Dynamics of a regularized and bistable Ericksen bar using an extended Lagrangian approach

Abstract

The motivation of this work is to better understand the dynamic behaviour of bistable structures presenting an analogy with reguralized 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 reguralized 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.

Hyperbolic approximation of the BBM equation

“…Equations ( 9) are thus equivalent to (8). The first equation can be written in the form of conservation law expressed in variables u and v :…”

“…In this section we resume the mathematical properties of the hyperbolic Benjamin-Bona-Mahony (BBMH) quasilinear system (8).…”

“…The question arises whether such a technique can be developed for conservative dispersive systems which are the Euler-Lagrange equations for a 'master' Lagrangian. Such a 'hyperbolic' approach was recently proposed in [9,23] for the Serre-Green-Naghdi (SGN) model [26,27,45] describing long dispersive waves, for the nonlinear defocusing Schrödinger equation [14], and for a bistable Ericksen bar [8]. The PDE systems were derived from an 'augmented' ('extended') Lagrangian and have a rather similar mathematical structure.…”

Hyperbolic modeling of gradient damage and one-dimensional finite volume simulations

“…This constraint allows recovering the gradient damage formulation commonly used. The same approach has been used in [28][29][30] for various classes of dispersive equations such as Serre-Green-Naghdi, Schrödinger equation and bistable Ericksen bar. The application of Hamilton's principle on that extended Lagrangian yields an additional equation corresponding to the variation of D. The governing system is then:…”

“…In this paper, a new formulation allowing the efficient explicit numerical solution of dynamic problems in gradient damage solids is proposed. The derivation is based on the "extended Lagrangian approach" originally developed by one of the author in [28] for the Serre-Green-Naghdi equation (shallow water equation with microinertia terms), in [29] for the defocusing Schrödinger equation (shallow water types equations with second gradient terms) or for bistable Ericksen bars [30]. Using this strategy, the original system with global minimization is recast as a purely local hyperbolic problem with source terms that can easily be solved using finite volumes.…”