A micromechanical constitutive model for unusual temperature-dependent deformation of Mg–NiTi composites

Accepted Manuscript

A micromechanical constitutive model for unusual temperature-dependent deformation of Mg-NiTi composites Chao Yu , Guozheng Kang , Daining Fang PII: DOI: Reference:

S0020-7683(19)30198-2 https://doi.org/10.1016/j.ijsolstr.2019.04.029 SAS 10357

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International Journal of Solids and Structures

Received date: Revised date:

15 August 2018 18 January 2019

Please cite this article as: Chao Yu , Guozheng Kang , Daining Fang , A micromechanical constitutive model for unusual temperature-dependent deformation of Mg-NiTi composites, International Journal of Solids and Structures (2019), doi: https://doi.org/10.1016/j.ijsolstr.2019.04.029

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A micromechanical constitutive model for unusual temperature-dependent deformation of Mg-NiTi composites Chao Yua, Guozheng Kanga*, Daining Fangb a

Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, Sichuan, China Institute of Advanced Technology, Beijing Institute of Technology, Beijing, China

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b

*Corresponding author: Dr. Prof. G.Z. Kang, Tel: +86-28-87634671; Fax: +86-28-87600797 E-mail address: [email protected] or [email protected]

Abstract

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Based on the mean-field approach, a micromechanical constitutive model is established to describe the unusual temperature-dependent deformation of Mg-NiTi composites. Firstly, the constitutive equations of two phases are proposed: for NiTi shape memory alloy (SMA), a simplified Hartl-Lagoudas model (Hartl and Lagoudas, 2009) which considering the martensite transformation

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and plastic deformation is adopted; while, for magnesium (Mg), an elastic-plastic model including a new nonlinear plastic hardening law is employed. The dependence of elastic moduli and yield

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surfaces of two phases at ambient temperature is addressed. Then, a non-isothermal incremental Mori-Tanaka homogenization method is employed and further extended to describe the interaction

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between two phases and calculate the macroscopic overall stress-strain response of Mg-NiTi composites. Finally, comparisons between the simulated results and the corresponding experimental

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ones show that the temperature-dependent deformation of the composites with different volume fractions of NiTi SMA phase can be well captured by the proposed model. Predicted results

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demonstrate that the unusual temperature-dependent deformation of Mg-NiTi composites originates from the change in the inelastic deformation mechanism of NiTi SMA with the variation of temperature.

Key words: NiTi shape memory alloy; Magnesium; Composites; Constitutive model; Temperature-dependence.

1. Introduction Magnesium (Mg) is the lightest structural metallic material which exhibits high specific

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strength/stiffness, damping capacity, electromagnetic shielding and biodegradability. Its density is about two-thirds of aluminum and less than one-quarter of iron. These excellent properties make Mg and its alloys become promising candidates in automotive and biomedical fields (Moedike and Ebert, 2001; Chino et al., 2006; Kim et al., 2008). However, the low ultimate strength (especially at high temperature) and poor creep resistance of Mg and Mg alloys limit their further applications. So far, much effort had been done to improve the mechanical property of Mg and Mg alloys. The

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design approach of composite materials has been recognized as a promising way to solve such shortcomings of Mg and its alloys, since the stress distribution and overall property of Mg and Mg alloy composites can be manipulated by introducing some suitable reinforcements, like SiC, TiC, Al2O3 (Ye and Liu, 2004), Cu (Hassan and Gupta, 2002a), Ni (Hassan and Gupta, 2002b), Ti (Perez

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et al., 2004), Ti6Al4V (Xi et al., 2005) and NiTi SMA (Mizuuchi et al., 2004; Yan and Li, 2005; Esen, 2012; Aydogmus, 2015). Among of them, NiTi SMA reinforced Mg/Mg alloy composites show an excellent mechanical property; that is, the introduction of NiTi reinforcement can result in a minimum decrease of ductility but a significant improvement of strength in the Mg and Mg alloy

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composites. Considering their different strengthening mechanisms, the existing NiTi SMA reinforced Mg and Mg alloy composites can be classified into two groups:

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(1) The composites strengthened by the shape memory effect of NiTi SMA (Mizuuchi et al., 2004; Yan and Li, 2005; Esen, 2012). In such a strengthening mechanism, the composites were

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pre-deformed at a temperature below the martensite finish temperature (Mf) of NiTi SMA. In the process of pre-deformation, the reorientation of martensite variants would occur and a large

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inelastic strain was produced in the NiTi SMA phase. Then, the composites were heated to a temperature above the austenite finish temperature (Af) to trigger the reverse transformation (i.e.,

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from the detwinned martensite to austenite phase). During the reverse transformation, the NiTi SMA phase tended to shrink back to its initial length and a high internal stress was generated, which resulted in a substantial improvement in the yield strength of the composites.

(2) The one strengthened by the super-elasticity of NiTi SMA (Aydogmus, 2015). In this case, the pre-deformation of the composites at low temperature (below the Mf) is not needed. Figs. 1a and 1b show the temperature-dependent stress-strain curves (obtained at the temperatures higher than the Af, and the NiTi SMA phase exhibited super-elasticity) of the Mg-NiTi composites with various volume fractions of NiTi SMA phase, as observed by Aydogmus (2015). It is seen that

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the yield strength of Mg is very low (<50MPa) and decreases monotonically with the increase of ambient temperature; while, the composites exhibit an unusual temperature-dependent deformation, i.e., its yield strength changes non-monotonically with the increase of temperature and a maximum yield strength of 130MPa can be obtained at 423K when the volume fraction of NiTi SMA phase reaches 30%. The unusual temperature-dependent deformation of Mg-NiTi composites originates from the different inelastic deformation mechanisms of NiTi SMA phase

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presented at various temperature regions (the inelastic deformation mechanism changes from the martensite transformation to dislocation slipping in austenite phase with the increase of ambient temperature). It will be discussed in details in Section 2.1.

Over the last three decades, many classical constitutive models were established to describe the

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unique shape memory effect and superelasticity of SMAs (Brinson, 1993; Leclercq and Lexcellent, 1996; Auricchio et al., 1997; Zaki and Moumni, 2007; Panico and Brinson, 2007; Lagoudas 2008, 2012; Hartl et al. 2010; Chemisky et al., 2011; Arghavani et al., 2011). Meanwhile, some advanced constitutive models were developed to describe the irreversible plastic deformation of SMAs

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occurred in the austenite phase (Jemal et al., 2009; Hartl and Lagoudas, 2009; Khalil et al., 2012), martensite phase (Yan et al., 2003; Paiva et al., 2005; Hartl and Lagoudas, 2009; Zaki et al., 2010) or

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austenite-martensite interface (Bo and Lagoudas 1999a, 1999b; Lagoudas and Entchev, 2004; Saint-Sulpice et al., 2009; Rebelo et al., 2011; Chemisky et al., 2014, 2018; Yu et al., 2015). In these

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models, various deformation mechanisms and their interactions were considered, simultaneously. Rebelo et al. (2004), Hartl and Lagoudas (2009) and Khalil et al. (2012) successfully implemented

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the constitutive models considering the plastic deformation into the finite element code and simulated the thermo-mechanical response of some typical devices and structures made by SMAs. It

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should be pointed out that only a brief review on the constitutive model is provided here, and more details about the state of the art can be found in the recent review papers of Cisse et al. (2016a, 2016b).

Based on the classical constitutive models, some micromechanical models had been established to investigate the heterogeneous deformation of SMA reinforced composites. As reviewed by Lester et al. (2015), the existing micromechanical models can be classified into two categories, i.e., a full-field approach and mean-field one. The full-field approach uses the finite element method to solve the stress equilibrium equation of the representative volume element of composites (Marfia, 2005; Freed

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and Aboudi, 2008, 2009; Chatzigeorgiou et al., 2015; Damanpack et al., 2015a, 2015b). The advantage of full-field approach is that the actual microstructures of composites can be reproduced by it in a closer way than that by the mean-field one. However, high computational cost limits the further application of full-field approach in the engineering structural analysis. The mean-field approach only considers the average information of the matrix and reinforcing phases in the composites. The overall property of composites and the interaction between the matrix

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and reinforcing phases can be described by adopting the Eshelby's equivalent inclusion theory and a suitable homogenization method. Yamada et al. (1993) proposed a micromechanical model to predict the Young’s modulus, yield stress and work hardening rate of SMA particle reinforced Al matrix composites. Boyd and Lagoudas (1994; 1996) established a model to predict the overall property of

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SMA fiber reinforced polymer matrix composites, and extended the Mori-Tanaka method (Mori and Tanaka, 1973) to a non-isothermal version since both the stress- and temperature-induced martensite transformation were considered. Cherkaoui et al. (2000) and Song and Sun (2000) developed a micromechanical model based on the Hill’s self-consistent homogenization method (Hill, 1965). The

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influences of the shape, volume fraction, orientation, and mechanical property of SMA inhomogeneity as well as the mechanical performance of matrix on the overall behavior of the

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composites were systematically studied. Lu and Weng (2000) proposed a two-level micromechanical model to describe the deformation of SMA reinforced composites. Lee and Taya (2004) established a

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micromechanical model to explain the strengthening mechanism of SMA fiber reinforced Al matrix composites. Zhu and Dui (2009) developed a three-phase micromechanical model to study the effect

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of fiber shape on the residual stress and overall mechanical response of SMA fiber reinforced composites. Lester et al. (2011) developed a constitutive model to investigate the effect of stress

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distribution on the martensite transformation of SMA reinforced ceramic composites. To sum up, the advantage of mean field approach is its low computational cost, and thus it can be readily used in the engineering structural analysis after being implemented into finite element codes. In order to better understand the strengthening mechanism and optimally design the microstructure of Mg-NiTi composites, a micromechanical constitutive model is urgently necessary. However, the existing micromechanical models cannot reasonably describe the usual temperature-dependent deformation of Mg-NiTi composites since the plastic deformation of NiTi SMA and the unique plastic hardening behavior of Mg have not been considered yet. Moreover, the homogenization

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methods used in the existing models are not applicable any more to model the deformation of Mg-NiTi composites (this will be discussed in details in Section 3). Therefore, in this work, based on the mean-field approach, a new micromechanical constitutive model is constructed. In the proposed model, the well-known Hartl-Lagoudas model (Hartl and Lagoudas, 2009) is adopted and simplified to describe the martensite transformation and plastic deformation of NiTi SMA phase. An elastic-plastic model with a newly developed plastic hardening

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law is proposed to describe the deformation of Mg matrix. The dependence of the elastic moduli and yield surfaces of two phases (i.e., Mg and NiTi SMA) on the ambient temperature is also addressed. To describe the interaction between two phases and calculate the overall mechanical response of the composites, a new incremental Mori-Tanaka homogenization method is proposed by modifying the

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original one established by Yu et al. (2018). The proposed micromechanical constitutive model is verified by comparing the predictions with the corresponding experiments done by Aydogmus (2015).

2.1 Constitutive model of NiTi SMA

Different inelastic deformation mechanisms of NiTi SMA at various temperature regions

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2.1.1

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2. Constitutive models of NiTi SMA and Mg

The unusual temperature-dependent yield strength of the composites can be explained by the

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super-elasticity of NiTi SMA phase and its temperature-dependence as follows: at the temperature above the finish temperature of austenite phase (Af), the start stress of martensite transformation

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increases with the increase of ambient temperature, which is represented by the well-known Clausius-Claperon relationship. However, more and more experimental results showed that when the

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ambient temperature reached to a critical value, dislocation glide could be activated in the austenite phase by the applied stress before the occurrence of martensite transformation (Miyazaki et al., 1981; Shaw and Kyriakides 1995; Qian et al., 2006; Hartl and Lagoudas, 2009). With the increase of temperature, the inelastic deformation mechanism of NiTi SMA changes from the martensite transformation to the dislocation glide of austenite phase, which is deduced from the experimental results shown in Figs. 2a and 2b. Meanwhile, the temperature-dependent martensite transformation and dislocation glide of NiTi SMA are summarized in Fig. 2c by referring to Hartl and Lagoudas

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(2009). It is seen from Fig. 2c that: in the region I (AfAd2), the critical stress of dislocation glide

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in the austenite phase becomes lower than the start stress of martensite transformation, and then, the inelastic deformation mechanism of NiTi SMA is dominated by the dislocation glide in the austenite phase. The temperatures Ad1 and Ad2 represent two critical temperatures for the plasticity of austenite phase. Thus, it seems that the unusual temperature-dependent deformation of Mg-NiTi composites

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originates from the different inelastic deformation mechanisms of NiTi SMA phase presented at the three different temperature regions. 2.1.2 Main equations

As mentioned above, when the ambient temperature reached to a critical value, dislocation glide

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could be activated in the austenite phase by the applied stress before the occurrence of martensite transformation. In NiTi SMAs, the crystalline structures of austenite and martensite phases are cubic

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and monoclinic ones, respectively. The resistance of dislocation glide in the martensite phase is much higher than that in the austenite one. For instance, the critical stress for the occurrence of plastic

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deformation in the martensite phase at room temperature was reported as 1200-1700MPa (Shaw and Kyriakides 1995; Qian et al. 2006). However, the maximum stress cannot reach to such a high level

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in the experimental results of Mg-NiTi composites (Aydogmus 2015). Moreover, it is known that the plastic deformation at the austenite-martensite interface is named as the transformation-induced

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plasticity and will accumulate during the cyclic deformation (Lagoudas and Entchev, 2004). However, in this work, we mainly focus on the deformation of Mg-NiTi composites in one loading-unloading cycle. Therefore, only the plastic deformation of austenite phase is considered and the ones of martensite phase and austenite-martensite interface are neglected. As mentioned in the Introduction, the constitutive models proposed by Jemal et al. (2009), Hartl and Lagoudas (2009) and Khalil et al. (2012) can reasonably describe the deformation of NiTi SMAs undergoing the martensite transformation and plastic deformation. The models proposed by Jemal et al. (2009) and Khalil et al. (2012) were established from a simplified micromechanical description.

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The interaction between martensite transformation and plastic deformation was addressed by decomposing the elastic energy into three parts, i.e., macroscopic elastic energy, inter-granular energy and intra-granular one. The model proposed by Hartl and Lagoudas (2009) comprehensively considered the martensite transformation, plastic deformation of austenite and martensite phases, simultaneously. The interaction between the martensite transformation and plastic deformation was addressed by introducing a plastic back stress tensor.

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The constitutive model of NiTi SMA phase adopted in this section can be regarded a simplified version of Hartl-Lagoudas model (Hartl and Lagoudas, 2009), i.e., the plastic deformation of martensite phase and the interactions among different inelastic deformation mechanisms are neglected for simplification. Meanwhile, some extensions are made, i.e., the dependences of plastic

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yield and elastic modulus of austenite phase on temperature are newly considered.

Considering the contributions of elastic deformation, thermal expansion, martensite transformation and plastic deformation, the total strain ε can be decomposed additively into four parts, i.e.,

ε  εe  εT  εtr  ε p ,

(1)

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where, ε e , εT , ε tr and ε p are the elastic, thermal expansion, transformation and plastic strain

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tensors, respectively.

The relationship between the elastic strain ε e and stress σ can be given by the well-known

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Hooke's law

εe  S1  , T  : σ  1    S A T    S M T  : σ ,

(2)

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where, S1 is the elastic compliance tensor and is a function of the volume fraction of martensite

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phase  and temperature T . S A and S M are the elastic compliance tensors of austenite and martensite phases, respectively. The relationship between the thermal expansion strain εT and temperature T is written as: εT  α1  T  Tr   1    α A   α M  T  Tr  ,

(3)

where, α1 is the thermal expansion tensor, which is a function of the volume fraction of martensite phase  . Tr is the room temperature. α A and α M are the thermal expansion tensors of austenite

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and martensite phases, respectively. Referring to Hartl and Lagoudas (2009) and Lagoudas et al. (2012), the rates of transformation strain ε tr and the volume fraction of martensite phase  can be linked by the following rule: εtr   Λtr ,

when   0

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 3 max dev  σ   H dev  σ   2 tr Λ  tr  εr when   0,   r

(4-a)

(4-b)

where, Λ tr is the transformation direction, H max is the maximum transformation strain, dev  σ 

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is the deviatoric stress tensor. ε trr and  r are the transformation strain and the volume fraction of martensite phase at the beginning of reverse transformation, respectively. Based on irreversible thermodynamics, the thermodynamics force of martensitic transformation

et al., (2012), which can be written as: 1 2

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was deduced from the Clausius’s dissipative inequality by Hartl and Lagoudas (2009) and Lagoudas

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  σ, T ,    σ : Λtr  σ : S : σ  σ : α T  T0   s0 T  T0  

f , 

(5)

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where, S  S M  S A , α  α M  α A .  is the density, s0 is the difference of configuration entropy between the austenite and martensite phases. T0 is the balance temperature. f is the

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hardening energy and is a function of  . Hartl and Lagoudas (2009) and Lagoudas et al., (2012) proposed a nonlinear hardening function which can well capture the smooth transition from the

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elastic to transformation response of SMAs, i.e., n 1  a1 1   n1  1    2   a3   f  2     1  n a2 1   n3  1    4   a3   2 

 0 (6)

 0

where, a1 , a2 , a3 , n1 , n2 , n3 and n4 are material parameters. The exponents n1 , n2 , n3 and

n4 are real number values in the interval (0,1]. It should be pointed out that a3 is not an

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independent parameter, but can be determined as follow (Lagoudas et al. 2012), i.e., a3  

a1  1 1  a2  1 1     1    1  4  n1  1 n2  1  4  n3  1 n4  1 

(7)

In this work, the hardening behaviors of NiTi SMA during forward (from austenite to martensite phase) and reverse (from martensite to austenite phase) transformation are assumed to be identical,

f 1  n  a1 1   n1  1    2     2

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i.e., a1  a2 , n1  n3 , n2  n4 . Then, the nonlinear hardening function (Eq. 6) can be simplified as: (8)

To ensure the compatibility of evolution equations with thermodynamics, the following constraints are introduced (Hartl and Lagoudas, 2009; Lagoudas et al., 2012):

  Y

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  Y when  >0,

when  <0,

no constraint when  =0,

(9-a) (9-b) (9-c)

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where, Y reflects the energy dissipation of martensite transformation and Y >0. By Eqs. (9-a) and (9-b), the forward and reverse transformation surfaces are defined as

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for   σ, T ,      Y  0 when   0   σ, T ,     rev .   σ, T ,      Y  0 when   0 tr

(10)

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Then, the evolution equations of martensite transformation can be obtained by the Kuhn-Tucker conditions:

(11-a)

Reverse transformation: if rev  0 ,   0 then  rev  0 ,

(11-b)

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Forward transformation: if  for  0 ,   1 then  for  0 ,

Elastic loading-unloading: other conditions then   0 .

(11-c)

A temperature-dependent yield surface is proposed to describe the plastic yield of austenite phase,

i.e.,

 p  σ, T ,   

3 dev  σ    T  Tr   R , 2

(12)

where,  is a material parameter which reflects the temperature-dependent plastic yielding. It

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should be noted that since the resistance of dislocation glide decreases with the increasing temperature (Otsuka and Shimizu, 1986), a temperature-dependent yield surface is introduced here.

R is the resistance of plastic deformation at room temperature. As observed by Miyazaki et al. (1981) and Hartl and Lagoudas (2009), the plastic deformation of NiTi SMA exhibited a strong nonlinearity, as shown in Fig. 2b. Meanwhile, some nonlinear hardening rules were proposed by and

following nonlinear hardening rule is adopted in this work, i.e.,

R  R0   Rsat  R0  1  eCH   ,

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Jemal et al. (2009) and Hartl and Lagoudas (2009). Referring to Hartl and Lagoudas (2009), the

(13)

where, R0 and Rsat are the initial and saturated resistances of plastic deformation, respectively. 

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2 p p ε :ε . 3

is the accumulated plastic strain, i.e.,  

The relationship between ε p and  can be given by the normality rule:  p  1    Λ p , σ

3 dev  σ  . 2 dev  σ 

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ε p  1    

(14-b)

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Λp 

(14-a)

The term 1    in Eq. (14-a) represents that dislocation glide only occurs in the austenite phase.

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Then, the evolution equations of plastic deformation can be obtained by the Kuhn-Tucker

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condition:

 0

 p  σ, T ,    0

 p  0 .

(15)

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It should be pointed out that J2 criterion is adopted in the Hartl-Lagoudas model since the martensite transformation and plasticity are assumed to be isotropic. Tension-compression asymmetry and anisotropic transformation are often observed in NiTi SMAs with an initial <111> texture. These phenomena can be well captured by advanced transformation criterions (J2-J3) proposed by Qidwai and Lagoudas (2000), Brocca et al. (2002), Bouvet et al. (2004), Lexcellent and Thiebaud (2008), Raniecki and Mróz (2008), Zaki (2010) and Chemisky et al. (2011). In the experiment of Aydogmus (2015), the initial texture of NiTi powder was not reported. Thus, the J2 criterion is adopted here for simplicity.

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2.1.3 Continuous tangent moduli In this subsection, the continuous tangent moduli of constitutive equations proposed in Section 2.1.1 will be derived. During the forward and reverse transformations, the Kuhn-Tucker condition requires that tr  0 .

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In this case, tr  0 , i.e.,

tr tr tr   σ, T ,    :σ  T   0, σ T 

(16)

tr Λtr  Λtr  σ :  σ : S  α T  T0  , σ σ

(17-a)

tr

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where,

(17-b)

tr 2 f 1 n 1   2   a1  n1 n1 1  n2 1    2  ,     2

(17-c)

 dev 2 Λ tr Λ tr  I     3 H max H max   dev  σ 

 0

(17-d)

  0.

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  Λ tr  3 H max  σ  2 0 

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tr 1 S  σ: : σ  σ : α  s0 , T 2 T

where, I dev is the fourth-ordered unit deviatoric tensor.

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By Eqs. (16) and (17-a) to (17-c), the relationship among σ , T and  can be obtained:

  P : σ  PT T

(18)

where,

P 

1 H tr

PT 

 tr  Λtr Λ  σ :  σ : S  α T  T0  ,  σ  

1  1 S  : σ  σ : α  s0  .  σ: H tr  2 T 

1 n 1 H tr  a1  n1 n1 1  n2 1    2    2

(19-a)

(19-b) (19-c)

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During the plastic deformation, the Kuhn-Tucker condition requires that  p  0 . In this case,  p  0 , i.e.,

 p  σ, T ,   

 p  p  p :σ  T  0, σ T 

(20)

3 dev  σ   Λp , 2 dev  σ 

 p  σ

 p . T

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where, (21-a)

(21-b)

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By Eqs. (20), (21-a) and (21-b), the relationship among σ , T and  can be obtained as

  Q : σ  QT T , where,

Λp , Hp

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Q 

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QT 



Hp

.

(22)

(23-a)

(23-b) (23-c)

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H p  CH  Rsat  R0  eCH 

By the definitions of elastic, thermal expansion, transformation and plastic strains (i.e., Eqs. (1),

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(2), (3), (4a) and (14a)), the total strain rate can be written as:

ε  ε e  εT  εtr  ε p (24)

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 S   S1 : σ   S : σ  α T  Tr      : σ  T  α1T  Λtr  1    Λ p .  T 

Substituting Eqs. (18) and (22) into Eq. (24), it yields:

 S  ε  S1 : σ   S : σ  α T  Tr      : σ  T  α1T  Λ tr  1    Λ p  T   S   S1 : σ   : σ  T  α1T   S : σ  α T  Tr   Λ tr   P : σ  PT T   1    Λ p  Q : σ  QT T   T 





 S1   S : σ  α T  Tr   Λ   P  1    Λ  Q : σ tr

p

 S    : σ  α1   S : σ  α T  Tr   Λ tr  PT  1    Λ p QT  T .  T 

(25)

ACCEPTED MANUSCRIPT

By Eq. (25), the linearized constitutive equation of NiTi SMA can be written as: σ  D : ε  DT T ,

(26)

where, D and DT are two continuous tangent moduli, i.e.,



D  S1   S : σ  α T  Tr   Λtr   P  1    Λ p  Q



1

,

(27-b)

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 S  DT  D :  : σ  α1  S : σ  α T  Tr   Λtr  PT  1    Λ pQT  .  T 

(27-a)

2.2 Constitutive model of Mg

The slip systems in Mg and Mg alloys are very limited due to their hexagonal close-packed

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(HCP) structures. Thus, twinning deformation is another important plastic deformation mechanism for Mg and Mg alloys, which results in a different stress-strain response from the face-centered cubic (FCC) and body-centered cubic (BCC) metallic materials. For instance, anisotropic yield stress and plastic hardening behavior are observed in Mg and Mg alloys rods, tubes and sheets (Lou et al., 2007;

M

Xiong et al., 2014; Li et al., 2016). Until now, it is well known that the unique mechanical property of Mg and Mg alloys is mainly caused by the texture formed in the machining processes (cold rolling

ED

and extrusion). After the machining processes, Mg exhibits a strong basal plane texture. In the last few decades, many constitutive models have been established for Mg and Mg alloys.

PT

Most of them are the crystal plasticity models, i.e., the models are constructed in the singe crystal scale by considering different plastic deformation mechanisms (dislocation slipping and twinning)

CE

and extended to the polycrystalline version with the help of finite element method or scale transition rule (Zhang et al., 2012; Wang et al., 2013, 2016, 2017). However, owing to the high computational

AC

cost, the crystal plasticity models are not suitable to describe the macroscopic deformation behaviors of Mg composites. Other type of model is the macroscopic phenomenological constitutive one. In these models, the underlying plastic deformation mechanisms are not considered any more. These models describe the stress-strain responses by introducing the anisotropic initial yield surface, kinemics and isotropic hardening rules (Cazacu et al. 2006; Shi and Mosler 2013; Shi et al. 2017; Lee et al. 2017) or adopting two yield surfaces (Lee et al. 2008). These models can successful capture the anisotropic deformation and the mechanical response occurred during the strain-path change of rolled/extruded Mg rods, tubes and sheets, which exhibit strong basal plane texture. It is

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known that the yield and hardening behaviors of Mg depend on its texture strongly. However, in the experiments of Aydogmus (2015), the Mg used in the composites is in a form of powder. Maybe, the texture in Mg powder is different from that observed in rods, tubes and sheets. Thus, the applicability of existing constitutive models to describe the deformation of Mg powder is still unknown. Moreover, large number of material parameters are involved in the existing models and should be calibrated based on the stress-strain curves in different loading directions. However, these experimental data are

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lack in Aydogmus (2015). In this work, we mainly focus on the temperature-dependent deformation of Mg-NiTi composites under compression. Thus, the J2 yield criterion is adopted and a new but very simple isotropic hardening rule is proposed for Mg powder here. The material parameters involved in the model can be easily determined from the stress-strain curves of Mg powder under compression.

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2.2.1 Main equations

Considering the contributions of elastic, thermal and plastic deformations, the total strain tensor

ε is decomposed into three parts, i.e.,

ε  εe  εT  ε p .

(28)

M

The elastic and thermal expansion strains, i.e., ε e and εT can be given as:

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ε e  S 2 T  : σ , εT  α 2 T  Tr  ,

(29-a) (29-b)

CE

expansion tensor.

PT

where, S 2 is the elastic compliance tensor and is a function of temperature, α 2 is the thermal

A temperature-dependent yield surface is introduced to describe the plastic deformation of Mg,

AC

i.e.,

 p  σ, T ,   

3 dev  σ   K T ,   , 2

(30)

where, K T ,   is the resistance of plastic deformation. It is a function of temperature T and accumulated plastic strain  .

K T ,   is further decomposed into two parts, i.e., K  T ,    K 0  T   K s T ,   ,

(31-a)

ACCEPTED MANUSCRIPT   K m1  K m2  K s  h T  1   s    s    ,   K1   K 2  

(31-b)

where, K 0 T  and K s T ,   are the resistances of initial and subsequent plastic deformation, respectively. Eq. (31-b) is a new proposed nonlinear hardening law to describe the unusual stress-strain response of Mg. h T  is the plastic hardening modulus. K1 , K 2 , m1 and m2 are

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material parameters.

The relationship between ε p and  can be given by the normality rule as  p  N p , σ

Np 

3 dev  σ  . 2 dev  σ 

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εp  

(32-a)

(32-b)

Then, the evolution equations of plastic deformation can be obtained by the Kuhn-Tucker condition:

 p  σ, T ,    0

 p  0 .

(33)

M

0 2.2.2 Continuous tangent moduli

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During the plastic deformation, the Kuhn-Tucker condition requires that  p  0 . So,  p  0 , i.e.,

PT

  K m1  K m2  K 0  p  p p   σ, T ,    :σ  K  N :σ  T  h T  1   s    s    . σ K T   K1   K 2  

CE

p

(34)

By Eq. (34), the relationship among σ , T and  can be obtained:

AC

  M : σ  M T T ,

(35)

where,

M 

Np   K m1 h T  1   s    K1 

K   s   K2 

m2

  

, (36-a)

ACCEPTED MANUSCRIPT K 0 T

MT 

  K m1  K m2  h T  1   s    s     K1   K 2  

. (36-b)

By the definitions of elastic, thermal expansion and plastic strains (i.e., Eqs. (28), (29-a), (29-b) and (32-a)), the total strain rate can be written as:  S  ε  S 2 : σ   2 : σ  T  α 2T  N p  .  T 

σ  D : ε  DT T ,

where,

D   S2  N p  M  ,

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1

CR IP T

Substituting Eqs. (35) into Eq. (37), it yields:

(37)

 S  DT  D :  2 : σ  α 2  N p M T  .  T 

(38)

(39-a) (39-b)

Eqs. (26) and (38) can be regarded as the linearized constitutive relationships of NiTi SMA and

M

Mg phases, respectively.

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3. Modified Mori-Tanaka homogenization method

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In Aydogmus (2015), the raw materials were spherical Mg and Ni-rich Ni50.6Ti49.4 powders. Spark plasma sintering was carried out and a 50MPa pressure was applied to obtain the interpenetrating

CE

Mg-NiTi composites with different volume fractions of NiTi SMA phase (0%, 10%, 20% and 30%). Fig. 3a shows a typical microstructure of Mg-NiTi (Mg-20%NiTi) composites observed by optical

AC

microscope. Based on such microstructural characteristics, the NiTi SMA and Mg phases are respectively modeled as the “matrix” and “inhomogeneity” in the proposed micromechanical model. It should be noted that in the existing micromechanical models (Yamada et al. (1993); Boyd and Lagoudas (1994; 1996); Cherkaoui et al. (2000); Song and Sun (2000); Lu and Weng (2000); Lee and Taya (2004); Zhu and Dui (2009); Lester et al. (2011)), the NiTi reinforcing phase is always treated as the inhomogeneity since the geometrical morphologies of NiTi SMA are sphere, ellipsoid or fibers, which are different from the interpenetrating Mg-NiTi composites manufactured by Aydogmus (2015).

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To reasonably estimate the overall response of composites, the Mori-Tanaka (M-T) homogenization method will be adopted in this work. The M-T method was first established to predict the elastic modulus and further extended to describe the inelastic deformation of composites, including the time-independent plasticity (Doghri and Ouaar, 2003; Guo et al., 2011; Peng et al., 2016), time-dependent plasticity (Pierard and Doghri, 2006; 2007; Mercier and Molinari, 2009; Doghri et al., 2010; Guo et al., 2013) and viscoelasticity (Berbenni et al., 2015; Lavergne et al.,

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2016). It should be noted that the traditional M-T method is based on the Eshelby's solution for an inclusion embedded in an infinite matrix (Eshelby, 1957). Thus, the accuracy of the prediction can be ensured only if the volume fraction of inhomogeneity is not very large (<50%). However, the traditional M-T method is not applicable any more in this work since the volume fraction of

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inhomogeneity (Mg) is much higher than 60%. Recently, Li et al. (2007a; 2007b) derived the new analytical solutions of Eshelby's tensors for a spherical inclusion in a finite spherical domain. Referring to Li et al. (2007a; 2007b), Yu et al. (2018) further extended the M-T method into an incremental case to describe the grain size dependent deformation of NiTi SMAs. However, in Yu et

M

al. (2018), the deformation of the matrix was assumed to be pure elastic and not dependent on temperature. Thus, in this work, the extended M-T method proposed by Yu et al. (2018) is further

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modified to predict the complex inelastic deformation (including the martensitic transformation and plasticity of NiTi SMA, the plastic deformation of Mg and the temperature-dependent properties of

PT

two phases) of Mg-NiTi composites.

In the M-T method, when the constitutive laws of inhomogeneity and matrix phases are nonlinear,

CE

the local constitutive equations of each phase must be linearized at first. Recalling the continuous tangent moduli derived in Section 2, the linearized constitutive equations of NiTi SMA (matrix) and

AC

Mg (inhomogeneity) phase can be, respectively, written as:

where, σ

M

and ε

σ

M

 DM : ε

M

 DTM T ,

(40-a)

σ I  DI : ε I  DTI T , M

(40-b)

are the average stress and strain rates in the matrix phase, respectively. DM

and DTM are the continuous tangent moduli of matrix phase. σ I , ε I , DI and DTI are the corresponding variables in the inhomogeneity phase. The operators



M

and 

I

represent the

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volume averages in the matrix and inhomogeneity, respectively, i.e., y

M

y

M

and ε

I

1 VM





1 VI

 y  x  dV .

I

VM

y  x  dV ,

(41-a)

(41-b)

VI

can be further decomposed into two parts (Li et al. 2007b), i.e.,

ε

M

 εb  ε d

M

,

ε I  εb  ε d , I where, ε b is the uniform background strain rate,

εd

M

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ε



and

εd

I

(42-a) (42-b)

are the average rates of

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disturbance strains in the matrix and inhomogeneity phases, respectively.

According to the Eshelby's equivalent inclusion theory (Eshelby, 1957), the tangent moduli of inhomogeneity ( DI and DTI ) can be replaced by that of matrix ( DM and DTM ) once an artificially eigenstrain is introduced. Then, the “inhomogeneity” is transformed to an equivalent “inclusion”, as

M

shown in Figs. 3c and 3d. Thus, the constitutive relationship of inhomogeneity phase (Mg) can be rewritten in the following equivalent form: I

 DI : ε I  DTI T  DM :  ε I  ε*   DTM T ,

ED

σ

(43)

PT

where, ε* is the rate of eigenstrain.

AC

CE

By Eqs. (43), the relationship between ε

I

and ε* can be obtained:

ε I  A : ε*  AT T ,

(44-a)

A   DM  DI  : DM ,

(44-b)

AT   DM  DI  :  DTI  DTM  .

(44-c)

1

1

The average rates of disturbance strains in the matrix and inclusion phases and the eigenstrain rate can be linked by two Eshelby's tensors, i.e.,

εd εd

M

I

 Π M : ε* ,

(45-a)

 Π I : ε* ,

(45-b)

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where, Π I and Π M are the interior and exterior Eshelby's tensors, respectively (Li et al., 2007a; 2007b). In the case of spherical inclusion, the analytical solutions for Π I and Π M are derived by Li et al. (2007a; 2007b), i.e.,

s1I

Π I  s1I I vol  s2I I dev ,

(46-b) 2

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s

(46-a)

21 f I2 1  f I2/3  2  4  5 M  f I , s   15 1  M  10 1  M  7  10 M 1  f I 

1  M  f I  3 1  M 

M 1

Π M  s1M I vol  s2M I dev ,

M 2

1  M 1  f I   3 1  M 

21 f I 1  f I2/3  2  4  5 M 1  f I  I , s2   15 1  M  10 1  M  7  10 M 

(46-c)

2

f I is the volume fraction of

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where,  M is the elastic Poisson's ratio of matrix phase.

(46-d)

inhomogeneity (Mg) phase. I vol and I dev are the fourth-ordered unit spherical and deviatoric tensors, respectively, i.e.,

1  1  1 , 3

M

I vol 

1 1  1 , 3

(47-b)

ED

I dev  I 

(47-a)

where, I and 1 are the fourth- and second-ordered unit tensors, respectively. From Eqs. (46-c) and

PT

(46-d), it is seen that when the representative volume element (RVE) is infinite, i.e., f I  0 , s1M ,

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s2M , s1I and s2I are reduced to 0, 0,

2  4  5 M  1  M and , respectively. In this case, Π M =0 15 1  M  3 1  M 

and Π I reduces to the traditional Eshelby's tensor.

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It should be noted that the analytical solutions of Eshelby's tensors in Eqs. (46-a) to (46-d) are

only suitable for the isotropic elastic solids. In the case of inelasticity, the Eshelby's tensors can only obtained by the numerical method which is very time consuming. To overcome this shortcoming, Doghri et al. (2003) proposed a simplified method, i.e., replacing the elastic Poisson's ratio  M by the inelastic Poisson's ratio  M , that is,

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kM

M

and

I vol :: DM   I vol 

D  M

ijkl



lkji

3kM  2M , 2  3kM  M 

(48-a)

kM 

1 vol I :: DM  ,  3

(48-b)

M 

1 dev I :: DM  ,  10

(48-c)

are

the

inelastic

bulk

and I dev :: DM   I dev 

D 

and

M

ijkl



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where,

M 

lkji

.

shear

moduli,

respectively.

Substituting Eqs. (44-a) and (45-a) into Eq. (42-a), the relationship between ε* and ε b can be



ε*  A ε  Π I

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obtained as

 : ε 1

b

 AT T  .

(49)

It is noted that the macroscopic strain and stress rates of two-phase composites can be written as the following rules of mixture:

M

,

(50-a)

σ  f I σ I  1  f I  σ

M

.

(50-b)

ED

M

ε  f I ε I  1  f I  ε

Substituting Eqs. (42-a), (42-b), (45-a), (45-b) and (49) into Eq. (50-a), the background strain rate

PT

can be expressed in terms of macroscopic strain rate, i.e., εb  B : ε  BT T ,





(51-a)





1

CE

B  Aε  Π I :  Aε  1  f I  Π I  Π M  ,





(51-b)

1

(51-c)

AC

BT   Aε  1  f I  Π I  Π M  :  fΠ I  1  f I  Π M  : AT .

Substituting Eqs. (45-a), (49) and (51-a) into Eq. (42-a), the average strain rate of matrix phase can

be expressed in terms of macroscopic strain rate, i.e.,

ε

M

 CM : ε  CTM T ,



CM  I  Π M : A  Π I 



CTM  I  Π M : A  Π I 



1



1

(52-a)

:B ,  



 : B  ΠM : A  ΠI   T

(52-b)



1

: AT .

(52-c)

Similarly, substituting Eqs. (45-b), (49) and (51-b) into Eq. (42-b), the average strain rate of

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inclusion phase can be expressed in terms of macroscopic strain rate, i.e.,

ε I  CI : ε  CTI T ,



CI  I  Π I : A  Π I 



CTI  I  Π I : A  Π I 



1



1

(53-a)

:B ,  



(53-b)

 : B  ΠI : A  ΠI   T



1

: AT .

(53-c)

relationship can be obtained as: σ  D : ε  DT T ,

D  f I DI : CI  1  f I  DM : CM ,

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Substituting Eqs. (40-a), (40-b), (52-a) and (53-a) into Eq. (50-b), the macroscopic stress-strain

(54-b) (54-c)

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DT  f I  DI : CTI  DTI   1  f I   DM : CTM  DTM  ,

(54-a)

where, D and DT are two macroscopic continuous tangent moduli.

In general, the tangent moduli of NiTi SMA and Mg, i.e., DM , DTM , DI and DTI depend on the stress state since the constitutive model is nonlinear. In the real material, the stress fields inside and

M

outside the inclusions are all inhomogeneous, leading to the heterogeneity of thermo-mechanical

ED

properties in the whole material. However, the Mori-Tanaka homogenization method used in this work assumes that the material property is uniform in each phase (i.e., matrix and inhomogeneity),

PT

and the disturbance is neglected for simplification. More accurate results can be obtained by adopting the full field solution (e.g., finite element method).

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4. Model verification and discussion 4.1 Temperature-dependent functions

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In Sections 2.1 and 2.2, the constitutive models of NiTi SMA and Mg phases are, respectively,

established. In this subsection, the specific forms of temperature-dependent functions, i.e., S A T  ,

S M T  , K0 T  and h T  will be proposed from the experimental observations of Aydogmus (2015). As mentioned by Aydogmus (2015), the elastic modulus of the austenite phase of NiTi SMA increases with the increasing temperature up to 423K; however, when the temperature is higher than 423K, the elastic modulus becomes almost constant. The differences of the elastic moduli between

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the austenite and martensite phases had been reasonably considered in the Hartl-Lagoudas model (Hartl and Lagoudas, 2009). However, in the experiment of Aydogmus (2015), the stress-strain curve of pure NiTi SMA was not provided. It is difficult to calibrate the temperature-dependent elastic modulus of martensite phase since three phases are coexisted in the NiTi-Mg composite (e.g., austenite, martensite and Mg phases). Thus, in this work, the elastic modulus differences between the

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austenite and martensite phases are neglected for simplicity. So, S A T  and S M T  are proposed as: S A T   S M T  

ENiTi T  NiTi E T  1  1  NiTi I, 1  NiTi 1  2 NiTi 1  NiTi 

(55-b)

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Tr  ENiTi T  423K 1  aT T  Tr   ENiTi T    Tr  ENiTi 1  aT  423  Tr   T  423K .

(55-a)

Tr where,  NiTi and ENiTi are the Poisson's ratio and elastic modulus of NiTi SMA. ENiTi is the elastic

modulus at room temperature, aT is a parameter.

S 2 T  

M

Similarly, the elastic compliance tensor of Mg, i.e., S 2 T  can be written as:

EMg T  Mg

1  2 1  

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Mg

Mg

11 

EMg T  1  Mg

I.

(56)

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where,  Mg and EMg are the Poisson's ratio and elastic modulus of Mg, respectively. Based on the experimental observations (Aydogmus 2015), the temperature-dependent elastic

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modulus EMg T  , initial plastic resistance K 0 T  and plastic hardening modulus h T  are

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proposed as:

EMg T   E

Tr Mg

  T  T n  1   * r   ,   T  

(57-a)

  T  T n  K 0 T   K 1   * r   ,   T  

(57-b)

  T  T n  h T   h 1   * r   ,   T  

(57-c)

Tr 0

Tr

Tr where, EMg , K 0Tr and hTr are the elastic modulus, initial plastic resistance and plastic hardening

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modulus of Mg at room temperature, respectively. T * is the reference temperature, n is a parameter.

4.2 Calibration of material parameters 4.2.1 Parameters related to Mg

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The thermal expansion tensor α 2 can be regarded as isotropic in the polycrystalline metallic materials, i.e., α 2   Mg 1 , where,  Mg is the isotropic coefficient of thermal expansion, and is set as 26.1  10-6/K by referring to Aydogmus (2015). The Poisson's ratio  Mg is set as 0.3. The elastic

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Tr modulus, initial plastic resistance and the parameters related to plastic hardening (i.e., EMg , K 0Tr ,

hTr , K1 , m1 , K 2 and m2 ) can be directly extracted from the stress-strain curve of Mg at room

temperature at a large strain, as shown in Fig. 4f (the black solid line). The parameters T * and n control the temperature-dependent initial plastic resistance of Mg and can be obtained by fitting the

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yielding points at various temperatures (423K and 523K).

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4.2.2 Parameters related to NiTi SMA

T0  Tr 

Y  X0 , s0

(58)

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The balance temperature T0 is chosen as

where, X 0 represents the transformation resistance at room temperature.

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Similarly, the thermal expansion tensors of austenite and martensite phases, i.e., α A and α M are

assumed to be isotropic. So, α A   A 1 and α M   M 1 . The isotropic coefficients of thermal expansion, i.e.,  A and  M are set as  A =11  10-6/K and 6.6  10-6/K, respectively, by referring to Aydogmus (2015). The Poisson's ratio  NiTi is set as 0.3. The parameter H max is set as 0.06 since many experiments have reported that the maximum transformation strain of NiTi SMA is about 4%-8% (Shaw and Kyriakides 1995; Yu et al., 2014; Chi et al. 2015). It should be noted that the

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stress-strain curve of NiTi SMA powder was not performed in Aydogmus (2015). Thus, the parameters related to the NiTi SMA phase are determined from the stress-strain curves of Mg-30%NiTi composites and referred to the existing experimental results of NiTi SMAs (Miyazaki Tr et al. 1981; Qian et al. 2006). The elastic modulus ENiTi can be determined from the elastic part of

stress-strain curve (segment o-a) of the composites at room temperature, as shown in Fig. 4a. The

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parameter aT controls the temperature dependence of elastic modulus and can be determined from the elastic part of stress-strain curve (segment o-a) of the composites at 423K, as shown in Fig. 4c. From Fig. 4a, it is seen that the deformation of Mg-30%NiTi composites at room temperature can be divided into three stages, i.e., o-a, a-b and b-c ones. In the segment o-a, only elastic deformation

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occurs. In the segment a-b, the NiTi SMA phase deforms elastically while the plastic deformation of Mg occurs. In the segment b-c, the inelastic hardening modulus further decreases, which is caused by the occurrence of martensite transformation in the NiTi SMA phase. The points b and c in Fig. 4a represent the start stresses of plastic deformation in Mg and martensite transformation in NiTi SMA

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phase, respectively. As mentioned above, the parameter X 0 controls the start stress of martensite transformation and the parameters a1 , n1 and n2 controls the transformation hardening behavior

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at room temperature. Thus, they can be obtained by fitting the stress-strain curve in the segment b-c.

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The parameter s0 is used to characterize the dependence of the start stress of martensite transformation on ambient temperature (Lagoudas 2008; 2012). Thus, it can be determined by fitting

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the start stress of martensitic transformation at another temperature (point c in Fig. 4b). From Fig. 4e, it is seen that the deformation of Mg-30%NiTi composites at 523K can also be

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divided into three stages, i.e., o-a, a-b and b-c ones. However, as mentioned above, the inelastic deformation of NiTi SMA phase is dominated by the dislocation glide of austenite phase at high temperature. Thus, the deformation mechanism in the segment b-c is the elastic and plastic deformations in both phases. The points b and c in Fig. 4e represent the start stresses of plastic deformation in Mg and NiTi SMA phases, respectively. Thus, the parameters  , R0 , Rsat and CH can be determined by fitting the stress-strain curve in the segment b-c.

The parameters used in the proposed model are listed in Table 1.

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4.3 Verification and discussion Using the parameters listed in Table 1, the proposed micromechanical model is used to predict the unusual temperature-dependent deformation of Mg-NiTi composites. At first, the predicted stress-strain responses of NiTi SMA phase are given in Figs. 5a and 5b. From Fig. 5a, it is seen that the NiTi SMA phase exhibits a typical super-elastic behavior. In fact, at the temperature lower than

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383K, the initial resistance of dislocation glide in the austenite phase is much higher than the finish stress of martensite transformation, and then the inelastic deformation mechanism of NiTi SMA phase is dominated by the martensite transformation (in the region I in Fig. 2c). The residual strain is caused by the residual martensite phase which cannot fully transform to the austenite phase even if

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the applied stress is completely unloaded. When the temperature reaches to 403K, the start and finish stresses of martensite transformation increase, but the initial resistance of dislocation glide decreases to a great degree. So, the martensite transformation and the dislocation glide in the austenite phase occur simultaneously (in the region II in Fig. 2c), and the residual strain is mainly originated from

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the plastic deformation in the austenite phase. With the further increase of temperature, the initial resistance of dislocation glide further decreases and is much lower than the start stress of martensite

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transformation. Therefore, the inelastic deformation mechanism of NiTi SMA phase at high temperatures (e.g., 473K and 523K) is dominated by the dislocation glide in the austenite phase and

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the martensite transformation is greatly suppressed by the plastic deformation of austenite phase (in the region III in Fig. 2c). Thus, it can be concluded that the “yield stress” (represented the martensite

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transformation or the dislocation glide in the austenite phase) of NiTi SMA changes non-monotonically with the variation of ambient temperature due to the change in inelastic

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deformation mechanism near two critical temperatures of austenite plasticity (i.e., Ad1 and Ad2). The experimental and predicted stress-strain responses of the composites at various temperatures

with a peak strain of 2% are given in Figs. 4a to 4e. It is seen that: (1) at any given temperature, the elastic modulus, yielding stress and inelastic hardening modulus of the composites increase with the increasing volume fraction of NiTi SMA phase; (2) the composites exhibit an unusual temperature-dependent deformation, i.e., the inelastic hardening modulus decreases with the increase of temperature, but the yield strength changes non-monotonically with the variation of temperature, and a maximum yield strength can be obtained at 423K. The predicted results at room temperature

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with large peak strains are shown in Fig. 4f. It should be noted that only the strain-hardening stage is predicted by the proposed model. The strain-softening is mainly caused by the damage of the composites (as shown in Fig. 1b), which is beyond the scope of this work. From Fig. 4a-4f, it is seen that all the experimental phenomena can be well captured by the proposed model since different inelastic deformation mechanisms and the temperature-dependent deformations of two phases, as well as the nonlinear transformation and plastic hardening behaviors have been comprehensively

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considered.

In Figs. 6a to 6e, the black solid curves show the predicted overall stress-strain responses of the composites; while the red dash and blue dash-dot curves show the predicted local responses of NiTi SMA and Mg phases, respectively. The critical points in the overall responses are denoted as o, a, b,

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c and d. The corresponding points are denoted as o', a', b', c' and d' in the local responses of Mg and o'', a'', b'', c'' and d'' in those of NiTi SMA. From Fig. 6a, it is seen that in the segment o-a, the overall stress-strain response of the composites exhibits linearity which means the Mg and NiTi SMA phases deform elastically (o'-a' and o''-a''). The local stress in the NiTi SMA or Mg phase is much higher or

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lower than the overall applied stress, which is caused by the stress redistribution in the composites since the elastic modulus of NiTi SMA is much higher than that of Mg. Once the Mg phase enters the

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stage of plastic deformation, the composites exhibit obvious plastic flow (e.g., in the segment a-b). When the applied stress reaches to a critical value (at point b), the martensite transformation occurs

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in the NiTi SMA phase, and a large transformation strain is produced (segment b''-c''). So, the overall hardening modulus of the composites further decreases (segment b-c). In the unloading process

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(segment c-d), only the elastic deformation occurs and a large residual strain can be observed after unloading. In fact, due to the large residual stress, the reverse transformation of NiTi SMA cannot

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occur even if the applied stress is fully unloaded, as shown in Fig. 6a. Thus, the residual strain originates from the residual martensite in the NiTi SMA phase and the plastic strain in the Mg one. It should be noted that at other temperatures, the initial strains and stresses of NiTi SMA and Mg phases are not equal to zero, as shown in Figs. 6b to 6e. This is caused by the differences of thermal properties (represented by the stress-temperature tangent moduli, i.e., DTM and DTI ) between the NiTi SMA and Mg phases and the resulting initial internal stress, which provides an additional strengthening mechanism. It should be pointed out that the understanding of the different inelastic

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deformation mechanisms in segments o-a-b-c-d is based on the assumption and model formulation, rather than the experimental measurement/observation. To sum up, through the calculations of our proposed model, the strengthening mechanism of Mg-NiTi composites can be summarized as the high elastic modulus, high “yield stress” (corresponding to the martensite transformation or the dislocation glide in the austenite phase) of

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NiTi SMA and the differences of thermal properties between the NiTi SMA and Mg phases.

5. Conclusions

(1) Based on the mean-field approach, a micromechanical constitutive model is constructed to describe the unusual temperature-dependent deformation of Mg-NiTi composites. The

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constitutive model of NiTi SMA phase adopted in this work is a simplified version of Hartl-Lagoudas model (Hartl and Lagoudas, 2009), and the dependences of the plastic yield and elastic modulus of austenite phase on temperature are newly considered. An elastic-plastic model including a new nonlinear plastic hardening law is employed to describe the

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stress-strain response of Mg phase.

(2) To describe the interaction between two phases and calculate the macroscopic overall

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stress-strain response of the composites, the incremental Mori-Tanaka homogenization method based on the Eshelby's tensors for a spherical inclusion embedded in a finite spherical domain

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(Li et al., 2007a; 2007b) is further modified. (3) Comparisons between the simulated results and the corresponding experimental ones

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(Aydogmus, 2015) show that the temperature-dependent deformation of the composites with different volume fractions of NiTi SMA phase can be reasonably described by the proposed

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model.

(4) Predictions of the proposed model prove that the unusual temperature-dependent deformation of the composites originates from the different inelastic deformation mechanisms of NiTi SMA presented at three different temperature regions. At the temperature lower than Ad2, the “yield stress” (dominated by the martensite transformation) increases with the increase of temperature; while, at the temperature higher than Ad2, the “yield stress” (dominated by the dislocation glide in the austenite phase) decreases with the increase of temperature.

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Acknowledgements Financial supports by the National Natural Science Foundation of China (11602203; 11532010; 11702060), Young Elite Scientist Sponsorship Program by CAST (No. 2016QNRC001) and

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Fundamental Research Funds for the Central Universities (2682018CX43) are appreciated.

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Wang, H., Capolungo, L., Clausen, B., Tomé, C. N., 2017. A crystal plasticity model based on

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transition state theory. Int. J. Plasticity 93, 251-268. Xi, Y. L., Chai, D. L., Zhang, W. X., Zhou, J. E., 2005. Ti-6Al-4V particle reinforced magnesium

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matrix composite by powder metallurgy. Mater. Lett. 59(14-15), 1831-1835. Xiong, Y., Yu, Q., Jiang, Y. 2014. An experimental study of cyclic plastic deformation of extruded

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ZK60 magnesium alloy under uniaxial loading at room temperature. Int. J. Plasticity 53, 107-124.

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Yamada, Y., Taya, M., Watanabe, R., 1993. Strengthening of metal matrix composite by shape memory effect. Mater. Trans. 34(3), 254-260.

Yan, B., Li, G., 2005. Mg alloy matrix composite reinforced with TiNi continuous fiber prepared by ball-milling/hot-pressing. Compos. Part A-Appl. S. 36(11), 1590-1594. Yan, W., Wang, C. H., Zhang, X. P., Mai, Y. W. 2003. Theoretical modelling of the effect of plasticity on reverse transformation in superelastic shape memory alloys. Mater. Sci. Eng. A, 354(1-2), 146-157. Ye, H. Z., Liu, X. Y., 2005. Microstructure and tensile properties of Ti6Al4V/AM60B magnesium

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ratchetting

of

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alloy:

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Zhu, Y., Dui, G., 2009. Effect of fiber shape on mechanical behavior of composite with elastoplastic

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matrix and SMA reinforcement. J. Mech. Behav. Biomed. 2(5), 454-459.

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Table 1 Material parameters for simulating the experiments of Aydogmus (2015). Parameters related to Mg: Tr EMg =10GPa;  Mg =0.3;  Mg =26.1  10-6/K; K 0Tr =39MPa; hTr =1GPa; K1 =65MPa; m1 =2;

K 2 =90MPa; m2 =4; T * =340K; n =4.25.

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Parameters related to NiTi SMA: Tr =30GPa;  NiTi =0.3;  A =11  10-6/K;  M =6.6  10-6/K; aT =6.67  10-3/K; H max =0.06; ENiTi

s0 =-6  104Pa; X 0 =8MPa; Y =9MPa; a1 =45MPa; n1 =0.4； n2 =0.6;

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M

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 =2.1MPa/K; R0 =620MPa; Rsat =720MPa; CH =400.

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(a) 373K

423K

Stress, MPa

RT

473K

Strain

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M

Stress, MPa

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(b)

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523K

Strain

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Fig. 1 (a) Stress-strain curves of Mg-NiTi SMA composites at various temperatures (room temperature RT, 373K, 423K, 473K and 523K) with a peak strain of 2%; (b) stress-strain curves at

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RT with large peak strains (cited from Aydogmus (2015)).

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700 393K

(a)

Stress (MPa)

365K

Stress, MPa

(b)

600

385K

345K 325K 309K 295K

223.7K 241.0K 273.2K 283.7K

500 400 300 200 100 0

0

1

2

3

4

5

6

7

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Strain (%)

Strain, %

Start stress of martensitic transformation

(c)

Finish stress of martensitic transformation

Stress

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Critical stress of dislocation glide in austenite phase

I

II

Ad1

Ad2

M

Af

III

Temperature

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Fig. 2 (a)Temperature-dependent stress-strain curves of Ni56.4Ti43.6 SMA (wt%, cited from Qian et al. 2006, Af=289K); (b) temperature-dependent stress-strain curves of Ni50.6Ti49.4 SMA (at%, cited from

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Miyazaki et al. 1981, Af=221K); (c) illustration for the temperature-dependent martensitic

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transformation and dislocation glide in the austenite phase of NiTi SMA (T>Af, referring to Hartl and Lagoudas, 2009).

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ε ,T

(a)

(b) Model

Mg

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NiTi

Modified Mori-Tanaka homogenization method

(d)

ε*

εb , T

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DM , DTM ,

(c)

εb , T

DM , DTM

DI , DTI

M

Eshelby equivalent

DM , DTM

Fig. 3 (a) the real microstructures of Mg-NiTi SMA composites (the volume fraction of NiTi

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SMA phase is 20%, cited from Aydogmus (2015)); (b) simplified micromechanical model (the NiTi SMA is modeled as the matrix and Mg is modeled as spherical inhomogeneity); (c)

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Mori-Tanaka homogenization method; (d) Eshelby’s equivalent inclusion principle.

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200

(a)

120

(b)

160 c

b 80 a 40

b

120 80

a

0.5

1.0

1.5

2.0

0 o 0.0

2.5

Strain (%) 200

(c)

Stress (MPa)

40

1.5

2.0

0%, Exp. 0%, Sim. 10%, Exp. 10%, Sim.

a

60

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b

0.5

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o 0 0.0

1.0

1.5

2.0

2.5

2.0

2.5

80

40

0 0.0

0.5

c

2.0

2.5

1.0

1.5

Strain (%) 400

20%, Exp. 20%, Sim. 30%, Exp. 30%, Sim.

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Stress (MPa)

90

2.5

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Strain (%)

(e)

1.5

20%, Exp. 20%, Sim. 30%, Exp. 30%, Sim.

0%, Exp. 0%, Sim. 10%, Exp. 10%, Sim.

M

1.0

1.0

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80

0.5

(d)

120

(f)

300

Stress (MPa)

Stress (MPa)

160

120

0 0.0

0.5

Strain (%)

20%, Exp. 20%, Sim. 30%, Exp. 30%, Sim.

0%, Exp. 0%, Sim. 10%, Exp. 10%, Sim.

160

30

c

40

0 o 0.0

120

20%, Exp. 20%, Sim. 30%, Exp. 30%, Sim.

0%, Exp. 0%, Sim. 10%, Exp. 10%, Sim.

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Stress (MPa)

20%, Exp. 20%, Sim. 30%, Exp. 30%, Sim.

0%, Exp. 0%, Sim. 10%, Exp. 10%, Sim.

Stress (MPa)

160

20%, Exp. 20%, Sim. 30%, Exp. 30%, Sim.

0%, Exp. 0%, Sim. 10%, Exp. 10%, Sim.

200

100

0

o 0

Strain (%)

3

6

9

12

15

Strain (%)

Fig. 4 Experimental and predicted stress-strain curves of Mg-NiTi composites at various

temperatures (the volume fractions of NiTi SMA phase are 0%, 10%, 20% and 30%): (a) RT; (b) 373K; (c) 423K; (d) 473K; (e) 523K; (f) RT with large peak strains (the experimental data are cited from Aydogmus (2015)).

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600

(a)

(b)

400 300

500 RT 323K 353K 383K

Stress (MPa)

Stress (MPa)

500

200 100 0 -100 -0.5

400

403K 423K 473K 523K

300 200 100 0

0.0

0.5

1.0

1.5

2.0

-100 -0.5

2.5

Strain (%)

0.0

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0.5

1.0

1.5

2.0

2.5

Strain (%)

Fig. 5 Predicted stress-strain responses of NiTi SMA at various temperatures: (a) RT, 323K, 353K

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M

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and 373K; (b) 393K, 423K, 473K and 523K.

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c'' Overall Mg NiTi SMA

200

b'' c

a'' a

50

b d''

c' b'

a'

o'' o o'

0

d

300

0.0

0.5

1.0

b''

2.0

100 o o'

500 c''

0 -100 -0.5

d''

a o o' 0.0

Stress (MPa)

c

b o''

c'

b'

a'

400

Overall Mg NiTi SMA

300

b''

200

d' 1.0

1.5

2.0

Strain (%)

-100 -0.5

2.5

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Stress (MPa)

2.5

c''

c

b

a

o'' o

d'' b' d

a'

o'

0.0

0.5

c' d'

1.0

1.5

2.0

2.5

(e)

300

CE

2.0

Strain (%)

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400

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1.5

a''

100 0

d

0.5

1.0

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a''

100

(d)

b''

300 200

0.5

M

Stress (MPa)

400

d'

Strain (%)

(c) Overall Mg NiTi SMA

c' d

a'

0.0

Strain (%) 500

b d'' b'

a o''

-100 -0.5

2.5

c

a''

0

1.5

c''

200

d'

-50 -100 -0.5

Overall Mg NiTi SMA

400

150 100

(b)

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250

Stress (MPa)

500

(a)

Stress (MPa)

300

Overall Mg NiTi SMA

b''

200

a''

100 0

-100 -0.5

c''

b

a

o''

c

d'' b'

o o'

a'

0.0

0.5

c' d d'

1.0

1.5

2.0

2.5

Strain (%)

Fig. 6 Predicted overall and local stress-strain responses of Mg-NiTi SMA composites at various

temperatures (the volume fractions of NiTi SMA phase are 30%): (a) RT; (b) 373K; (c) 423K; (d) 473K; (e) 523K.