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Phase field simulation on the cyclic degeneration of one-way shape memory effect of NiTi shape memory alloy single crystal Bo Xu , Guozheng Kang , Qianhua Kan , Chao Yu , Xi Xie PII: DOI: Reference:
S0020-7403(19)33016-4 https://doi.org/10.1016/j.ijmecsci.2019.105303 MS 105303
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
13 August 2019 23 October 2019 3 November 2019
Please cite this article as: Bo Xu , Guozheng Kang , Qianhua Kan , Chao Yu , Xi Xie , Phase field simulation on the cyclic degeneration of one-way shape memory effect of NiTi shape memory alloy single crystal, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105303
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Highlights The martensite reorientation mode is gradually converted Reverse transformation is restrained by the plastic deformation Plastic deformation and residual martensite lead to the decrease of recoverable
strain Degeneration of OWSME of NiTi SMA single crystal occurs during the cycling
Phase field simulation on the cyclic degeneration of one-way shape memory effect of NiTi shape memory alloy single crystal Bo Xu1, Guozheng Kang1,2*, Qianhua Kan1,2, Chao Yu1,2, Xi Xie1 1
Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province,
School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China 2
Institute of Applied Mechanics, State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, 610031, China
*Correspondent author: Prof. Kang, Tel: 86-28-87603794; Fax: 86-28-87600797 E-mail address:
[email protected] ABASTRCT In the framework of thermodynamics and based on the Ginzburg-Landau’s theory and crystal plasticity, a three-dimensional phase field model was constructed to predict the cyclic degeneration of the one-way shape memory effect (OWSME) of NiTi shape memory alloy (SMA) single crystal. Two inelastic deformation mechanisms, i.e., austenite plasticity and martensite plasticity were newly introduced into the proposed model in order to describe the martensite transformation- and reorientation-induced plasticity and the stress-strain response of NiTi SMA single crystal more reasonably. From the phase field simulations, it is found that: during the thermo-mechanical cyclic deformation of NiTi SMA single crystal involving the OWSME, the Mode-I martensite reorientation (i.e., achieved by the migration of twinned interfaces) is gradually converted to the Mode-II one (i.e., achieved by the nucleation and growth of martensite variants with favorable orientations) due to the influence of plastic deformation; the reverse transformation is restrained by the plastic deformation, leading to the accumulation of residual martensite phase; moreover, the twinned martensite phase is gradually transited into the re-oriented one due to the interaction among the plastic deformation,
residual martensite phase and
temperature-induced martensite transformation; the irreversible plastic deformation
and the accumulation of residual martensite phase lead to the decrease of recoverable strain, eventually resulting in the predicted cyclic degeneration of the OWSME of NiTi SMA single crystal. Key words: NiTi shape memory alloy; One-way shape memory effect; Cyclic degeneration; Plasticity; Phase field
1. Introduction Owing to the reversible thermoelastic martensite transformation occurred in NiTi shape memory alloys (SMAs) subjected to thermal and (or) mechanical loadings, the alloys exhibit unique super-elasticity (SE) and shape memory effect (SME). In the last decades, NiTi SMAs had been extensively employed in various engineering fields [1]. During the service of NiTi SMA products, they are inevitably subjected to thermal and (or) mechanical cyclic loadings, and the cyclic deformation can cause the function deterioration and fatigue failure of the products. Thus, it is imperative to investigate the cyclic deformation of such alloys. So far, a lot of experimental and theoretical investigations have been carried out on the cyclic deformation and fatigue failure of super-elastic NiTi SMAs, as summarized by Lagoudas [2], Kang et al. [3], Cisse et al. [4,5] and Kang and Kan [6]. The SE of NiTi SMAs and its cyclic degeneration have been well studied, the cyclic degeneration of SE is attributed to the interaction of martensite transformation and dislocation slipping, and the accumulated residual strain originates from the transformation-induced plasticity (TRIP) and the residual martensite phase due to the incomplete reverse martensite transformation [7-9]. However, different from that for the SE, the researches for the SME of NiTi SMAs are not so comprehensive. Much attention has been paid to the thermo-mechanically coupled cyclic deformation of the NiTi SMAs subjected to a cyclic temperature loading with a constant stress, which is denoted as two-way shape memory effect (TWSME) [10-16]; but little to the one-way shape memory effect (OWSME) of NiTi SMAs.
It is well-known that the uniaxial OWSME of NiTi SMAs with initial twinned martensite phase can be generally divided into three stages, i.e., uniaxial tension-unloading (involving the reorientation and detwinning of B19' twinned martensite phase), heating (involving the thermal-induced reverse transformation of re-oriented martensite phase) and cooling (involving the thermal-induced martensite transformation from the B2 austenite phase to a self-accommodated and twinned B19' martensite one). However, only the stress-strain response of martensitic SMAs at the stage of tension-unloading has been addressed now in most of related references: for instance, Ng and Sun [17] experimentally researched the effect of loading rate on the tension-unloading deformation of martensitic NiTi SMA tubes; Kang et al. [7] investigated the ratchetting of martensitic NiTi SMA bars by performing uniaxial stress-controlled cyclic tests; more recently, Laplanche et al. [18] studied the effect of temperature and texture on the reorientation of martensite variants in the martensitic NiTi SMA wires and sheets under tension-unloading conditions by using digital image correlation (DIC) method. Moreover, in terms of microscopic experiments, Xie et al. [19], Liu et al. [20], Zhang et al. [21], Ezaz et al. [22] and Tadayyon et al. [23] observed the microstructure evolution of martensitic NiTi SMAs during the monotonic tensile loading by transmission electron microscope (TEM) and revealed the multiple mechanisms of martensite deformation involved in such a process, i.e., the elastic deformation and reorientation of twinned martensite domains, martensite detwinning, and the plastic deformation of martensite phase. It was found that dislocations and defects always occurred and accumulated during the inelastic deformation of martensite phase. On the other hand, the martensite transformation of NiTi SMAs during the temperature loading has also been studied by some microscopic observations, such as: Simon et al. [24] observed the evolution of dislocations during the temperature-induced martensite transformation and its reverse in a NiTi SMA single crystal by TEM, and it was found that the temperature-induced martensite transformation was generally accompanied by an increase in dislocation density; Pelton et al. [25] conducted a cyclic temperature loading test of
polycrystalline NiTi SMAs from the temperature below M f
(i.e., the end
temperature of martensite transition) to the one above A f (i.e., the end temperature of austenite transition); it was found that a large decrease (about 25K) in M s (i.e., the start temperature of martensite transition) occurred after 100 temperature cycles with an accumulation of dislocations induced by martensite transformation via the test of differential scanning calorimetry (DSC) and the observation of TEM, which was attributed to the increase in the frictional and elastic energies; Soejima et al. [26] investigated the temperature-induced martensite transformation from B2 to B19' phase in the cyclic heating-cooling process of polycrystalline NiTi SMAs by using scanning electron microscopy (SEM), and the observations showed that the habit plane
variant
clusters
nucleated,
grew
and
disappeared
during
the
temperature-induced martensite transformation and its reverse, and the residual martensite phase accumulated with the increase in the number of cycles. Although the microscopic mechanisms of the tensile deformation of martensite phase and temperature-induced martensite transformation have been revealed in the aforementioned researches, the complete process reflecting the OWSME of NiTi SMAs under thermo-mechanically coupled cyclic loading conditions is rarely addressed. Moreover, the experiments for the cyclic degeneration of OWSME have not been extensively reported now, except for that done by Zhao et al. [27]. Zhao et al. [27] investigated the cyclic degeneration of the uniaxial OWSME of NiTi SMAs by the cyclic uniaxial tension-unloading-heating-cooling tests (named as the OWSME cycling simply), and the results showed that the residual strain accumulated, but the start stress of martensite reorientation and the dissipation energy per cycle decreased with increasing the number of cycles; the cyclic degeneration of OWSME aggravated with the increase of stress/strain amplitude and the cyclic stress-strain response depended strongly on the phase-angle differences between the mechanical and thermal cyclic loadings. On the other hand, based on the experimental observations, a macroscopic multi-mechanism based constitutive model [28] and a crystal plasticity one [29] were constructed to describe and predict the cyclic degeneration of the
OWSME of NiTi SMAs; and a multi-mechanism constitutive model was developed by Xu et al. [30] to describe the degeneration of OWSME of NiTi SMAs observed in the OWSME tests at different peak strains. Nevertheless, the microstructure evolution during the cyclic deformation reflecting the OWSME (i.e., during the “OWSME cycling”) cannot be captured by the macroscopic experiments and constitutive models. Although the microstructure evolution of NiTi SMAs has a major impact on both macro- and microscopic properties of such alloys, the microscopic mechanism of the cyclic degeneration of OWSME is still not clear enough. Because of the lack of microscopic observations during the OWSME cycling, the phase field method was used to predict the cyclic degeneration of OWSME in NiTi SMA single crystal. The phase field method has been widely employed to simulate the microstructure evolution of materials in recent decades, and many researchers have also used it to investigate the microstructure evolution of NiTi SMAs. However, most of the phase field studies on NiTi SMAs have only concerned with the stress- or strain-induced martensite transformation [31-35] and SE [36-40] of such alloys, but the ones addressing the OWSME of NiTi SMAs are still very few. Although Xu et al. [41] recently proposed a phase field model and initially realized the simulation on the OWSME of NiTi SMA single crystal, the model did not consider the effect of plastic deformation on the OWSME and cannot predict the cyclic degeneration of OWSME. Therefore, in this work, based on the Ginzburg-Landau’s and crystal plasticity theories, a novel phase field model was proposed by considering four inelastic deformation mechanisms, i.e., martensite transformation, martensite reorientation, austenite plasticity and martensite plasticity, simultaneously. The constructed phase field model was used to simulate the OWSME cycling of NiTi SMA single crystal. By analyzing the obtained stress-strain-temperature response and the evolutions of microstructure morphology, equivalent stress and plastic strain fields in the simulation cell, the microscopic mechanism of the cyclic degeneration of OWSME in NiTi SMA single crystal was predicted, which can provide a reference and basis for the establishment of related constitutive models and the evaluation to the fatigue life and reliability of NiTi SMA components.
2. Crystal plasticity based phase field model As the OWSME of NiTi SMAs involves the mutual transition between austenite phase and martensite one, the dislocation slipping can occur inside both the two phases. Since the crystal structure of austenite phase is body-centered cubic (BCC) while the one of B19' martensite phase is monoclinic, and the involved slip systems of dislocation slipping inside two phases are different, it is necessary to consider both austenite plasticity and martensite plasticity when the plastic deformation mechanism is introduced in the proposed model. At present, the crystal plasticity theory has been maturely developed and the related constitutive models have been widely employed. In the crystal plasticity models, inelastic strains can be measured by the dislocation slipping in the slip systems with certain crystallographic orientations. Therefore, in this work, the crystal plasticity theory is combined with the phase field theory on the scale of single crystal grain, and then describes the microscopic plastic deformation. 2.1 Definitions of inelastic strains As reported by Otsuka and Ren [42], there are 12 martensite variants in NiTi SMAs. Therefore, the volume fraction of martensite variant is used as order parameter and then 12 order parameters, i.e., i i 1, 2,
,12 are introduced in the phase
field model to measure the martensite transformation and its reverse occurring during the OWSME cycling of NiTi SMA single crystal. In the simulation system, the material point is the austenite phase if 1 =2 =
=12 0 ; while it becomes to be the
i-th martensite variant if i =1 and j =0 for all i j . The 12 order parameters should
satisfy
the
constraint
conditions
of
0 i 1 i 1, 2,
,12
and
12
0 i 1 . In addition, to measure the martensite reorientation involved during the i 1
OWSME cycling, other order parameters, i.e., , 1, 2,
,11; 1,
,12
are introduced to measure the transition from the -th martensite variant to the
-th
one
at
each
1 , 1 1, 2,
material
point,
,11; 1,
and
the
constraint
condition
of
,12 should be satisfied.
Since the OWSME of NiTi SMA mainly involves the elastic deformation of martensite phase, the reorientation of martensite variants, martensite transformation and its reverse, and the plastic deformation of austenite and martensite phases [23, 25, 42], the total strain ε at each material point is decomposed into five parts [29], i.e., elastic strain ε e , transformation strain ε tr , reorientation strain ε reo , austenitic plastic strain ε Ap and martensitic plastic strain ε Mp from the assumption of small deformation. That is,
ε=εe εtr εreo ε Ap εMp
(1)
During the martensite transformation and its reverse, the transformation strain ε tr generated at each material point is set as a function of order parameters i [43], which can be expressed as: 12
εtr εi0i
(2)
i 1
where, ε i0 is the stress-free transformation strain corresponding to the complete transformation from the austenite phase to the i-th martensite variant. During the martensite reorientation, since a strain can generate from the mutually transition between every two of 12 martensite variants (e.g., the strain generates from the transition between -th and -th variants can be denoted as ε0 ε0 , ), the sum of such strains is defined as the reorientation strain (i.e., ε reo ) at each material point according to [41], i.e.,
ε reo ε0 ε 0 , 12
11
(3)
1
where, ε0 and ε 0 are the stress-free transformation strains corresponding to the
-th and -th martensite variants, respectively. Moreover, the relationship
between i (i.e., the order parameter corresponding to the i-th martensite variant during the martensite reorientation) and , is constructed by referring to [41] as: 12
11
i iM k i , 1
where, iM i 1, 2,
i 1, 2,
,12
(4)
,12 are the initial order parameters corresponding to the
twinned martensite phase obtained by temperature-induced martensite transformation; the second term in the right side of Eq. (4) is the increment of i produced in the process of martensite reorientation; k i is the interaction matrix which can be defined by referring to [44] as: k i
i 1 1 i and i 0 otherwise
(5)
Based on the crystal plasticity theory, different plastic deformation mechanisms are considered at each stage in the OWSME of NiTi SMA single crystal. In the stages of temperature-induced martensite transformation and its reverse, only the dislocation slipping inside austenite phase is considered. The rate of austenitic plastic strain, i.e., ε Ap is defined as [40, 45]:
12 nA ε Ap 1 i jA PjA i 1 j 1
(6)
where, jA is the dislocation slipping rate; n A denotes the number of slip systems in 12
austenite phase; 1 i is the volume fraction of austenite phase at each material i 1
point; P jA is the orientation tensor which represents the deformation gradient produced by a unit amount of dislocation slipping on the j-th slip system in austenite phase [45], and it can be written as [46, 47]: PjA
1 m Aj n Aj n Aj m Aj 2
(7)
where, m Aj and n Aj are the slipping direction and slipping plane normal vectors of
j-th slip system in austenite phase, respectively. On the other hand, in the stage of stress-induced martensite reorientation, only the dislocation slipping inside martensite phase is considered. By referring to the definition of ε Ap in Eq. (6), the rate of martensitic plastic strain, i.e., ε Mp is newly defined as: 12 nM ε Mp i kM PkM i 1 k 1
(8)
where, i is the order parameter corresponding to the i-th martensite variant as expressed by Eq. (4); kM and PkM are the dislocation slipping rate and orientation tensor of k-th slip system in martensite phase, respectively; nM is the number of slip systems in martensite phase. According to Wang et al. [47], PkM can be written as: PkM
1 M sk IkM I kM skM 2
(9)
where, s kM and I kM are the slipping direction and slipping plane normal vectors of k-th slip system in martensite phase, respectively. 2.2 Framework of thermodynamics The Helmholtz free energy density of the simulation system is decomposed into four parts, i.e., elastic, thermal [2, 44], chemical [48, 49] and plastic [46] energy densities, and its specific expression can be written as:
1 2
T 1 12 2 f i p 0 2 i 1 T0 plastic
ε e : C : ε e c T T0 T ln
elastic
thermal
(10)
chemical
where, C and c are the elasticity tensor and heat capacity, respectively; T0 is the phase equilibrium temperature which can be calculated by T0 = M s As 2 (where
M s and As are the start temperatures of martensite transformation and its reverse, respectively) [50]; the first and second terms of chemical energy density denote the local free energy and gradient energy densities, respectively; is the gradient
energy coefficient. The local free energy density f 0 is given by a Landau’s polynomial: 2
1 12 2 1 12 3 1 12 2 1 12 4 f 0 i a i b i d i e i 2 i 1 3 i 1 4 i 1 4 i 1
(11)
where, the coefficients a, b, d and e can be expressed as: a 32f 0 ,
b 96f0 12f 0
and
d e 32f0 6f 0 , respectively [51];
f 0
is the
difference of the local free energy densities between austenite and martensite phases and depends on temperature T , and f 0 is the energy barrier opposing the mutual transition between such two phases. Since the plastic work is partly dissipated as heat and partly stored in the material as the defect (such as dislocation) energy, the rate of plastic energy density in Eq. (10) is proposed in this work as: 12 nA A A 12 nM M M 1 i R j j i Qk k i 1 k 1 i 1 j 1 p
(12)
where, R jA and QkM are the plastic resistances corresponding to the j-th slip system in austenite phase and the k-th one in martensite phase, respectively. Further, the balance equation of microscopic forces [52, 53] and that of internal power consumption [53] are, respectively, introduced as:
di pi g i 0
(i 1, 2,
12
12
i 1
i 1
,12)
Pin σ : ε di i pii
(13) (14)
where, d i , p i and g i indicate the microstress vector, internal and external configurational forces corresponding to the i-th martensite variant, respectively. The well-known second law of thermodynamics can be mathematically expressed by the Clausius dissipation inequality as [2]: q T T P in sT 0 T
(15)
where, s and q denote the entropy and heat flux, respectively. Substitution of Eqs. (1),
(6), (8), (10), (12) and (14) into Eq. (15) yields: σ e ε
12 12 i f 0 i e i tr reo ε s T d p i +σ : ε i σ : ε i i i i T i 1 i 1
12 12 nM nA + 1 i σ : PjA jA R jA jA i σ : PkM kM QkM kM i 1 k 1 i 1 j 1
q T 0 T
(16) Since Eq. (16) should be satisfied in the whole process, the elastic stress-strain and entropy-temperature relationships and the expression of microstress can be derived as: ε e s T σ
di
i i 1, 2, i
(17) (18)
,12
(19)
Considering the Fourier’s law of heat flux, i.e., q κ T , where κ is the heat conductivity coefficient, a second-ordered positive definite tensor, the heat flux dissipation inequality distinctly satisfies that: flux
q T 0 T
(20)
On the other hand, since several inelastic processes are involved during the OWSME of NiTi SMAs and the thermodynamic dissipation of each inelastic mechanism cannot be less than zero, stronger thermodynamic constraints related to the martensite transformation and reorientation dissipation and the plasticity one are, respectively, introduced as [46]: 12 f M = pi 0 i i +σ : εtr i σ : ε reo i 0 i i 1 12 12 nM nA p 1 i σ : PjA jA R jA jA i σ : PkM kM QkM kM i 1 k 1 i 1 j 1
(21)
0 (22)
According to Tadayyon et al. [23] and Pelton et al. [25], the temperature-induced martensite transformation and its reverse occur with the generation of dislocations in the process of heating-cooling; and the martensite reorientation occurs with the
generation of dislocations during the stage of tension-unloading. To simplify, it is assumed that:
k 1,
ε reo 0; kM 0 εtr 0; jA 0
j 1,
, nM
, nA
Heating-cooling
(23-a)
Tension-unloading
(23-b)
Substituting Eq. (2) into Eq. (21) with the condition of Eq. (23-a) and Eqs. (3) and (4) into Eq. (21) with the condition of Eq. (23-b), the dissipations related to the martensite transformation and reorientation can be, respectively, obtained as 12 f tr pi σ : εi0 0 i i 0 i i 1
(24-a)
12 11 12 f reo pi 0 i k i σ : ε0 ε 0 , 0 i 1 i 1
(24-b)
Combining Eqs. (13) and (19) and introducing the condition of g i 0 i 1, 2, [52], the internal configuration force pi 2i i 1, 2,
,12
,12 can be obtained. In
addition, a Langevin noise term i x, t [43] that plays a role in thermal fluctuation is
introduced to
assist
the
nucleation
of
martensite
phase
during
the
temperature-induced martensite transformation, and it is turned off when the martensite variants begin to nucleate. Through Eqs. (24-a) and (24-b), the thermodynamic driving force of i (denote as tri ) during the temperature-induced martensite transformation and its reverse and that of , (denote as reo ) during the
martensite reorientation, which are, respectively, conjugated to i and , , can be defined as [41]:
tri 2i σ : εi0 12
i 1
reo 2i
f 0 i i x, t i
f 0 i i 0 0 k σ : ε ε i
1, 2,
i 1, 2,
,12
,11; 1,
(25-a)
,12
(25-b)
For the plastic dissipative inequality, Eq. (22), since the inelastic processes involved in martensite transformation and martensite reorientation are independent of
each other, the thermodynamic dissipation of plasticity in such two processes cannot be, respectively, less than zero. Therefore, stricter thermodynamic conditions are introduced with considering the conditions of Eqs. (23-a) and (23-b) as:
12 trp 1 i σ : PjA jA R jA jA 0 i 1 nM
p reo i σ : PkM kM QkM kM 0 k 1
j 1,
i 1,
, nA
,12; k 1,
(26-a)
, nM
(26-b)
To make the thermodynamic constraints (i.e., Eqs. (26-a) and (26-b)) be always true (i.e., the thermodynamic dissipation of plasticity is non-negative), it is assumed that [29]: sign σ : PjA sign jA
(27-a)
sign σ : PkM sign kM
(27-b)
Substituting Eq. (27-a) into Eq. (26-a) and Eq. (27-b) into Eq. (26-b), the thermodynamic driving forces of
jA
kM
and
p (denote as trp and reo ,
respectively) which are conjugated to jA and kM , respectively, can be defined as follows [46]:
trp σ : PjA R jA
(28-a)
p reo σ : PkM QkM
(28-b)
Therefore, the dissipative inequalities, i.e., Eqs. (24-a), (24-b), (26-a) and (26-b), can be always satisfied when the following thermodynamic conditions are introduced:
tri i 0 reo , 0
i 1,
1,
,12
,11; 1,
(29-a)
,12
(29-b)
trp jA 0
j 1,
, nA
(29-c)
p reo kM 0
k 1,
, nM
(29-d)
2.3 Phase field kinetic equations and evolution equations of slipping rate Considering the constraints of thermodynamics, the control conditions for the
martensite transformation and its reverse can be obtained from Eq. (29-a) as:
i 0 if tri 0
(30-a)
i 0 if tri 0
(30-b)
i =0
other conditions
where, the evolutions of i i 1, 2,
(30-c)
,12 can be controlled by the following phase
field kinetic equations, i.e., time-dependent Ginzburg-Landau (TDGL) equations:
tri 2i σ : εi0
f 0 i i x, t i
i 1, 2,
,12
(31)
where, tr is the kinetic coefficient corresponding to the martensite transformation and its reverse, and tr 0 . Similarly, the control conditions for the martensite reorientation can be obtained from Eq. (29-b) as: , 0 if reo Yreo σ reo : ε0 ε0 0
, Yreo if reo 0
, =0
(32-a) (32-b)
other conditions
(32-c)
where, σ reo is the start stress of martensite reorientation, and an equivalent stress eq reo 224 MPa is set by referring to the experimental data obtained by Xu et al. [30].
The evolutions of , 1, 2,
,11; 1,
,12 can be controlled by other
TDGL equations, i.e., 12
i 1
reo , 2i
f 0 i i 0 0 k σ : ε ε 1, 2, i
,11; 1,
,12 (33)
where, reo is the kinetic coefficient corresponding to the martensite reorientation, and reo 0 . For simplification, it is assumed that tr reo in the simulation in this work. Moreover, it is worth noting that the TDGL equations, i.e., Eqs. (31) corresponding to the martensite transformation and its reverse and Eqs. (33)
corresponding to the martensite reorientation are mutually independent. During the martensite transformation and its reverse, the evolution equation of the austenitic dislocation slipping rate jA corresponding to the j-th slip system in austenite phase is defined in a power-law form, which depends on the local internal stress at the material points and the plastic deformation resistance in austenite phase. The specific expressions [40] are given as following by considering Eq. (29-c):
A j
σ : PjA
mA
R jA
sign σ : PjA if trp 0
jA 0 if trp 0
(34-a) (34-b)
where, mA denotes the viscosity coefficient of austenite phase, and the evolution equation of plastic deformation resistance in austenite phase, i.e., R jA is defined as [40]: R jA H A jA
(35)
where, H A is the hardening parameter of plastic deformation in austenite phase. Regarding the martensite reorientation, and referring to the definitions of jA and
R jA , the evolution equation of martensitic dislocation slipping rate kM corresponding to the k-th slip system in martensite phase is newly defined by considering Eq. (29-d) as:
M k
σ : PkM QkM
mM
p sign σ : PkM if reo 0
p kM 0 if reo 0
(36-a) (36-b)
where, mM is the viscosity coefficient of martensite phase. The evolution equation of plastic deformation resistance in martensite phase, i.e., QkM is newly defined as: QkM H M kM
(37)
where, H M is the hardening parameter of plastic deformation in martensite phase.
2.4 Numerical implementation The numerical implementation of the proposed phase field model is completed by using the MATLAB software and performed for a three-dimensional (3D) cubic cell which is uniformly divided into 64 × 64 × 64 discrete grids. Firstly, a part of the parameters and the Langevin noise term x,t are non-dimensionalized as follows:
x
t f 0 x, t x ;t ; ; x , t l0 f 0 l02 f 0
where, the symbol “
(38)
” indicates the operation of non-dimensionalization;
is the coordinate vector; l0 is the size of grid mesh. x,t is set to
x x1 , x2 , x3
T
meet the requirement of the fluctuation-dissipation theorem [54], i.e., i x, t , j x, t 2kBT ij x x t t
(39)
where, k B and are the Boltzmann constant and Dirac delta function, respectively. Thus, the dimensionless Langevin noise term x,t satisfies the normal distribution with a mean-value of zero and a variance of 2kBT
f l . 0
3 0
To solve the TDGL equations, i.e., Eqs. (31) and (33), the semi-implicit Fourier-spectral method [55] with periodic boundary conditions is used. After substituting Eq. (38) into Eqs. (31) and (33), respectively, the obtained partial differential equations are discretized and subjected to a Fourier’s transformation. Thus, it yields:
1 r
ˆin1 ˆin 12
i 1
k j
12
11
j 1
t ˆ local t ˆ el ˆ Fi Fi i x, t t f 0 f 0
ˆn,1 r k i ˆ njk1k ijk ˆn ,
t f 0
(40-a)
12 t ˆ el local i ˆ F k F r ˆin k i i f i 1 i 1 0 12
(40-b) where, the symbol “
” denotes the operation of Fourier’s transformation;
r 4 2 s12 s22 s32 t ; s s1 , s2 , s3
T
is the coordinate vector in Fourier’s space
corresponding to x; Fi local f0 in in is the driving force of i associated
with the local free energy density f 0 ; Fi el σ : εi0 is the driving force of i during martensite transformation and its reverse while Fel σ : ε0 ε0 is that of
, during martensite reorientation associated with the elastic energy density. 2.5 Parameter determination 12 stress-free transformation strains εi0 i 1,
,12 corresponding to the 12
variants in B19′ martensite phase considered in the proposed phase field model can be given as: ε 0 1
- - ε - -
ε
- ε - - -
ε
- ε - - -
0 5
0 9
- ε - - -
0 2
0 6
0 10
0 3
- - ε - - - 0 7
- ε - - - 0 11
- ε - - - 0 4
- ε - - - 0 8
- - ε - - 0 12
(41) where, 0.0437 , 0.0427 , 0.0580 and 0.0243 by referring to Hane and Shield [56]. Since two different families of slip systems, i.e., 100 011 and 100 001 in austenite phase have been observed by Gall and Maier [57] and Gall et al. [58], the total number of slip systems is 12, i.e., nA 12 . Furthermore, as discussed by Otsuka and Ren [42], since the B19' martensite phase in NiTi SMA is monoclinic, the possible slipping planes are so few and the most possible slip system is only one, i.e.,
100 001 , thus,
nM 1 . However, as the established reference coordinate system in
the whole simulation process is set to be aligned with the cubic axes of B2 austenite single crystal, the slip system corresponding to each martensite variant should be
obtained by a rotation. The difference between the local free energy densities of austenite and martensite phases f 0 (i.e., the so-called transformation driving force) [43] and the energy barrier f 0 [59] can be expressed as:
f0 T Q T T0 T 0.4Q 32 f 0 = 0.8 0.06 T T0 Q 32
(42-a)
when T T0 when T T0
(42-b)
where, Q is the transformation latent heat. By referring to Cahn and Hilliard [60], the interface energy density related to the gradient energy coefficient is defined as
= 4 2 f 0 3 and that of type-I twin is adopted directly as =187 mJm2 [61] in this work. Considering as l0 3
4 f
0
f l 0
2 0
in Eq. (38), the grid size l0 can be obtained
2 .
The values of all the parameters involved in the numerical simulation are summarized in Table 1. Table 1 The values of parameters used in the simulations. Non-zero elasticity constants [62]: C11 183GPa ; C12 146GPa ; C44 46GPa Poisson’s ratio: v 0.33 Transformation latent heat: Q 110MJm-3 Phase equilibrium temperature: T0 337.5K [41] Dimensionless gradient energy coefficient: 120 Dimensionless time step: t 0.008 Parameters related to austenite plasticity: nA 12 ; mA 6 ; H A =800 MPa Parameters related to martensite plasticity: nM 1 ; mM 6 ; H M =1200MPa
3. Simulations and discussion In the previous work [41], the twinned martensite structures with different variant
pairs were obtained by phase field simulation, and the mechanical response in different tensile directions of two representative ones (i.e., the twinned martensite structures with variant pairs of 1:11 and 1:2, respectively) was investigated. Since different twinned martensite structures exhibit the same microstructure evolution during martensite reorientation (i.e., the reorientation is achieved by the migration of twinned interfaces), the one with the variant pair of 1:11 is used as the representative to investigate the OWSME cycling of NiTi SMA single crystal in this work. In order to obtain a twinned martensite phase without plastic deformation and overall strain, the temperature-induced martensite transformation with the 1-st and 11-th variants is simulated first without considering the plasticity. Subsequently, the obtained twinned martensite phase with the variant pair of 1:11 is set as the initial configuration and a cyclic mechanical and thermal loading is applied to it: a maximum tensile strain of 2.5% is applied firstly in the direction of x 10 0B2 (i.e., the 100 direction of B2 austenite single crystal) at room temperature Troom 298K which is lower than the martensite finish temperature M f 313K [29]; secondly, the simulation cell is continuously heated from Troom to Tmax 400 K after unloading it to zero stress; finally, the cell is cooled from Tmax to Tmin , where the number of time steps is set to be 2000, thus, ttotal 20 . Such a cyclic tension-unloading-heating-cooling loading is carried out for 8 cycles. It should be pointed out that it is difficult to obtain a stable twinned martensite phase with a continuous cooling condition through the established model, so not practical cooling process (i.e., continuously from Tmax to Tmin ) but f 0 44 MJm-3 [33] is directly adopted to approximately represent the cooling process (i.e., the simulation cell is quenched to Tmin 241K , which is much lower than M f to get a sufficient driving force for martensite transformation) in each OWSME cycle. A homogenization method is employed to calculate the overall stress and strain of the simulation cell, i.e., volumetric averaging to the stresses and strains at all material
points: σ
1 V
V
σ point d V , ε
1 V
V
ε pointd V . The equivalent stress (i.e., eq ) and
p plastic strain (i.e., eq ) at each material point can be calculated through the following
equations:
eq
eqp
1 2
11 22
2
2 22 33 33 11 6 122 132 23 2
2
2 p 4 p 2 p 2 p p 2 p p 2 p 2 p 2 11 22 22 33 33 11 12 13 23 3 9
(43)
(44)
where, ij i 1, 2,3; j 1, 2,3 and ijp i 1, 2,3; j 1, 2,3 are the components of stress and plastic strain tensors at each material point, respectively. Further, the overall equivalent plastic strain of the simulation cell can be calculated by
eqp
1 V
V
p eq
dV .
3.1 Predicted cyclic degeneration of OWSME for NiTi SMA single crystal Figs. 1a, 1b and 1c depict the stress-strain-temperature (i.e., 11 11 T ) curves, stress-strain (i.e., 11 11 ) ones at the stage of martensite reorientation and the strain-temperature ones for temperature-induced reverse transformation obtained in the simulation, respectively. It is worth noting that in Figs. 1a and 1c, the temperature axis is set to start from Troom 298K because the twinned martensite phase does not change when the temperature increases from Tmin 241K to Troom 298K ; meanwhile, since the cooling processes during the OWSME cycling are not practically continuous decrease of temperature and 11 and 11 would not be changed therein, such a process in each OWSME cycle is denoted by dash line. It is seen from Fig. 1a that as the number of cycles increases, the residual strain accumulates gradually; the peak stress decreases gradually and becomes stabilized finally; and the 11 11 T curves gradually alter from an “open” type to a “closed” one. A nearly stable response will be obtained when the 11 11 T curve becomes
completely “closed” in the 8-th cycle. Fig. 1d shows the experimental cyclic stress-strain-temperature curves of polycrystalline NiTi SMA obtained in the cyclic OWSME tests with a peak stress of 480MPa, and it is seen that the simulated results are qualitatively consistent with such experimental observations.
Fig. 1 (a) Stress-strain-temperature (i.e., 11 11 T ) curves, (b) stress-strain (i.e.,
11 11 ) curves in the tensile direction and (c) residual strain-temperature curves obtained during the simulation for the OWSME cycling of NiTi SMA single crystal; (d) the experimental cyclic stress-strain-temperature curves of polycrystalline NiTi SMA obtained in the cyclic OWSME tests with peak stress of 480MPa (from Zhao et al. [27]) It is found from Fig. 2 that with increasing the number of cycles: (a) the start stress of martensite reorientation in the 2-nd cycle decreases sharply by comparing with that in the 1-st cycle, and then decreases slowly and becomes stabilized finally; (b) the volume fraction of residual martensite phase gradually increases; (c) all the total irrecoverable strain, overall plastic strain in the tensile direction (i.e.,
11p
1 V
V
p 11
d V , where 11p is the component in the tensile direction of plastic strain
tensor at each material point) and the irrecoverable strain caused by the residual martensite phase increase and gradually become stabilized; (d) the recoverable strain decreases; (e) the total increment of overall equivalent plastic strain (i.e., eqp ) and that at the stages of martensite reorientation and forward martensite transformation decrease and tend to be zero, while the eqp during the reverse transformation is closed to zero and almost unchanged.
Fig. 2 Evolutions of (a) the start stress of martensite reorientation, (b) the volume fraction of residual martensite phase, (c) the irrecoverable strains (i.e., the total irrecoverable strain, the overall plastic strain in tensile direction and the irrecoverable strain caused by residual martensite phase), (d) the recoverable strain and (e) the total increment of overall equivalent plastic strain (i.e., eqp ) and the ones at different stages (i.e., martensite reorientation, temperature-induced reverse martensite transformation and forward martensite transformation) obtained during the OWSME cycling of NiTi SMA single crystal. (Note that the “M” in (c) denotes the martensite phase, and the “MR” in (e) denotes the martensite reorientation) According to the simulation data shown in Figs. 1a, 1b, 1c and 2, it is found that the OWSME of NiTi SMA single crystal gradually degenerates with the increase in the number of cycles. In order to reveal the microscopic mechanism of the cyclic degeneration of OWSME, corresponding microstructure, equivalent stress and plastic strain fields obtained in some representative cycles (i.e., the 1-st, 2-nd, 5-th and 8-th cycles) are selected for an in-depth analysis and discussion in the next subsection.
3.2 Evolutions of predicted microstructure, equivalent stress and plastic strain fields in representative cycles In this subsection, to show the predicted microstructure evolution more clearly, the distributions of two variables (i.e., 1 211 and 1 11 ) and their evolutions during the OWSME cycling are addressed. As discussed in Section 2.1, the material point is the austenite phase if 1 =11 =0 ; while it becomes to be the 1-st martensite variant if 1 =1 and 11 =0 , or the 11-th one if 11 =1 and 1 =0 . Therefore, the value of 1 211 can represent the phase state of each material point, i.e., the value of “1” indicates that the material point is the 1-st variant while “2” means the 11-th one, and the value between “1” and “2” means that the two variants coexist; moreover, the value of 1 11 can measure the degree of martensite transformation at a material point, i.e., the value of “1” denotes a completely martensite transformation while “0” denotes no martensite transformation or a completely reverse transformation, and the value between “0” and “1” means that the martensite transformation and its reverse are not fully developed. 3.2.1 1-st cycle Before simulating the OWSME cycling of NiTi SMA single crystal, the twinned martensite phase without any plastic deformation and overall strains is obtained by cooling the single crystal in an austenite phase to a temperature lower than the finish temperature of martensite transformation
M f . The evolutions of predicted
microstructure and equivalent stress field at t ( ttotal 20 ) during the cooling are shown in Fig. 3. It is seen from Fig. 3a that during the temperature-induced martensite transformation, a twinned martensite phase with the (101) type twinned interfaces between the 1-st and 11-th variants is formed in a self-coordinating manner. Fig. 3b shows that a local internal stress is induced during the martensite transformation and eventually concentrated at the twinned interfaces (see Fig. 3b, D) due to the
unmatched inelastic deformation of the two variant layers.
Fig. 3 (a) Microstructure morphology (the color legends indicate the values of
1 211 ) and (b) equivalent stress field (unit: GPa) obtained during the temperature-induced martensite transformation before simulating the OWSME cycling (i.e., the process obtaining initial twinned martensite phase). Subsequently, the simulation cell of NiTi SMA single crystal shown in Fig. 3a D is subjected to a tension-unloading-heating-cooling cyclic loading (i.e., the OWSME cycling) for 8 cycles. The 3D 11 11 T curve obtained in the 1-st cycle and the microstructure morphologies corresponding to the selected key points are shown in Fig. 4.
Fig. 4 3D stress-strain-temperature ( 11 11 T ) curve and microstructure morphologies corresponding to the selected key points (i.e., points A1 to H1) obtained in the 1-st cycle for the NiTi SMA single crystal, where the color legends in pictures A1 to E1 indicate the values of 1 211 but that in pictures F1 to H1 indicate the values of 1 11 . The twinned martensite phase shown in Fig. 4 A1 elastically deforms first with no variation of microstructure morphology under the tensile loading. When a critical stress (about 200MPa) is reached, with the help of local internal stress, the driving force of martensite reorientation becomes sufficient enough to result in the occurrence of the reorientation of variants. Figs. 4 B1 and 4 C1 show that as the twinned interfaces migrate, the regions of 11-th variant (i.e., in red color) gradually expand and take over those of 1-st variant (i.e., in blue color), and the reorientation strain concomitantly
generated leads to the appearance of a stress plateau on the 3D curve in Fig. 4. At the point D1 (the peak strain), the martensite reorientation is completed, and the entire simulation cell is occupied by a single-oriented martensite variant with a favorable orientation (i.e., the 11-th variant). The elastic deformation of the re-oriented martensite phase occurs during the unloading and no variation of microstructure morphology occurs, but a large residual strain is retained when the cell is unloaded to zero stress (point E1). Subsequently, a uniform temperature field is applied to the simulation cell for heating. The residual strain varies slightly until the temperature reaches the As (the start temperature of reverse martensite transformation, as marked in the 3D curve in Fig. 4). The value of 1 11 at each material point decreases (which implies that the martensite phase transforms gradually into the austenite one) with increasing the temperature after the As is reached, as shown in Figs. 4 F1 to 4 G1. At point H1 where the reverse martensite transformation is terminated, but the values of 1 11 do not completely decrease to zero, which means that a small amount of residual martensite phase is retained at the end of heating process. Fig. 5a illustrates the equivalent stress fields corresponding to the key points selected in the 1-st cycle. After the martensite reorientation begins, the internal stress concentrated at the twinned interfaces is relaxed as the interfaces migrate (see Fig. 5a B1). However, there are few local regions in the simulation cell (such as the circled regions shown in Fig. 5a D1) where the internal stress is not completely relaxed and is still significantly higher than that in other regions even the reorientation is ended. When unloading the cell to zero stress, a residual internal stress (the maximum is about 420MPa) is retained in the circled regions, as shown in Fig. 5a E1, which is caused by the existence of a small number of residual interfaces after the reorientation. Then, during the reverse martensite transformation induced by heating, the residual interfaces gradually disappear, resulting in the relaxation of corresponding local internal stress (see Fig. 5a F1). Nevertheless, the local internal stress cannot be completely relaxed after the reverse martensite transformation due to the existence of
plastic deformation; thus, some local regions with residual internal stress (the maximum is about 50MPa) are found in the simulation cell (see Fig. 5a H1).
(a) equivalent stress field (unit: GPa)
0 (b) distributions of 1 +11 at point E1 and σ : ε11 at points E1, F1 and H1
(c) equivalent plastic strain field (unit: %)
Fig. 5 (a) Equivalent stress fields corresponding to the key points selected in the 1-st cycle, (b) distributions of the values of 1 +11 at point E1 and that of the stress part 0 of transformation driving force (i.e., σ : ε11 , unit: MPa) at points E1, F1 and H1 and (c)
equivalent plastic strain fields corresponding to the key points selected in the 1-st cycle. It is noted that the values of 1 11 in the circled region (see Fig. 4 F1) are always lower than that in other regions during the reverse martensite transformation in the 1-st cycle. From the equivalent stress field (Fig. 5a E1) and the distribution of
1 11 (Fig. 5b E1) at point E1, it is found that the circled region in Fig. 4 F1 corresponds to the residual interfaces after unloading. During the simulation in this work, since the twinned interfaces appear always in the locations where the austenite phase and martensite one coexist, the values of 1 11 in the residual interfaces are slightly lower than that in other regions due to the existence of a small amount of austenite phase. On the other hand, the driving force of reverse martensite transformation (i.e., Eq. (25-a)) mainly consists of two parts, i.e., the stress part (i.e., σ : εi0 ) and the temperature one (i.e., f 0 i i , which is also related to the values
of i ). The applied uniform temperature field makes the f 0 i i at each material point identical since the values of 1 11 are almost the same there; while the non-uniformly distributed internal stress can result in a difference in the σ : εi0 . Further, from the driving force (Eq. (25-a)) and thermodynamic constraint (Eq. (30-b)) for reverse martensite transformation, it can be concluded that the internal stress can hinder the reverse martensite transformation if the value of σ : εi0 is positive; conversely, it can promote the reverse martensite transformation if the value of σ : εi0 is negative. Since the 11-th variant is obtained after the reorientation, the distribution 0 of σ : ε11 is concerned and the obtained results corresponding to points E1, F1 and H1
are provided in Fig. 5b E1 , F1 and H1, respectively. It is found that the values of 0 in the circled regions (see Fig. 5b E1 ) are all positive, which implies that the σ : ε11
reverse martensite transformation is hindered in such regions. In addition, the values 0 of σ : ε11 at points F1 (during the reverse martensite transformation) and H1 (at the
end of reverse martensite transformation) are either positive or negative and their distributions are both scattered. Nevertheless, the positive values are dominant in a whole sense, which indicates that the reverse martensite transformation occurring in the simulation cell is hindered and eventually leads to the occurrence of a residual 0 martensite phase. Since the residual internal stress (i.e., the σ in σ : ε11 ) is
correspondent to the accumulated plastic deformation occurring in the OWSME cycling, it implies that the plastic deformation can restrain the reverse martensite transformation. The equivalent plastic strain fields in the 1-st cycle are given in Fig. 5c. During the reorientation occurring in the original twinned martensite phase, the equivalent plastic strains in the regions of two variants vary differently (see Fig. 5c B1). Because the slip systems of two variants are different (i.e., different PkM in Eq. (8)) in the same austenitic coordinate system, significantly different equivalent plastic strains can be obtained, even if the difference of internal stresses in the regions of two variants is slight. As the reorientation proceeds, the equivalent plastic strain field gradually evolves to be homogeneous, as shown in Fig. 5c B1 to D1. The plasticity generated during the martensite reorientation is called as the reorientation-induced plasticity as addressed by Yu et al. [29]. And the simulated results here are qualitatively consistent with the experimental ones observed by Liu et al. [20], Ezaz et al. [22] and Tadayyon et al. [23] (i.e., the reorientation of martensite variants occurs through the migration of twinned interfaces when the martensitic NiTi SMAs are stretched, and it is generally accompanied by a generation of dislocations, even if the applied stress is much lower than the martensite yield stress). In the stage of reverse martensite transformation, due to the low internal stress level, almost no new plastic strain is accumulated and the
equivalent plastic strain field remains unchanged (as shown in Fig. 5c E1 to H1). It is worth noting that the equivalent plastic strain (i.e., eqp ) at each material point is calculated by Eq. (44), which is different from the overall plastic strain (i.e., 11p ) of the simulation cell in the tensile direction (given in Fig. 2c); hence, the values of eqp (as shown in Fig. 5c H1, ranging from about 0.6% to 1%) differ greatly from that of
11p (about 0.34%) in the 1-st cycle. 3.2.2 2-nd cycle After the reverse martensite transformation at the heating stage of the 1-st cycle, an austenite NiTi SMA single crystal with a plastic deformation and a small amount of residual martensite phase is obtained. To obtain a twinned martensite phase again, the simulation cell (Fig. 4 H1) is cooled to the temperature lower than M f . Thus, the predicted microstructure morphologies, equivalent stress and plastic strain fields at different t ( ttotal 20 ) are presented in Fig. 6.
Fig. 6 (a) Microstructure morphology (the color legends indicate the values of
1 211 ), (b) equivalent stress (unit: GPa) and (c) plastic strain fields obtained during the temperature-induced martensite transformation after the reverse martensite transformation at the heating stage of the 1-st cycle. As seen from Fig. 6a, with the increase of t , two variants gradually nucleate and grow and an alternative lamellar twinned martensite phase is eventually formed. Different from Fig. 3a D, the Miller index of the twinned interfaces here is not (101) and parts of the interfaces are irregular (see Fig. 6a D) due to the effect of plastic deformation. The equivalent stress fields in Fig. 6b shows that, before the variants nucleate (Fig. 6b A), different from the case without considering plasticity (i.e., the equivalent stress field is almost homogeneous, as shown in Fig. 3b A), there are certain local regions in the cell where the equivalent stress is obviously higher than that in other regions, which is caused by the interaction between the plastic deformation and residual martensite phase accumulated after the 1-st cycle. Moreover,
the internal stress is concentrated only at the twinned interfaces when the martensite transformation is completed in the case without considering plasticity (Fig. 3b D); while with the influence of plastic deformation, besides the internal stress accumulated at the twinned interfaces, a high internal stress is found in certain local regions inside the obtained twinned martensite phase (see Fig. 6b D). It is seen from Fig. 6c that, there are regions with distinctly high equivalent plastic strains within the simulation cell before the nucleation of martensite phase ( t 6 ), and a new plastic strain is gradually accumulated owing to the local internal stress. The plasticity generated during the martensite transformation (essentially caused by dislocations and their slipping) is called as the transformation-induced plasticity (TRIP) as observed by Simon et al. [24] and Pelton et al. [25] (the experimental observations found that the temperature-induced martensite transformation was generally accompanied by an increase in dislocation density). Finally, the equivalent plastic strains at twinned interfaces and in certain local regions with high internal stresses are significantly higher than that in other regions (as shown in Fig. 6c D). Furthermore, the comparison of Figs. 6a B and 6b A demonstrates that the nucleation sites of two variants correspond to the locations with high equivalent stresses, which means that the local internal stress can assist the nucleation of martensite variants during the temperature-induced martensite transformation. Fig. 7 gives the 11 11 T curve and predicted microstructure morphologies for the key points selected in the 2-nd cycle. During the martensite reorientation in the 2-nd cycle, it is interesting that the variant with a favorable orientation (i.e., the 11-th variant in red color) directly nucleates in certain locations (such as “S 1” in Fig. 7 B2) inside the variant with an unfavorable one (i.e., the 1-st variant in blue color) when the twinned interfaces are migrating. Subsequently, the twinned interfaces continue to migrate, and the 11-th variant nucleated previously (see “S1” in Fig. 7 B2) grows gradually; meanwhile, the 11-th variant nucleates and grows in some new locations inside the 1-st variant (such as “S2” in Fig. 7 C2). It is demonstrated that the martensitic reorientation mode is changed, i.e., besides the migration of twinned
interfaces (called as Mode-I), a new reorientation mode, i.e., the nucleation and growth of the variants with favorable orientations (called as Mode-II) emerges (a detailed discussion will be presented in Section 3.3). In general, the Mode-I martensite reorientation is still dominant in the 2-nd cycle. At point E2 in Fig. 7 (the point with the peak strain), the reorientation is completed, leading to the monotonic 11-th variant. At the stages of subsequent unloading and reverse martensite transformation, the microstructure evolution is similar to that in the 1-st cycle. After the reverse martensite transformation (Fig. 7 H2), the residual martensite phase is further accumulated in the simulation cell, compared to that in the 1-st cycle (Fig. 4 H1).
Fig. 7 Stress-strain-temperature ( 11 11 T ) curve and microstructure morphologies for the selected key points (i.e., points A2 to H2) in the 2-nd cycle, where the color
legends in pictures A2 to F2 indicate the values of 1 211 , while that in pictures G2 and H2 indicate that of 1 11 . Figs. 8a and 8b report the equivalent stress and plastic strain fields corresponding to the selected points in the 2-nd cycle, respectively. As indicated in Fig. 8a, the equivalent stresses in the regions of two variants (such as the circled and framed regions in Fig. 8a B2) are distinctly different after the reorientation starts. It is because the plastic strains in the regions of two variants and their evolutions are different, which leads to a different influence on the local internal stress. From the equivalent stress fields shown in Fig. 8a F2 (unloading to zero stress) and H2 (at the end of reverse martensite transformation), it is found that the residual internal stress levels are clearly higher than that in the 1-st cycle (as shown in Fig. 5a E1 and H1, respectively), correspondingly. From Fig. 8b A2 to E2, it is seen that the equivalent plastic strains in two variants gradually accumulate. During the reverse martensite p transformation (Fig. 8b F2 to H2), the values of equivalent plastic strains (i.e., eq ) in p the regions with higher eq (such as the circled regions in Fig. 8b F2) gradually p decrease; while those in other regions with lower eq (such as the framed region in
Fig. 8b F2) increase gradually. It implies that the whole equivalent plastic strain field evolves to be nearly homogeneous.
(a) equivalent stress field (unit: GPa)
(b) equivalent plastic strain field (unit: %) Fig. 8 (a) Equivalent stress and (b) plastic strain fields corresponding to the key points selected in the 2-nd cycle. 3.2.3 5-th cycle The simulated results obtained in the 5-th cycle are shown in Fig. 9. It is found in Fig. 9 A5 that the proportion of 11-th variant in the obtained twinned martensite phase is significantly reduced by comparing with that in the first two cycles (as shown in Fig. 4 A1 and Fig. 7 A2, respectively). During the martensite reorientation (see Figs. 9 B5 to 9 D5), when the twinned interfaces migrate, the 11-th variant continuously nucleates and grows inside the 1-st variant. Compared to that in the 2-nd cycle, the reorientation mode in the 5-th cycle is still the combination of Mode-I and Mode-II, whereas the Mode-II martensite reorientation becomes dominant here. When loading the cell to the peak strain (point D5), the reorientation is not completed and both variants exist in the simulation cell; moreover, the local regions where two variants coexist (i.e., the corresponding values on the color legend are between “1” and “2”) are found. More residual martensite phase is accumulated after the reverse martensite transformation (Fig. 9 H5).
Fig. 9 Stress-strain-temperature ( 11 11 T ) curve and microstructure morphologies for the key points (i.e., points A5 to H5) selected in the 5-th cycle, where the color legends in pictures A5 to E5 indicate the values of 1 211 ; while that in pictures F5 to H5 indicate that of 1 11 . The equivalent stress and plastic strain fields at points B5 and H5 in the 5-th cycle are shown in Fig. 10. It is illustrated from Fig. 10 B5 that the local internal stress is mainly concentrated around the 11-th variant when the martensite reorientation starts. Comparison of Fig. 9 B5 and Fig. 10 B5 indicates that the nucleation sites of 11-th variant inside 1-st one (such as the circled regions in Fig. 9 B5) correspond to the regions with high equivalent plastic strains. In addition, the equivalent stress (Fig. 10 H5) and plastic strain (Fig. 10 H5 ) fields at the end of reverse martensite
transformation (i.e., point H5 in Fig. 9) are compared with that in the 2-nd cycle (as shown in Figs. 8a H2 and 8b H2, respectively); as a result, it is found that the residual internal stress level in the austenite single crystal with a residual martensite phase increases gradually during the cyclic loading, and the plastic deformation is accumulated continuously.
Fig. 10 Equivalent stress and plastic strain fields at points B5 and H5 in the 5-th cycle. 3.2.4 8-th cycle Fig. 11 shows the microstructure evolution in the process obtaining twinned martensite phase (where ttotal 20 ) prior to the tensile loading stage in the 8-th cycle. It is seen that the microstructure evolutions before the variants nucleate (see Fig. 11 A) and during its nucleation and growth (see Fig. 11 B) are similar to that in the previous cycles (Fig. 6a). Furthermore, it is found from Figs. 11 C and 11 D that the formation of 11-th variant is suppressed and the formed twinned martensite phase at t 16 is unstable, which leads to the absence of 11-th variant in the stable microstructure obtained finally.
Fig. 11 Microstructure evolution during the temperature-induced martensite transformation prior to the tensile loading stage in the 8-th cycle, where the color legends indicate the values of 1 211 .
The 11 11 T curve and the microstructure morphologies of the simulation cell corresponding to the selected points in the 8-th cycle are shown in Fig. 12. Although there is almost no 11-th variant in the simulation cell (Fig. 12 A8), it nucleates and grows simultaneously in various locations inside 1-st variant after the reorientation begins, as shown in Figs. 12 B8 to 12 D8. Here, the Mode-I martensite reorientation is completely converted to the Mode-II one. When loading the cell to the peak strain (Fig. 12 D8), the reorientation is not completed to more degree than that in the 5-th cycle. Currently, both variants exist in the simulation cell, and there are more local regions where two variants coexist than that in the 5-th cycle. After the reverse martensite transformation (Fig. 12 H8), the volume fraction of residual martensite phase has reached about 13%.
Fig. 12 Stress-strain-temperature ( 11 11 T ) curve and microstructure
morphologies for the key points (i.e., points A8 to H8) selected in the 8-th cycle, where the color legends in pictures A8 to E8 indicate the values of 1 211 while that in pictures F8 to H8 indicate the values of 1 11 . 3.3 Discussions The simulated results indicate that during the OWSME cycling of NiTi SMA single crystal, the plastic deformation occurs and can affect all the stages of OWSME cycling (i.e., temperature-induced martensite transformation, martensite reorientation and reverse martensite transformation). The main variations in the thermo-mechanical responses occurring during the cyclic loading are that: (1) the twinned martensite phase is gradually transited into the re-oriented one at the stage of repeated temperature-induced martensite transformation; (2) the start stress of martensite reorientation gradually decreases and the reorientation mode is converted gradually from the Mode-I to Mode-II one; (3) the residual martensite phase is gradually accumulated and the irrecoverable strain increases gradually. The influence of plastic deformation on such variations during the OWSME cycling will be discussed in detail below based on the predicted results of phase field simulations. 3.3.1 Effect of plastic deformation on temperature-induced martensite transformation After the reverse martensite transformation in each cycle, a self-balanced residual internal stress exists in the simulation cell, which evolves with the increase in the number of cycles. Subsequently, at the stage of repeated temperature-induced martensite transformation, the local internal stress induced by the martensite transformation causes a plastic deformation which can in turn affect the martensite transformation. On the other hand, since the driving force of martensite transformation (Eq. (25-a)) mainly consists of stress and temperature parts, the forces for different variants at each material point are not identical even an isotropic temperature field is applied due to the directionality of local internal stress. Consequently, with the influence of the internal stress evolving during the cyclic loading, the nucleation and growth of different variants vary correspondingly.
Generally, the formation of certain variants is suppressed (e.g., the formation of 11-th variant is suppressed, as shown in Fig. 11). In addition, the nucleation of martensite variants during the temperature-induced martensite transformation can also be affected by the accumulated residual martensite phase. More precisely, the twinned martensite phase is gradually transited into the re-oriented one due to the interaction among the plastic deformation, residual martensite phase and temperature-induced martensite transformation; however, the variants in the re-oriented martensite phase obtained after the temperature-induced martensite transformation are not always the ones with favorable orientations when an external stress is applied in certain direction, so the martensite reorientation can still occur (such as the simulated results in the 8-th cycle in this work). 3.3.2 Effect of plastic deformation on martensite reorientation As discussed by Hamilton et al. [63], Delville et al. [64,65] and Pelton et al. [25], the dislocation density gradually increases during the cyclic deformation of NiTi SMAs, and a local internal stress is induced and enhanced by the increase of dislocation density. Further, the induced local internal stress can assist the nucleation of martensite phase and lead to a decrease in the start stress of martensite transformation [66] during the cyclic deformation of super-elastic NiTi SMAs. Similarly, from the simulations in this work, the level of self-balanced internal stress in the initial twinned martensite phase obtained in the 2-nd cycle is obviously higher than that in the 1-st cycle, and the high internal strain energy in the local regions with high internal stress can drive the migration of twinned interfaces during the martensite reorientation. Therefore, it can be concluded that the local internal stress can assist the reorientation of martensite variants, resulting in a decrease in the reorientation start stress in the 2-nd cycle; then, with increasing the number of cycles, since the level of self-balanced internal stress in the initial twinned martensite phase tends to be stabilized, the reorientation start stress gradually becomes stabilized (Fig. 2a). It is seen from Fig. 1b that the 11 11 curve obtained in the 1-st cycle increases rapidly at “Stage I” (as marked in Fig. 1b), so that the whole curve is much higher
than others. It is because the occurrence of martensite reorientation is accompanied by a rapid relaxation of the internal stress at the twinned interfaces (see Fig. 5a), and the insufficient driving force leads to the stagnation of reorientation until the applied stress level increases (after “Stage I”). However, since the internal stress level before the reorientation in the 2-nd cycle is significantly higher than that in the 1-st cycle, the driving force of reorientation is still sufficient, even if it is relaxed when the reorientation starts. At the stages of martensite reorientation in the subsequent cycles, the applied stress and local internal one can provide a sufficient driving force for the reorientation of variants. On the other hand, when the accumulated plastic strain is high enough in certain local regions, the high internal strain energy in such regions can assist the nucleation and growth of the variants with favorable orientations inside the regions with unfavorable orientations during the martensite reorientation. Consequently, the reorientation mode is gradually converted with increasing the number of cycles, i.e., from the Mode-I (i.e., achieved by the migration of twinned interfaces) Mode-I & Mode-II Mode-II (i.e., achieved by the nucleation and growth of the variants with favorable orientations). 3.3.3 Effect of plastic deformation on reverse martensite transformation As observed by Gall et al. [58], Simon et al. [24] and Brinson et al. [67], the reverse martensite transformation can be pinned by dislocations (i.e., the transition from martensite phase to austenite one is hindered). Also, the simulated results obtained in this work demonstrate that the reverse martensite transformation can be pinned by the plastic deformation; as a result, the martensite phase cannot be completely transformed into the austenite one, even if the applied temperature is higher than the A f . As the number of cycles increases, the plastic deformation is continuously accumulated, resulting in the accumulation of residual martensite phase after the reverse martensite transformation per cycle, which can further lead to an irrecoverable strain. However, the irreversible plastic strain is clearly dominant, compared to the
irrecoverable strain caused by residual martensite phase. Since the Mode-II martensite reorientation gradually replaces the Mode-I one, the internal stress level gradually decreases and becomes stabilized; and the plastic deformation resistances increase with increasing the number of cycles. Therefore, the increment of plastic strain gradually decreases (Fig. 2e), the total accumulated plastic strain tends to be saturated, and the total irrecoverable strain increases and becomes stabilized as the number of cycles increases (Fig. 2c). Conversely, the recoverable strain gradually decreases (Fig. 2d), which implies that the OWSME of NiTi SMA single crystal is progressively degenerated. Although the plastic deformation can be induced by the applied stress and the local
internal
stress
generated
during
the
temperature-induced
martensite
transformation and martensite reorientation per cycle, it is mainly come from the latter here due to a low applied stress level (the maximum is about 580 MPa). Besides, the plastic deformation generated at the stage of martensite reorientation (i.e., the reorientation-induced plasticity) is obviously more significant than that generated at the stage of temperature-induced martensite transformation (i.e., the TRIP) (Fig. 2e). On the other hand, the plastic deformation can in turn affect the martensite transformation and martensite reorientation during the OWSME cycling. In summary, it can be anticipated from the phase field simulations in this work that the interaction among the martensite transformation, martensite reorientation and plastic deformation during the cyclic deformation leads to the cyclic degeneration of the OWSME of NiTi SMA single crystal.
4 Conclusions In the framework of thermodynamics and based on the Ginzburg-Landau’s and crystal plasticity theories, a new phase field model is proposed in this work, where four inelastic deformation mechanisms, i.e., martensite transformation, martensite reorientation, austenite plasticity and martensite plasticity are simultaneously considered. Through the Helmholtz’s free energy and Clausius’s dissipative inequality, the thermodynamic driving forces of phase field kinetic equations and inelastic
deformation processes, as well as the thermodynamic constraints of phase field equations and the evolution equations of plasticity were derived. The OWSME cycling of NiTi SMA single crystal is simulated by the constructed phase field model. The simulated results indicate that such a model can reasonably predict the cyclic degeneration of the OWSME for NiTi SMA single crystal. Some conclusions are summarized from the simulations as follows: (1) At the stage of temperature-induced martensite transformation (i.e., the process obtaining twinned martensite phase) during the OWSME cycling of NiTi SMA single crystal, the residual local internal stress produced after the previous reverse martensite transformation can promote the nucleation of martensite variants. However, the twinned martensite phase is gradually transformed into the re-oriented one at the stage of repeated cooling due to the interaction among the plastic deformation, residual martensite phase and temperature-induced martensite transformation. (2) At the stage of stress-induced martensite reorientation, the local internal stress can assist the reorientation of variants and lead to a decrease in the start stress of reorientation. The martensite reorientation develops through the migration of twinned interfaces (i.e., the Mode-I reorientation) when the plastic deformation is small; while the high internal strain energy caused by the large plastic deformation can assist the nucleation and growth of variants with favorable orientations inside the variants with unfavorable ones, and then a new reorientation mode (i.e., the Mode-II one) occurs. With increasing the number of cycles, the Mode-I martensite reorientation is gradually replaced by the Mode-II one. (3) At the stage of reverse martensite transformation by heating, the plastic deformation caused by local internal stress can restrain the reverse martensite transformation, resulting in the production of a residual martensite phase. Moreover, the plastic deformation is continuously accumulated during the OWSME cycling, which leads to the increase in the volume fraction of residual martensite phase after reverse martensite transformation. (4) During the OWSME cycling of NiTi SMA single crystal, both the irrecoverable
strain caused by residual martensite phase and the irreversible plastic strain gradually increase; while, the recoverable strain (reflecting the capability of the OWSME of NiTi SMAs) decreases gradually. The interaction among the martensite transformation, martensite reorientation and plastic deformation during the cyclic deformation leads to the cyclic degeneration of the OWSME of NiTi SMA single crystal.
Acknowledgements The financial supports of National Natural Science Foundation of China (11532010) and the Doctoral Innovation Fund Program of Southwest Jiaotong University (D-CX201738) are greatly acknowledged.
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Conflict of interest There is no conflict of interest in this work.
Graphical abstract