Closed-form solutions for inhomogeneous states of a slender 3-D SMA cylinder undergoing stress-induced phase transitions
Introduction
Shape memory alloys (SMAs) such as Ni–Ti have been used for numerous real-world applications in various fields such as aerospace, automotive and biomedical industry (Lagoudas, 2008). They are well known for two remarkable properties: shape memory effect and pseudo-elasticity. Pseudo-elasticity means that SMAs can undergo large recoverable strain (up to 8%) without permanent damage by loading and unloading at high temperatures (Huo & Müller, 1993). The mechanism is associated with the stress-induced phase transitions between the parent austenite phase and the product martensite phase (Shaw & Kyriakides, 1995).
A large number of experiments (He et al., 2010, Shaw and Kyriakides, 1995, Shaw and Kyriakides, 1997, Sun et al., 2012, Zhang et al., 2010) on pseudo-elasticity have been conducted on SMA wires at different temperatures under tension. For a class of SMAs (like Ni–Ti) exhibiting strain softening behaviors, the stress-induced phase transitions are realized by nucleation of martensite band in some particular region and subsequent growth along the wire by propagation of transformation fronts (Shaw and Kyriakides, 1997, Tobushi et al., 1993, Tse and Sun, 2000). The distinction and roles of nucleation and propagation stresses have been observed in experiments, and they are essential for the above Lüders-like behavior. At nucleation, initial homogeneous deformations will be replaced by localized inhomogeneous deformations, where the strain varies rapidly across the transformation front or the phase transition region (PTR). Thus, capturing the structure of the inhomogeneous deformation and determining the width of the transformation front are very important, and also their dependence on geometrical and material parameters is crucial both theoretically and practically. However, systematic analytical investigations on the inhomogeneous deformations with general interfaces under a 3-D framework are few due to the complexity of the governing equations.
In the literature, various constitutive models have been proposed to investigate the inhomogeneous deformations during phase transitions. The phase field (Ginzburg–Landau) models (Artemev et al., 2001, Artemev et al., 2000, Levitas et al., 2010, Wang and Khachaturyan, 1997) concentrate on the formation and evolution of inhomogeneous deformations at the microscopic level, which are described by a set of kinetic equations for the order parameters. There is no doubt that the evolution of microstructure can uncover the origin of some distinctive properties of SMAs, however, the size of simulated specimen is limited (e.g. 100–1000 nm). As the specimen treated in experiments are relatively large (e.g. 10–100 mm) and the size of the inhomogeneous region is comparable with the radius of the wire (e.g. 0.1–1 mm), it is more suitable to treat the above Lüders-like behavior at the macroscopic level. At this level, the formation of localized inhomogeneous deformations is due to material instability, i.e. multiple solutions exist in a certain range of the total elongation. At the continuum level, the models are subdivided into two categories: sharp interface models and diffuse interface models.
In sharp interface models, the phase interface is treated as a sharp interface, where the strain is discontinuous and some jump conditions are proposed (e.g. the continuity of displacements and traction). Abeyaratne and Knowles, 1990, Abeyaratne and Knowles, 1991 treated the boundary of two distinct phases as a singular surface called equilibrium shock, and proposed a thermomechanical framework with suitable jump conditions and kinetic relations that govern the velocity of singular surfaces. Subsequently in Abeyaratne and Knowles, 1993, Knowles, 1999, both quasi-static and dynamic processes of phase transitions of SMAs were studied in a 1-D setting based on a specific nonconvex Helmholtz free energy. With non-equilibrium jump conditions at a moving phase interface, Berezovski and Maugin, 2004, Berezovski and Maugin, 2005 conducted some simulations for the dynamic process of front propagation. Some models (Sun and Zhong, 2000, Zhong et al., 2000) treated the martensite band as an inclusion with uniform eigenstrain in an inclusion–matrix system, and obtained some analytical results in a 3-D setting. It is easy to handle the sharp interface in a 1-D setting, while in a 3-D setting the shape of the interface, which is a free boundary, needs also to be considered.
In diffuse interface models, the phase interface (transformation front) is identified with a small transition zone, where the strain will be continuous (or smooth) with large gradients. Some thermoelastic models (Chang et al., 2006, He and Sun, 2010, Shaw, 2002, Sun and He, 2008) introduced the gradient of strain, then a characteristic length scale for the width of the transformation front was set by the coefficient. It is expected that the width is proportional to the radius of the specimen, however it is not an easy task to determine the coefficient (or the coefficient tensor in a 3-D setting) beforehand. Some phenomenological models (Idesman et al., 2005, Rajagopal and Srinivasa, 1999, Zaki and Moumni, 2007) introduced the martensite volume fraction as an internal variable into the Helmholz (or Gibbs) free energy, then both the strain and martensite volume fraction were continuous across the transformation front. Based on such models, some recent works (Wang and Dai, 2010, Wang and Dai, 2012a, Wang and Dai, 2012b) analytically studied the inhomogeneous deformations together with the phase field for SMA wires and layers. One advantage of these phenomenological models is that there are no artificial coefficients and the influence of each parameter on the inhomogeneous deformation can be easily identified.
Based on an internal-variable model, the previous work (Song, Dai, & Sun, 2013) studied piecewise homogeneous deformations of an SMA wire, where analytical formulas for nucleation and propagation stresses were derived. Based on the same constitutive model, the present study aims at seeking inhomogeneous deformations (in particular the steady state during propagation) of a slender SMA cylinder in a 3-D setting, and clarifying the roles of material parameters on the optimal solution. Specifically, the mechanical system is first simplified for three different regions (i.e. austenite, martensite, and phase transition regions) by utilizing the smallness of the characteristic axial strain and aspect ratio (Dai et al., 2009, Dai and Wang, 2009). Then, we concentrate on the special case that the cylinder contains only one middle PTR far away from the two ends. Some analytical results are obtained for inhomogeneous states with planar interfaces and some semi-analytical results are shown for those with general interfaces to justify the analytical results.
Now we make some comparisons between the current work and the works in the literature. Compared with the previous works of Wang and Dai, 2012a, Wang and Dai, 2012b on inhomogeneous deformations of a layer in a 2-D setting, there are mainly two distinctions: (1) Instead of reducing the mechanical system to a 1-D rod-like equation for the axial strain, the axial and radial strains are coupled together in the present system but without nonlinearity. (2) Instead of proposing certain jump conditions on the interface, some connection conditions regarding the continuity of the martensite volume fraction and the deformation gradient are adopted. In this work, the high-dimensional effect and the coupling of the two strains are essential for obtaining continuous strain fields by proposing proper connection conditions. Most other analytical results in the literature are based on 1-D models, accounting for various aspects of SMAs, e.g. stress–strain–temperature response of specific structures (wires (Brinson et al., 1996, Marfia and Rizzoni, 2013), layer structures (Wu, Gordaninejad, & Wirtz, 1996) and actuators (Kosel and Videnic, 2007, Shaw and Churchill, 2009)) and bending of SMA beams (Auricchio and Sacco, 1997, Mirzaeifar et al., 2013). However, the high-dimensional effect and strain softening behavior are not fulled taken into account, and they are crucial to determining the width of the transformation front (or PTR). Therefore, it is desirable to adopt a 3-D model with strain softening behavior to capture some unusual features of SMAs as in this work.
In a 1-D setting, (Ericksen, 1975) studied the equilibrium theory of bars with a nonmonotone stress–strain relation. The main results for a displacement-controlled process are: (1) The stable states are piecewise homogeneous deformations with strain discontinuities. (2) The stress is exactly the Maxwell stress in the stable states. (3) There are infinitely many stable states, with the number of strain discontinuities being arbitrary. In the present work, based on a 3-D model with a Helmholtz free energy and a dissipation function, the stress–strain relation with softening behavior is derived with meaningful physical parameters. The subsequent analytical results reveal that: (1) The optimal state (with minimal energy) is a localized inhomogeneous deformation with a continuous strain field. (2) The average axial stress on the cross-section of the cylinder in the optimal state is exactly the Maxwell stress of the 1-D stress–strain curve, although the axial stress is inhomogeneous on the cross-section in the 3-D setting. (3) The optimal inhomogeneous state is unique with only one transformation front (PTR), since more transformation fronts will contribute more energy. In a certain sense, the present result is more conclusive by considering the high-dimensional and strain-coupling effects.
In the literature, the models (Chang et al., 2006, Sun and He, 2008) with gradient terms are popular to capture the localized inhomogeneous deformations both analytically and numerically. Nevertheless, the gradient parameter is difficult to determine and its relation with material parameters is unclear. In this work, the localized inhomogeneous deformation (gradient effect) is determined by a local theory with a local Helmholtz free energy and dissipations. The occurrence of such inhomogeneous deformations is due to the structural effect and the strain softening behavior of the stress–strain relation. The width of the transformation front is closely related to the stress drop of the stress–strain curve (see experiments in Sun & Li, 2002), which is determined by the competition between the interfacial energy and some dissipation terms. Therefore, such terms might be considered as the underlying mechanism of localized inhomogeneous deformations, and under the 3-D setting they naturally play the same role as the extra gradient terms (with artificial coefficients).
Finally, we mention some potential applications based on the current analytical results. First, the formulas for the width of the transformation front can be used to calibrate material constants. In Song et al. (2013), we have utilized the nucleation and propagation stresses to calibrate material constants, however, in general the measured nucleation stresses are not accurate. Therefore, if the two widths of the transformation front for loading and unloading processes are available from experiments, they can replace the two nucleation stresses for calibration. Second, by comparing the present result with that of a 1-D gradient model, the relation between the gradient parameter and material parameters is found. So, the formula can provide some guidance on the choice of the gradient parameter in numerical calculations. Third, multiple solutions exist for certain fixed total strain, thus it can shed light on the difficulties of direct numerical simulations (Mirzaeifar et al., 2013) (e.g. mesh sensitivity and convergence difficulty). The analytical results of the optimal state might be used to verify the numerical simulations or to provide some initial guess for the inhomogeneous deformations in calculations.
This paper is arranged as follows. Section 2 briefly recalls the 3-D constitutive model and Section 3 builds up the mechanical system. Section 4 derives the recursive formulas for the higher-order terms in the series expansions of strains, and simplifies the original system to three 1-D linear systems for three different regions. In Section 5, we consider the special case of a cylinder with three regions and two planar interfaces. Closed-form solutions of strains are given in Section 6. In Section 7, we compare the present result with that of a 1-D gradient model. In Section 8, some semi-analytical results are shown based on a general formulation to justify the previous analytical results. Finally conclusions are drawn in Section 9.
Section snippets
Constitutive model
In this section, we describe the constitutive model for SMAs as adopted in the previous paper (Song et al., 2013). Following the constitutive model of Rajagopal and Srinivasa (1999), a Helmholtz free energy will be adopted to model SMAs. By the energy balance law and entropy production equation we have the following systemwhere is the nominal stress tensor, is the entropy, is the rate of mechanical dissipation, T is the absolute temperature
The mechanical system
In this section we consider a slender SMA circular cylinder with radius a and length L subjected to a uniaxial force (see Fig. 1). Then we present the mechanical system of the cylinder, based on the previous Helmholtz free energy .
Suppose that the cylinder is homogeneous in its reference configuration and entirely made up of austenite phase, and it is assumed that . We use the cylindrical polar coordinate system and denote by and the coordinates of a point of the
Recursive formulas and governing equations
Two kinds of small parameters arise in the present system. Firstly, the total axial strain (as in experiments) is small, thus the displacements are small quantities. Then, to leading order, the mechanical system becomes piecewise linear. Secondly, from the geometry of the slender cylinder the ratio is small, and so is the scaled radial variable . Then, we can readily expand the axial and radial strains in series of , where the higher-order terms are further determined by the
The case that the PTR is far away from the ends
For the inhomogeneous deformations observed in experiments, the SMA cylinder contains multiple phase regions. The interfaces between the AR and the PTR and between the PTR and the MR are free boundaries, in the sense that their positions and shapes are not known beforehand. Their interaction with the end boundary imposes an enormously difficult problem. In the following sections, we restrict our attention on the special case (the steady state during propagation) that the PTR is far away from
Analytical results
In the previous section, we have obtained a nonlinear algebraic system for determining the ten unknowns: constants and eigen pair . In this subsection, we present closed-form expressions for them.
First we observe that the unknowns appear in linear form in the algebraic system. From the eight conditions regarding and at (cf. (63)) and , we can express the constants in terms of and . In the AR and MR, we have
The 1-D case with a gradient term
In this section, we conduct a pure 1-D analysis for a 1-D gradient model. The width of the PTR is determined by a similar procedure, and by comparing with the 3-D results one can relate the gradient parameter to material constants.
The case with general interfaces
In the previous sections, we have analyzed the case with planar interfaces. Now, we examine the situation with general interfaces. Some numerical results are provided to verify the correctness of the analytical results for planar interfaces, and the optimal solution is selected by the energy criterion.
Conclusions
Based on a 3-D constitutive model, we have studied the inhomogeneous deformations of a slender SMA cylinder subjected to uniaxial tension/extension. By taking advantage of two small parameters, the mechanical system was reduced to three linear systems for three different phase regions. Then the special inhomogeneous deformations with three regions (the steady state during propagation) were considered. For the case of planar interfaces, infinitely many solutions for strains were obtained
Acknowledgments
We thank Professor Qing-Ping Sun of HKUST for valuable discussions. The work described in this paper is supported by a strategic grant from City University of Hong Kong (Project No.: 7004066).
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