Biot's Theory Research Articles

Torsional vibration of open-ended pipe piles in saturated soil considering the construction disturbance

A soil-pipe pile-inner soil torsional dynamic interaction model is developed accounting for the radial inhomogeneity of the outer saturated soil as well as the soil plug (inner soil) due to the construction disturbance. Both the outer and inner saturated soils are divided into a series of annular zones in the radial direction with linearly varying soil properties from zone to zone. The governing equation of motion for each soil zone is established based on the Biot's two-phase linear theory and is solved to derive the circumferential shear stress at the pipe pile-soil interface in conjunction with the continuous conditions at the interface of adjacent soil zones. Further, the governing equation for the open-ended pipe pile is established and is solved to derive an analytical solution for the torsional vibration of the pile in the frequency domain. The developed solution is validated against the results of available analytical solutions. The validated solution is then used to examine the torsional vibration characteristics of the open-ended pipe pile under different saturated soil parameters and construction disturbance conditions. The results show that: (1) The pile resistance to torsional dynamic load is enhanced as the soil strengthening range or degree increases, while is reduced as the soil porosity, soil weakening range or degree increases; (2) The inner soil contributes a lot to the enhancement of pile torsional resistance.

Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method

In this paper, the singular boundary method (SBM) in conjunction with the exponential window method (EWM) is firstly extended to simulate the transient dynamic response of two-dimensional saturated soil. The frequency-domain (Fourier space) governing equations of Biot theory is solved by the SBM with a linear combination of the fundamental solutions. In order to avoid the perplexing fictitious boundary in the method of fundamental solution (MFS), the SBM places the source point on the physical boundary and eliminates the source singularity of the fundamental solution via the origin intensity factors (OIFs). The EWM is carried out for the inverse Fourier transform, which transforms the frequency-domain solutions into the time-domain solutions. The accuracy and feasibility of the SBM-EWM are verified by three numerical examples. The numerical comparison between the MFS and SBM indicates that the SBM takes a quarter of the time taken by the MFS.

Biot Theory-Based Finite Element Modeling of Continuous Ultrasound Propagation Through Microscale Articular Cartilage.

Low-intensity ultrasound has shown promise in promoting the healing and regeneration of articular cartilage degraded by osteoarthritis. In this study, a two-dimensional finite element method (FEM) model was developed for solving the Biot theory equations governing the propagation of continuous ultrasound through the cartilage. Specifically, we computed the ultrasound-induced dilatations and displacements in the microscale cartilage that is represented as consisting of four zones, namely, the chondrocyte cell and its nucleus, the pericellular matrix (PCM) that forms a layer around the chondrocyte, and the extracellular matrix (ECM). The chondrocyte-PCM complex, referred to as the chondron, is embedded in the ECM. We model multiple cartilage configurations where in the ECM layer contains chondrons along the depth, as well as laterally. The top surface of the ECM layer is subjected to specified amplitude and frequency of continuous ultrasound. The resulting wave propagation is modeled by numerically solving the two-dimensional Biot equations for seven frequencies in the 0.5 MHz-5 MHz range. It is seen that ultrasound is attenuated in the ECM and the attenuation increases monotonically with frequency. In contrast, manyfold augmentation of the ultrasound amplitude is observed inside the cytoplasm and the nucleus of the chondrocyte. Chondrocytes act as a major sink of ultrasound energy, thereby reducing the depthwise propagation of ultrasound fluctuations. Regions of high dilatations and displacements were found at the ECM-PCM interface, PCM-chondrocyte interface, as well as in the cytoplasm and nucleus of the chondrocyte. We observe that the ultrasound field around a chondron interacts with that around a neighboring chondron located at the same depth in the ECM layer. The qualitative and quantitative insights gained from our study may be relevant to designing ultrasound-based therapies for osteoarthritis.

Lateral dynamics of a monopile partially embedded in layered saturated soil with radial disturbance

A bidirectional transfer matrix method is presented to investigate the impedances of a laterally loaded monopile partially embedded in layered saturated soil with radial disturbance. To characterize the radial variation of soil properties induced by construction excitations or cyclic lateral excitations, the radial transfer matrix of the saturated soil is developed to determine the horizontal resistance of soil to the pile shaft based on Biot's theory. Lateral, rocking and lateral-rocking coupled impedances at the head of the monopile are obtained by solving the differential equations for transverse vibration of the pile shaft based on the Timoshenko beam theory, combined with an axial transfer matrix to treat the layered property of soil along the partially embedded monopile. The proposed model is validated by comparing it with some existing results. The theoretical rigour and broad engineering applicability of the present model are highlighted by a series of parametric analyses for the influences of soil and pile properties on the impedances of the laterally loaded monopile.

Monochromatic Wave of Fluid Pressure in a Porous Rock.

From the perspective of rock mechanics, transmission of fluid pressure that drives groundwater flow is not instantaneous; instead it is a slow compressional (P) wave in porous rocks saturated with a fluid. In this paper, the approach of monochromatic wave is used to analyze the slow P-wave. Permeability is derived as a function of the frequency of the wave. The permeability appears to be a complex number which asymptotes to Darcy permeability at the low frequency end but tends to vanish at the high frequency end. With Biot theory, the permeability predicts phase velocity and the quality factor of the monochromatic wave. Using Berea sandstone as an illustrative example, the complex-number permeability can yield a lower phase velocity (within frequency of 105 -108 Hz) and a larger attenuation (for frequency above 106 Hz) than Darcy permeability does. This study extends the recent research of the low-frequency waves of fluid pressure to the regime of very high frequency.

Generalized finite difference method based meshless analysis for coupled two-phase porous flow and geomechanics

This paper applies GFDM to analyze coupled two-phase flow and geomechanics for the first time, to our knowledge, is also the first meshless solver for two-phase fluid-solid coupling problems in porous media. Firstly, the implicit GFDM-based discrete schemes of the hyperbolic two-phase flow equation and elliptic stress equilibrium equation coupled by Biot's poroelastic theory are derived. Then the nonlinear solver based on Newton's method is used to solve the fluid pressure, water saturation, and displacement at each node simultaneously. Two technical details are found critical to the implementation of this method. The one is using different radii of the node influence domain to discretize the flow equation and the stress equilibrium equation respectively, and the other one is the selection criterion of the two radii of the node influence domain, which improves the implementation flexibility of the multi-physics coupling calculation, reduces the dissipation error of the calculation results of the convection-dominated water saturation distribution, and ensure a low computational cost. Several numerical test cases are implemented to analyze the computational efficiency and accuracy, and illustrate that GFDM can achieve good computational performance in case of regular domains, irregular domains, and different point clouds.Overall, this work may provide an important reference for building a GFDM-based general-purpose numerical simulator for multi-physics flow problems in porous media.

Time-domain poroelastic full-waveform inversion of shallow seismic data: methodology and sensitivity analysis

SUMMARY Full-waveform inversion (FWI) is considered as a high-resolution imaging technique to recover the geophysical parameters of the elastic subsurface from the entire content of the seismic signals. However, the subsurface material properties are less well estimated with elastic constraints, especially for the near-surface structure, which usually contains fluid contents. Since Biot theory has provided a framework to describe seismic wave propagations in the poroelastic media, in this work, we propose an algorithm for the 2-D time-domain (TD) poroelastic FWI (PFWI) when the fluid-saturated poroelastic equations are applied to carve the physical mechanism in the shallow subsurface. To detect the contribution of the poroelastic parameters to shallow seismic wavefields, the scattered P-SV&SH wavefields corresponding to a single model parameter are derived explicitly by Born approximation and shown numerically afterward. The Fréchet kernels are also derived and exhibited in P-SV&SH schemes to analyse the sensitivities of the objective function to different poroelastic parameters. Furthermore, we verify the accuracy of the derivations through model parameter reconstructions. We perform a series of numerical tests on gradients with respect to different model parameters to further evaluate inter-parameter trade-offs. PFWI holds potential possibilities to directly invert fluid-related physical parameters of the shallow subsurface.

Dynamic response of sheet‒pile groin under tidal bore considering pile‒pile mutual interaction and hydrodynamic pressure

Sheet‒pile groin appearing as two rows of piles, with both pile heads connected by a concrete sheet, are widely applied to resist the tidal bores, although its dynamic soil‒structure mechanism is still under research. This paper utilizes the Euler beam and one‒dimensional rod to simulate the pile and sheet constituting the framework of the sheet‒pile groin, respectively, and Biot's poroelastic theory is introduced to simulate the layered saturated soil in which the sheet‒pile groin is embedded. To account for the pile‒pile mutual interactions, interaction factors considering the secondary wave effect are adopted. Radiation wave theory is adopted to simulate the hydrodynamic pressure. The theoretical solutions to the dynamic response of the sheet‒pile groin in the frequency and time domains are derived by applying the matrix calculations and the Fourier transform. The correctness and accuracy of the present solution have been verified through FEM results. An extensive parametric analysis is then conducted to investigate the influence of various factors on the dynamic horizontal complex impedance of the sheet‒pile groin.

Coupled horizontal and rocking vibrations of a rigid circular disc on the surface of a transversely isotropic and layered poroelastic half‐space

The coupled horizontal and rocking vibrations of a rigid circular disc resting on a transversely isotropic (TI) and layered poroelastic half-space is solved by a newly developed semi-analytical method for the first time. The poroelastic medium is modelled via Biot's poroelastodynamic theory. The Green's function corresponding to the horizontal uniform and vertical linearly changing patch loads on the surface of the layered half-space are first derived by virtue of the cylindrical system of vector functions and dual variable and position method. To solve the dynamic interaction, the influence function by the ring load is then introduced and calculated via the method of superposition. Together with the mixed boundary-value conditions, the dynamic horizontal, rocking, and coupling compliances are finally solved by a novel discretization scheme and an integral least-square approach. The developed fundamental solutions are verified by comparing with existing solutions in the purely elastic and poroelastic media. Numerical examples are presented to investigate the influence of poroelastic properties on the coupled vibration characteristics, which could be useful to the dynamic design of foundation on the saturated soil. It should be emphasized that the effect of the coupling compliance must be considered since its contribution can be as large as 26%, compared to the diagonal compliances.

Numerical simulation of rock erosion performance of a high-speed water jet using an immersed-body method

Water jet drilling (WJD) is an effective technique for drilling micro-holes in the subsurface for reservoir stimulation . This study aims to investigate numerically the failure mechanism of rock during WJD and to assess the WJD performance before drilling. A 3D fluid-solid coupling model is developed for simulating WJD by coupling a mechanical solver based on the combined finite-discrete element method (FDEM) with a fluid solver using an immersed-body method. The new numerical model is capable of simulating crack initiation and propagation and fragment removal under the impact load of a high-speed water jet. The poroelastic effect is implemented via Biot's theory of poroelasticity . The numerical results show that: (1) rock failure is only observed in the Gildehaus sandstone with the lowest strengths among the three types of rock tested, (2) most of the cracks are tensile failures and pure shear cracks are rare, mixed mode cracks account for 15%∼40% of the total crack number depending on the mechanical boundary conditions , (3) increased water back pressure significantly suppresses jet erosion. The poroelastic effect on the rock failure is insignificant in the mesoscale simulations and will be further investigated using a microscale model in future.

Homogenization of discrete mesoscale model of concrete for coupled mass transport and mechanics by asymptotic expansion

Mass transport phenomenon in concrete structures is strongly coupled with their mechanical behavior. The first coupling fabric is the Biot's theory according to which fluid pressure interacts with solid stress state and volumetric deformation rate of the solid induces changes in fluid pressure. Another coupling mechanism emerges with cracks which serve as channels for the fluid to flow through them and provide volume for fluid storage. Especially the second coupling mechanism presents a challenge for numerical modeling as it requires detailed knowledge about cracking process. Discrete mesoscale mechanical models coupled with mass transport offer simple and robust way to solve the problem. On the other hand, however, they are computationally demanding. In order to reduce this computational burden, the present paper applies the asymptotic expansion homogenization technique to the coupled problem to deliver (i) continuous and homogeneous description of the macroscopic problem which can be easily solved by the finite element method, (ii) discrete and heterogeneous mesoscale problem in the periodic setup attached to each integration point of the macroscale along with (iii) equations providing communication between these two scales. The transient terms appear at the macroscale only, as well as the Biot's coupling terms. The coupling through cracking is treated at the mesoscale by changing conductivity of the conduit elements according to the mechanical solution, otherwise the two mesoscale steady state problems are decoupled and can be therefore solved in a sequence. This paper presents verification studies showing performance of the homogenized solution.

Simulation study of the sensitivity of mechanical parameters on the reflected signal at the second interface of human cancellous bone—Application of Biot theory

A simulation study of mechanical parameters sensitivity to reflected ultrasound waves at the second interface of a hypothetical human cancellous bone sample is proposed. The sample is considered as a biphasic porous medium saturated by a fluid. The visco-inertial exchanges between the structure and the saturating fluid are described by the Biot theory modified by the Johnson model. An analytical expression of the reflection coefficient at the second interface is obtained in the frequency domain. This expression depends on three physical parameters, which are the porosity, the tortuosity, and the viscous characteristic length, as well as on mechanical parameters, Young's modulus and Poisson's ratio of the solid and the skeletal frame of the medium. The reflected signal is calculated in the frequency domain by the product of the spectrum of the incident signal by the reflection coefficient. In the time domain, the reflected wave is obtained by taking the inverse Fourier transform of the reflected frequency signal. The effect of mechanical parameters on the reflected signal inside the medium is studied. The results obtained are discussed and compared to those given in the literature.

Sensitivity of acoustic parameters on the reflected signal from a rigid porous medium in the low-frequency range of ultrasound

Porous materials are two-phase media consisting of pores saturated with a fluid emerging from a solid skeleton. There are several parameters describing the porous medium. These parameters are classified into two categories: the high-frequency parameters and the low-frequency parameters. During the excitation by an acoustic wave, two cases can occur; the vibration of both phases simultaneously in the case where the structure is flexible, this case is well studied by Biot theory. For a rigid structure, it is the movement of the fluid that is taken into account; this last case is treated by the equivalent fluid theory which is a particular case of Biot theory. Previous work [Proc. Mtgs. Acoust. 45, 045004 (2021)] has shown the influence of the parameters involved in the corrected Johnson-Allard model recently introduced by Sadouki [Phys. Fluids 33, (2021)] on the transmitted signal from a rigid-porous material in the low-frequency regime of ultrasound. The objective of this work is to show, once again, the influence of the same parameters on the reflected signal by providing a comparative study between transmitted and reflected modes.

Salinity, force chains, and creep in muddy sediments

The salinity of sediments is not usually measured. Experience with sandy sediments shows that it has no significant effect on the acoustic properties. In muddy sediments, salinity is critical to the skeletal frame, because it causes the clay particles to flocculate, forming an aggregate of larger particles with significant water fraction. It behaves like a granular medium, in which stress is transmitted along random force chains. Mud is known to suffer from creep, and the force chain model fits neatly into creep theory. It may be modeled as a form of stationary creep, which is linear in many respects. No net strain-hardening is involved. The result is a creep model of the skeletal frame that naturally couples into the Biot theory of porous media. It predicts an attenuation that increases linearly with frequency at low frequencies, which is overtaken by viscous attenuation that increases as the second power of frequency and high frequencies. [Work supported by ONR, Ocean Acoustics Program.]

Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus

In this paper, the influence of the transverse wave on sound propagation in a porous medium with a flexible structure is considered. The study is carried out in the time domain using the modified Biot theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The viscous exchanges between the fluid and the structure are described by fractional calculus. When a sound pulse arrives at normal incidence on a porous material with a flexible structure, the transverse waves interfere with the longitudinal waves during propagation because of the viscous interactions that appear between the fluid and the structure. By performing a calculation in the Laplace domain, the reflection and transmission operators are derived. Their time domain expressions depend on the Green functions of the longitudinal and transverse waves. In order to study the effects of the transverse wave on the transmitted longitudinal waves, numerical simulations of the transmitted waves in the time domain by varying the characteristic parameters of the medium are realized whether the transverse wave is considered or not.

Vibration and damping analysis of a thin finite-size microperforated plate

Microperforated plates (MPP) are commonly used to absorb acoustic waves in sound control technologies. However, less is known concerning the added damping exhibited by thin finite-size MPPs. This added damping effect is known to be induced by viscous and thermal exchanges in the boundary layers near the fluid–solid interface of the perforations, which results in energy dissipation. The present work actually shows that MPPs can feature a significant added viscous damping, especially at low frequencies of vibration. An analytical approach based on an alternative form of the Biot theory for finite-size porous plates is developed. Parametric studies highlight the existence of a characteristic damping frequency at which the added damping of the perforations reaches a maximum. For specified boundary conditions, the damping level can be maximized by adjusting the geometrical parameters of the perforations (diameter, perforation ratio) so that the resonance frequency of the plate coincides with the characteristic damping frequency. The damping reaches a maximum at a particular frequency, and is also effective over a non-negligible bandwidth. Formulations on the mass density and Young’s modulus as a function of the perforation ratio are provided to address the sensitivity of mass and stiffness to the microperforations. The structural damping capabilities of MPPs are validated by measurements on finite-size MPPs which confirm the significant damping increase in the low-frequency range.

Tunable sound packages consisting of granular aerogels and fibrous media

In this study, two granular aerogel materials, both consisting of particles with sizes on the scale of 1 to 50 mm, were first investigated both theoretically and experimentally to improve our understanding of their acoustical behavior. Specifically, layers of the aerogels were evaluated in terms of their sound absorption coefficients, and loss mechanisms in the frequency region below 2000 Hz were quantified for both aerogels by using a previously-developed modeling tool based on the Biot theory. Further studies were then conducted in order to optimize the acoustical performance of potential sound packages featuring these aerogel particle stacks. For example, one of the aerogel materials was placed in series with a fibrous layer to take advantage of the low and high frequency sound absorption offered by the aerogel and the fibrous layers, respectively. The two types of aerogels were also combined in parallel to show that absorption peaks of one of the materials could be used to compensate for the absorption “dips” of the other material. After experimentally validating the design concepts mentioned above, the aerogel granule stacks' properties were tuned to realize a very low frequency and wideband sound absorption treatment when combined with the fibrous layer.

Influence of defects on the lateral dynamic characteristics of offshore piles considering hydrodynamic pressure

This paper develops an analytical framework to reflect the associative effect of dynamic soil-pile interaction (DSPI) and dynamic water-pile interaction (DWPI) on the lateral dynamic characteristics of offshore concrete piles. The pile is assumed to be an Euler-Bernoul20beam and the soil is described by the Biot's poroelastic theory. The hydrodynamic pressure applying on the pile shaft is calculated based on the radiation wave theory with the consideration of water compressibility. The influence of defects along with the hydrodynamic pressure on the lateral dynamic impedance and natural frequency of offshore piles is thoroughly investigated with the proposed solution. The results show that the hydrodynamic pressure cannot be ignored when analyzing the lateral dynamic characteristics, especially for those offshore piles with deep water depth subjected to high frequency vibration. The damage caused by necking defects is much larger than that of bulging defects, and even a small necking defect can lead to a significant decrease of dynamic stiffness and natural frequency of piles. The lateral dynamic characteristics of piles are more sensitive to the variation of defect radius rather than the defect length.

Numerical Analysis of Soil Consolidation Coupled Biot Theory and Hansbo Flow

Numerical Analysis of Soil Consolidation Coupled Biot Theory and Hansbo Flow

Analytical solution for kinematic response of offshore piles under vertically propagating S-waves

This study develops an analytical framework to investigate the kinematic response of offshore piles under vertically propagating S-waves considering hydrodynamic pressure. The hydrodynamic pressure is calculated by using the radiation wave theory. The offshore pile and saturated soil are described by the Euler-Bernoulli beam theory and Biot's poroelastic theory, respectively. The motion of water, scattered field soil and free field soil are derived from governing equations separately. Neglecting water-soil interaction, the rigorous solution for the translational kinematic response factor and curvature ratio along the pile shaft are then obtained based on the boundary conditions of pile-soil and pile-water interaction. The rationality and accuracy of the proposed mathematical model have been verified by comparing the calculated results with the existing studies. Parametric analyses are finally conducted to investigate the influence of hydrodynamic pressure, water depth, embedment depth, slenderness ratio and soil porosity on the kinematic response of offshore piles. The results show that the frequency corresponding to the strongest kinematic response of piles will be overestimated while curvature ratio along the pile shaft will be underestimated if the hydrodynamic pressure is not taken into consideration.