Poroelastic behaviors of the osteon: A comparison of two theoretical osteon models

In the paper, two theoretical poroelastic osteon models are presented to compare their poroelastic behaviors, one is the hollow osteon model (Haversian fluid is neglected) and the other is the osteon model with Haversian fluid considered. They both have the same two types of impermeable exterior boundary conditions, one is elastic restraint and the other is displacement constrained, which can be used for analyzing other experiments performed on similarly shaped poroelastic specimens. The obtained analytical pressure and velocity solutions demonstrate the effects of the loading factors and the material parameters, which may have a significant stimulus to the mechanotransduction of bone remodeling signals. Model comparisons indicate: (1) The Haversian fluid can enhance the whole osteonal fluid pressure and velocity fields. (2) In the hollow model, the key loading factor governing the poroelastic behavior of the osteon is strain rate, while in the model with Haversian fluid considered, the strain rate governs only the velocity. (3) The pressure amplitude is proportional to the loading frequency in the hollow model, while in the model with Haversian fluid considered, the loading frequency has little effect on the pressure amplitude.     

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.     

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.     

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.     

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.      

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.     

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).     

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.     

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。     

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.     

Three-dimensional steady-state closed form solution for multilayered fluid-saturated anisotropic finite media due to surface/internal point source

A three-dimensional(3 D)steady-state solution of fluid saturated anisotropic finite media is presented.The eigenequation method and the pseudo-Stroh formalism are used to obtain the exact solution for homogeneous saturated finite media.The propagator matrix method is introduced to deal with the corresponding multilayered poroelastic media.The poroelastic solutions due to surface or internal point fluid source are obtained.The comparison of the results of the saturated isotropic media in a half space and those obtained by the finite element method is given to illustrate the accuracy of the solution in a finite domain.Numerical solutions of a sandwich poroelastic medium are presented to analyze its hydromechanical behaviors.Two ratios of the horizontal permeability to vertical permeability and different source positions are investigated.The results show that the fluid parameters and source positions have great influence on the hydromechanical behaviors of the layered media.     

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.     

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.     

移动简谐力作用下三维多孔饱和半空间的动力问题

研究了移动荷载作用下多孔饱和地基的动力问题.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地基动力响应的影响,重点研究了移动速度达地基表面波速时多孔饱和半空间的振动问题(马赫效应),并与相应的弹性介质的解答进行了比较.结果显示当移动速度与多孔饱和半空间的表面波速相近时,地基会产生很大的振动;当移动速度大于表面波速时,多孔饱和半空间的动力响应与弹性半空间的动力响应有较大的差别.     

Acoustical characterization of fluid-saturated porous media with local heterogeneities: Theory and application

A linear dynamic model of fully saturated porous media with local (either microscopic or mesoscopic) heterogeneities is developed within the context of Biot’s theory of poroelasticity. Viscoporoelastic behavior associated with local fluid flow is characterized by the notion of the dynamic compatibility condition on the interface between the solid and the fluid. Complex, frequency-dependent material parameters characterizing the viscoporoelasticity are derived. The complex properties can be obtained through determining the quasi-static poroelastic parameters, the properties of individual constituents, and the relaxation time of the dynamic compatibility condition on the interface. Relationships among various quasi-static poroelastic parameters are developed. It is shown that local fluid flow mechanism is significant only in the porous media with local heterogeneities. The relaxation time of the compatibility condition on the interface depends upon the details of local structure of porous media that control local fluid pressure diffusion. The new model is used to describe the velocity dispersion and attenuation in fully saturated porous media. The proposed model provides a theoretical framework to simulate the acoustical behavior of fully saturated porous media over a wide range of frequencies without making any explicit assumption about the structure of local heterogeneities.     

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.     

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.     

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.     

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.     

DYNAMIC RESPONSE OF A DOUBLE-LAYERED SUBGRADE WITH ROCK SUBSTRATUM TO A MOVING POINT LOAD

In this paper, an analytical solution for the dynamic response of a double-layered subgrade with rock substratum to a moving point load is derived. The subgrade profile is divided into two layers. The upper layer is modeled by an elastic medium and the lower layer by a fully saturated poroelastic medium governed by Biot’s theory. In the meanwhile, the subgrade is resting on the rock substratum. The analytical solutions for stress, displacement and pore pressure are derived by using the Fourier transform. Numerical results obtained by using the inverse fast Fourier transform (IFFT) are used to analyze the influence of the moving load velocity, the thickness of an elastic medium layer and a fully saturated poroelastic medium layer on the dynamic response.