Abstract The motivation of this work is to better understand the dynamic behaviour of bistable structures presenting an analogy with regularized Ericksen bars. The archetype of such structures is the bistable tape spring, which exhibits a particular scenario of deployment, from the stable coiled configuration to the straight stable configuration: at each time of the deployment, the geometry of the tape is similar to a two-phase bar with a coiled part and a straight part separated by a transition zone that moves along the tape. One goal of this work is to show that a regularized and bistable Ericksen bar model contains all the properties to reproduce such a dynamic behaviour. The mathematical structure of this model presents a locally non-convex potential with two minima and a dependence of higher order terms. Some similarities exist between this model and the Euler-Korteweg system with a Van der Waals equation. To study numerically the dynamic behaviour of such models, it is necessary to solve a dispersive and conditionally hyperbolic system. For this purpose, the Lagrangian of the regularized bistable Ericksen model is extended and penalized. Variable boundary conditions are deduced from Hamilton’s principle and are used to control the evolution of the system. Dispersion analysis allows to determine the numerical parameters of the model. The obtained non–homogeneous hyperbolic system can be solved by standard splitting strategy and finite-volume methods. Numerical simulations illustrate how the parameters of the model influence the width and the propagation speed of the transition zone. The effect of energy dissipation is also examined. Finally, comparisons with an exact kink wave solution indicate that the extended Lagrangian solution reproduces well the dynamics of the original Lagrangian.

中文翻译：

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使用扩展拉格朗日方法的正则化和双稳态 Ericksen 棒的动力学

摘要 这项工作的动机是更好地理解双稳态结构的动态行为，该结构呈现出与正则化 Ericksen 条的类比。这种结构的原型是双稳态胶带弹簧，它展示了一种特殊的部署场景，从稳定的盘绕配置到直线稳定配置：在每次部署时，磁带的几何形状类似于两相杆卷曲部分和直线部分由沿胶带移动的过渡区隔开。这项工作的一个目标是证明一个正则化和双稳态的 Ericksen bar 模型包含重现这种动态行为的所有属性。该模型的数学结构呈现出具有两个最小值和高阶项相关性的局部非凸势。该模型与具有范德瓦尔斯方程的 Euler-Korteweg 系统之间存在一些相似之处。为了数值研究此类模型的动态行为，有必要求解色散和条件双曲系统。为此，对正则化双稳态埃里克森模型的拉格朗日量进行了扩展和惩罚。可变边界条件是从哈密顿原理推导出来的，用于控制系统的演化。色散分析允许确定模型的数值参数。得到的非齐次双曲系统可以通过标准分裂策略和有限体积方法求解。数值模拟说明了模型参数如何影响过渡区的宽度和传播速度。还检查了能量耗散的影响。最后，