不可压流体饱和多孔弹性梁的变分原理及有限元方法  

VARIATIONAL PRINCIPLES AND FINITE ELEMENT METHOD OF INCOMPRESSIBLE SATURATED POROELASTIC BEAM

作　　者：杨骁[1] 王佩菁[2] 

机构地区：[1]上海大学土木工程系,上海200072 [2]上海大学、上海市应用数学和力学研究所,上海200072

出　　处：《固体力学学报》2009年第1期54-60,共7页Chinese Journal of Solid Mechanics

基　　金：国家自然科学基金项目(10872124);上海市重点学科建设项目(Y0103)资助

摘　　要：基于不可压饱和多孔弹性梁动力弯曲的数学模型,建立了以多孔弹性梁挠度和孔隙流体压力等效力偶为宗量的Gurtin型变分原理,并给出了特殊边界条件下解耦时的仅以挠度为宗量的变分原理.同时,作为动力响应的退化情形,讨论了拟静态情形下的相应变分原理.根据所建立的变分原理,导出了一个有限元离散公式.由于Gurtin型变分原理是关于时间的卷积型的泛函,空间的有限元离散导致一个关于时间的对称微分—积分方程组,此方程组可进一步转化为常微分方程组.利用隐式Euler法,给出了时间区域的计算格式.作为一个数值例子,分析了饱和多孔弹性悬臂梁在自由端简谐载荷作用下的动力响应,分析了流相与固相相互作用对饱和多孔弹性悬臂梁动力响应的影响.Based on the mathematical model for dynamical bending of incompressible saturated poroelastic beam, a Gurtin variational principle is developed, in which the basic unknowns are deflection of the solid skeleton and the equivalent couple of the pore fluid pressure, and a variational principle for uncoupled problem in which the unknown is only the beam deflection is also presented for special boundary conditions. Furthermore, as the degenerated case of the dynamical response, the corresponding variational principles for quasi-static responses of the saturated poroelastic beam are discussed. At last, a finite element discretization scheme is derived based on the established variational principle. Since Gurtin-type functional of the variational principle contains a convolution formulation, the finite element discretization in space results in symmetrical differential-integral equations in time domain, which can be reduced to symmetrical differential equations further. Scheme in time domain is derived with Implicit Euler method. As a numerical example, the dynamical behavior of a saturated poroelastic cantilever beam subjected to a concentrated harmonic load at its free end is investigated, and the influence of the interaction between the solid skeleton and the pore fluid on the dynamical responses is analyzed numerically.

关 键 词：饱和多孔弹性梁 变分原理 动静力响应 有限元方法 

分 类 号：O343[理学—固体力学] TU473.2[理学—力学]