Modeling of the stress–strain responses and deformation patterns of superelastic NiTi tubes subjected to biaxial loadings

Abstract

Nearly equiatomic NiTi shape memory alloy exhibits superelasticity, i.e., it can be strained up to ~ 7% and recover completely upon unloading, and consequently, the stress–strain response forms a closed hysteresis. The mechanical behavior of superelastic NiTi is characterized by significant tension–compression asymmetry, which leads to complexity in the stress–strain responses and deformation patterns of thin-walled superelastic NiTi tubes loaded by axial force and internal pressure simultaneously. In the reported biaxial experiments, the NiTi tube exhibits hardening responses and essentially homogeneous deformation in a neighborhood of equibiaxiality. In other cases, its stress–strain responses trace stress plateaus associated with localized deformation patterns, and the level of plateaus, magnitude of transformation strains, and orientation of the localization bands are strongly dependent on the axial-to-hoop stress ratio. In this paper, finite element modeling is performed to analyze numerically the mechanical response of biaxially loaded superelastic NiTi tube. A numerical feedback control scheme is developed to maintain the stress ratio to follow the target value. The simulations reproduce successfully the observed phenomena in the experiments, such as the localization of helical bands, the variation of band angles with stress ratio, as well as the hardening and uniform deformation near the state of equibiaxial stress. In addition, the variation of axial and hoop stress–strain responses with different stress ratios are also studied, which are reasonably close to the experimental ones. The presented work demonstrates the validity of the developed finite element analysis framework and paves the way for analysis of superelastic shape memory alloy structures under multiaxial loading.

Appendix

1.1 Constitutive model

The constitutive model is based on the classical kinematic hardening model of plasticity. The total strain is decomposed into elastic and transformation strains:

$$\varepsilon_{ij} = \varepsilon_{ij}^{e} + \varepsilon_{ij}^{t} .$$

(4)

The elastic strain is related to the stress tensor by elasticity:

$$\sigma_{ij} = C_{ijkl} \varepsilon_{kl}^{e} ,$$

(5)

where

$$C_{ijkl} = \frac{E}{{2\left( {1 + \nu } \right)}}\left( {\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} + \frac{2\nu }{{1 - 2\nu }}\delta_{ij} \delta_{kl} } \right)$$

(6)

is the elastic stiffness tensor and E and ν are the Young’s modulus and Poisson’s ratio of the material, respectively.

The stress state is enclosed by a J₂-type transformation surface described by

$$\Phi = \frac{3}{2}\left( {s_{ij} - s_{ij}^{B} } \right)\left( {s_{ij} - s_{ij}^{B} } \right) - \sigma_{o}^{2} = 0,$$

(7)

where $s_{ij} = \sigma_{ij} - \sigma_{kk} \delta_{ij} /3$ and $s_{ij}^{B} = \sigma_{ij}^{B} - \sigma_{kk}^{B} \delta_{ij} /3$.

The evolution of transformation strain abides by the following flow rule:

$$\dot{\varepsilon }_{ij}^{t} = \Lambda \left( {s_{ij} - s_{ij}^{B} } \right),$$

(8)

where Λ is a non-negative multiplier.

The back stress $\sigma_{ij}^{B}$ is defined as the derivative of a potential function $\psi^{t}$ with respect to the transformation strain tensor:

$$\sigma_{ij}^{B} = \frac{{\partial \psi^{t} }}{{\partial \varepsilon_{ij}^{t} }}.$$

(9)

To introduce the tension/compression asymmetry of superelastic NiTi, the following form of the potential function is adopted:

$$\psi^{t} = \xi \psi_{C}^{t} (\overline{\varepsilon }^{t} ) + \left( {1 - \xi } \right)\psi_{T}^{t} (\overline{\varepsilon }^{t} ),$$

(10)

where $\psi_{C}^{t}$ and $\psi_{T}^{t}$ are two potential functions calibrated to the compressive and tensile stress–transformation strain responses, respectively. They are assumed to depend on an equivalent transformation strain $\overline{\varepsilon }^{t}$, which is defined as

$$\overline{\varepsilon }^{t} = J_{2}^{e} f\left( {J_{r} } \right).$$

(11)

In Eq. (11), $J_{2}^{e}=(2e_{ij}^{t} e_{ij}^{t} /3)^{1/2}$ and $J_{3}^{e} = \left( {4e_{ij}^{t} e_{jk}^{t} e_{ki}^{t} /3} \right)^{1/3}$ are, respectively, the second and third invariants of the deviatoric transformation strain and $J_{r} = {{J_{3}^{e} } \mathord{\left/ {\vphantom {{J_{3}^{e} } {J_{2}^{e} }}} \right. \kern-0pt} {J_{2}^{e} }}$. The ratio $J_{r}$ represents the transformation state, since it reaches − 1 under uniaxial contraction and 1 under uniaxial extension, and always lies in the interval [− 1, 1] in other cases. The function f scales $J_{2}^{e}$ to the equivalent transformation strain $\overline{\varepsilon }^{t}$ and it takes the following form:

$$f = \cos \left\{ {\frac{1}{3}\arccos \left[ {1 - a\left( {J_{r}^{3} + 1} \right)} \right]} \right\}.$$

(12)

One should note that $f( - 1) = 1$ and $f(1) < 1$. Resultantly, f (1) is equals to the ratio between the maximum transformation strain under compression versus that under tension.

The weight function ξ(J_r) aims to interpolate between the compressive and the tensile responses, by requiring ξ(− 1) = 1 and ξ(1) = 0. It takes the following form:

$$\xi = \frac{{f\left( {J_{r} } \right) - f\left( 1 \right)}}{{f\left( { - 1} \right) - f\left( 1 \right)}}.$$

(13)

The back stress is then derived as

$$\sigma_{ij}^{B} = \xi \frac{{{\text{d}}\psi_{C}^{t} }}{{{\text{d}}\overline{\varepsilon }^{t} }}\frac{{\partial \overline{\varepsilon }^{t} }}{{\partial \varepsilon_{ij}^{t} }} + \left( {1 - \xi } \right)\frac{{{\text{d}}\psi_{T}^{t} }}{{{\text{d}}\overline{\varepsilon }^{t} }}\frac{{\partial \overline{\varepsilon }^{t} }}{{\partial \varepsilon_{ij}^{t} }} + \left( {\psi_{C}^{t} - \psi_{T}^{t} } \right)\frac{\partial \xi }{{\partial \varepsilon_{ij}^{t} }}.$$

(14)

In uniaxial settings, as shown in Appendix A in [29] and [45], Eq. (14) can be reduced to

$$\sigma_{11,C}^{B} = \frac{{{\text{d}}\psi_{C}^{t} }}{{{\text{d}}\overline{\varepsilon }^{t} }}\frac{{\partial \overline{\varepsilon }^{t} }}{{\partial \varepsilon_{11}^{t} }}\;\;{\text{and}}\;\;\sigma_{11,T}^{B} = \frac{{{\text{d}}\psi_{T}^{t} }}{{{\text{d}}\overline{\varepsilon }^{t} }}\frac{{\partial \overline{\varepsilon }^{t} }}{{\partial \varepsilon_{11}^{t} }},$$

(15)

for compression and tension, respectively. Resultantly, the transformation function in Eq. (7) becomes

$$\left| {\sigma_{11,C} } \right| - f( - 1)\frac{{{\text{d}}\psi_{C}^{t} }}{{{\text{d}}\overline{\varepsilon }^{t} }} = \sigma_{oC} \;\;{\text{and}}\;\;\sigma_{11,T} - f(1)\frac{{{\text{d}}\psi_{T}^{t} }}{{{\text{d}}\overline{\varepsilon }^{t} }} = \sigma_{oT} ,$$

(16)

for loading, and a minus sign will appear in the right-hand sides for unloading. Obviously, the back stress under uniaxial compression and tension is governed by the derivative of the potential functions $\psi_{C}^{t}$ and $\psi_{T}^{t}$, respectively, which are assigned the same form:

$$\frac{{{\text{d}}\psi^{t} }}{{{\text{d}}\overline{\varepsilon }^{t} }} = h_{0} \overline{\varepsilon }^{t} + \left( {h_{1} - h_{0} } \right)\left[ {\overline{\varepsilon }^{t} - \frac{1}{b}\left( {1 - e^{{ - b\overline{\varepsilon }^{t} }} } \right)} \right] + \left( {h_{2} - h_{1} } \right)\left\{ {\begin{array}{*{20}c} 0 \\ {\left( {\varepsilon_{2} - \varepsilon_{1} } \right)\left( {{{\zeta^{3} } \mathord{\left/ {\vphantom {{\zeta^{3} } 4}} \right. \kern-0pt} 4} + {{\zeta^{2} } \mathord{\left/ {\vphantom {{\zeta^{2} } 8}} \right. \kern-0pt} 8}} \right)} \\ {{{3\left( {\varepsilon_{2} - \varepsilon_{1} } \right)} \mathord{\left/ {\vphantom {{3\left( {\varepsilon_{2} - \varepsilon_{1} } \right)} 8}} \right. \kern-0pt} 8} + \left( {\overline{\varepsilon }^{t} - \varepsilon_{2} } \right)} \\ \end{array} } \right. \, \begin{array}{*{20}c} {0 \le \overline{\varepsilon }^{t} < \varepsilon_{1} } \\ {\varepsilon_{1} \le \overline{\varepsilon }^{t} \le \varepsilon_{2} } \\ {\varepsilon_{1} \le \overline{\varepsilon }^{t} } \\ \end{array} ,$$

(17)

where $\zeta = {{\left( {\overline{\varepsilon }^{t} - \varepsilon_{1} } \right)} \mathord{\left/ {\vphantom {{\left( {\overline{\varepsilon }^{t} - \varepsilon_{1} } \right)} {\left( {\varepsilon_{2} - \varepsilon_{1} } \right)}}} \right. \kern-0pt} {\left( {\varepsilon_{2} - \varepsilon_{1} } \right)}}$. The parameters b, h₀, h₁, h₂, ε₁, and ε₂ for compression and those for tension are calibrated separately.

The size of the transformation strain σ_o in Eq. (7) also shows asymmetry between tension and compression and the following form is adopted:

$$\sigma_{o} = \left\{ {\begin{array}{*{20}c} {\sigma_{oC} } \\ {\sigma_{oC} + \left( {\sigma_{oT} - \sigma_{oC} } \right)\left( {10\eta^{3} - 15\eta^{4} + 6\eta^{5} } \right)} \\ {\sigma_{oT} } \\ \end{array} } \right. \, \begin{array}{*{20}c} {\eta < 0} \\ {0 \le \eta \le 1} \\ {\eta > 1} \\ \end{array} ,$$

(18)

where $\eta = \left( {J_{3}^{e} + 0.01} \right)/0.0098$.

The calibrated parameters in the constitutive model are listed in Table 1. The uniaxial compressive and tensile stress–strain responses produced by the calibrated model are shown in Fig. 14.

Table 1 Model parameters for compressive and tensile responses

Fig. 14

The resultant tensile and compressive stress–strain responses together with the measured ones

1.2 Mesh sensitivity of the simulation results

Incorporation of softening in the material’s tensile stress–strain response implies mesh dependence of the finite element solution, which can be removed by adding a mild rate dependence into the constitutive model (see [46, 47] for example) or regularizing the model by gradient-related energy terms, like [25, 48, 49]. In this paper, the adopted constitutive model for superelastic NiTi is rate independent; hence, we must examine the mesh sensitivity of the finite element simulations. Therefore, we adopted three mesh densities to compare their performances on simulating the uniaxial tension test. Mesh 1 has 1 element in the radial (through-the-thickness) direction, 90 elements in the circumferential direction, and 225 elements in the length direction. Mesh 2 (the mesh density used in this study) has 1, 112, and 280 elements in the radial, circumferential, and axial directions, respectively. Mesh 3 has 1, 160, and 400 elements in the radial, circumferential, and axial directions, respectively. All three meshes contain only one element through the thickness based on the consensus that one element is enough for successful simulation of the localization phenomena in thin walls since the variation of strains and other quantities through the thickness is nearly negligible [40, 44, 50].

The stress–strain responses calculated using the three meshes are shown in Fig. 15. It is seen that Meshes 2 and 3 produce almost identical stress–strain curves, regarding the levels of the upper and lower plateaus and the strain at completion of martensitic transformation, while the stress plateaus obtained by Mesh 1 are obviously higher than those by the two denser meshes. As the mesh density increases from Mesh 1 to Mesh 3, the calculated stress–strain curve exhibits a trend of convergence. The calculated stress–strain curve using Mesh 2 is much closer to the experimental one compared with Mesh 1, but the improvement brought by further refinement into Mesh 3 is not that significant.

Fig. 15

Simulation results of stress–strain responses under uniaxial tension with the three meshes

Figure 16 shows a set of axial strain contours in the FE simulation results using the three meshes. At the onset of localization, all three meshes exhibit similar localization bands with an angle of about 55° with respect to the axial direction, which is consistent with experimental observations. However, the localization band of Mesh 1 quickly turns perpendicular to the axial direction and shows distinct features to Mesh 2 and Mesh 3, both of which produce a finger-like propagating front that travels all the way to the bottom of the tube until the completion of the forward transformation. During unloading, the localization band of Mesh 1 keeps perpendicular to the axis while propagating in the reverse direction to that during loading, while Mesh 2 and Mesh 3 exhibit a spiral band that extends along the axis. The finger-like and spiral localization bands under uniaxial tension and the orientation angle of approximately 55° have been predicted theoretically [42] and reported widely in the literature, see [13, 19, 50] and other publications. Mesh 1 fails but both Meshes 2 and 3 successfully reproduce such important features of localized deformation in superelastic NiTi.

Fig. 16

Simulation results of strain contours during uniaxial tension with the three meshes

Based on the results of FE simulations on uniaxial tension of the superelastic NiTi tube using the three mesh densities, we can conclude that Mesh 2 is sufficiently fine to reproduce the experimentally recorded stress–strain response and finer features of localization of deformation, which are very similar to the simulations by the finer mesh. Thus, this paper chose Mesh 2 for finite element modeling in order to reach a tradeoff between accuracy and computational cost, as was done in other works on superelastic shape memory alloy thin-walled tubes [37, 38, 40]. Main effects of mesh sensitivity lie on the width of the transition zones between the high and low strain regions in the localization band, which can be eliminated by introducing rate dependence or gradient energies into the constitutive model in the future.