A macroscopic multi-mechanism based constitutive model for the thermo-mechanical cyclic degeneration of shape memory effect of NiTi shape memory alloy

Abstract

A macroscopic based multi-mechanism constitutive model is constructed in the framework of irreversible thermodynamics to describe the degeneration of shape memory effect occurring in the thermo-mechanical cyclic deformation of NiTi shape memory alloys (SMAs). Three phases, austenite A, twinned martensite $M^{\mathrm{t}}$ and detwinned martensite $M^{\mathrm{d}}$, as well as the phase transitions occurring between each pair of phases ($A\rightarrow M ^{\mathrm{t}}$, $M^{\mathrm{t}}\rightarrow A$, $A\rightarrow M ^{\mathrm{d}}$, $M^{\mathrm{d}}\rightarrow A$, and $M^{\mathrm{t}}\rightarrow M ^{\mathrm{d}})$ are considered in the proposed model. Meanwhile, two kinds of inelastic deformation mechanisms, martensite transformation-induced plasticity and reorientation-induced plasticity, are used to explain the degeneration of shape memory effects of NiTi SMAs. The evolution equations of internal variables are proposed by attributing the degeneration of shape memory effect to the interaction between the three phases (A, $M^{\mathrm{t}}$, and $M^{\mathrm{d}})$ and plastic deformation. Finally, the capability of the proposed model is verified by comparing the predictions with the experimental results of NiTi SMAs. It is shown that the degeneration of shape memory effect and its dependence on the loading level can be reasonably described by the proposed model.

Appendix

Integrating Eq. (23) yields:

$$\begin{aligned} \sqrt{\rho }=\frac{k_1 }{k_2 }\left[ 1-\hbox {e}^{-\frac{k_2 }{2}\left( {\gamma _{\mathrm{tp}} +\gamma _{\mathrm{rp}} } \right) }\right] . \end{aligned}$$

(A1)

Equation (A1) is the explicit relationship between the slippage amount $\gamma _{\mathrm{tp}} +\gamma _{\mathrm{rp}}$ and the dislocation density $\rho$. Thus, the dislocation densities after the first and tenth cycles can be obtained as

$$\begin{aligned} \sqrt{\rho _1 }= & {} \frac{k_1 }{k_2 }\left( 1-\hbox {e}^{-\frac{k_2 \varepsilon _\mathrm{p}^\mathrm{1} }{2}}\right) , \end{aligned}$$

(A2a)

$$\begin{aligned} \sqrt{\rho _{10} }= & {} \frac{k_1 }{k_2 }\left( 1-\hbox {e}^{-\frac{k_2 \varepsilon _\mathrm{p}^{{10}} }{2}}\right) , \end{aligned}$$

(A2b)

where $\rho _1$ and $\rho _{10}$ are the dislocation densities after the first and tenth cycles, respectively.

In the first loading cycle, the reorientation condition (Eq. 18b) at point $\sigma _{\mathrm{reo}}^1$ can be written as (note that ${\varvec{B}}_{\mathrm{reo}}$ and $f_3$ are both zero)

$$\begin{aligned} g\sigma _{\mathrm{reo}}^1 =Y_{M_\mathrm{t} \rightarrow M_\mathrm{d} } . \end{aligned}$$

(A3)

The parameter $Y_{M_\mathrm{t} \rightarrow M_\mathrm{d} }$ can be determined by Eq. (A3), as shown in Eq. (34a).

At the inflection point $\left( {\sigma ^{*},\varepsilon ^{*}} \right)$, since the applied stress is very low, the plastic deformation can be neglected. In this condition, the total strain consists of the elastic strain and the transformation strain, i.e.,

$$\begin{aligned} \varepsilon ^{*}=\frac{\sigma ^{*}}{E}+g\lambda _{\mathrm{ref}} . \end{aligned}$$

(A4)

The parameter $\lambda _{\mathrm{ref}}$ can be obtained by Eq. (A4), as shown in Eq. (34b).

The reorientation condition at the inflection point can be written as

$$\begin{aligned} g\sigma ^*-H_{M_\mathrm{t} \rightarrow M_\mathrm{d} }^\mathrm{I} \lambda _{\mathrm{ref}} =Y_{M_\mathrm{t} \rightarrow M_\mathrm{d} } . \end{aligned}$$

(A5)

The parameter $H_{M_\mathrm{t} \rightarrow M_\mathrm{d} }^\mathrm{I}$ can be obtained by Eq. (A5), as shown in Eq. (34c).

Considering the transition between the austenite and twinned martensite phases under the stress-free conditions, the transformation condition at the finish point of the forward transformation and the start point of the reverse transformation can be written as (by Eqs. (15a), (16b), (16c), and (20a))

$$\begin{aligned}&-\left[ {\beta \left( {M_\mathrm{f} -M_\mathrm{s} } \right) -Y_{A\rightarrow M_\mathrm{t} } +H_{A\rightarrow M_\mathrm{t} } } \right] = Y_{A\rightarrow M_\mathrm{t} } , \end{aligned}$$

(A6a)

$$\begin{aligned}&-\left[ {\beta \left( {A_\mathrm{s} -M_\mathrm{s} } \right) -Y_{A\rightarrow M_\mathrm{t} } +H_{A\rightarrow M_\mathrm{t} } } \right] = -Y_{A\rightarrow M_\mathrm{t} } . \end{aligned}$$

(A6b)

Two parameters $H_{A\rightarrow M_\mathrm{t} }$ and $Y_{A\rightarrow M_\mathrm{t} }$ can be obtained by Eq. (A6a) and  (A6b), as shown in Eq. (34d) and (34e).

Rewrite Eqs. (26a), (26b), (29), and (31) in the following form

$$\begin{aligned} \dot{Z}_i =d\left( {c_i \sqrt{\rho }-Z_i } \right) \left( {\sum _{i=1}^3 {\left| {\dot{\lambda }_i } \right| } } \right) ,\quad i=1,2,3,4,5, \end{aligned}$$

(A7)

where $Z_1, Z_2, Z_3, Z_4$, and $Z_5$ represent $B_{\mathrm{tr}}^\mathrm{n}$, $B_{\mathrm{reo}}^\mathrm{n}$, $H_{A\rightarrow M_\mathrm{d} }^{\uprho }$, $H_{M_\mathrm{t} \rightarrow M_\mathrm{d} }^{\mathrm{I\uprho }}$, and $Y_{A\rightarrow M_\mathrm{d} }^{\uprho }$, respectively. Note that Eq. (A7) cannot be integrated unless the $\sqrt{\rho }$ is regarded as a constant in the first cycle. Thus, for simplicity, it is assumed that $\sqrt{\rho }$ can be replaced by $\sqrt{\rho _1 }$ in Eq. (A7). Then, integrating Eq. (A7) yields:

$$\begin{aligned} Z_i =c_i \sqrt{\rho _1 }\left[ {1-\exp \left( {-d\sum _{i=1}^3 {\bar{{\lambda }}_i } } \right) } \right] ,\quad i=1,2,3,4,5,\nonumber \\ \end{aligned}$$

(A8)

where $\dot{\bar{{\lambda }}}_i =\left| {\dot{\lambda }_i } \right|$.

From Eqs. (21a) and (22a), it can be concluded that the amount of dislocation slippage in the loading part is much larger than that in the unloading and cooling/heating processes. Thus, in the first and tenth cycles, the plastic strains at maximum stress points are approximately equal to $\varepsilon _\mathrm{p}^1$ and $\varepsilon _\mathrm{p}^{10}$, respectively. Thus, the maximum strains in the first and tenth cycles can be given as

$$\begin{aligned} \varepsilon _{\max }^1= & {} \frac{\sigma _{\max }^1 }{E}+g\xi _\mathrm{d}^{\max ,1} +\varepsilon _\mathrm{p}^1 , \end{aligned}$$

(A9a)

$$\begin{aligned} \varepsilon _{\max }^{10}= & {} \frac{\sigma _{\max }^{10} }{E}+g\xi _\mathrm{d}^{\max ,10} +\varepsilon _\mathrm{p}^{10} . \end{aligned}$$

(A9b)

By Eq. (A9), the maximum volume fraction of the detwinned martensite in the first cycle, i.e., $\xi _\mathrm{d}^{\max ,1}$ can be obtained as

$$\begin{aligned} \xi _\mathrm{d}^{\max ,1} =\frac{\varepsilon _{\max }^1 -\frac{\sigma _{\max }^1 }{E}-\varepsilon _\mathrm{p}^1 }{g}, \end{aligned}$$

(A10a)

$$\begin{aligned} \xi _\mathrm{d}^{\max ,10} =\frac{\varepsilon _{\max }^{10} -\frac{\sigma _{\max }^{10} }{E}-\varepsilon _\mathrm{p}^{10} }{g}. \end{aligned}$$

(A10b)

By Eq. (A8), the variable $Z_i$ at the start and finish points, respectively, of reverse transformation in the first cycle can be written as

$$\begin{aligned} Z_{i,A_{\hbox {s}}}^1= & {} c_i \sqrt{\rho _1 }\left[ {1-\exp \left( {-d\xi _\mathrm{d}^{\max ,1} } \right) } \right] ,\quad i=1,2,3,4,5, \nonumber \\ \end{aligned}$$

(A11a)

$$\begin{aligned} Z_{i,A_{\hbox {f}}}^1= & {} c_i \sqrt{\rho _1 }\left[ {1-\exp \left( {-2d\xi _\mathrm{d}^{\max ,1} } \right) } \right] ,\quad i=1,2,3,4,5.\nonumber \\ \end{aligned}$$

(A11b)

For the transition between the austenite and detwinned martensite phases, the transformation conditions at the start and finish points of reverse transformation in the first cycle can be written as, respectively (by Eqs. (15b), (17b), (17c), (20b), (26)–(31))

$$\begin{aligned}&gc_1 R-\beta \left( {T_{A_{\hbox {s}}}^1 -M_\mathrm{s} } \right) +Y_{A\rightarrow M_\mathrm{d} }^\mathrm{f} -c_5 R\nonumber \\&\quad -\left( {H_{A\rightarrow M_\mathrm{d} }^\mathrm{f} +c_3 R} \right) \xi _\mathrm{d}^{\max ,1} =-Y_{A\rightarrow M_\mathrm{d} }^\mathrm{f} +c_5 R, \end{aligned}$$

(A12a)

$$\begin{aligned}&gc_1 P-\beta \left( {T_{A_{\hbox {f}}}^1 -M_\mathrm{s} } \right) +Y_{A\rightarrow M_\mathrm{d} }^\mathrm{f} -c_5 P\nonumber \\&\quad =-Y_{A\rightarrow M_\mathrm{d} }^\mathrm{f} +c_5 P, \end{aligned}$$

(A12b)

where $R=\sqrt{\rho _1 }\left[ {1-\exp \left( {-d\xi _\mathrm{d}^{\max ,1} } \right) } \right]$, $P=\sqrt{\rho _1 }\times \left[ {1-\exp \left( {-2d\xi _\mathrm{d}^{\max ,1} } \right) } \right]$.

In the tenth cycle, the term $\exp \left( {-d\sum \limits _{i=1}^3 {\bar{{\lambda }}_i } } \right)$ approaches zero, since $\sum \limits _{i=1}^3 {\bar{{\lambda }}_i }$ is a very large number. Thus, the variable $Z_i$ at the start and finish points, respectively, of the reverse transformation in the tenth cycle can be written as

$$\begin{aligned} Z_{i,A{\mathrm{s}}}^{10} =c_i \sqrt{\rho _{10} },\quad i=1,2,3,4,5, \end{aligned}$$

(A13a)

$$\begin{aligned} Z_{i,A{\mathrm{f}}}^{10} =c_i \sqrt{\rho _{10} },\quad i=1,2,3,4,5. \end{aligned}$$

(A13b)

Similarly, the transformation conditions at the start and finish points of the reverse transformation can be written as:

$$\begin{aligned}&gc_1 \sqrt{\rho _{10} }-\beta \left( {T_{A\mathrm{s}}^{10} -M_\mathrm{s} } \right) +Y_{A\rightarrow M_\mathrm{d} }^\mathrm{f} -c_5 \sqrt{\rho _{10} }\nonumber \\&\qquad -\left( {H_{A\rightarrow M_\mathrm{d} }^\mathrm{f} +c_3 \sqrt{\rho _{10} }} \right) \xi _\mathrm{d}^{\max ,10} \nonumber \\&\quad =-Y_{A\rightarrow M_\mathrm{d} }^\mathrm{f} +c_5 \sqrt{\rho _{10} }, \end{aligned}$$

(A14a)

$$\begin{aligned}&gc_1 \sqrt{\rho _{10} }-\beta \left( {T_{A\mathrm{f}}^{10} -M_\mathrm{s} } \right) +Y_{A\rightarrow M_\mathrm{d} }^\mathrm{f} -c_5 \sqrt{\rho _{10} }\nonumber \\&\quad =-Y_{A\rightarrow M_\mathrm{d} }^\mathrm{f} +c_5 \sqrt{\rho _{10} }. \end{aligned}$$

(A14b)

After the cyclic deformation, the residual strain $\varepsilon _\mathrm{r}^{\mathrm{10}}$ consists of two parts, the transformation strain and the plastic strain, i.e.,

$$\begin{aligned} \varepsilon _\mathrm{r}^{10} =g\xi _\mathrm{d}^{M_{\hbox {f}},10} +\varepsilon _\mathrm{p}^{10} . \end{aligned}$$

(A15)

By Eq. (A15), the volume fraction of the detwinned martensite phase at the finish point of the forward transformation in the tenth cycle, i.e., $\xi _\mathrm{d}^{M_{\hbox {f}},10}$ can be obtained as

$$\begin{aligned} \xi _\mathrm{d}^{M_{\hbox {f}},10} =\frac{\varepsilon _\mathrm{r}^{10} -\varepsilon _\mathrm{p}^{10} }{g}. \end{aligned}$$

(A16)

The transformation condition at this point can be written as

$$\begin{aligned} gc_1 \sqrt{\rho _{10} }-\beta \left\langle {T_{M_{\hbox {f}}}^{10} -M_\mathrm{s} } \right\rangle -\left( {H_{A\rightarrow M_\mathrm{d} }^\mathrm{f} +c_3 \sqrt{\rho _{10} }} \right) \xi _\mathrm{d}^{M_{\hbox {f}},10} =0.\nonumber \\ \end{aligned}$$

(A17)

By Eqs. (A12a), (A12b), (A14a), (A14b), and (A17), the five parameters $H_{A\rightarrow M_\mathrm{d} }^\mathrm{f}$, $Y_{A\rightarrow M_\mathrm{d} }^0$, $c_1$, $c_3$, and $c_5$ can be obtained, as shown in Eq. (34f).

In the tenth cycle, the reorientation condition at point $\sigma _{\mathrm{reo}}^{10}$ can be written as

$$\begin{aligned} g\left( {\sigma _{\mathrm{reo}}^{10} +c_2 \sqrt{\rho _{10} }} \right) =Y_{M_\mathrm{t} \rightarrow M_\mathrm{d} } . \end{aligned}$$

(A18)

The parameter $c_2$ can be determined by Eq. (A18), as shown in Eq. (34g).