Biot's Theory Research Articles

A Poroelastic Solution for Dynamics of Laterally Loaded Offshore Monopiles

In this study, a rigorous poroelastodynamic solution is proposed for monopiles which accounts for 3D (three dimensional) dynamic interaction of the soil and pile. The dynamic Biot's theory is used to model the soil, while Timoshenko beam is used to model the monopile. The perfect contact conditions between the soil and monopile are realized by decomposing the pile's horizontal translation to radial, hoop and vertical displacements. Correspondingly, 3D radial, hoop and vertical stresses on the pile surface are used to formulate the governing equations of the interaction problem. Using the integral equation methods, the governing differential equations are reduced to coupled integral equations, which are solved numerically. Selected numerical results for coupled lateral and rocking dynamic impedances together with the 3D dynamic contact stresses at the soil-pile interface are studied for different pile and poroelastic material parameters and frequencies of excitation. The distributed soil reaction force and moment along the pile are also obtained by integrating the 3D radial, hoop and vertical stresses along the circumference of the pile at the soil-pile interface. Finally, bending moment, shear force, horizontal translation and rocking rotation of the monopile are obtained based on the classical dynamic Timoshenko beam theory.

Study on Dynamic Response Characteristics of Saturated Asphalt Pavement under Multi-Field Coupling.

To study the dynamic response of saturated asphalt pavement under moving load and temperature load, 3-D finite element models for asphalt pavements with hydro-mechanical coupling and thermal-hydro-mechanical coupling were built based on the porous media theory and Biot theory. First, the asphalt pavement structure was considered as an ideal saturated fluid–solid biphasic porous medium. Following this, the spatial distribution and the change law of the pore-water pressure with time, the transverse stress, and the vertical displacement response of the asphalt pavement under different speeds, loading times, and temperatures were investigated. The simulation results show that both the curves of the effective stress and the pore-water pressure versus the external loads have similar patterns. The damage of the asphalt membrane is mainly caused by the cyclic effect of positive and negative pore-water pressure. Moreover, the peak value of pore-water pressure is affected by the loading rate and the loading time, and both have positive exponential effects on the pore-water pressure. In addition, the transverse stress of the upper layer pavement is deeply affected by the temperature load, which is more likely to cause as transverse crack in the pavement, resulting in the formation of temperature cracks on the road surface. The vertical stress at the middle point in the upper layer of the saturated asphalt pavement, under the action of the temperature load and the driving load, shows a single peak.

Periodic pile barriers for Rayleigh wave isolation in a poroelastic half-space

A detailed investigation is carried out to investigate the use of periodic piles as wave barriers for Rayleigh wave isolation from the perspective of periodic theory in solid state physics. Specifically, this study tries to figure out two questions: At what frequency can the Rayleigh wave be blocked by multi-rows of piles? And in what way does the Rayleigh wave decay? From above point of view, the Biot's theory and Floquet-Bloch theory are employed to simplify the analysis of the pile-soil system. Based on the derived formulation, the attenuation zone of Rayleigh waves is calculated by dispersion analysis using finite element method in three-dimensional contexts. Through comparison with harmonic response, it is found that the frequency range of Rayleigh wave reduction and the screening capability agree well with the theoretical attenuation zone and the attenuation coefficient, respectively. It is concluded that the dispersion analysis provides basic information for Rayleigh wave isolation using piles, including the frequency range and the form of decay. The present investigation could open up new horizons for the design of wave barriers.

A fully-relaxed variationally-consistent framework for inelastic micro-sphere models: Finite viscoelasticity

The micro-sphere modeling framework provides a popular means by which one-dimensional mechanical models can easily and quickly be generalized into three-dimensional stress-strain models. The essential notion of the framework, similar to homogenization theory, is that one allows the microstructural kinematic fields to relax subject to a constraint connected to a macroscopic deformation measure. In its standard presentation, the micro-sphere modeling framework is strictly applicable to elastic materials. Presentations considering inelastic phenomena invariably, and inconsistently, assume an affine relation between inelastic macroscopic and microscopic phenomena. In this work we present a methodology by which one can lift this modeling restriction using two formally different approaches. In particular, we show how one can construct and apply a homogenization with Biot theory to generate fully-relaxed variationally-consistent macroscopic models for inelastic materials within the context of the micro-sphere model. The primary application example will be finite deformation viscoelasticity.

Scattering of a plane wave by shallow buried cylindrical lining in a poroelastic half-space

The shallow buried tunnel is frequently encountered in underground engineering. The dynamic response of a tunnel under incident wave is of great importance for guiding the safety design in tunnel engineering. In this paper, a model for predicting the dynamic response of a shallow buried tunnel in saturated soil is proposed based on nonlocal Biot theory. The analytical solution is obtained using the wave function expansion method. To consider practical engineering problem, a set of material parameters for saturated soil and tunnel lining are selected for the numerical analysis. The influence of nonlocal parameter, which is introduced in nonlocal Biot theory to consider the pore size effect and pore dynamic effect, on dynamic stress concentrate factor in the lining is investigated in detail. The dynamic responses affected by the other factors, such as incident wave angle, frequency of incident wave and buried depth of the tunnel, have also been implemented. The dynamic stress concentrate factor distributed in the lining is also shown and the position and orientation appearing maximum concentrate factor can be easily determined from the contour plot, which can provide a visual guideline for safety design of a tunnel.

Boundary element modeling and simulation of biothermomechanical behavior in anisotropic laser-induced tissue hyperthermia

The main aim of this article is to propose a novel boundary element model for describing thermomechanical interactions in anisotropic laser-induced tissue hyperthermia. The two-dimensional governing equations are studied based on the thermal wave model of bioheat transfer (TWMBT) and Biot's theory. These uncoupled governing equations are solved independently using the boundary element method (BEM), which is a versatile and powerful method because it is dealing with more complex shapes of soft tissues and not needing the discretization of the internal domain, also, it has low RAM and CPU usage. Two numerical examples for square and elliptical cylinder tissues are solved to demonstrate the validity, efficiency and accuracy of the proposed model.

Axisymmetric buckling of saturated circular porous-cellular plate based on first-order shear deformation theory

We studied the mechanical equilibrium and stability of the circular poroelastic plate using first-order shear deformation and Biot theories. We considered the saturated and unsaturated pores media within three types of distribution include poro/nonlinear nonsymmetric with shear modulus, poro/nonlinear symmetric with shear modulus, and poro/monotonous with shear modulus. According to our results, an increase of the pore size, porosity, and plate thickness result in stability decrement. The plate with a symmetrical distribution of pores exhibited more stable behaviour than the other cases. Likewise, decreasing the pore fluid compressibility cause to stability increment of the plate. Finally, our results are validated by comparison to data from the literature.

A Macroscopic Analytical Model for Pressure Wave Propagation in the Water of a Variably Saturated Porous Medium

Core Ideas Pressure waves in pore water can affect transport phenomena. A macroscopic analytical model follows pressure waves traveling in the pore water. Pressure attenuation is predicted through soil hydraulic and mechanical properties. Soil water pressure measurements indicate that soil texture impacts the attenuation. A macroscopic one‐dimensional analytical model has been developed for the attenuation of low‐frequency pressure waves in the water of a variably saturated porous medium. The developed model associates pressure amplitude attenuation with pore scales. Pressure propagation in a single pore size is described by a linear wave equation, and the composition of characteristic pore sizes are combined for representing the porous medium. The capillary bundle approach allows for a geometrical abstraction of the pore sizes and the definition of the water‐filled capillaries under different water content states. This approach allows the pore water pressure attenuation to be addressed directly with readily available soil properties and parameters. Theoretical findings backed by experimental observations in three soils under saturated conditions indicate that the pressure wave attenuates more rapidly in the water of fine‐textured soils than coarser ones. Furthermore, the results indicate that the effect of larger pores on pressure attenuation is significant at high water contents, but the pore‐size distribution and the smaller pores dominantly control the pressure attenuation as the water content decreases. The developed model is consistent with previous studies and with the theory of Biot for waves in composite media.

Sound transmission in two-dimensional periodic poroelastic structures

To obtain excellent sound reduction performance, in this paper we introduce a novel periodic poroelastic composite structure, which combines poroelastic material and periodic structure and aims at using the remarkable acoustic performance of these two. This periodic composite structure comprises three parts, i.e. the poroelastic domain, the elastic domain (thin plate), and the periodic resonators, which can be simple single-degree-of-freedom resonators (SRs) or composite two-degree-of-freedom resonators (CRs). A theoretical model is established by using Biot theory for the poroelastic domain, and by using the effective medium method for the resonator-plate coupling system, which is considered as an isotropic plate with an equivalent dynamic density. This method is validated with degenerated model in the literature; the results obtained by this method are in excellent consistence with the results in the literature. Parameter analyses are performed to test the influences of poroelastic addition and periodic resonator on the sound transmission loss (STL) of this periodic composite structure under two kinds of boundary conditions. The poroelastic addition is found to increase the STL while the influences of resonators are complicated. The STL increases notably in the frequency range bounded by the characteristic frequencies of these resonators, however, a decrease just follows when it exceeds these frequencies, which can be observed in both SR case and CR case under the two boundary conditions. In the meantime, when multiple SR is placed in a periodic lattice, it is found that different resonators with ascending mass and characteristic frequencies have superior STL to those with ascending characteristic frequencies but have equal mass. The case with CR, which is more complicated as expected, shows less STL decrease than the case with SR, but wider frequency range where the STL increases than a poroelastic composite structure without resonators. This results from the fact that the frequency band of vibration suppression in the CR case is wider than in the SR case. As a result, to achieve the desired STL performance in a frequency range, the proposed composite structure using SR with tuned characteristic frequencies is enough; however, if a wider frequency band is expected even if there is a slight STL tradeoff, the CR case is a better option. Though the method proposed is only valid in the low-to-medium frequency range, the results obtained can benefit theoretical development of low-to-medium sound modulation applications, they are also valuable and illuminating for investigating the broadband sound modulation.

Hydro mechanical coupling using a hybrid dimensional approach and interface element formulations

AbstractCharacteristics of underground fluid storage and properties related to heat transport and underground matter are of high interest in the community of porous and fractures media. Especially once hydro‐mechanical phenomena in fluid filled fractures are of interest, such as inverse waterlevel fluctuations during well stimulations (Noordbergum effect), the numerical coupling of the fluid and the porous domain is challenging. Existing strategies such as direct numerical simulations or coarse‐grained approaches (Biot's theory) either fail due to technical issues (number of Degrees of Freedom (DoF)) or are not capable of intuitively taking permeability changes based on the deformation state into account. This work proposes a fully coupled scheme using hybrid‐dimensional element formulations and zero‐thickness interface elements that naturally describes flow in deformable fractures surrounded by porous media.

Modeling of wave reflection in gas hydrate-bearing sediments

The content and distribution pattern of the gas hydrate are the key factors affecting the free surface reflection of the gas hydrate-bearing sediment. In this study, a theoretical model of wave reflection in gas hydrate-bearing sediment is discussed for the incidence of P1-wave and S1-wave to analyze the influences of the gas hydrate. The gas hydrate-bearing sediment is modeled by a Biot-type three-phase theory and assumed to be composed of sediment grains with connected pore occupied by the mixture of fluid and hydrate. The incoming wave is split into three reflected compressional waves and two reflected shear waves at the free surface of a gas hydrate-bearing sediment half-space. The analytical solutions for various reflected waves at open-pore and sealed-pore boundaries are obtained in terms of displacement potentials. Then, two-phase type models based on Biot theory are proposed by assuming the gas hydrate and sediment frame are weakly coupled or essentially consolidated to verify the rationality of the theoretical model. A third two-phase type model is given by assuming there is no gas hydrate in the pore. At last, the influences of the content and distribution pattern of gas hydrate, the frequency of the incident wave, and the permeability condition on the horizontal displacements and displacement ratios are analyzed respectively through numerical calculation. It is revealed that the solutions based on the Biot-type three-phase theory are very consistent with the solutions based on Biot theory if the gas hydrate and sediment frame are weakly coupled or essentially consolidated. Furthermore, the model proposed in this study can be used to analyze the influence of the distribution pattern of the hydrate.

A coupled hydraulic-mechanical-damage geotechnical model for simulation of fracture propagation in geological media during hydraulic fracturing

This paper proposed a coupled hydraulic-mechanical-damage (HMD) geotechnical model for simulating the entire failure process in geological media. In this model, elastic damage mechanics describes the damage process of elements. Darcy's Law and unsteady seepage flow equation describe fluid flow in elements, where compression of fluid and rock are considered. The coupled process is fulfilled by Biot's consolidation theory describing how pore-pressure affects the stress field and the stress/aperture/mesh-dependent permeability that represents the influence of stress on seepage field. The HMD model is implemented in the calculation module of Rock-Failure-Process-Analysis on Petroleum-problems (RFPA-Petrol) codes. The RFPA suite of software codes is based on the idea that the heterogeneity leads to non-linearity and causes progressive failure behavior observed in brittle rock. The capability of RFPA-Petrol simulator is demonstrated by a typical coupled problem related to double fractures propagation during hydraulic fracturing. This simulation successfully reproduces the asynchronous initiation, asymmetric growth and non-equal deflection of fractures in the numerical model. Then this simulator was used to investigate how fracture spacing and horizontal stress ratio influence propagation and reorientation of three hydraulic fractures from a horizontal well. At last, a horizontal well drilled by SHENGLI oilfield is simulated to certificate RFPA-Petrol's ability on field-scale models. It is found that larger fracture spacing and higher initial fracture height will form longer and wider fractured zones. HMD-based RFPA-Petrol is suitable for simulation of laboratory scale and field scale fracture propagation problems.

Sound transmission of periodic composite structure lined with porous core: Rib-stiffened double panel case

Porous materials are effective for the isolation of sound with medium to high frequencies, while periodic structures are promising for low to medium frequencies. In the present work, we study the sound insulation of a periodically rib-stiffened double-panel with porous lining to reveal the effect of combining the two characters above. The theoretical development of the periodic composite structure, which is based on the space harmonic series and Biot theory, is included. The system equations are subsequently solved numerically by employing a precondition method with a truncation procedure. This theoretical and numerical framework is validated with results from both theoretical and finite element methods. The parameter study indicates that the presence of ribs can lower the overall sound insulation, although a direct transfer path is absent. Despite the unexpected model results, the method proposed here, which combines poroelastic modeling and periodic structures semi-analytically, can be promising in broadband sound modulation.

Research on the vibration transmissibility of impedance modulating structure based on magnetorheological fluid

Due to the fact that wave propagation has superior penetration at low frequencies, its attenuation in a limited range is a challenge. To solve this issue, a novel impedance modulating structure based on magnetorheological (MR) fluid is proposed, and the vibration transmissibility of the proposed structure is investigated. The modulation of impedance is realized by controlling the applied magnetic field, which can result in liquid–solid conversion of MR fluid. The wave equation in MR fluid is established based on the Biot theory. Subsequently, the wave equation is solved and the expression of vibration level difference (VLD) is obtained. Furthermore, the vibration transmissibility of different modulation parameters of impedance is analyzed. Finally, experiments are conducted to evaluate the variation in the vibration transmissibility depending on the modulation parameters of impedance. It is demonstrated that the modulation of MR fluid impedance is validly showing greater attenuation for wave propagation than that without modulation. Due to its great structural variability and adjustability parameter, the proposed structure has excellent controllability in vibration transference, and it is promising for further applications in isolating vibration at the platform of the large power systems.

Study on dynamic responses of unsaturated railway subgrade subjected to moving train load

A new track-multilayer ground model was established to investigate railway subgrade dynamic responses induced by moving train load in this paper. The ground structure was modeled by an elastic layer overlying an unsaturated porous elastic half space. The dynamic governing equations of the three-phase porous medium, which are easy to be degraded to Biot's theory, were derived with consideration of both viscous drag and inertial coupling between pore fluids and soil skeleton as well as capillary pressure. The model solutions were obtained based on dynamic stiffness matrix method and Fourier transform and introducing corresponding boundary condition. The vertical displacement on the subgrade top surface and the pore water pressure along the depth were studied. The numerical results show that the effects of saturation, vehicle speed, train axle load and subgrade bed material properties on the subgrade responses are significant.

Acoustic wave propagation in orthotropic porous silicon with specific pore shapes

In the literature, porous materials are normally treated as isotropic media. In this case, analytical solutions for wave velocities and attenuations as a function of frequency have been given by M. A. Biot. Nevertheless, porous silicon (PSi) synthesized using electrochemical manufacturing is generally highly anisotropic. Then, it can be considered as orthotropic. In order to optimize acoustic applications using PSi, it is necessary to take account to this anisotropy. In early works, by using FEM simulations, skeleton elastic constants have been computed for five PSi pore shapes and different porosities. In this work, the "Gedanken experiments" have been implemented and compliance tensors have been calculated. By inverting the compliance tensor, the elastic parameters of fluid-saturated PSi have been calculated. Moreover, geometrical parameters such as tortuosity, permeability and characteristic length of pores have been simulated. The wave equations can be deduced by combining Biot's theory and Christoffel's equations, and have been solved by using the simulation results. Then, the phase velocity and attenuation variations, according to frequency, were studied. The results for five different pore shapes have been compared and the differences show that the shape strongly influences wave propagation.

Ocean sediments and the Biot theory

The surficial ocean sediment is usually a granular medium, therefore it is a porous medium and follows Biot’s theory. It predicts one shear wave and two compressional waves, called the fast and slow waves. This sets it apart from the acoustic theories of uniform media (solids or fluids) which permit only one compressional wave. The fast wave is similar to the compressional wave in a solid or fluid. The slow wave is unique to porous media, it is difficult to detect and its presence in the ocean sediment is controversial. There is indirect evidence of the Biot slow wave: The deviation of the acoustic reflection coefficient of the ocean sediment from Rayleigh’s reflection equation, can be explained in terms of the Biot slow wave. Direct detection of the Biot slow wave has been reported in some porous media, particularly sintered glass beads. However, its detection in unconsolidated media such as water saturated sand needs more work. Direct measurement of its properties is desirable. Simulations suggest that it could explain some unusual behavior of finely layered sediments. Finally, due to natural granularity, there should be a continuous conversion between the fast and slow waves that is missing from current models. The surficial ocean sediment is usually a granular medium, therefore it is a porous medium and follows Biot’s theory. It predicts one shear wave and two compressional waves, called the fast and slow waves. This sets it apart from the acoustic theories of uniform media (solids or fluids) which permit only one compressional wave. The fast wave is similar to the compressional wave in a solid or fluid. The slow wave is unique to porous media, it is difficult to detect and its presence in the ocean sediment is controversial. There is indirect evidence of the Biot slow wave: The deviation of the acoustic reflection coefficient of the ocean sediment from Rayleigh’s reflection equation, can be explained in terms of the Biot slow wave. Direct detection of the Biot slow wave has been reported in some porous media, particularly sintered glass beads. However, its detection in unconsolidated media such as water saturated sand needs more work. Direct measurement of its properties is desirable. Simulations suggest that ...

A micro-mechanical model for the Biot theory of acoustic waves in a fully saturated granular material

In the context of the classical Biot theory for acoustic waves in a fully-saturated granular material, we improve upon the constitutive relation of the solid phase by means of micro-mechanical modeling. This is needed to explain discrepancies on the dependency of the frequency with the sound speed attenuation and dispersion between present models and experiment. As first step, we provide a more detailed description of the interaction between the particles and water. A Standard Linear Solid model is proposed to represent the behavior at contact level between particles immersed in a compressible fluid.

An overview of the sediment acoustic models and their experimental validations

The Biot theory of acoustic wave propagation in porous media was first adapted to marine sediments by Stoll (1969). To describe the response of a slightly inelastic skeletal frame over a wide frequency range, Stoll introduced the notion of constant complex bulk and shear moduli with small imaginary parts. Stoll’s model predicts f1 frequency-dependence of attenuation at low frequencies and nearly f2 and f1/2 frequency-dependence at frequencies where maximum velocity dispersion occurs. Although the theory well reproduces the velocity and attenuation measurements in marine sediments, the assumption of constant complex moduli slightly violates the causality (Turgut, 1990). The more recent Grain-Shearing (GS) and Viscous Grain-Shearing (VGS) models (Buckingham, 2000 and 2007) are causal and the VGS model also predicts a frequency-dependence of attenuation like that of the Stoll model. With the selection of proper parameter values, both models predict compressional and shear-wave dispersions that are in agreement with those of previous in-situ and laboratory measurements. In addition, the Stoll model predicts the existence of slow compressional waves that have been observed in synthetic porous media (Plona, 1980) but not in natural marine sediments. Several reflection and in-sediment transmission experiments are discussed to facilitate the detection of slow compressional waves in marine sediments. [Work supported by ONR.]

A Simplified Method for Calculating Multilayer Sound Insulation with Layers of Fibrous Porous Material

Abstract—The proposed method is based on the Biot theory of wave propagation in porous saturated elastic media. To simplify the cumbersome full calculation procedure, specific features of typical fibrous porous materials are used: a large density and moderate stiffness of the elastic frame in comparison to air. In this case, the transfer properties of an elastic porous layer are represented by two second-order matrices, which characterize sound transfer by airborne and frame-borne waves. Combined with the boundary conditions, such a representation in standard schemes makes it possible to form a single transfer matrix for the layer, by considering it a linear four-pole that relates the pressures and normal velocity components at the inlet and exit. When calculating the sound insulation of a multilayer structure, the formed matrix of the elastic porous layer is introduced via a cofactor into a chain of second-order transfer matrices determined by the other layers. Examples and comparative calculations are presented that demonstrate the high correspondence of the method’s results to the full calculation data.