矩形磁流变弹性薄板的动力稳定性分析

摘要

磁流变弹性体材料制成的构件在机械荷载和强磁场共同作用下时,材料的弹性模量随外加磁场的变化而变化,构件的变形也使得构件内、外的磁场随之改变,因此这是一个耦合场问题,而且本构关系是非线性的.在载流薄板的运动方程、物理方程及磁流变本构关系的基础上,导出了由磁流变弹性体制成的薄板在电磁场与机械荷载共同作用下的磁弹性动力屈曲方程,应用Galerkin原理将屈曲方程整理为Mathieu方程的标准形式,并将动力屈曲问题归结为对Mathieu方程的求解.利用Mathieu方程解的稳定性,及系数λ和∞之间的本征关系,导出了磁流变弹性薄板动力屈曲临界状态的判别方程.讨论了磁场强度、薄板厚度、颗粒体积分数、薄板长度等参量对四边简支磁流变弹性薄板临界失稳荷载的影响.%When the magnetorheological member is under mechanical load and strong magnetic field simutaneously,the magnetic induction effect will take place which will cause the variation of the modulus of elasticity.Meanwhile,the variation of modulus has also effects on the deformation of the component which affects the magnetic field distribution.So it is a coupling system and the constitutive relation is nonlinear.In this paper,based on the equations of motion,physical equations and magnetorheological constitutive relations,the magneto-elastic dynamic buckling equation of a thin magnetorheological elastic plate under the action of mechanical load in a magnetic field is derived.Then the buckling equation is transformed into a standard Mathieu equation by using Galerkin method.Thus,the solving of buckling problem is changed to the solving of Mathieu equation.According to the solution's stability of Mathieu equation and the relation of the coefficients λ and ηin Mathieu equation,the criterion equation of the buckling problem is also presented.The influence of the magnetic field intensity,the thickness of the plate,particle volume fraction,plate length parameters on the critical buckling load of simply supported thin magnetorheological elastic plate are discussed.