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A multiscale analytical model for superelastic deformation of gradient nano-grained NiTi shape memory alloys
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Bo Xu1, Xingyu Zhou2, Chao Yu2, *
Acta Mechanica Sinica | 2025, 41(12) : 124342
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Acta Mechanica Sinica | 2025, 41(12): 124342
• RESEARCH PAPER •
A multiscale analytical model for superelastic deformation of gradient nano-grained NiTi shape memory alloys
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Bo Xu1, Xingyu Zhou2, Chao Yu2, *
Affiliations
  • 1Failure Mechanics and Engineering Disaster Prevention, Key Laboratory of Sichuan Province, Sichuan University, Chengdu 610065, China
  • 2Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Aerospace Engineering, Southwest Jiaotong University, Chengdu 610031, China
Published: 2025-12-01 doi: 10.1007/s10409-024-24342-x
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A multiscale nonlocal continuum model is proposed to describe the superelastic deformation of gradient nano-grained NiTi shape memory alloys (SMAs). At the mesoscopic scale, the polycrystalline aggregate is regarded as a composite, i.e., the grain-interior (GI) phase is assumed to be a cuboidal inclusion embedded in a matrix of grain-boundary (GB) phase. An intrinsic energetic length and a gradient energy are introduced into the Helmholtz free energy of the GI phase. The criterion of martensite transformation (MT) is derived based on the principle of virtual power and second law of thermodynamics. The hindering effect of GB on MT in GI phase is addressed. By deriving the analytical solution of the proposed model and introducing a scale transition rule, the overall and local stress-strain responses of the specimen at the macroscopic scale are obtained. The prediction capability of the proposed model is verified by comparing the analytical solution with the experiment. The influences of the distribution form for the grain size (GS) on the superelastic deformation of gradient nano-grained NiTi SMAs are further predicted and discussed. The analytical form and low computational cost of the proposed model make it an appropriate theoretical tool to design the gradient nano-grained SMAs with desired mechanical property.

Gradient nano-grained NiTi SMA  /  Multiscale continuum model  /  Analytical solution  /  Martensite transformation  /  Superelasticity
Bo Xu, Xingyu Zhou, Chao Yu. A multiscale analytical model for superelastic deformation of gradient nano-grained NiTi shape memory alloys[J]. Acta Mechanica Sinica, 2025 , 41 (12) : 124342 - . DOI: 10.1007/s10409-024-24342-x
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1.　Introduction
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Shape memory alloys (SMAs) exhibit unique superelasticity and shape memory effect, and have been successfully utilized as core-component materials in various engineering fields such as aerospace, biomedical, mechanical engineering, and microelectromechanical systems [1]. The diverse engineering applications of SMA structures and devices pose distinct requirements on their service performances. For instance, in the field of solid-state refrigeration, it is expected that SMAs can exhibit significant isothermal entropy change or adiabatic temperature change during martensite transformation (MT) [2-4]. As a seismic damper, SMAs with large stress-strain hysteresis loop are needed to ensure the sufficient energy absorption capacity [5,6]. As an actuator, SMAs should exhibit obvious transformation hardening and small strain-temperature hysteresis loop to achieve the accurate driving control [7-10]. Moreover, in long time service, the cyclic stability of SMAs is an important evaluation indicator [11-13]. Since these service performances at the macroscopic scale are determined by the microstructure characteristics, the mechanical properties of SMAs can be tailored by changing the grain size (GS).
Grain refinement has been commonly employed in conventional metallic materials as a means to enhance the strength of material. As the GS decreases, conventional metallic materials can exhibit higher yield strength, which is often inversely proportional to the square root of the GS, i.e., the well-known Hall-Petch relation [14,15]. GS also significantly influences the stress-induced MT in SMAs. Through a severe plastic deformation process such as cold rolling or cold drawing, as well as subsequent annealing, the GS of NiTi SMA can be reduced to the nanometer scale (< 100 nm). Due to the strong effect of GS on the MT when the GS < ~80 nm, as the GS decreases, the superelastic stress-strain curve of NiTi SMAs changes from the “plateau-type” to a “hardening-type”, with an increase in transformation hardening modulus and a significant reduction in dissipation energy [16-19]. The GS-dependent super-elasticity of NiTi SMAs has been widely investigated in the last decade [10-12,20-29].
Gradient-grained metals and alloys have manifested themselves with excellent mechanical properties as compared to their homogeneous counterparts, which have attracted much attention [30]. Interestingly, due to the GS dependence of MT in SMAs, the gradient nano-grained SMAs can show unprecedented mechanical and functional properties [31-35]. Huang et al. [31] fabricated the gradient nano-grained NiTi SMA sheet with the GS ranging from 25 to 175 nm along the axial direction in an approximately linear distribution. The quasi-static loading experiments indicated that, compared to the specimens with uniform GS, the superelasticity of gradient nano-grained NiTi SMA exhibited a wider range of response stress. Chen et al. [32] prepared a series of uniformly fine-grained, uniformly nano-grained and gradient nano-grained NiTi SMA sheets, and further investigated the rate-dependent superelastic deformation of these materials. Experimental results showed that the transformation hardening modulus of coarse-grained specimen increased with the strain rate due to the strong thermomechanical coupling effect. However, such rate dependence is significantly weakened in nano-grained specimen owing to the weaker temperature variation together with the weaker transformation temperature dependence. Meanwhile, the rate dependence of superelasticity in the gradient nano-grained specimens fell between those of fine-grained and nanocrystalline ones. Recently, Chen et al. [13,33,34,36] prepared the NiTi SMA plates with the GS gradients in the thickness and width directions, which showcased superior elastocaloric effect and cyclic stability, compared to the coarse-grained NiTi SMAs. So far, it has been revealed that the mechanical properties of NiTi SMAs can be significantly improved by introducing the gradient nano-grained microstructures.
To assess the service performance of SMA devices, it is essential to establish appropriate theoretical models. In the last decades, significant efforts have been devoted to developing the continuum models of SMAs. Comprehensive reviews of this topic can be found in Refs. [37-39], and recent representative works can be referred to Refs. [40-58], and so on. However, few works incorporated the GS effect. Sun and He [59] constructed a multiscale continuum model and successfully described the dependence of energy dissipation on the GS. Li and Sun [60] developed a one-dimensional (1D) core-shell model for nanocrystalline SMAs. The predicted results showed that with the reduction of GS, the energy of grain-boundary (GB) phase gradually became dominant during MT, and brought a fundamental change in the growth mode of martensite. Qiao and Radovitzky [61] and Qiao et al. [62] constructed 1D and three-dimensional (3D) non-local models by introducing two intrinsic length scales. The increased transformation hardening modulus with the decrease of GS and specimen size was well captured. Meanwhile, a 3D constitutive model based on micromechanics was also developed, and reasonably captured the GS-dependent stress-strain responses of SMAs [63]. However, the aforementioned models are only applicable for SMAs with uniformly distributed GS. Although few phase field models have been developed in recent years for the gradient nano-grained SMAs and revealed the synergic deformation mechanism of the coarse-grain and nano-grain [64,65], it was difficult to reflect the real gradient microstructure in the materials due to the huge computational cost for such type of model. As a result, the experimental stress-strain responses of the gradient nano-grained NiTi SMAs cannot be accurately reproduced.
In this work, we will develop a multiscale nonlocal continuum model aimed at describing the superelastic deformation of gradient nano-grained NiTi SMAs and providing a theoretical tool to design this type of material with desired mechanical property. At the mesoscopic scale, the polycrystalline aggregate is regarded as a composite, i.e., the grain-interior (GI) phase is assumed to be a cuboidal inclusion embedded in a matrix of GB phase. An intrinsic energetic length and a gradient energy are introduced into the Helmholtz free energy of the GI phase. The criterion of MT is derived based on the principle of virtual power and second law of thermodynamics, and the hindering effect of GB on MT in GI phase is addressed. By deriving the analytical solution of the proposed model and introducing a scale transition rule, the overall and local stress-strain responses of the specimen at the macroscopic scale are obtained. The prediction capability of the proposed model is verified by comparing the analytical solution with the experiments.
2.　Multiscale nonlocal continuum model
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The gradient nano-grained NiTi SMA specimen can be prepared by gradient heat treatment after severe plastic deformation [32]. As shown in Fig. 1(a), the average GS varies from a few nanometers at one end of the specimen to one hundred or even a few ten hundred nanometers at the other end along the axial direction. Thus, at the macroscopic scale, we can divide the specimen into a series of sub-domains with an identical length, i.e., lΩ=l/N, where l is the gauge length of the specimen, N is the total number of the subdomains, lΩ denotes the length of the Ω-th sub-domain. The GS in a given sub-domain can be regarded as uniform if N is large enough. In a smaller spatial scale (mesoscopic scale), the microstructure of a sub-domain emerges. It means that a sub-domain in the macroscopic scale becomes a polycrystalline aggregate containing a large number of grains at the mesoscopic scale. From a modelling point of view, the polycrystalline aggregate can be described as a core-shell type “GI-GB composite”, i.e., the GI phase is assumed to be a cuboidal inclusion embedded in a matrix of GB phase, as shown in Fig. 1(b) (this approach was also adopted in Refs. [60,62]). However, in order to obtain the analytical solution of the proposed model, further simplification is necessary. Firstly, the polycrystalline aggregate can be regarded as the repeated arrangement of the grains in the space. Thus, the deformation response of the polycrystalline aggregate can be replaced by a representative grain. Secondly, generally speaking, the strain field along the axial direction in such a grain is 3D. However, when the grain is subjected to a uniaxial mechanical loading as shown in Fig. 1(b), inspired by the work of Li and Sun [60], we can assume that the axial strain is uniformly distributed in the square cross section perpendicular to the axial direction, i.e., ε = ε(x) (to avoid confusion, the spatial coordinates at the macroscopic and mesoscopic scales are denoted as X and x, respectively). In this case, the 3D grain is reduced into an effective 1D grain (see Fig. 1(b)). Such 3D→1D effective modelling is expected to retain the key features of the 3D polycrystalline aggregate such as the volume fraction of GB and its deformation constraint to the GI phase.
Consider an effective 1D grain in the Ω-th sub-domain of the specimen. The local stress applied on this sub-domain is denoted as σΩ. The volume fraction of the GB phase can be calculated as fGB,Ω = 1-(1-tb / lg,Ω)3, where lg,Ω represents the grain size of the Ω-th sub-domain, tb is the thickness of GB, which is a constant. According to the geometric and deformation features of the 1D grain, we further divide the GB phase into two categories, the type I GB and type II GB, as shown in Fig. 1(b). For a point x at the mesoscopic scale, when -lg,Ω/2≤x<-(lg,Ω-tb)/2 or (lg,Ω-tb)/ 2 < x≤lg,Ω/2, it belongs to the type I GB phase; when -(lg,Ω-tb)/2≤x≤lg,Ω-tb)/2, it belongs to the GI-type II GB mixture phase. The volume fractions of these two GB phases can be, respectively, calculated as fGBI,Ω = tb / lg,Ω and fGBII,Ω = (1-tb / lg,Ω)(2-tb / lg,Ω)tb / lg,Ω. The Helmholtz free energies of type I GB (ψGBI,Ω), type II GB (ΨGBII,Ω) and GI (ψGI,Ω) phases can be formulated as
where EGB and EGI represent the elastic moduli of GB and GI phases, respectively. εΩ(x) represents the total strain at point x. and ξΩ(x) are the transformation strain and the volume fraction of martensite phase at point x in the GI phase (noted that 0≤ξΩ(x)≤1), and there is a linear relationship between and ξΩ(x), i.e., = gtrξΩ(x), gtr is maximum transformation strain. is the entropy difference between austenite and martensite phases. θ0 is the balance temperature. le is the intrinsic energetic length. S0 is a parameter with the dimension of stress. It should be pointed out that in the GI phase, both the elastic and transformation strains are taken into account, i.e., , and the term in Eq. (1c) represents the elastic strain. However, owing to the non-transformable feature of GB, such a phase is modelled as a linear elastic body.
By Eqs. (1b) and (1c), the Helmholtz free energy of the GI-type II GB mixture phase can be formulated as the mixture rule
where
EGI-GB,Ω, and are the effective elastic modulus, transformation strain and entropy difference between austenite and martensite of the GI-type II GB mixture phase, respectively. Noted that AΩ and BΩ are always positive. The hindering effect of GB on the MT in GI phase is partly reflected by Eqs. (3b) and (3d). On one hand, it can be easily proven that is a monotonic decreasing function of fGBII,Ω, which means the maximum transformation strain in the GI-type II GB mixture phase decreases with increasing the volume fraction of type II GB phase; on the other hand, the term in Eq. (2) represents an additional transformation hardening energy, which is also governed by fGBII,Ω.
Considering any segment of the GI-type II GB mixture phase , the internal power in this segment can be defined as [62,66]
where kΩ and wΩ are the work conjugates to the martensite volume fraction ξΩ and its gradient ξΩ,x, respectively. The external power expended on this segment can be formulated as
where is the applied boundary traction, is work conjugate to the martensite volume fraction ξΩ at the boundary, uΩ is the displacement. We further introduce the principle of virtual power, i.e.,
where and are the virtual displacement, strain, transformation strain and martensite volume fraction, respectively, and they satisfy the constraint of deformation compatibility, i.e., (where represents the effective maximum transformation strain, according to Eq. (3b), we have ). Substituting Eqs. (4) and (5) into Eq. (6), it yields
Since Eq. (7) should be always satisfied for any integral interval [x1, x2], it requires
Eqs. (8a) and (8b) are the force and micro-force balance equations, Eqs. (8c) and (8d) represent the boundary conditions. Considering the second law of thermodynamics
where is the rate of energy dissipation. Substituting Eqs. (2) and (5) into Eq. (9), we can obtain
where is the elastic strain of GI-type II GB mixture phase. Since there is no energy dissipation in the process of elastic deformation, we have
Owing to the arbitrariness of , to satisfy Eq. (11), it requires
Equation (12) gives the relationship between the stress and elastic strain of GI-type II GB mixture phase. As a result, Eq. (10) is reduced as
To satisfy Eq. (13), it is assumed that
where Y is a positive constant, which represents the energy dissipation of MT and controls the width of the hysteresis loop in stress-strain response, sign(·) is the sign function. By Eqs. (14a) and (14b), it is seen that when = 0, = 0; when ≠ 0, > 0. Thus, the second law of thermodynamics can be always satisfied. Combining Eqs. (14a), (14b) and (8b), the criterion of MT can be obtained
For simplification, let
In the process of MT , considering the boundary conditions: = 0 and = 0 (such boundary conditions are adopted to reflect that two ends of the GI-type II GB mixture phase are obstacles to MT), the analytical solution of Eq. (15) can be formulated as
Equation (17) gives the distribution of ξΩ(x), and the maximum value of ξΩ(x) occurs at the point x = 0. Noted that ξΩ(x) is always non-negative. Thus, when FΩ < 0, ξΩ(x) should equal zero. Moreover, ξΩ(x) must be less than or equal to one, i.e., ξΩ(x)≤1. Thus, Eq. (17) is only applicable when the maximum value of ξΩ(x = 0)≤1, that is
By Eq. (18), the critical value of FΩ is determined
When Fc≤Fc,Ω, the distribution of ξΩ(x) can be fully described by Eq. (17). However, when Fc > Fc,Ω, Eq. (17) is not applicable. In this case, there are two real roots for the equation ξΩ(x) = 1, which can be expressed as
As a result, in the regions and , the distribution of ξΩ(x) can also be described by Eq. (17); in the region , ξΩ(x) = 1. In summary, in the process of MT, ξΩ(x) is formulated as
when 0≤FΩ≤Fc,Ω,
when FΩ > Fc,Ω.
The overall strains of the GI- type II GB mixture phase and the type I GB phase are defined as
Moreover, the overall strain of the whole grain at the mesoscopic scale is defined as
As mentioned above, only the elastic deformation of GB phase is considered in this work. Without energy dissipation, the stress-strain relationship of the type I GB phase can be directly obtained by the Helmholtz free energy (Eq. (1a))
It is noted that since σΩ is uniformly distributed, the strain field in the type I GB phase does not depend on the location x. Substituting Eqs. (12), (21), (22) and (24) into Eq. (23), it yields
Equation (25) gives the analytical expression for the overall strain of the whole grain εG,Ω at the mesoscopic scale. It should be pointed out that εG,Ω also represents the local strain of the Ω-th sub-domain at the macroscopic scale. The overall stress and strain of the specimen at the macroscopic scale can be obtained by the following volume average relationship
According to the geometric feature of the specimen, it is assumed that the local stresses in sub-domains are identical, i.e., σ1 = σ2 =, ..., σN. In this case, according to Eq. (26a), we have
Equation (27) is the scale transition rule from mesoscopic to macroscopic scale. Combining Eqs. (25), (26b) and (27), the analytical expression for overall strain of the specimen can be obtained
So far, the multiscale analytical model has been constructed. In the subsequent calculations, for a given applied stress Σ at the macroscopic scale, we can respectively use Eqs. (25) and (28) to get the local strain in the Ω-th subdomain and overall strain of the specimen.
3.　Model validation and discussion
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The proposed model is verified by comparing the predicted results with the corresponding experimental data of Chen et al. [32]. In Chen et al. [32], the superelastic deformation for three types of NiTi SMA specimens are investigated, including the coarse-grained (GS = 235 nm), nano-grained (GS = 10 nm) and gradient nano-grained specimens. The gauge length of the specimens is 18 mm, and the loading rate is set as 1 × 10-4/s. The GSs in the coarse-grained and nano-grained specimens are uniformed distributed. For the gradient nano-grained specimen, the GS changes from 10 nm at one end to 180 nm at the other end, and the GS at the central point of the specimen is 60 nm. Thus, in simulations, a quadratic function lg(X) = a(X/l)2 + b(X / l) + c (a = 140 nm, b = -310 nm, c = 180 nm) is employed to describe the GS distribution along the axial direction. The material parameters used in the following simulations are EGI = 60 GPa, EGB = 50 GPa, gtr = 0.0425, = 11.1MPa, Y = 4.0 MPa, = 6.0 × 10-10 Pa m2. The thickness of GB is set as 2 nm by referring to the existing literature [67]. The details for the determination of material parameters are provided in the Appendix A.
Figure 2(a) and (b) shows the experimental and predicted stress-strain responses of the coarse-grained (235 nm) and nano-grained (10 nm) NiTi SMA specimens. Figure 2(c)-(f) shows the predicted results of NiTi SMA with other typical GSs (1000, 200, 50 and 20 nm). It is seen that the superelastic deformation of NiTi SMAs depends on the GS strongly, i.e., for the coarse-grained specimen, the stressstrain response exhibits transformation plateau and large hysteresis loop. However, the transformation hardening modulus/hysteresis loop increases/decreases monotonically with decreasing GS. When the GS decreases to 10 nm, a quasi-linear response with very small hysteresis loop can be observed. These phenomena were also observed in some other experiments [16,20] and were well captured by the proposed model since (1) the increased volume fraction of GB phase with decreasing the GS was reflected and (2) the hindering effect of GB phase on the MT in GI phase has been reasonably considered. Figure 3(a) shows the overall stress-strain response of the gradient nano-grained specimen. Figure 3(b) gives the strain distribution along the axial direction when the overall strain reaches 3%. It is seen that the stress-strain response of the gradient nano-grained specimen exhibits high nonlinearity. Meanwhile, the strain field in the specimen is inhomogeneous, i.e., the strain in the region with large GS is much larger than that with small GS. Comparing the predicted results with the experimental data, it can be concluded that both the overall and local deformations of the gradient nano-grained specimen are reasonably captured by the proposed model.
The influence of the distribution form for the GS on the superelastic deformation of gradient nano-grained NiTi SMAs is further predicted. Figure 4(a) shows the first five different distributions, i.e., the GSs at the two ends of the specimen are fixed (180 and 10 nm), and the distributions satisfy the quadratic function relations. Figure 5(a) shows the latter five different distributions, i.e., the GS at the right end of the specimen is fixed (10 nm), and the distributions satisfy the linear function relations. Figures 4(b) and 5(b) show the predicted overall stress-strain responses. Figures 4(c) and 5(c) show the predicted distributions of strain field when the overall strain reaches 3%. It is concluded that the superelastic deformation of gradient nano-grained NiTi SMAs can be tailored by changing the distribution form of GS, and the proposed model provides a theoretical tool to design the gradient nano-grained NiTi SMAs with desired mechanical property.
It should be pointed out that in this work, we only focus on the isothermal superelastic deformation of gradient nano-grained NiTi SMAs in one loading-unloading cycle. The shape memory effect, superelasticity degradation during cyclic deformation, and the rate-dependent deformation originated from the thermo-mechanical coupling effect are also important phenomena observed in NiTi SMAs. A more comprehensive constitutive model is needed to describe these phenomena in gradient nano-grained NiTi SMAs, and this will be done in our future work.
4.　Conclusion
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(1) Based on the nonlocal continuum theory, a multiscale analytical model is proposed to describe the superelastic deformation of gradient nano-grained NiTi SMAs. The proposed model is first constructed at the mesoscopic scale by addressing the hindering effect of GB on the MT in GI phase, and then transited to the macroscopic scale by employing a scale transition rule. The main advantages of the proposed model are its analytical form and low computational cost.
(2) For the coarse-grained NiTi SMA specimen, the superelastic deformation exhibits transformation plateau and large hysteresis loop. In contrast, a quasi-linear response with a very small hysteresis loop can be observed for the nano-grained specimen. However, different from the coarse-grained and nano-grained NiTi SMAs, the stress-strain response of the gradient nano-grained specimen exhibits high nonlinearity. All these phenomena are well predicted by the proposed model since the underlying mechanisms for the GS-dependent deformation of NiTi SMA are reasonably captured.
(3) The influence of the distribution form for the GS on the deformation of gradient nano-grained NiTi SMAs is further predicted by the proposed. It is concluded that the superelasticity can be tailored by changing the distribution form of GS. The proposed model provides a theoretical tool to design this type of materials.
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Year 2025 volume 41 Issue 12
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doi: 10.1007/s10409-024-24342-x
  • Receive Date：2024-06-03
  • Online Date：2026-03-24
  • Published：2025-12-01
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  • Received：2024-06-03
  • Accepted：2024-09-10
Affiliations
    1Failure Mechanics and Engineering Disaster Prevention, Key Laboratory of Sichuan Province, Sichuan University, Chengdu 610065, China
    2Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Aerospace Engineering, Southwest Jiaotong University, Chengdu 610031, China

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* E-mail address: (Chao Yu)
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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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