Mathematical osteon model for examining poroelastic behaviors

An extended and reasonable stress boundary condition at an osteon exterior wall is presented to solve the model proposed by Rémond and Naili. The obtained pressure and fluid velocity solutions are used to investigate the osteonal poroelastic behaviors. The following results are obtained. (i) Both the fluid pressure and the velocity amplitudes are proportional to the strain amplitude and the loading frequency. (ii) In the physiological loading state, the key role governing the poroelastic behaviors of the osteon is the strain rate. (iii) At the osteon scale, the pressure is strongly affected by the permeability variations, whereas the fluid velocity is not.     

Hierarchical model for strain generalized streaming potential induced by the canalicular fluid flow of an osteon

A hierarchical model is developed to predict the streaming potential(SP) in the canaliculi of a loaded osteon. Canaliculi are assumed to run straight across the osteon annular cylinder wall, while disregarding the effect of lacuna. SP is generalized by the canalicular fluid flow. Analytical solutions are obtained for the canalicular fluid velocity, pressure, and SP. Results demonstrate that SP amplitude(SPA) is proportional to the pressure difference, strain amplitude, frequency, and strain rate amplitude. However, the key loading factor governing SP is the strain rate, which is a representative loading parameter under the specific physiological state. Moreover, SPA is independent of canalicular length. This model links external loads to the canalicular fluid pressure, velocity, and SP, which can facilitate further understanding of the mechanotransduction and electromechanotransduction mechanisms of bones.     

一种骨小管中液体流动产生的流量及切应力模型

骨组织受力变形后其内部液体就会流动,同时在其微观结构——骨单元壁中扩散,并进一步产生一系列与骨液流动相关的物理效应,如流体剪切应力、流动电位等,这些物理效应被细胞感知并做出破骨或成骨等反应,来使骨适应外部载荷环境.鉴于骨组织产生的内部液体流动很难实验测定,理论模拟是目前的主要研究手段.基于骨单元的多孔弹性性质建立了骨小管内部液体的流动模型,该模型将骨单元所受的外部载荷与骨小管内部液体的压力、流速、流量和切应力联系起来,并进一步可以研究其力传导与力电传导机制.骨小管模型的建立分别基于中空和考虑哈弗液体的骨单元模型,并考虑了骨单元外壁的弹性约束和刚性位移约束两种边界条件.最终得到骨单元在外部轴向载荷作用下,骨小管内部液体的流量及流体切应力的解析解.结果表明:骨小管中的液体流量与流体切应力都正比于应变载荷幅值和频率,并由载荷的应变率决定.因此应变率可以作为控制流量和流体切应力的一种生理载荷因素.流量随着骨小管半径的增大而非线性增大,而流体切应力则随着骨小管半径的增大而线性增大.此外,在相同的载荷下,含哈弗液体的骨单元的模型中,骨小管中液体的流量和切应力均大于中空骨单元模型.     

Transverse isotropic poroelastic osteon model under cyclic loading

The poroelastic problem associated with a hollow cylinder under cyclic loading is solved. This cylinder models an osteon, basic unit of cortical bone. Both fluid and solid phases are supposed compressible. Solid matrix is modeled as an elastic transverse isotropic material. An explicit close-form solution for the steady state is obtained. Fluid flow distribution as a function of poroelastic properties and cyclic loading is discussed as it could influence bone remodeling. Strain rate of loading is shown to play a significant role in mass flux in the porous material.     

An analytical poroelastic model for laboratorial mechanical testing of the articular cartilage (AC)

The articular cartilage (AC) can be seen as a biphasic poroelastic material. The cartilage deformation under compression mainly leads to an interstitial fluid flow in the porous solid phase. In this paper, an analytical poroelastic model for the AC under laboratorial mechanical testing is developed. The solutions of interstitial fluid pressure and velocity are obtained. The results show the following facts. (i) Both the pressure and fluid velocity amplitudes are proportional to the strain loading amplitude. (ii) Both the amplitudes of pore fluid pressure and velocity in the AC depend more on the loading amplitude than on the frequency. Thus, in order to obtain the considerable fluid stimulus for the AC cell responses, the most effective way is to increase the loading amplitude rather than the frequency. (iii) Both the interstitial fluid pressure and velocity are strongly affected by permeability variations. This model can be used in experimental tests of the parameters of AC or other poroelastic materials, and in research of mechanotransduction and injury mechanism involved interstitial fluid flow.     

Cylindre transverse isotrope poroélastique chargé cycliquement : application à un ostéonCyclic loading of a transverse isotropic poroelastic cylinder: a model for the osteon

The poroelastic problem associated with a hollow cylinder under cyclic loading is solved. Both fluid and solid phases are supposed compressible. Solid matrix is modeled as an elastic transverse isotropic material. An explicit close-form solution for the steady state is obtained. This cylinder is considered as a model for an osteon, the basic unit of cortical bone. The fluid flow distribution as a function of poroelastic properties and cyclic loading is discussed, as this could influence bone remodeling. To cite this article: A. Rémond, S. Naili, C. R. Mecanique 332 (2004).     

Fluid flow and fluid shear stress in canaliculi induced by external mechanical loading and blood pressure oscillation

The paper studies the problem of fluid flow and fluid shear stress in canaliculi when the osteon is subject to external mechanical loading and blood pressure oscillation. The single osteon is modeled as a saturated poroelastic cylinder. Solid skeleton is regarded as a poroelastic transversely isotropic material. To get near-realistic results, both the interstitial fluid and the solid matrix are regarded as compressible. Blood pressure oscillation in the Haverian canal is considered. Using the poroelasticity theory, an analytical solution of the pore fluid pressure is obtained. Assuming the fluid in canaliculi is incompressible, analytical solutions of fluid flow velocity and fluid shear stress with the Navier-Stokes equations of incompressible fluid are obtained. The effect of various parameters on the fluid flow velocity and fluid shear stress is studied.     

骨陷窝-骨细胞形状和方向对骨单元内液体流动行为的影响

骨组织内的流体流动不仅为骨细胞的生存提供了充足营养供应及代谢物排放途径,也在骨重建过程中起到关键作用. 为了更精确地阐明骨内液体流动的具体形式,这项研究利用骨陷窝-骨细胞的密度,形态和方向等参数来计算骨单元内液体的流动行为. 首先,计算出不同形状和方向的骨陷窝周围骨小管的数量及分布情况,其次利用算出的参数以及骨组织其他微结构数据来估计骨组织的渗透率和孔隙率等参数,最后根据计算所得的参数建立骨单元的多孔弹性力学有限元模型,并分析了在轴向位移载荷作用下骨陷窝形状和方向对骨单元内液体渗流行为的影响. 结果表明,在所研究的参数范围内不同骨单元模型的相同区域上,骨陷窝形状影响下的骨单元最大压力和流速比最小的分别增加了86%和18%;骨陷窝方向影响下的最大压力和流速比最小的分别增加了125%和56%. 伸长形骨陷窝对单个骨单元局部压力的影响远大于扁平形和圆形骨陷窝. 骨陷窝从0°绕$x$轴旋转到90°过程中压力是逐渐降低的,且30°,45°和60°的模型对骨单元内局部流速有显著影响. 该模型表示骨陷窝的形状和方向以及骨小管的三维分布对骨单元内液体压力和流速幅值及沿不同方向的流动差异有显著的影响. 这项研究将有助于精确量化描述骨内液体的流体行为.      

Transient response of fluid pressure in a poroelastic material under uniaxial cyclic loading

Poroelasticity is a theory that quantifies the time-dependent mechanical behavior of a fluid-saturated porous medium induced by the interaction between matrix deformation and interstitial fluid flow. Based on this theory, we present an analytical solution of interstitial fluid pressure in poroelastic materials under uniaxial cyclic loading. The solution contains transient and steady-state responses. Both responses depend on two dimensionless parameters: the dimensionless frequency Ω that stands for the ratio of the characteristic time of the fluid pressure relaxation to that of applied forces, and the dimensionless stress coefficient H governing the solid-fluid coupling behavior in poroelastic materials. When the phase shift between the applied cyclic loading and the corresponding fluid pressure evolution in steady-state is pronounced, the transient response is comparable in magnitude to the steady-state one and an increase in the rate of change of fluid pressure is observed immediately after loading. The transient response of fluid pressure may have a significant effect on the mechanical behavior of poroelastic materials in various fields.     

Fluid pressure response in poroelastic materials subjected to cyclic loading

When cyclic loading is applied to poroelastic materials, a transient stage of interstitial fluid pressure occurs, preceding a steady state. In each stage, the fluid pressure exhibits a characteristic mechanical behavior. In this study, an analytical solution for fluid pressure in two-dimensional poroelastic materials, which is assumed to be isotropic, under cyclic axial and bending loading is presented, based on poroelasticity. The obtained analytical solution contains transient and steady-state responses. Both of these depend on three dimensionless parameters: the dimensionless stress coefficient; the dimensionless frequency; and, the axial-bending loading ratio. We focus particularly on the transient behavior of interstitial fluid pressure with changes in the dimensionless frequency and the axial-bending loading ratio. The transient properties, such as half-value period and contribution factor, depend largely on the dimensionless frequency and have peak values when its value is about 10. This suggests that, under these conditions, the transient response can significantly affect the mechanical behavior of poroelastic materials.     

Stress and pressure fields around two wellbores in a poroelastic medium

The problem of two circular wellbores of different size in a poroelastic medium is considered in the present work. The constitutive behaviour of the poroelastic medium is assumed to comply with the classical Biot model for isotropic porous materials infiltrated by compressible fluid. The wellbores are assumed infinitely long and the fluid flow is taken stationary, thus making it possible to perform a plane strain analysis. Owing to the geometrical layout of the system, bipolar cylindrical coordinates have been adopted. Three different sets of BCs on the pressure field and on the fluid flux have been considered, founding the corresponding forms of the pressure field. Based on Helmholtz representation, a displacement potential has been introduced, and the corresponding stress field in the poroelastic medium has been assessed. However, such a solution does not satisfy the BCs at the edges of the wells. Then, an auxiliary stress function, which allows accomplishing the BCs, is introduced, leading to the complete solution of the problem. The cases of two coaxial wellbores (eccentric annulus), a single hole bored in a poroelastic half plane and two intersecting holes have been considered also. The proposed approach allows evaluating the pore pressure and the stress and strain fields in the system varying the amplitude of the wells and the physical parameters of the porous material. In particular, the evaluation of the peak values of the stress components around the circular boreholes plays a key role in a variety of engineering contexts, with particular reference to the stability analysis of wellbores and tunnels and failure of vascular vessels in biological tissues.     

基于多孔弹性模型的脊椎关节生物力学特性和体液流动特性研究

建立了L4/L5段人体腰椎关节的非线性多孔弹性有限元模型,并对其施加1000N振动载荷1h,考察在不同的振动频率(1Hz、4Hz、8Hz、11.5Hz、15Hz)下腰椎关节的变形、应力分布和体液流动情况;并对不同频率作用下脊椎组织的生物力学特性进行了对比分析。结果表明,在不同频率振动载荷下,脊椎模型的应力分配、体液流量都呈现与振动载荷不同的周期性波动变化。振动载荷频率等于腰椎关节的固有频率11.5Hz时,椎间盘应力分配和体液流量波动的幅值最短;而振动频率为4Hz、8Hz、15Hz时各项指标波动的幅值比11.5Hz时小。振动过程中,椎间盘内外压力梯度的变化引起体液的流动,振动时间越长,总流失量越大。     

Dynamic effects of moving load on a poroelastic soil medium by an approximate method

The problem of the dynamic response of a fully saturated poroelastic soil stratum on bedrock subjected to a moving load is studied by using the theory of Mei and Foda under conditions of plane strain. The applied load is considered to be the sum of a large number of harmonics with varying frequency in the form of a Fourier expansion. The method of solution considers the total field to be approximated by the superposition of an elastodynamic problem with modified elastic constants and mass density for the whole domain and a diffusion problem for the pore fluid pressure confined to a boundary layer near the free surface of the medium. Both problems are solved analytically in the frequency domain. The effects of the shear modulus, permeability and porosity of the soil medium and the velocity of the moving load on the dynamic response of the soil layer are numerically evaluated and compared with those obtained by the exact solution of the problem. It is concluded that for fine poroelastic materials, the accuracy of the present method against the exact one is excellent.     

受移动简谐力作用的多孔弹性半平面问题

研究了匀速移动的振动荷载作用下半无限多孔饱和固体中产生的应力和孔隙水压力．应用Fourier变换求解该问题的控制偏微分方程,考虑了荷载的移动速度及振动频率对多孔饱和固体中应力与孔隙水压力的影响,并与相应的弹性介质的解答进行了比较．结果显示多孔饱和半平面中应力和孔隙水压力随荷载的移动速度与振动频率的增加而增大,多孔饱和固体在移动荷载下的动力响应与相应的单相弹性固体的动力响应有较大的差别。     

人的密质骨的力学性能

本文把骨组织看作是一种材料和骨骼系统的粘弹性结构元件,评述了它们的力学性能.首先描述了骨的结构,包括宏观结构、微结构和超微结构.随后是骨的力学性能.第三部分介绍了哈弗氏骨的力学模型.笔者指出,单个哈弗氏骨可考虑为正交铺设的胶原纤维增强的粘弹性厚壁筒.最后讨论了哈弗氏骨的应力-应变关系.显然,简单的弹性关系是不可用于实际的.      

Drag reduction by reconfiguration of a poroelastic system

Because of their flexibility, trees and other plants deform with great amplitude (reconfigure) when subjected to fluid flow. Hence the drag they encounter does not grow with the square of the flow velocity as it would on a classical bluff body, but rather in a less pronounced way. The reconfiguration of actual plants has been studied abundantly in wind tunnels and hydraulic canals, and recently a theoretical understanding of reconfiguration has been brought by combining modelling and experimentation on simple systems such as filaments and flat plates. These simple systems have a significant difference with actual plants in the fact that they are not porous: fluid only flows around them, not through them. We present experimentation and modelling of the reconfiguration of a poroelastic system. Proper scaling of the drag and the fluid loading allows comparing the reconfiguration regimes of porous systems to those of geometrically simple systems. Through theoretical modelling, it is found that porosity affects the scaling of the drag with flow velocity. For high porosity systems, the scaling is the same as for isolated filaments while at low porosity, the scaling is constant for a large range of porosity values. The scalings for the extreme values of porosity are also obtained through dimensional analysis.     

A simple BEM formulation for poroelasticity via particular integrals

A simple particular integral formulation is presented for poroelastic analysis. The elastostatics and steady-state potential flow equations are used as the complementary solution. A set of global shape functions is considered to approximate the pore pressure loading term in the poroelastic equation, the transient terms of pore pressure and displacements in the pore fluid flow equation to obtain the particular integrals for displacement, traction, pore pressure and flux.Numerical results for four plane problems of soil consolidation are given and compared with their analytical solutions to demonstrate the accuracy of the present formulation. Generally, agreement among all of those results is satisfactory if a few interior points are added to the usual boundary elements.     

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。     

Response of railway track system on poroelastic half-space soil medium subjected to a moving train load

Based on the dynamic poroelastic theory of Biot, dynamic responses of a track system and poroelastic half-space soil medium subjected to moving train passages are investigated by the substructure method. The whole system is divided into two separately formulated substructures, the track and the ground, and the rail is described by introducing the Green function for an infinitely long Euler beam subjected to the action of moving axle loads of the train and the reactions of the sleeper. Sleepers are represented by a continuous mass and the effect of the ballast is considered by introducing the Cosserat model for granular medium. Using the double Fourier transform, the governing equations of motion are then solved analytically in the frequency-wave-number domain. The time domain responses are evaluated by the inverse Fourier transform computation for a certain train speed. Computed results show that the shape of the rail displacements of the elastic and poroelastic soil medium are in good agreement with each other of the low train velocity, but the result of the poroelastic soil medium is significantly different to that of the elastic soil medium for the high train velocity which is higher than Rayleigh-wave speed in the soil. The influence of the soil intrinsic permeability on soil responses is discussed with great care in both time domain and frequency domain. The dynamic responses of the soil medium are considerably affected by the fluid phase as well as the load velocity.     

Liquefaction and displacement of saturated sand under vertical vibration loading

In order to investigate the influence of the vertical vibration loading on the liquefaction of saturated sand, one dimensional model for the saturated sand with a vertical vibration is presented based on the two phase continuous media theory. The development of the liquefaction and the liquefaction region are analyzed. It is shown that the vertical vibration loading could induce liquefaction. The rate of the liquefaction increases with the increase of the initial limit strain or initial porosity or amplitude and frequency of loading, and increases with the decrease of the permeability or initial modulus. It is shown also that there is a phase lag in the sand column. When the sand permeability distribution is non-uniform, the pore pressure and the strain will rise sharply where the permeability is the smallest, and fracture might be induced. With the development of liquefaction, the strength of the soil foundation becomes smaller and smaller. In the limiting case, landslides or debris flows could occur. The project supported by the National Natural Science Foundation of China (40025103, 10202024)