to verify the consistency of the developed model, the following simulations are performed at the single crystal scale on a CuZnAl SMA with Zr spherical inclusions. The temperature is assumed to be uniform in the material. Tension, shear and rolling tests are simulated at room temperature (30 °C). The corresponding applied stresses, whose components are different from zero are σ11 for tensile test, σ12 = σ21 for shear test, σ11 and σ33 = −σ11 for rolling test. Inclusions are assumed to be
Conclusion and prospects
This paper presented a thermo-mechanical constitutive model for polycrystalline SMA containing elastic precipitates. It is based on a micromechanical approach and considers two scale transition techniques in order to lead to the macroscopic effective behavior starting from the ones of an SMA matrix and elastic precipitates within each grain. Firstly, the Mori–Tanaka scale transition technique gave the equivalent behavior of each grain by considering a composite material constituted of elastic
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