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.     

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.     

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.     

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.     

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.     

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.     

Wave propagation modeling in cylindrical human long wet bones with cavity

The wave propagation modeling in cylindrical human long wet bones with cavity is studied. The dynamic behavior of a wet long bone that has been modeled as a piezoelectric hollow cylinder of crystal class 6 is investigated. An analytical solutions for the mechanical wave propagation during a long wet bones have been obtained for the flexural vibrations. The average stresses of solid part and fluid part have been obtained. The frequency equations for poroelastic bones are obtained when the medium is subjected to certain boundary conditions. The dimensionless frequencies are calculated for poroelastic wet bones for various values for non-dimensional wave lengths. The dispersion curves of the dry bone and wet bone for the flexural mode n=2 are plotted. The numerical results obtained have been illustrated graphically.     

循环剪切荷载作用下软土的力学性状研究

利用从现场取得的原状土样,通过室内动三轴试验对循环荷载作用下的变形、孔压和强度特性进行研究。试验研究主要考虑了周围固结压力、循环剪切应力比、荷载频率和循环次数等因素的影响。研究表明,在循环荷载作用下,孔隙水压力和轴向应变均是一个波动上升的过程。当循环应力比增大时,动孔隙水压力变化幅值显著增加,而残余孔压也较大。同时,孔隙水压力值也随着周围压力的增大而明显增大。当作用荷载频率比较大时,需要更多的循环次数才能达到小频率作用荷载能达到的孔隙水压力值。但是,随着循环荷载作用次数的增加,频率对孔隙水压力的影响有减小的趋势。     

Axisymmetric deformation of poroelastic shells of revolution

A linear theory is developed for axisymmetric deformation of thin poroelastic shells of revolution. With fluid solid coupling included through Biot's consolidation theory, results are presented for cylindrical shells with an oscillating internal pressure and various surface boundary conditions on the fluid. First, the effects of fluid flow and shell inertia on the stretching behavior are studied through a separation of variables solution. Then, the bending behavior near a clamped edge is examined through an asymptotic solution of a matrix form of the governing equations. The results show that the asymptotic solution is accurate in the low frequency range, when the loading time is large compared to the consolidation time. In addition, for the examples studied, the fluid flow influences the membrane more than the bending behavior, but damping due to flow resistance is limited near resonance.     

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.     

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.     

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.     

Dimensionless parameters in symmetric double lap joints: An orthotropic solution for thermomechanical loading

Two thermomechanical analytical models are developed for orthotropic double lap joints with a view to identifying key dimensionless parameters that describe the behavior of the joint under combined thermal–mechanical loads. The solutions, based on the principle of virtual work, differ in the complexity of the assumed stress field. The first solution is similar to Volkersen [Volkersen, O., 1938. Die niektraftverteilung in zugbeanspruchten mit konstanten laschenquerschritten. Luftfahrtforschung 15, 41–47] with the addition of orthotropic and thermal effects. The second solution, extending the work of Davies [Davies, G.A.O., 1982. Virtual Work in Structural Analysis, John Wiley & Sons, New York] captures the peel stress as well as the traction free boundary condition at the adhesive edge. Relevant non-dimensional parameters are identified in terms of geometric, material, and load quantities. A dimensionless load ratio is identified which dictates the shape of the stress distribution. This ratio can also be used to quickly determine the dominant loading mechanism. Dimensionless stress plots are presented for representative lap joints.     

Poromechanical response of a finite elastic cylinder under axisymmetric loading

Drained or undrained cylindrical specimens under axisymmetric loading are commonly used in laboratory testing of soils and rocks. Poroelastic cylindrical elements are also encountered in applications related to bioengineering and advanced materials. This paper presents an analytical solution for an axisymmetrically-loaded solid poroelastic cylinder of finite length with permeable (drained) or impermeable (undrained) hydraulic boundary conditions. The general solutions are derived by first applying Laplace transforms with respect to the time and then solving the resulting governing equations in terms of Fourier–Bessel series, which involve trigonometric and hyperbolic functions with respect to the z-coordinate and Bessel functions with respect to the r-coordinate. Several time-dependent boundary-value problems are solved to demonstrate the application of the general solution to practical situations. Accuracy of the numerical solution is confirmed by comparing with the existing solutions for the limiting cases of a finite elastic cylinder and a poroelastic cylinder under plane strain conditions. Selected numerical results are presented for different cylinder aspect ratios, loading and hydraulic boundary conditions to demonstrate the key features of the coupled poroelastic response.     

DP钢板应变路径多步演变下的弹塑性力学行为

针对DP高强双相钢板在复杂载荷作用下的弹塑性力学特征,提出利用三步拉伸力学实验,对比分析单轴循环加载和非等轴加载下材料的各向异性硬化、永久软化和弹性模量衰减特性等力学行为,揭示应变路径多步演变下的弹塑性力学特性。研究结果表明：材料再加载初期的瞬态行为与应变路径有关,在初期瞬态阶段显示出明显的各向异性,且再加载角度、预应变量以及马氏体含量的增加都会加剧两种DP钢过渡期后的应力软化行为。同时对不同加载下DP钢弹性模量的演化进行了对比,发现弹性模量的衰减程度也随着再加载角度,预应变量和马氏体含量的增加而增加,弹性模量衰减程度在多步非等轴加载比单轴循环载荷作用下表现更明显。     

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.     

Mixed-mode fracture analysis of orthotropic functionally graded materials under mechanical and thermal loads

Mixed-mode fracture problems of orthotropic functionally graded materials (FGMs) are examined under mechanical and thermal loading conditions. In the case of mechanical loading, an embedded crack in an orthotropic FGM layer is considered. The crack is assumed to be loaded by arbitrary normal and shear tractions that are applied to its surfaces. An analytical solution based on the singular integral equations and a numerical approach based on the enriched finite elements are developed to evaluate the mixed-mode stress intensity factors and the energy release rate under the given mechanical loading conditions. The use of this dual approach methodology allowed the verifications of both methods leading to a highly accurate numerical predictive capability to assess the effects of material orthotropy and nonhomogeneity constants on the crack tip parameters. In the case of thermal loading, the response of periodic cracks in an orthotropic FGM layer subjected to transient thermal stresses is examined by means of the developed enriched finite element method. The results presented for the thermally loaded layer illustrate the influences of the material property gradation profiles and crack periodicity on the transient fracture mechanics parameters.     

Dynamic response of an axially loaded rigid sphere embedded in a saturated poroelastic medium

In this paper, an analytical solution for the response of a rigid sphere embedded in a full space poroelastic medium subjected to a dynamic lateral load is derived. The solution is obtained using Biots theory for acoustic waves. In this solution, the displacements of the solid skeleton and the pore pressure are expressed in terms of three scalar potentials. These potentials correspond to the wave velocities of the slow and fast compressional waves and to the shear wave. The governing equation for the dynamic motion is expressed in the frequency domain using Fourier transformation. Different boundary and load conditions were investigated. Curves showing variation in the fluid pressure and solid displacements with the loads frequency were plotted in non-dimensional forms.     

裂纹闭合和循环塑性预应变及循环载荷压缩部分对裂纹扩展速率的影响

本文针对三种国产材料 Ｌｙ１１ｃｚ、 Ｌｙ１２ｃｚ 铝合金和 １８ Ｍｎ Ｈ Ｐ钢,通过实验初步考察了循环塑性预应变和循环载荷压缩部分对疲劳裂纹扩展的影响;采用电测法,测定了两种铝合金材料疲劳裂纹扩展的张开应力和有效应力强度因子幅值比 Ｕ。结果表明：（１）材料循环塑性预应变和循环载荷压缩部分,都使疲劳裂纹扩展速率提高;（２）常幅载荷下,在疲劳裂纹稳定扩展阶段,有效应力强度因子幅值比 Ｕ 与应力比 Ｒ 有关,与裂纹长度ａ 无关,并依赖于材料的力学性能。