饱和多孔弹性Timoshenko梁动力响应的广义多辛数值实现

Generalized Multi-Symplectic Numerical Implementation of Dynamic Responses for Saturated Poroelastic Timoshenko Beam

采用广义多辛数值算法研究不可压饱和多孔弹性Timoshenko梁的流固耦合动力响应特性,构造梁动力响应方程的广义多辛形式,给出其Preissmann Box离散格式及各种广义多辛局部守恒律误差离散格式。数值模拟两端可渗透多孔弹性Timoshenko悬臂梁的动力响应过程,并分析其动力响应特性。发现两相耦合作用系数增大,梁各横截面的孔隙流体压力等效力偶、固相挠度和固相有效应力达到稳态值所需的时间缩短;梁长细比增大,所需时间加长,且挠度稳态值越接近相应经典单相弹性Euler-Bernoulli梁的静挠度值;随时间的推移,梁固相骨架承担所有外荷载,孔隙流体压力等效力偶最终将为零。表征耗散效应的参数取值减小,各种广义多辛数值误差的数量级也减小。

Based on the porous media theory and Timoshenko beam theory,properties of dynamic responses in fluid-solid coupled incompressible saturated poroelastic Timoshenko beam are investigated by generalized multi-symplectic method.Dynamic response equation set of incompressible saturated poroelastic Timoshenko beam is presented at first.Then a first order generalized multi-symplectic form of this dynamic response equation set is constructed,and errors of generalized multi-symplectic conservation law,generalized multi-symplectic local momentum and generalized multi-symplectic local energy are also derived.A Preissmann Box generalized multi-symplectic scheme of the dynamic response equation set is presented,the discrete errors of generalized multi-symplectic conservation law,generalized multi-symplectic local momentum conservation law and generalized multi-symplectic local energy conservation law are also obtained.In view of the dynamic responses of incompressible saturated poroelastic Timoshenko cantilever beam with two ends permeable and free end subjected to the step load,the transverse dynamic response process of the solid skeleton is simulated numerically,the evolution processes of solid effective stress and the equivalent moment of the pore fluid pressure over time are also presented numerically.The effects of fluid-solid coupled interaction parameter and slenderness ratio of the beam on the solid dynamic response process are revealed,as well as the effects on all generalized multi-symplectic numerical errors are checked simultaneously.From results obtained,the processes for solid deflection,solid effective stress and the equivalent moment of the pore fluid pressure approaching to their steady response values are all shortened with increasing of fluid-solid coupled interaction parameter,while the response process of solid deflection and the pore fluid equivalent moment are lengthened with increasing of slenderness ratio of the beam.Moreover,the steady value of solid deflection is much closer to the static deflection value of classic single phase elastic Euler-Bernoulli beam with increasing of the slenderness ratio.As time goes on,the solid skeleton of the beam will support all outside load,so equivalent moment of the pore fluid pressure becomes zero at last.In addition,it is presented all generalized multi-symplectic numerical errors decrease with the decreasing of parameters representing the dissipation effect for the dynamic system.