Analytic Studies on QuasiStatic and Dynamic Phase Transitions in Shape Memory Alloys
對形狀記憶合金的擬靜態和動態相變的解析研究
Student thesis: Doctoral Thesis
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Award date  14 May 2018 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(a7cbc8c3f21247fe8592683e583709e6).html 

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Abstract
This thesis focuses on both the quasistatic and the dynamic phase transitions in shape memory alloys. Analytic studies on these two aspects are shown by three parts.
In the first part, a pretwisted shape memory alloy tube under tension is considered. A constitutive model with the specific forms of Helmholtz free energy and mechanical dissipation rate is employed to formulate the governing system. Exact solution of the tube under pure torsion is first derived and the shear stressshear strain response is determined, which reveals the hardening effect. For the pretwisted tube under uniaxial tension, the onedimensional asymptotic tensile stresstensile strain relations for the austenite, the phase transition and the martensite regions are derived by using the asymptotic expansion method. By properly defining an elastic energy potential, the present system can be viewed as an elastic problem, which can be related to the problem of Ericksen's bar. The analytical formulas for the nucleation and propagation stresses in terms of the preshear strain (caused by the pretwist) are obtained. Tension tests with fixed pretwists on SMA thinwalled tube are conducted, with a focus on the stressstrain response. The measured values and the analytical formula for the propagation stress are used to determine the material parameters, which, in turn, yields the updownup response of a shape memory alloy tube under pure tension. The tendency and turning point of the phase transformation in the pretwisted tube from localization to homogeneous deformation are also determined, which suggests a plausible way to avoid the instability in actuation applications.
In the second part, we consider dynamical phase transitions in a shape memory alloy bar. A difficulty is to determine the solution for a propagating phase boundary uniquely, which is not completely overcome yet. By adopting the same constitutive model, we derive the governing equations for the three phase regions from the threedimensional dynamical equations by a reduction through the coupledseries asymptotic expansion method. First, the threedimensional dynamical equations are truncated to the leading order with respect to the sufficiently small strain. Then we expand the axial and radial strains as a series of a small geometrical quantity (i.e. the scaled radial variable). By using the field equations, the higher terms can be expressed in terms of the leading term. As a result, the governing equations are obtained involving only the two leading terms of the axial and radial strains. By using the evolution law of the martensite volume fraction, the governing equations for each of the three regions are obtained. To relate the systems of the three phase regions, some connection conditions for the two phase interfaces based on the continuity of the martensite volume fraction and the deformation gradient are further proposed. By considering the travelingwave solution, we eventually derive a uniqueness condition for the unique solution in the sharpinterface model of shape memory alloys.
In the third part, the derived uniqueness condition is applied to study some concrete problems. Phase boundary propagation in a shape memory alloy bar/wire subjected to a displacementcontrolled tensile loading is first considered. It is found that at a sufficiently slow loading rate, the stress along the bar is approximately the Maxwell stress, with some modification by the loading rate. Also, the relation between the propagation speed of the phase boundary and the loading rate is obtained, which is, remarkably, the same as the one in the literature obtained by fitting the experimental data. This gives a solid validation of the obtained uniqueness condition. Furthermore, the uniqueness condition is used to study two dynamical problems: a simple impact problem and the Riemann problem. The results provide a deep insight to the wave structures of shape memory alloys under dynamical loading and give some motivation for a future investigation.
In the first part, a pretwisted shape memory alloy tube under tension is considered. A constitutive model with the specific forms of Helmholtz free energy and mechanical dissipation rate is employed to formulate the governing system. Exact solution of the tube under pure torsion is first derived and the shear stressshear strain response is determined, which reveals the hardening effect. For the pretwisted tube under uniaxial tension, the onedimensional asymptotic tensile stresstensile strain relations for the austenite, the phase transition and the martensite regions are derived by using the asymptotic expansion method. By properly defining an elastic energy potential, the present system can be viewed as an elastic problem, which can be related to the problem of Ericksen's bar. The analytical formulas for the nucleation and propagation stresses in terms of the preshear strain (caused by the pretwist) are obtained. Tension tests with fixed pretwists on SMA thinwalled tube are conducted, with a focus on the stressstrain response. The measured values and the analytical formula for the propagation stress are used to determine the material parameters, which, in turn, yields the updownup response of a shape memory alloy tube under pure tension. The tendency and turning point of the phase transformation in the pretwisted tube from localization to homogeneous deformation are also determined, which suggests a plausible way to avoid the instability in actuation applications.
In the second part, we consider dynamical phase transitions in a shape memory alloy bar. A difficulty is to determine the solution for a propagating phase boundary uniquely, which is not completely overcome yet. By adopting the same constitutive model, we derive the governing equations for the three phase regions from the threedimensional dynamical equations by a reduction through the coupledseries asymptotic expansion method. First, the threedimensional dynamical equations are truncated to the leading order with respect to the sufficiently small strain. Then we expand the axial and radial strains as a series of a small geometrical quantity (i.e. the scaled radial variable). By using the field equations, the higher terms can be expressed in terms of the leading term. As a result, the governing equations are obtained involving only the two leading terms of the axial and radial strains. By using the evolution law of the martensite volume fraction, the governing equations for each of the three regions are obtained. To relate the systems of the three phase regions, some connection conditions for the two phase interfaces based on the continuity of the martensite volume fraction and the deformation gradient are further proposed. By considering the travelingwave solution, we eventually derive a uniqueness condition for the unique solution in the sharpinterface model of shape memory alloys.
In the third part, the derived uniqueness condition is applied to study some concrete problems. Phase boundary propagation in a shape memory alloy bar/wire subjected to a displacementcontrolled tensile loading is first considered. It is found that at a sufficiently slow loading rate, the stress along the bar is approximately the Maxwell stress, with some modification by the loading rate. Also, the relation between the propagation speed of the phase boundary and the loading rate is obtained, which is, remarkably, the same as the one in the literature obtained by fitting the experimental data. This gives a solid validation of the obtained uniqueness condition. Furthermore, the uniqueness condition is used to study two dynamical problems: a simple impact problem and the Riemann problem. The results provide a deep insight to the wave structures of shape memory alloys under dynamical loading and give some motivation for a future investigation.
 Shape memory alloys, phase transitions