双程形状记忆效应的唯象动力学模型

构造可以用于描述一维结构的形状记忆合金（SMA）的双程形状记忆效应的唯象动力学模型. 该模型基于与形状记忆合金中热弹性相变有关的唯象理论，将应力场和热场下的滞回环曲线视为马氏体相变和马氏体变体重构在宏观层面上的表现. 为了模拟温度诱发的相变，构造非凸自由能函数，使得函数的每个局部平衡对应于相变过程中的一个相. 在外部负载（力或者热）的作用下，可以通过模拟系统状态（应变）在不同平衡态之间的转变，研究温度诱发的相变. 相变动力学的控制方程采用拉格朗日方程，以非线性微分方程来表示. 利用非线性常微分方程描述单程形状记忆效应，通过对不同相变过程的加权组合描述双程形状记忆效应. 开展有关力和热负载下的数值实验，模拟热和应力诱发的相变以及热负载下与单程形状记忆效应和双程形状记忆效应有关的滞回环，模拟马氏体重构所导致的单滞回环以及超弹性效应所引起的双滞回环. 从实验结果可以看出，双程形状记忆效应及超弹性效应均可以被提出的模型成功捕捉，验证了该模型的描述能力.

Phenomenological dynamic model on two-way shape memory effects of shape memory alloy

A phenomenological dynamic model was constructed for the modeling of two-way shape memory effect in one-dimensional shape memory alloy (SMA) structure. The model was based on the phenomenological theory of thermoelastic phase transformations in SMAs. Hysteresis loops in both mechanical and thermal fields were treated as macroscopic illustrations of martensite transformations and martensite variant re-orientations. A non-convex free energy function was constructed to characterize the phase transformations induced by temperature. Then each of its local equilibriums can be used to represent a phase in the transformations. System states (strain) can be transformed upon external loadings (mechanical or thermal) from one stable equilibrium to another. Then the dynamics of phase transformations can be modeled by simulating the system state transformations. Governing equations for the transformation dynamics were formulated by employing the Lagrange's equation, and were expressed as nonlinear differential equations. One-way shape memory effect was described by a nonlinear ordinary differential equation, and the model for two-way shape memory effect was constructed by taking the weighted combination of different phase transformations. A series of numerical experiments were conducted. Phase transformations induced by both mechanical and thermal loadings were simulated. Hysteresis loops associated with both one-way shape memory effect and two-way shape memory effect under thermal loadings were presented. A single hysteresis loop associatedwith mechanical-induced martensite variant re-orientations and double hysteresis loops associated with the pseudo-elastic effects were presented. The numerical results showed that two-way shape memory effect and pseudo-elastic effect were successfully modeled, which demonstrated the capability of the current model.