AN ANALYTICAL MODEL for PHASE TRANSITIONS of an SMA WIRE UNDER UNIAXIAL TENSION Zilong Song∗ 1 and Hui-Hui Dai1

XXIV ICTAM, 21-26 August 2016, Montreal, Canada

AN ANALYTICAL MODEL FOR PHASE TRANSITIONS OF AN SMA WIRE UNDER UNIAXIAL TENSION Zilong Song∗ 1 and Hui-Hui Dai1

1Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong Summary This paper considers isothermal stress-induced phase transitions of shape memory alloys (SMAs) based on a two-variant constitutive model. Specifically, inhomogeneous deformations of a slender circular cylinder under uniform axial stress are investigated analytically. By utilizing two small parameters, the complex three-dimensional (3-D) mechanical system is reduced to a 1-D coupled system of two strains for each of three phase regions. Then, for given load parameter, inhomogeneous deformations with two and three regions are constructed, and many analytical period-k solutions are found for the first time. For displacement-controlled process, the transition from homogeneous states to localized inhomogeneous ones is identified, capturing the possible instabilities at nucleation. The analytical results reveal that phase transition starts from a small band rather than a point. This work can also shed light on the difficulties of direct numerical simulations.

INTRODUCTION

Shape memory alloys (SMAs) such as Ni-Ti have been employed for numerous real-world applications in various in- dustries. Many experiments on SMA wires under tension show that the stress-induced phase transitions exhibit Luders-like¨ behavior, i.e, they are realized by nucleation of martensite band and subsequent propagation of transformation front [1]. At nucleation, initial homogeneous deformation will be replaced by localized inhomogeneous deformations. Various models have been proposed to study such inhomogeneous deformations, including phenomenological models [2], phase field models (e.g. Levitas, Lee & Preston 2010) and strain gradient models (e.g. Chang, Shaw & Iadicola 2006). The original 3-D problem can be considered a free boundary problem, where the location and shape of interfaces are not known beforehand. To the authors’ knowledge, analytical results for such a complicated problem are rare and how the material and geometric parameters influence the transition process is unclear. Based on [2], this work will examine analytically the inhomogeneous deformations of an SMA wire with finite length, subjected to uniaxial tension in a 3-D setting. Analytical solutions with two or three phase regions will be constructed, with explicit dependence on material and geometric parameters. The transition process and the associated instabilities will be clarified. Also, this work can potentially provide certain guidance for direct numerical simulations.

THE CONSTITUTIVE MODEL AND GOVERNING SYSTEM

A two-variant constitutive model For phase transitions of SMAs, we adopt a two-variant constitutive model [2], which is based on two scalar functions, Helmholtz free energy Φ and rate of mechanical dissipation ξ. Let the volume fraction of martensite phase be α (0 ≤ α ≤ 1), and the natural conﬁguration of martensite phase be G = diag[1 − s1, 1 − s1, 1 + s2]. Then, Φ is adopted as | | − − − − −1 Φ(F, α) = Gα [(1 α)Φ1(Fα)+αΦ2(Fα)]+Bα(1 α)+[(1 α)ϕ1 +αϕ2], Gα = (1 α)I+αG, Fα = FGα , (1) where F is the deformation gradient, Φ1, Φ2 and ϕ1, ϕ2 are strain energy functions and thermal free energies of austenite and martensite phases, and B is the interfacial constant. And for ξ, we adopt a rate-independent form ξ = A+(α)|α˙ | := (k+α + Y +)|α˙ |, ifα ˙ ≥ 0; ξ = A−(α)|α˙ | := [k−(1 − α) + Y −]|α˙ |, ifα ˙ ≤ 0, (2) where k±,Y ± are dissipative constants. Whenever phase transition happens, the evolution of α is determined by −∂Φ/∂α = ±A±(α) for loading and unloading processes.

The mechanical system Now, we consider a slender SMA circular cylinder with radius a and length L subjected to a uniform axial stress γ. It is assumed that a/L ≪ 1, and we set L = 1 in the following. The reference and deformed positions are denoted by coordinates (R, Θ,Z) and (r, θ, z), then the concerned deformation is of the form r(R,Z) = R + V (R,Z), θ = Θ, z(R,Z) = Z + W (R,Z), 0 ≤ R ≤ a, 0 ≤ Z ≤ 1. (3) Accordingly, F can be easily obtained, and subsequently the nominal stress Σ is determined by the formula Σ = ∂Φ/∂F. The mechanical system is given by

DivΣ = 0, ΣRr(a, Z) = ΣRz(a, Z) = 0, ΣZz(R,Z)|Z=0,1 = γ, ΣZr(R,Z)|Z=0,1 = 0, (4) which will be coupled with the evolution of α in the previous section. ∗Corresponding author. Email: [email protected] The reduced system Two kinds of small parameters arise in the present system. First, the strains are small, thus we keep only leading terms, or equivalently, we set Φ1, Φ2 to be the elastic energies of linear isotropic materials. And for simplicity, we assume Φ1 = Φ2. Second, it follows from∑ the geometry that a is small, and so is the∑ radial variable R. Thus, we expand the displacements in ∞ 2k ∞ 2k the form V (R,Z) = R k=0 Vk(Z)R /2(1 + k),W (R,Z) = k=0 Wk(Z)R . By the technique in [3], the recursive ′ T formulas for coefficients uk := [Wk,Vk] can be deduced from the field equation (4)1 together with evolution of α, and 2 the conditions in (4)2 furnish the governing equations for u0. By keeping up to O(a ) terms, we get a final vector equation ′′ u0 + Au0 = f for each of three regions: austenite region (AR), martensite region (MR) and phase transition region (PTR, 2 0 < α < 1). Proper boundary conditions for u0 at Z = 0, 1 can be deduced from (4)3,4 with consistent order O(a ). Note that the coupling of axial and radial strains, and hence the high-dimensional effect, is kept in the reduced system.

INHOMOGENEOUS SOLUTIONS AND THE TRANSITION PROCESS

Inhomogeneous solutions with given load parameter γ The general solutions in each phase region can be readily obtained from the preceding linear equation of u0, e.g. in PTR

d3Z −d3Z u0(Z) = u0p + C1q1e + C2q1e + C3q2 cos(d4Z) + C4q2 sin(d4Z), (5) where u0p is the particular solution, Ci are integrating constants, and ±d3, ±d4i and q1, q2 are eigenvalues and eigenvectors. Experiments show that phase transition often takes place in the middle part of the wire. Suppose it starts exactly in ′′ ′ the middle, then the symmetric conditions for the two strains are adopted: W0 (0.5) = 0,V0 (0.5) = 0. First, we analyze two-region solutions composed of AR and PTR, with a planar interface in between. Given γ, the general solutions contain eight undetermined integrating constants, and the location of interface Z0 is also unknown. The unknown constants can be determined by four conditions at Z = 0, 0.5 and ﬁve connection conditions at Z0 derived from the continuity of F and α. (k) − Analytically, other constants can be expressed by Z0, which has period-k solutions of the form Z0 = 0.5 (arctan d5 + kπ)/d4,(k = 1, 2..). By a similar procedure, one can obtain inhomogeneous solutions with three regions AR, PTR and MR, and analytical period-k solutions are found for γ near Maxwell stress. There are also other connecting solutions, when the interface interacts with the middle surface (or boundary), see [3] for more details and for the nonsymmetric case.

Analysis on displacement-controlled process In this section, we analyze the stress-elongation curves, as shown in many experiments. Based on the preceding (period-1) solutions, Figure 1(a) shows the curves near nucleation for a relatively small a = 0.01. There are three solutions for each ﬁxed elongation in a small interval [∆1, ∆2], and by minimum energy criteria, the solution will jump from homogeneous state to an inhomogeneous one at ∆ = ∆1 and soon goes along a stress plateau (Maxwell stress) with three-region solutions, as in Figure 1(b). The complete loading curve is shown in Figure 1(c), as observed in experiments. The transition process does depend on the parameters, e.g. for a = 0.03 the transition will be smooth and stable without such a jump.

Γ÷ Γ÷ Γ÷ ÷ ÷ ΓNM ΓNM 0.01

0.009 0.009 0.009

0.0085 0.0085 0.008 Γm Γm 0.008 0.008 0.007

0.008 0.01 0.011 0.012 1 2 0.008 1 2 0.01 0.011 0.012 0 0.01 0.02 0.03 0.04 0.05 0.06 (a) for all period-1 solutions (b) real curve near nucleation (c) complete loading curve Figure 1: The stress-elongation curves with a = 0.01, γ∗ = γ/E and E = 4 ∗ 104MPa, see [3] for other material parameters.

CONCLUSIONS

In this paper, inhomogeneous deformations of an SMA wire with finite length are investigated analytically, and the transition process is identified. Phase transition starts from a finite band rather than a point, since the transition region in inhomogeneous deformations is always of finite width. This work can also shed light on the difficulties in direct numerical simulations (e.g. mesh sensitivity and convergence difficulty), since multiple solutions exist for certain fixed elongation near nucleation.

References

[1] Tse K.K.K., Sun Q.P.: Some Deformation Features of Polycrystalline Superelastic NiTi Shape Memory Alloy Thin Strips and Wires Under Tension. Key Eng. Mater., 177: 455-460, 2000. [2] Rajagopal K.R., Srinivasa A.R.: On the Thermomechanics of Shape Memory Wires. ZAMP 50(3):459-96, 1999. [3] Song, Z.: Analytic Studies on Martensitic Transformations of a Ni-Ti Shape Memory Alloy Wire Under Uniaxial Tension. PhD Thesis, CityU, 2013.