饱和多孔弹性Timoshenko梁的大挠度分析

基于微观不可压饱和多孔介质理论和弹性梁的大挠度变形假设,考虑梁剪切变形效应,在梁轴线不可伸长和孔隙流体仅沿轴向扩散的限定下,建立了饱和多孔弹性Timoshenko梁大挠度弯曲变形的非线性数学模型.在此基础上,利用Galerkin截断法,研究了两端可渗透简支饱和多孔Timoshenko梁在突加均布横向载荷作用下的拟静态弯曲,给出了饱和多孔 Timoshenko梁弯曲变形时固相挠度、弯矩和孔隙流体压力等效力偶等随时间的响应.比较了饱和多孔Timoshenko梁非线性大挠度和线性小挠度理论以及饱和多孔 Euler-Bernoulli梁非线性大挠度理论的结果,揭示了他们间的差异,指出当无量纲载荷参数q＞l0时,应采用饱和多孔Timoshenko梁或Euler-Bernoulli梁的大挠度数学模型进行分析,特别的,当梁长细比λ＜30时,应采用饱和多孔Timoshenko梁大挠度数学模型进行分析.     

饱和多孔弹性Timoshenko悬臂梁的拟静力弯曲

在经典单相Timoshenko梁变形和孔隙流体仅沿多孔梁轴向运动的假定下,基于不可压饱和多孔介质的三维理论,本文首先建立了横观各向同性饱和多孔弹性Timoshenko悬臂梁拟静力弯曲的一维数学模型,并给出了相应的边界条件。其次,利用Laplace变换及其数值逆变换,分析了端部不同渗透条件下,饱和多孔弹性Timoshenko悬臂梁在端部梯载荷作用下的拟静力响应,给出了饱和多孔Timoshenko悬臂梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶等随时间的响应曲线,并与饱和多孔Euler-Bernoulli悬臂梁的响应进行了比较,考察了梁长细比对弯曲的影响。数值结果表明：固相骨架与孔隙流体的相互作用具有粘性效应,梁弯曲的拟静态挠度具有蠕变行为,端部渗透条件对梁的弯曲响应有显著的影响,并且,饱和多孔弹性Timoshenko悬臂梁的拟静态响应亦存在Mandel-Cryer现象。     

简支饱和多孔弹性梁的非线性弯曲

基于饱和多孔介质理论和弹性梁的大挠度弯曲假设,在多孔弹性梁轴线不可伸长,孔隙流体仅沿轴向方向扩散的限制下,建立了微观不可压饱和多孔弹性梁大挠度拟静态响应的一维非线性数学模型.在此基础上,利用Galerkin截断法,分析了两端可渗透的简支多孔弹性梁在突加横向均布载荷作用下的非线性弯曲,给出了梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶随时间的响应曲线.数值结果表明:当载荷较小时,大挠度非线性与小挠度线性理论的结果相差很小,而当载荷较大时,非线性大挠度理论的结果小于相应线性小挠度理论的结果,并且这种差异随着载荷的增大而增大.同时,在载荷突加于梁上时,多孔弹性梁骨架起初不变形,孔隙流体压力等效力偶由零突增为非零,其值与外载荷保持平衡.随着时间的增加,梁的挠度增加,等效力偶逐渐减小为零,最终多孔梁骨架承担全部的外载荷.     

不可压饱和多孔弹性简支梁的动力响应

在杆件弯曲小变形的假定下,考虑杆件的侧向变形因素,根据多孔介质理论,本文首先建立了不可压饱和多孔弹性梁弯曲变形时动力响应的控制方程。其次,基于所建立的控制微分方程,利用变量分离法,研究了两端可渗透的饱和多孔弹性简支梁在梁中间集中载荷作用下的动力响应,得到了不同物性参数下简支梁动态弯曲时挠度和孔隙流体压力等效力偶等随时间的响应曲线。研究发现由于孔隙流体和固相骨架的相互作用,不可压饱和多孔弹性梁挠度的动力响应具有粘性特征,同时,随着时间的增加,饱和多孔弹性梁的挠度、弯矩等最终趋于经典弹性梁的静挠度、弯矩,此时,孔隙流体压力为零,梁的固相骨架承担所有的外载荷。     

轴向扩散下简支饱和多孔弹性梁的大挠度分析

基于多孔介质理论和弹性梁的大挠度理论,并考虑轴向变形,在孔隙流体仅沿轴向扩散的假设下,建立了微观不可压饱和多孔弹性梁大挠度弯曲变形的一维非线性数学模型.在此基础上,忽略饱和多孔弹性梁的轴向应变,并利用Galerkin截断法,研究了两端可渗透的简支饱和多孔弹性梁在突加横向均布载荷作用下的拟静态弯曲,给出了饱和多孔梁弯曲时挠度、弯矩和轴力以及孔隙流体压力等效力偶等沿轴线的分布曲线.揭示了大挠度非线性和小挠度线性模型的结果差异,指出大挠度非线性模型的结果小于相应小挠度线性模型的结果,并且这种差异随着载荷的增大而增大.计算表明:当无量纲载荷参数q>5时,应该采用大挠度非线性数学模型进行研究.     

简支饱和多孔弹性梁的非线性动力响应

基于饱和多孔弹性梁大挠度变形的数学模型,利用Galerkin截断法,本文研究了两端可渗透的简支饱和多孔弹性梁分别在突加横向均布常载荷和简谐载荷作用下的动力响应,得到了梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶等随时间的响应,考察了不同载荷下多孔弹性梁弯曲的响应特征.结果表明:随着载荷的增加,在常载荷作用下多孔弹性梁非线性大挠度响应与线性小挠度的差别愈加明显,而在简谐载荷作用下,多孔弹性梁的动力响应呈现较丰富的性态,相图由最初的单一椭圆曲线不断变形,形状随载荷幅值的增加而逐渐复杂,同时,时程曲线也由简单正弦曲线变为具有多峰值特征的一个周期曲线.     

不可压饱和多孔Timoshenko梁动力响应的数学模型

基于饱和多孔介质理论,假定孔隙流体仅沿梁的轴向运动,本文建立了横观各向同性饱和多孔弹性Timoshenko梁动力响应的一维数学模型,通过不同的简化,该模型可分别退化为饱和多孔梁的Euler-Bernoulli模型、Rayleigh模型和Shear模型等。研究了两端可渗透Timoshenko简支梁自由振动的固有频率、衰减率和阶梯载荷作用下的动力响应特征,给出了梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶等随时间的响应曲线,并与饱和多孔Euler-Bernoulli简支梁响应进行了比较,考察了固相与流相相互作用系数、梁长细比等的影响。可见,固相骨架与孔隙流体的相互作用具有粘性效应,随着作用系数的增加,梁挠度振动幅值衰减加快,并最终趋于静态响应,Euler-Bernoulli梁的挠度幅值和振动周期小于Timoshenko梁的挠度幅值和周期,而Euler-Bernoulli梁的弯矩极限值等于Timoshenko梁的弯矩极限值。     

轴向扩散不可压饱和多孔弹性梁的拟静态弯曲

建立了横观各向同性不可压饱和多孔弹性梁拟静态弯曲的数学模型,并给出了一般的求解方法。作为例子,研究了端部不同渗透条件对梁中点承受突加常集中载荷作用的饱和多孔悬臂梁拟静态弯曲的影响,给出了挠度和孔隙流体压力等效力偶沿梁轴线的分布以及随时间的响应曲线。结果表面:端部渗透条件对饱和多孔弹性梁的弯曲行为有显著的影响,梁的弯曲挠度既可随时间单调递增、亦可单调递减,其性态依赖于梁端部的渗透条件。同时发现不同于经典单相弹性梁,由于孔隙流体压力的作用,不承受载荷作用的梁段亦发生弯曲,并且Man-del-Cryer效应亦存在于不可压饱和多孔弹性梁的拟静态响应中,这些结果有助于揭示传热管道、植物根茎等力学行为的机理。     

饱和不可压多孔弹性板在面内扩散下的动力弯曲理论

根据多孔介质理论,在Kirchhoff假定和小变形前提下,针对流体的面内扩散情形,建立了饱和不可压多孔弹性板动力弯曲的数学模型.然后,利用Fourier展开法研究分析了阶梯载荷作用下四边简支透水矩形多孔弹性板的拟静定和动力弯曲响应,考察了不同参数下多孔弹性板的挠度、孔隙流体压力等效弯矩和固相有效应力等效弯矩的变化规律和特征.同时,通过基于Biot三维固结理论所建立的动力弯曲模型,比较了可压与不可压情况下其结果的差异.     

不可压流体饱和多孔弹性梁的变分原理及有限元方法

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一积分方程组,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.     

Dynamic and quasi-static bending of saturated poroelastic Timoshenko cantilever beam

Based on the three-dimensional Gurtin-type variational principle of the incompressible saturated porous media, a one-dimensional mathematical model for dynamics of the saturated poroelastic Timoshenko cantilever beam is established with two assumptions, i.e., the deformation satisfies the classical single phase Timoshenko beam and the movement of the pore fluid is only in the axial direction of the saturated poroelastic beam. Under some special cases, this mathematical model can be degenerated into the Euler-Bernoulli model, the Rayleigh model, and the shear model of the saturated poroelastic beam, respectively. The dynamic and quasi-static behaviors of a saturated poroelastic Timoshenko cantilever beam with an impermeable fixed end and a permeable free end subjected to a step load at its free end are analyzed by the Laplace transform. The variations of the deflections at the beam free end against time are shown in figures. The influences of the interaction coefficient between the pore fluid and the solid skeleton as well as the slenderness ratio of the beam on the dynamic/quasi-static performances of the beam are examined. It is shown that the quasi-static deflections of the saturated poroelastic beam possess a creep behavior similar to that of viscoelastic beams. In dynamic responses, with the increase of the slenderness ratio, the vibration periods and amplitudes of the deflections at the free end increase, and the time needed for deflections approaching to their stationary values also increases. Moreover, with the increase of the interaction coefficient, the vibrations of the beam deflections decay more strongly, and, eventually, the deflections of the saturated poroelastic beam converge to the static deflections of the classic single phase Timoshenko beam.     

A nonlinear mathematical model for large deflection of incompressible saturated poroelastic beams

Nonlinear governing equations are established for large deflection of incom- pressible fluid saturated poroelastic beams under constraint that diffusion of the pore fluid is only in the axial direction of the deformed beams.Then,the nonlinear bend- ing of a saturated poroelastic cantilever beam with fixed end impermeable and free end permeable,subjected to a suddenly applied constant concentrated transverse load at its free end,is examined with the Gaierkin truncation method.The curves of deflections and bending moments of the beam skeleton and the equivalent couples of the pore fluid pressure are shown in figures.The results of the large deflection and the small deflection theories of the cantilever poroelastic beam are compared,and the differences between them are revealed.It is shown that the results of the large deflection theory are less than those of the corresponding small deflection theory,and the times needed to approach its stationary states for the large deflection theory are much less than those of the small deflection theory.     

QUASI-STATIC ANALYSIS FOR VISCOELASTIC TIMOSHENKO BEAMS WITH DAMAGE

Based on convolution-type constitutive equations for linear viscoelastic materials with damage and the hypotheses of Timoshenko beams, the equations governing quasi-static and dynamical behavior of Timoshenko beams with damage were first derived. The quasi-static behavior of the viscoelastic Timoshenko beam under step loading was analyzed and the analytical solution was obtained in the Laplace transformation domain. The deflection and damage curves at different time were obtained by using the numerical inverse transform and the influences of material parameters on the quasi-static behavior of the beam were investigated in detail.