A microplane constitutive model for shape memory alloys considering tension–compression asymmetry

According to the approach proposed by Poorasadion et al [33], considering the tension–compression asymmetry, the 1D stress–strain relation of a SMA can be described using the following equation:

$$\begin{eqnarray}&&\sigma =E\left( \varepsilon -\varepsilon _{L}^{+}\xi _{s}^{+}-\varepsilon _{L}^{-}\xi _{s}^{-} \right)\end{eqnarray} \tag{ 1 }$$

in which ${{\xi }_{s}}$ and ${{\varepsilon }_{L}}$ are the stress-induced martensite volume fraction and maximum recoverable strain, respectively. The superscripts '+' and '–' denote tension and compression, respectively. σ and are respectively the uniaxial stress and strain, and E the elastic modulus. It is shown that the elastic modulus of SMAs is a function of the martensite volume fraction and can be written using the Reuss model as follows:

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \frac{1}{E}=\frac{1}{{{E}_{{\rm A}}}}+\xi _{s}^{+}\left( \frac{1}{E_{{\rm M}}^{+}}-\frac{1}{{{E}_{{\rm A}}}} \right)+\xi _{s}^{-}\left( \frac{1}{E_{{\rm M}}^{-}}-\frac{1}{{{E}_{{\rm A}}}} \right) \\ \quad \;\;\;+{{\xi }_{{\rm T}}}\left( \frac{1}{E_{{\rm M}}^{T}}-\frac{1}{{{E}_{{\rm A}}}} \right) \\ \end{array}\end{eqnarray} \tag{ 2 }$$

where E_A and E_M are the elastic moduli of full austenite and full martensite phases, respectively, and ${{\xi }_{{\rm T}}}$ the temperature-induced martensite volume fraction. The superscript 'T' is attributed to temperature-induced quantities.

To specify the evolution of $\xi _{s}^{\pm },$ and ${{\xi }_{{\rm T}}},$ the evolution function presented by eqution (3) is utilized. This evolution function was firstly proposed by Brinson [6] and then modified by Poorasadion et al [33] to account for the tension–compression asymmetry. In this paper, this evolution function is enhanced to model the start and finish of the reverse transformation more accurately with the use of the stress-temperature phase diagram depicted in figure 1 [34].

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {\rm if}\;T\leqslant {{M}_{{\rm s}}}\;{\rm and}\;\sigma _{{\rm s}}^{{\rm cr}}\lt \hat{\sigma }\lt \sigma _{{\rm f}}^{{\rm cr}} \\ Y={\rm cos} \left( \frac{\pi }{\sigma _{{\rm f}}^{{\rm cr}}-\sigma _{{\rm s}}^{{\rm cr}}}\left( \hat{\sigma }-\sigma _{{\rm s}}^{{\rm cr}}-{{C}_{{\rm M}}}(T-{{M}_{{\rm s}}} \right) \right) \\ \xi _{s}^{r}=\frac{1-y}{2}+\xi _{s0}^{r}\frac{1+y}{2},\xi _{s}^{p}=\xi _{s0}^{p}\frac{1-\xi _{s}^{r}}{1-\xi _{s0}^{r}}, \\ {{\xi }_{{\rm T}}}=\left( {{\xi }_{{\rm T}0}}+\Delta \right)\frac{\left( 1-\xi _{s}^{r} \right)}{1-\xi _{s0}^{r}} \\ {\rm if}\;T\lt {{M}_{{\rm f}}}:\;\Delta =0 \\ {\rm else}:\;\;\Delta =\frac{1-{{\xi }_{0}}}{2}\left( 1-{{Y}_{{\rm MT}}} \right) \\ {\rm if}\;T\gt {{M}_{{\rm s}}}\;{\rm and}\;\sigma _{{\rm s}}^{{\rm cr}}+{{C}_{{\rm M}}}\left( T-{{M}_{{\rm s}}} \right)\lt \hat{\sigma }\lt \sigma _{{\rm f}}^{{\rm cr}} \\ \quad \ \;\ \;\ \;+{{C}_{{\rm M}}}\left( T-{{M}_{{\rm s}}} \right): \\ Y={\rm cos} \left( \frac{\pi }{\sigma _{{\rm f}}^{{\rm cr}}-\sigma _{{\rm s}}^{{\rm cr}}}\left( \hat{\sigma }-\sigma _{{\rm s}}^{{\rm cr}}-{{C}_{{\rm M}}} \right)\left( T-{{M}_{{\rm s}}} \right) \right) \\ \xi _{s}^{r}=\frac{1-Y}{2}+\xi _{s0}^{r}\frac{1+Y}{2},\;\xi _{s}^{p}=\xi _{s0}^{p}\frac{1-\xi _{s}^{r}}{1-\xi _{s0}^{r}}, \\ {{\xi }_{{\rm T}}}={{\xi }_{{\rm T}0}}\frac{1-\xi _{s}^{r}}{1-\xi _{s0}^{r}} \\ {\rm if}\;T\gt {{A}_{{\rm s}}}\;{\rm and}\;{{C}_{{\rm Af}}}\left( T-{{A}_{{\rm f}}} \right)\lt \hat{\sigma }\lt {{C}_{{\rm As}}}\left( T-{{A}_{{\rm s}}} \right): \\ Y={\rm cos} \left( \frac{\pi }{{{C}_{{\rm As}}}\left( T-{{A}_{{\rm s}}} \right)-{{C}_{{\rm Af}}}\left( T-{{A}_{{\rm f}}} \right)} \right. \\ \quad \ \;\ \;\ \;\times \left. \left( {{C}_{{\rm As}}}\left( T-{{A}_{{\rm s}}} \right)-\hat{\sigma } \right) \right) \\ \xi _{s}^{r}=\frac{\xi _{s0}^{r}}{2}\;\left( 1+Y \right),\xi _{s}^{p}=\frac{\xi _{s0}^{p}}{2}\;\left( 1+Y \right), \\ {{\xi }_{{\rm T}}}=\frac{{{\xi }_{{\rm T}0}}}{2}\;\left( 1+Y \right) \\ {\rm if}\;{{M}_{{\rm f}}}\lt T\lt {{M}_{{\rm s}}}\;{\rm and}\;\hat{\sigma }\lt \sigma _{{\rm s}}^{{\rm cr}} \\ \xi _{s}^{r}=\xi _{s0}^{r},\;\;\xi _{s}^{p}=\xi _{s0}^{p},{{\xi }_{{\rm T}}}=\frac{1-{{\xi }_{0}}}{2}\;\left( 1-{{Y}_{{\rm MT}}} \right)+{{\xi }_{{\rm T}0}} \\ \end{array}\end{eqnarray} \tag{ 3 }$$

in which M_f, M_s, A_s, A_f, and T are martensite finish, martensite start, austenite start, austenite finish, and temperature, respectively. Superscripts 'r' and 'p' are as (r = +, p = −) in tension and (r = −, p  = +) in compression, $\sigma _{{\rm s}}^{{\rm cr}}$and $\sigma _{{\rm f}}^{{\rm cr}}$ the critical start and finish stresses for martensite detwinning, $\xi _{s0}^{r},$ $\xi _{s0}^{p},$ and ${{\xi }_{{\rm T}0}}$ the initial values of the corresponding martensite volume fraction, C_M, C_As, and C_Af the slopes of the transformation strips as shown in figure 1, and ${{Y}_{{\rm MT}}}=\;{\rm cos} \left( \pi \left( T-{{M}_{{\rm s}}} \right)/\left( {{M}_{{\rm f}}}-{{M}_{{\rm s}}} \right) \right).$ $\hat{\sigma }$ is the modified stress to account for the asymmetry in tension and compression as follows:

$$\begin{eqnarray}&&\hat{\sigma }=\frac{1\pm \beta }{1+\beta }\sigma \end{eqnarray} \tag{ 4 }$$

where '+' and '–' in the numerator indicate tension and compression, respectively, and β is a parameter to determine the value of asymmetry, which is defined using the following relation:

$$\begin{eqnarray}&&\beta =\left\{ \begin{array}{ccccccccccccccc} {{\beta }_{1}}\;{\rm if}\;\dot{\sigma }\gt 0 \\ {{\beta }_{2}}\;{\rm if}\;\dot{\sigma }\lt 0 \\ \end{array} \right.\end{eqnarray} \tag{ 5 }$$

in which β₁ and β₂ are chosen to distinguish the asymmetry level of loading from that of unloading. The effect of β on the asymmetry level will be discussed in the following section.

Zoom In Zoom Out Reset image size

When using the phase diagram at high temperatures, if the values of ${{C}_{{\rm As}}}$ and ${{C}_{{\rm Af}}}$ are higher than ${{C}_{{\rm M}}},$ it should be noted that the corresponding lines may cross each other. This issue would be problematic when the critical stress of austenite finish is higher than that of martensite start or if the critical stress of austenite start is higher than that of martensite finish. Figures 2(a) and (b) demonstrate these cases, respectively. It implies some restrictions on the transformation stresses as presented by equations (6) and (7):

$$\begin{eqnarray}&&{{C}_{{\rm As}}}\left( T-{{A}_{{\rm s}}} \right)\lt \sigma _{{\rm cr}}^{{\rm f}}+{{C}_{{\rm M}}}\left( T-{{M}_{{\rm s}}} \right)\;\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{{C}_{{\rm Af}\left( T-{{A}_{{\rm f}}} \right)}}\lt \sigma _{{\rm cr}}^{{\rm s}}+{{C}_{{\rm M}}}\left( T-{{M}_{{\rm s}}} \right)\end{eqnarray} \tag{ 7 }$$

Zoom In Zoom Out Reset image size

According to the above-mentioned explanations and considering ${{{\boldsymbol{M}} }_{{\bf d}}}$ as the martensite dead temperature, the phase diagram should be used for temperatures with the constraint below:

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} T\lt {\rm min} \left( \frac{\left( {{C}_{{\rm Af}}}{{A}_{{\rm f}}}+\sigma _{{\rm cr}}^{{\rm s}}-{{C}_{{\rm M}}}{{M}_{{\rm s}}} \right)}{{{C}_{{\rm Af}}}-{{C}_{{\rm M}}}}, \right. \\ \left. \quad \quad \quad \frac{\left( {{C}_{{\rm As}}}{{A}_{{\rm s}}}+\sigma _{{\rm cr}}^{{\rm f}}-{{C}_{{\rm M}}}{{M}_{{\rm s}}} \right)}{{{C}_{{\rm As}}}-{{C}_{{\rm M}}}},\;{{M}_{{\rm d}}} \right) \\ \end{array}\end{eqnarray} \tag{ 8 }$$

To develop a 3D constitutive model using the microplane theory, three main steps should be followed. These steps are the projection of the macroscopic stress tensor on each microplane, the definition of a 1D constitutive law for each microplane, and the generalization of 1D constitutive laws to 3D using a homogenization process [28–31]. In this regard, a static constraint formulation with a volumetric-deviatoric split is employed in the present work. As shown in figure 3, the macroscopic stress tensor may be projected as normal and shear vectors on a microplane. The normal and shear stresses on each microplane can be described as:

$$\begin{eqnarray}&&{{\sigma }_{{\rm N}}}={{N}_{ij}}{{\sigma }_{ij}}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{{\sigma }_{{\rm T}}}={{T}_{ij}}{{\sigma }_{ij}}\end{eqnarray} \tag{ 10 }$$

where ${{\sigma }_{ij}}$ is the macroscopic stress tensor, ${{\sigma }_{{\rm N}}}$ normal stress, ${{\sigma }_{{\rm T}}}$ shear stress, ${{N}_{ij}}={{n}_{i}}{{n}_{j}},$ ${{T}_{ij}}=\left( {{t}_{i}}{{n}_{j}}+{{t}_{j}}{{n}_{i}} \right)/2$ in which ${{t}_{i}}=\left( {{\sigma }_{ik}}{{n}_{k}}-{{\sigma }_{{\rm N}}}{{n}_{i}} \right)/\sqrt{{{\sigma }_{jr}}{{\sigma }_{js}}{{n}_{r}}{{n}_{s}}-\sigma _{{\rm N}}^{2}}\;$ is the unit vector parallel to the resultant shear stress on the plane, and n_i are the components of the unit normal vector, n, to a microplane. The normal stress is split into volumetric and deviatoric parts as:

$$\begin{eqnarray}&&{{\sigma }_{{\rm N}}}={{\sigma }_{{\rm V}}}+{{\sigma }_{{\rm D}}}\end{eqnarray} \tag{ 11 }$$

in which ${{\sigma }_{{\rm V}}}={{\delta }_{ij}}{{\sigma }_{ij}}/3,$ ${{\sigma }_{{\rm D}}}=\left( {{N}_{ij}}-{{\delta }_{ij}}/3 \right){{\sigma }_{ij}},$ and ${{\delta }_{ij}}$ is the Kronecker delta. It is shown that using the volumetric-deviatoric split, the micro-level elastic moduli are equal to the macroscopic ones [27, 28]. It is also supposed that the martensite transformation is just associated with the shear component of microplane stresses [28, 29]. Accordingly, the volumetric and deviatoric parts are described using Hooke's law, and the shear stress can be described using eqution (1) as follows:

$$\begin{eqnarray}&&{{\varepsilon }_{{\rm V}}}=\frac{1-2\nu }{E}{{\sigma }_{V}}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{{\varepsilon }_{{\rm D}}}=\frac{1+\nu }{E}{{\sigma }_{D}}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{{\varepsilon }_{{\rm T}}}=\frac{1+\nu }{E}{{\sigma }_{T}}+\varepsilon _{L}^{+}\xi _{s}^{+}+\varepsilon _{L}^{-}\xi _{s}^{-}\end{eqnarray} \tag{ 14 }$$

where ν is the SMAs Poisson ratio, ${{\varepsilon }_{{\rm V}}}$ the volumetric strain, ${{\varepsilon }_{{\rm D}}}$ the deviatoric strain, and ${{\varepsilon }_{{\rm T}}}$ the shear strain. By applying the principle of complementary virtual work [28, 29] and some simplifications, the following relation is obtained for the strain tensor:

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\varepsilon }_{ij}}=-\frac{\nu }{E}\;{{\sigma }_{mm}}{{\delta }_{ij}}+\frac{1+\nu }{E}{{\sigma }_{mn}}\cdot \frac{3}{2\pi }\mathop \int \limits_{\Omega } \left( {{N}_{mn}}{{N}_{ij}}+{{T}_{mn}}{{T}_{ij}} \right){\rm d}\Omega \\ \quad \quad +\left( \varepsilon _{L}^{+}\xi _{s}^{+}+\varepsilon _{L}^{-}\xi _{s}^{-} \right)\cdot \frac{3}{2\pi }\mathop \int \limits_{\Omega } {{T}_{ij}}{\rm d}\Omega . \\ \end{array}\end{eqnarray} \tag{ 15 }$$

Zoom In Zoom Out Reset image size

To consider the material asymmetry in tension and compression, it is necessary to define an appropriate equivalent stress to be used in equation (3) for evaluating the martensite volume fractions in equation (14). In this paper, an equivalent stress based on second and third invariants of deviatoric stress tensor (J₂ and J₃) is used as demonstreated below [15, 19, 22]:

$$\begin{eqnarray}&&\hat{\sigma }=\frac{1}{1+\beta }\left\{ \sqrt{3{{J}_{2}}}+\frac{9}{2}\beta \frac{{{J}_{3}}}{{{J}_{2}}} \right\}\end{eqnarray} \tag{ 16 }$$

in which J₂ and J₃ are the second and third invariants of the deviatoric stress tensor, respectively. Note that, equation (16) reduces to equation (4) for 1D cases. Figure 4 shows the transformation surfaces for different typical values of β in the plane stress state. As it is obvious, when β = 0, the symmetric transformation surface is achieved. By increasing the value of β, the level of asymmetry will increase. According to experimental observations, the transformation surface of martensite-to-austenite and austenite-to-martensite might be of a different level of asymmetry. This can be taken into account by using different values of β presented in equation (5).

Zoom In Zoom Out Reset image size

In this subsection, the experimental tension–compression responses of superelastic NiTi samples provided by Zhu and Dui [15] (case 1) and Thamburaja and Anand [35] (case 2) are used for verification of the model. For these cases, the material is 50.8at.% NiTi and 55.96 wt.% NiTi, respectively. Tables 1 and 2 show the material parameters utilized for the determination of the stress–strain response in cases 1 and 2, respectively. These material parameters are obtained from the literature [15, 36]. Since the test temperature is above the austenite finish, the material is in the austenite phase; so, the initial value for $\xi _{s0}^{+},$ $\xi _{s0}^{-},$ and ${{\xi }_{T0}}$ is zero for these two cases.

Table 1.  Material parameters for case 1 [15].

${{E}_{{\rm A}}}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{+}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{-}\;\left( {\rm MPa} \right)$	ν	${{M}_{{\rm f}\,}}\left( {\rm K} \right)$	${{M}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm f}}}\;\left( {\rm K} \right)$	$\sigma _{{\rm s}}^{{\rm cr}}\;\left( {\rm MPa} \right)$
24 000	22 000	25 000	0.42	200.3	232.3	262.5	295.1	0
$\sigma _{{\rm f}}^{{\rm cr}}\;\left( {\rm MPa} \right)$	${{C}_{{\rm M}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm As}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm Af}}}\;\left( {\rm MPa}/{\rm K} \right)$	$\varepsilon _{L}^{+}$	$\varepsilon _{L}^{-}$	${{\beta }_{1}}$	${{\beta }_{2}}$	$T\;\left( {\rm K} \right)$
180	3.37	3.7	2	0.052	−0.023	0.165	0.165	299.1

Table 2.  Material parameters for case 2 [36].

${{E}_{{\rm A}}}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{+}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{-}\;\left( {\rm MPa} \right)$	ν	${{M}_{{\rm f}\,}}\left( {\rm K} \right)$	${{M}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm f}}}\;\left( {\rm K} \right)$	$\sigma _{{\rm s}}^{{\rm cr}}\;\left( {\rm MPa} \right)$
71 000	35 000	42 000	0.33	213	251.3	260.3	268.5	256
$\sigma _{{\rm f}}^{{\rm cr}}\;\left( {\rm MPa} \right)$	${{C}_{{\rm M}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm As}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm Af}}}\;\left( {\rm MPa}/{\rm K} \right)$	$\varepsilon _{L}^{+}$	$\varepsilon _{L}^{-}$	${{\beta }_{1}}$	${{\beta }_{2}}$	$T\;\left( {\rm K} \right)$
288	4.28	5.73	4.07	0.048	-0.031	0.13	0.13	298

Figures 5 and 6 show the stress–strain response of NiTi samples in tension and compression obtained using the proposed approach, the Zhu and Dui model, and the experimental measurements [15, 35]. In all cases, the absolute values of the stress and strain are reported in compression tests. The results show a good agreement between the empirical findings and the numerical results of the present approach. In comparison with the Zhu and Dui model, the proposed model correlates better to the experimental findings especially during unloading. It is because of using the modified phase diagram (figure 1) which allows one to adjust the start and finish martensite–austenite transformation stresses more adaptively. As demonstrated in figure 5, since the elastic modulus of martensite is considered to be different in tension and compression, the proposed model is able to reproduce the experimental stress–strain response in compression more accurately. However, there is a considerable mismatch between the numerical and experimental tensile stress–strain responses at the first portion of the loading curve in figure 5. This is due to different experimental values of the austenite elastic modulus in tension and compression. In figure 5, the value of the austenite elastic modulus is calibrated using the compressive stress–strain response to be able to compare the results with those reported in [15]. Zhu and Dui [15] suggest the use of a compressive stress–strain curve to calibrate the model parameters because the transformation is more stable during unloading in the case of compression compared with tension. In addition, in all the experimental cases, there is a small amount of residual strain due to the transformation induced plasticity (TRIP) phenomenon, which can be reduced by training the SMA specimen through cyclic loading. Since the present model is not capable of considering the effect of TRIP, there is not any residual strain in the predicted stress–strain response.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In order to show the capability of the model to consider the effects of tension–compression asymmetry in ferroelastic behavior, the stress–strain responses obtained using the proposed model are compared with the numerical results reported by Jaber et al [36] (case 3) and the experimental findings for Au-47.5at.%Cd by Nakanishi et al [37] (case 4). The material parameters used for the former and the latter cases are provided in tables 3 and 4, respectively. In case 3, the temperature is below ${{A}_{{\rm s}}}$ and the material is initially fully austenitic, which means the initial values of $\xi _{s0}^{+}=\xi _{s0}^{-}={{\xi }_{{\rm T}0}}=0.$ For case 4, the temperature is below the martensite finish temperature and the material is initially in the martensite phase which causes $\xi _{s0}^{+}=\xi _{s0}^{-}=0$ and ${{\xi }_{{\rm T}0}}=1.$ The result of the present model are compared with the previously reported findings by Jaber et al [36] and the experimental data by Nakanishi et al [37] in figures 7(a) and (b), respectively. The obtained stress–strain responses show that the present approach is in good agreement with the previously proposed model and experiments. Since the proposed model by Jaber et al [36] is not capable of considering different values for martensite elastic moduli in tension and compression as well as different values for C_A, it is assumed that $E_{{\rm M}}^{+}=E_{{\rm M}}^{-}$ and ${{C}_{{\rm As}}}={{C}_{{\rm Af}}}$ to be able to compare the results.

Table 3.  Material parameters for case 3 [36].

${{E}_{{\rm A}}}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{+}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{-}\;\left( {\rm MPa} \right)$	ν	${{M}_{{\rm f}\,}}\left( {\rm K} \right)$	${{M}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm f}}}\;\left( {\rm K} \right)$	$\sigma _{{\rm s}}^{{\rm cr}}\;\left( {\rm MPa} \right)$
80 000	30 000	30 000	0.33	273	283	293	303	105
$\sigma _{{\rm f}}^{{\rm cr}}\;\left( {\rm MPa} \right)$	${{C}_{{\rm M}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm As}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm Af}}}\;\left( {\rm MPa}/{\rm K} \right)$	$\varepsilon _{L}^{+}$	$\varepsilon _{L}^{-}$	${{\beta }_{1}}$	${{\beta }_{2}}$	$T\;\left( {\rm K} \right)$
225	5	5	5	0.0615	−0.046	0.186	0.186	285

Table 4.  Material parameters for case 4 [37].

${{E}_{{\rm A}}}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{+}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{-}\;\left( {\rm MPa} \right)$	ν	${{M}_{{\rm f}\,}}\left( {\rm K} \right)$	${{M}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm f}}}\;\left( {\rm K} \right)$	$\sigma _{{\rm s}}^{{\rm cr}}\;\left( {\rm MPa} \right)$
5000	5000	5000	0.33	308	331	345	358	11
$\sigma _{{\rm f}}^{{\rm cr}}\;\left( {\rm MPa} \right)$	${{C}_{{\rm M}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm As}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm Af}}}\;\left( {\rm MPa}/{\rm K} \right)$	$\varepsilon _{L}^{+}$	$\varepsilon _{L}^{-}$	${{\beta }_{1}}$	${{\beta }_{2}}$	$T\;\left( {\rm K} \right)$
44	2	2	2	0.032	−0.034	−0.16	−0.16	300

Zoom In Zoom Out Reset image size

The stress–strain response presented in figure 7(b) shows that the model is capable of modeling subloops due to incomplete phase transformations for test temperature below martensite finish temperature. To show the ability of the model in reproducing this behavior for temperatures above austenite finish, the experimental results for 50.95at.%NiTi reported by Takeda et al [38] (case 5) are used. Figure 8 shows a comparison of the stress–strain response reproduced by the proposed model and the experimental results [38]. The material parameters utilized for this simulation are presented in table 5 and the initial conditions are as $\xi _{s0}^{+}=\xi _{s0}^{-}={{\xi }_{{\rm T}0}}=0.$ Since the test is just performed in tension, the material parameters related to compression are not provided. Referring to figures 7(b) and 8, the present model can be used for modeling the internal subloops of an SMA in a wide range of temperatures.

Zoom In Zoom Out Reset image size

Table 5.  Material parameters for case 5.

${{E}_{{\rm A}}}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{+}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{-}\;\left( {\rm MPa} \right)$	ν	${{M}_{{\rm f}\,}}\left( {\rm K} \right)$	${{M}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm f}}}\;\left( {\rm K} \right)$	$\sigma _{{\rm s}}^{{\rm cr}}\;\left( {\rm MPa} \right)$
27 600	24 000	NA	0.3	100	220	254	281	100
$\sigma _{{\rm f}}^{{\rm cr}}\;\left( {\rm MPa} \right)$	${{C}_{{\rm M}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm As}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm Af}}}\;\left( {\rm MPa}/{\rm K} \right)$	$\varepsilon _{L}^{+}$	$\varepsilon _{L}^{-}$	${{\beta }_{1}}$	${{\beta }_{2}}$	$T\;\left( {\rm K} \right)$
240	2.31	3.7	3.8	0.062	NA	NA	NA	298

In this subsection, the proposed model is used to simulate four-point bending of NiTi tubes. First, the finite element modeling of the four-point bending is described. Then, the procedure of finding material parameters for two different cases, named cases 6 and 7, is explained. Finally, the results of the numerical simulations are presented and discussed for both cases.

Consider a tube with length L, gage length L_e, outer diameter D, and thickness t as shown in figure 9(a) which depicts the schematic of the four-point bending test. The four loading points are circular rollers which hold the tube in the bending plane. Each pair of rollers is fixed to a loading wheel. These loading wheels are loaded by two cables and rotate around their center which cause bending of the tube. The magnitude of the applied moment can be calculated by measuring the force in the cable, F. A gage with length L_e is attached at the middle of the tube to measure the rotation angle due to the applied moment. More information about the test apparatus and method are provided [39, 40]. As shown in figure 9(b), the mean curvature of the tube, κ, can be calculated by measuring the angles of rotation at the tube's ends (θ) and using the following equation:

$$\begin{eqnarray}&&\kappa =\frac{\theta }{{{L}_{e}}}\end{eqnarray} \tag{ 17 }$$

Because the problem is symmetric about the z = 0 plane, only half of the tube is modeled and appropriate boundary conditions are imposed. Referring to figure 10, a tube with length L_e/2 is considered for modeling purposes because the response of the tube in the gage region is of concern. Due to the symmetry about the z = 0 plane, this plane should not move toward the z direction and the slope of the tube would be zero at this point. Accordingly, at one end (symmetry plane) of the half tube, the translational degree of freedom in the z direction and all rotational degrees of freedom are fixed. Referring to figure 9(a), the tube at the end of the gage length rotates with the angle of θ due to the applied moment. To model this rotation, at this end, a reference point is defined and all nodes of that end are tied to the reference point. A rotation about the x-axis, UR₁, is applied to the reference point using the amplitude variation shown in figure 10(b). In addition, the history of the reaction moment in response to the applied amount of rotation is requested to be able to depict the moment-rotation diagrams.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In this paper, the experimental results reported by Reedlunn et al [40] and Bechle and Kyriakides [39] are used for comparison. These experimental measurements are performed isothermally using a cold-drawn, slightly Ni-rich NiTi tube from the Memry Corporation [39, 40]. For the sake of simplicity, in the rest of the paper, the former and the latter cases are denoted by case 6 and case 7, respectively. Table 6 shows the geometric details of these two cases.

Similar to cases 1 and 2, the test temperature is higher than the austenite finish temperature and the initial phase is austenite for cases 6 and 7. Therefore, the same initial conditions are used for these simulations. To attribute suitable material parameters to the finite element model, the experimental stress–strain response in uniaxial tension–compression is used to calibrate the required quantities. Table 7 shows the calibrated material parameters for case 6, which was performed isothermally at 298 K. Figure 11 depicts the predicted stress–strain response that correlates fairly well and is fitted to the experiment.

Table 7.  Material parameters obtained by calibration of stress–strain response in tension and compression for case 6.

${{E}_{{\rm A}}}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{+}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{-}\;\left( {\rm MPa} \right)$	ν	${{M}_{{\rm f}\,}}\left( {\rm K} \right)$	${{M}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm f}}}\;\left( {\rm K} \right)$	$\sigma _{{\rm s}}^{{\rm cr}}\;\left( {\rm MPa} \right)$
65 300	28 000	87 000	0.45	126	210	248	292	90
$\sigma _{{\rm f}}^{{\rm cr}}\;\left( {\rm MPa} \right)$	${{C}_{{\rm M}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm As}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm Af}}}\;\left( {\rm MPa}/{\rm K} \right)$	$\varepsilon _{L}^{+}$	$\varepsilon _{L}^{-}$	${{\beta }_{1}}$	${{\beta }_{2}}$	$T\;\left( {\rm K} \right)$
170	3.65	4.5	32	0.054	−0.035	0.23	0.23	298

Zoom In Zoom Out Reset image size

Since the experiments of case 7 are performed at several different temperatures, it is reasonable to develop the phase diagram for the tube used by Bechle and Kyriakides [39]. The reported uniaxial stress–strain responses at 286, 296, and 306 K are considered, and the phase diagram illustrated in figure 12 is obtained. According to the phase diagram as well as the stress–strain response at 306 K, the material parameters reported in table 8 are determined and are utilized to study case 7. Figures 13(a)–(c) show the uniaxial stress–strain responses using the material parameters presented in table 8. The comparison of the presented numerical predictions with the corresponding experimental findings demonstrates a good correlation between the results. It is worth mentioning that the phase diagram reported in figure 12 is obtained using the stress–strain response in tension. Therefore, the predictions of the model are closer to the experimental stress–strain curve during tension in comparison with compression.

Zoom In Zoom Out Reset image size

Table 8.  Material parameters obtained by calibration of the stress–strain response in tension and compression for case 7.

${{E}_{{\rm A}}}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{+}\;\left( {\rm MPa} \right)$	$E_{{\rm M}}^{-}\;\left( {\rm MPa} \right)$	ν	${{M}_{{\rm f}\,}}\left( {\rm K} \right)$	${{M}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm s}}}\;\left( {\rm K} \right)$	${{A}_{{\rm f}}}\;\left( {\rm K} \right)$	$\sigma _{{\rm s}}^{{\rm cr}}\;\left( {\rm MPa} \right)$
65 000	26 000	62 000	0.3	213	253	265.6	271.6	130.2
$\sigma _{{\rm f}}^{{\rm cr}}\;\left( {\rm MPa} \right)$	${{C}_{{\rm M}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm As}}}\;\left( {\rm MPa}/{\rm K} \right)$	${{C}_{{\rm Af}}}\;\left( {\rm MPa}/{\rm K} \right)$	$\varepsilon _{L}^{+}$	$\varepsilon _{L}^{-}$	${{\beta }_{1}}$	${{\beta }_{2}}$	$T\;\left( {\rm K} \right)$
155	5.5	6	6	0.057	−0.027	0.28	0.4	286, 296, 306

Zoom In Zoom Out Reset image size

The rest of the paper is allotted to the modeling of four-point bending in the above-mentioned cases. The models are meshed using 20-node quadratic brick reduced integration elements denoted by C3-D20R in ABAQUS. To reduce the effects of the mesh, the number of elements was refined until a negligible change was observed in the output results. Figures 14(a) and (b) show the final meshed model for cases 6 and 7 containing 375 and 615 elements, respectively.

Zoom In Zoom Out Reset image size

Figure 15(a) shows variations of the normalized moment, MC/I, with the dimensionless curvature, $C\kappa ,$ where M is the reaction moment about the x-axis, I the corresponding area moment of inertia of the cross section, C = D/2 the outer radius of the tube, and $C\kappa =2C\left( U{{R}_{1}} \right)/{{L}_{e}}$ the dimensionless curvature. Figure 15(b) compares the position of the neutral axis of the tube obtained using the proposed model with the experimentally measured one [40]. In this figure, the position of the neutral axis (Y₀) is measured from the centerline of the tube. In figures 15(a) and (b), a good agreement is observed for both moment and neutral axis position. The conformity is very good for small values of curvature. However, the conformity is good for higher values of curvature as well as in the unloading cycle; the predictions of the model are overestimated in these regions. This may be due to the effects of strain localization in experimental measurements as reported in [40]. Since the present model is not capable of capturing this phenomenon, it over-predicts the amount of stress in high values of curvature. In addition, the length of the tube is not long enough in comparison with its diameter, and this causes the cross sectional planes not to remain planar. However, in the utilized model, all the degrees of freedom for the loading plane are tied to a reference point meaning that the plane remains planar. Therefore, the model is over-constrained in comparison with the real sample, and the required moment is over-predicted. As another reason, the transformation induced plasticity may cause some errors in finding the appropriate material parameters, especially maximum recoverable strain. Qidwai and Lagoudas [18] stated that plastic deformations occur before full Martensitic transformation takes place. Therefore, it is not possible to obtain the maximum recoverable strain using the experimental data. This issue can be one of the reasons for the difference between experimental and numerical observations in the unloading cycle. Auricchio and Taylor [17] reported that, for the case of four-point bending, the part of the curve corresponding to the unloading and the size of the hysteresis loop differ from the experimental findings.

Zoom In Zoom Out Reset image size

To compare the numerical predictions with the experimentally reported results, the moment and curvature are normalized with the same method as the one used in experiments [39]. Referring to figure 13(b), suppose that ${{\sigma }_{PM}}$ is the stress of the upper plateau, and ${{\varepsilon }_{0}}$ is the amount of strain at the beginning of the martensite transformation in tension at 296 K. Using these values, two quantities are defined for normalizing the moment and curvature, respectively, as ${{M}_{o23}}={{\sigma }_{PM}}D_{0}^{2}t$ and ${{\kappa }_{o23}}=2{{\varepsilon }_{0}}/D.$ In the first quantity, D₀ = D−2t is the inner diameter of the tube. Variations of the dimensionless moment with the dimensionless curvature for case 7 are depicted in figures 16(a)–(c) at 286 K, 296 K, and 306 K. The demonstrated results in this figure show the ability of the proposed model to predict the tension–compression asymmetry of shape memory alloys with good correlations with the experiment.

Zoom In Zoom Out Reset image size

Referring to figures 16(a)–(c), deviation of predicted response from the experiment increases, especially in the first linear region, as the temperature decreases. In all these simulations, the material is initially in the austenite phase and this deviation would be due to variations of austenite elastic modulus. Figure 17 shows the variations of austenite elastic modulus with temperature according to the experimental stress–strain curves provided in figure 13. As is obvious, E_A differs between tension and compression, which may have arisen due to the existence of some micro- (nano-) pores as well as defects in the microstructure of test samples [41, 42]. In addition, the experimental value of austenite elastic modulus increases with temperature. According to these findings, the difference between the numerical and experimental results in figure 16 can be explained. However, it should be stated that the present model cannot take different elastic moduli of austenite in tension and compression into account. Accordingly, only one fixed value is used through all the simulations. As shown in figure 17, the average value of E_A of tension and compression is smaller than the value used for the simulation at the temperatures of 286 and 296 K. Therefore, the numerically predicted value of the dimensionless moment is higher than the experimental one. However, for the temperature of 306 K, the average value of E_A is higher but not too far from that used for the simulation, which causes a better correlation with experimental observations. These findings demonstrate that, in order to obtain reasonable results, it is necessary to calibrate the elastic modulus of austenite based on the average value in tension and compression.

Zoom In Zoom Out Reset image size

A comparison of the moment–curvature response for cases 6 and 7 reveals that the difference between the experiment and the presented model is larger for case 6. It might be owing to less accurate material parameters for case 6. Since there is just a uniaxial stress–strain curve at one temperature in case 6, it is not possible to find exact values of the required material parameters. However, in case 7, the existing stress–strain responses at three different temperatures makes it possible to develop a phase diagram and to find more accurate material parameters, which lead to a better correlation with experimental findings.