Analytic studies on Martensitic transformations of a ni-ti shape memory alloy wire under uniaxial tension
柱狀 Ni-Ti 記憶合金在單軸拉伸下所發生的馬氏體相變的解析研究
Student thesis: Doctoral Thesis
Related Research Unit(s)
This thesis focuses on the isothermal stress-induced martensitic transformations of a Ni-Ti wire, and some analytical studies on the macroscopic inhomogeneous deformations and the unstable mechanical behaviors are shown. In the first part, we focus on homogeneous or piecewise homogeneous deformations. Based on the constitutive model in the literature with the specific Helmholtz free energy and the rate of mechanical dissipation, the three-dimensional (3-D) model is formulated. Identifying the characteristic axial strain as the small parameter, we arrive at the asymptotic one-dimensional (1-D) equation which involves the stress, the axial strain and the phase state variable. By considering the evolution law of the phase state variable, the stress-strain relations corresponding to austenite, martensite and phase transition (phase mixture) regions are obtained. By referring to Ericksen's bar problem, we successfully deduce the analytical formulas for the nucleation and propagation stresses. The analytical results reveal explicitly how such important quantities depend on the material constants and the temperature. As an important application, we show that these formulas can be used for calibration of the material constants by comparing with the measured stress-strain curves. In the second part, the localized inhomogeneous deformations for an infinite cylinder are studied. Based on the same model, we truncate the mechanical system up to the leading order (linearized elasticity) and expand the strains as a series of the scaled radial variable. The recursive formulas are derived for higher order terms in the series, then the governing system involves only the two leading terms of radial and axial strains. For three different regions (i.e. austenite, martensite and phase transition regions), three linear systems are obtained. We seek inhomogeneous solutions with all three regions and general interfaces (free boundaries) between different regions, and some proper connection conditions are proposed by the continuity of martensite volume fraction and deformation gradient. Numerically eight solutions are easily found by Newton's method, and the optimal solution is selected by the energy criteria. We find that in the optimal solution the interfaces happen to be planar interfaces in the reference configuration. For the reduced case of planar interfaces, infinitely many solutions for the strains are obtained analytically. The analytical results reveal explicitly how the optimal solution and the width of the transition region depend the material and geometric parameters. The analytical results for the optimal state in the 3-D setting can be considered as a generalization of Ericksen's 1-D result. In the third part, the localized inhomogeneous deformations for a finite cylinder are studied. The constitutive formulation and the governing equations in three different regions are the same as the second part, the boundary conditions at infinity are replaced by those of uniform axial stress at two end surfaces. Both nonsymmetric and symmetric cases with two or three regions are considered, and analytical solutions for the strains are obtained for planar interfaces. For the symmetric case, the interaction of the interface with the middle surface is considered, and intermediate solutions are found to connect the two-region and three-region solutions. Subsequently we obtain the stress-elongation curve, which can determine the possible nucleation point for the displacement-controlled process. Thus the transition from homogeneous deformations to localized inhomogeneous deformations is identified, and finally the stress stays at Maxwell stress with the three-region solutions. Therefore, the essential features of the nucleation and propagation processes of SMAs are captured by the above inhomogeneous solutions.
- Martensitic transformations, Mathematical models, Nickel-titanium alloys