Permeable Boundary Conditions Research Articles

Mode-III fracture of four cracks emanating from an elliptical hole in one-dimensional orthorhombic quasicrystals with piezoelectric effect

In this study, a new generalized conformal mapping is introduced using the generalized complex variable method, focusing on the investigation of the anti-plane problem of an elliptical hole with four cracks in one-dimensional (1D) orthogonal piezoelectric quasicrystals. Under the consideration of electrically impermeable and electrically permeable boundary conditions, the governing equations of the anti-plane problem in 1D orthogonal piezoelectric quasicrystals are successfully simplified into an eighth-order partial differential equation and a fourth-order partial differential equation by introducing a potential function. Based on this, analytical solutions for the stress field and electric field are obtained, and the exact solution for the stress intensity factor (SIF) is derived. In addition, formulas for the electric displacement intensity factor (EDIF) and energy release rate are provided. Subsequently, the influence of geometric parameters and external loads on the field intensity factors and energy release rate is analyzed through numerical examples, providing a theoretical basis for engineering mechanics analysis.

Dynamics of fixed-volume pinned films – dealing with a non-self-adjoint thin-film problem

The use of thin liquid films has expanded beyond lubrication and coatings, and into applications in actuators and adaptive optical elements. In contrast to their predecessors, whose dynamics can be typically captured by modelling infinite or periodic films, these applications are characterized by a finite amount of liquid in an impermeable domain. The global mass conservation constraint, together with common boundary conditions (e.g. pinning), create quantitatively and qualitatively different dynamics than those of infinite films. Mathematically, this manifests itself as a non-self-adjoint problem. This work presents a combined theoretical and experimental study for this problem. We provide a time-dependent closed-form analytical solution for the linearized non-self-adjoint system that arises from these boundary conditions. We highlight that, in contrast to self-adjoint problems, here, special care should be given to deriving the adjoint problem to reconstruct the solution based on the eigenfunctions properly. We compare these solutions with those obtained for permeable and periodic boundary conditions, representing common models for self-adjoint thin-film problems. We show that, while the initial dynamics is nearly identical, the boundary conditions eventually affect the film deformation as well as its response time. To experimentally illustrate the dynamics and to validate the theoretical model, we fabricated an experimental set-up that subjects a thin liquid film to a prescribed normal force distribution through dielectrophoresis, and used high-frame-rate digital holography to measure the film deformation in real time. The experiments agree well with the model and confirm that confined films exhibit a different behaviour which could not be predicted by existing models.

Micromechanical analysis of suffusion in gap-graded granular soils considering soil heterogeneity and non-uniform seepage flow

The dams and their foundations can exhibit different forms of heterogeneity and experience non-uniform seepage flow, significantly affecting the suffusion. Although experimental studies on the effect of heterogeneities have been performed, their micro-mechanics remain unclear. For this reason, systematic simulations considering various soil heterogeneity configurations and flow fields are performed using the coupled CFD-DEM approach. Layered samples with different initial fines content and density are generated and analysed. The effect of flow paths is investigated for 3D conditions in conjunction with partially permeable boundary conditions and initially blocked zone in the samples. The obtained macro-responses are presented and interpreted from the micromechanical perspectives. The primary influential factors of the sample's susceptibility to suffusion are identified as the migration distance, flow field characteristics, cutoff wall position, and the number of fine particles in the weak force chain network. The sample is more prone to suffusion when there are more fine particles in the weak force chain or near the flow outlet. The flow rate has a positive effect on the cumulative eroded mass for different flow fields. The cutoff wall downstream is more effective in retaining the fine particles.

기체 형상이 로터-기체 상호작용 소음에 미치는 영향에 관한 전산해석

Urban Air Mobility(UAM) or Advanced Air Mobility(AAM) has emerged as a next-generation transportation system based on electric-powered vertical takeoff and landing aircraft. Distributed Electric Propulsion(DEP) system is applied to most UAM aircraft, which inevitably accompanies an increase in the number of rotors or propellers, thus leading to a strong interaction between the rotor and the airframe. This study conducted a computational investigation of the effects of rotor-airframe interaction on aerodynamic performance and noise generation depending on the airframe shapes. Aerodynamic analysis was performed using Lattice Boltzmann Method(LBM) simulation, and the noise level was predicted using the Ffowcs Williams-Hawkings(FW-H) acoustic analogy with permeable boundary condition. Two different types of airframe geometry were considered: cylinder and conical airframes. The thrust coefficient of rotor blades with the conical airframe is higher than that of rotor blades with the cylinder airframe. Tip vortex breakdown phenomenon and transition into turbulent wake state were captured in both airframes. Moreover, it was observed that high-intensity noise was radiated over the broad surface of the airframe for the conical airframe case. The comparison of overall sound pressure level(OASPL) on the hemisphere showed the strong interaction between the rotors and conical airframe causes a considerable increase in the noise level, compared to the isolated rotor and rotor-cylinder airframe.

Diffusion with stochastic resetting screened by a semipermeable interface

In this paper we consider the diffusive search for a bounded target with its boundary totally absorbing. We assume that the target is surrounded by a semipermeable interface given by the closed surface with . That is, the interface totally surrounds the target and thus partially screens the diffusive search process. We also assume that the position of the diffusing particle (searcher) randomly resets to its initial position x 0 according to a Poisson process with a resetting rate r. The location x 0 is taken to be outside the interface, , which means that resetting does not occur when the particle is within the interior of . (Otherwise, the particle would have to cross the interface in order to reset to x 0.) Hence, the semipermeable interface also screens out the effects of resetting. We illustrate the theory by explicitly solving the boundary value problem for a three-dimensional (3D) spherically symmetric interface and a concentric spherical target. We calculate the mean first passage time (MFPT) to find (be absorbed by) the target and explore its behavior as a function of the permeability κ 0 of the interface and the radius of the interface R. In particular, we find that increasing R for a fixed target size reduces the MFPT and increases the optimal resetting rate at which the MFPT is minimized. We also find that the sensitivity of the MFPT to changes in κ 0 is a decreasing function of R. Finally, we introduce a stochastic single-particle realization of the search process based on a generalization of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of reflecting BM on either side of the interface. The main challenge is establishing that the probability density generated by the snapping out BM satisfies the permeable boundary conditions at the interface. We show how this can be achieved using renewal theory.

Dynamic fracture of a partially permeable crack in a functionally graded one-dimensional hexagonal piezoelectric quasicrystal under a time-harmonic elastic SH-wave

The dynamic fracture of a crack in a functionally graded one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) subjected to a time-harmonic elastic SH-wave is studied by using integral transform technique and Copson method. It is assumed that the crack surface is a partially permeable boundary condition and the material properties vary continuously as an exponential function. With the help of the Fourier transform, the boundary value problem of partial differential equation describing fracture problem is formulated to three pairs of dual integral equations, which are numerically solved by Copson method. Explicit expressions for the electroelastic field including phonon and phason stresses and electric field on the crack face are determined. The dynamic intensity factors of the electroelastic field are obtained in closed form, and some special cases of obtained results are discussed. On the basis of theoretical analysis and numerical simulation of the established models, numerical analysis was then conducted to discuss crack length, gradient parameter, electric boundary condition, electric loading, incident angle, amplitude, and wave number on the fracture characteristics of material. The research of this paper will provide a theoretical basis for nondestructive testing, optimal design, and reliability analysis of materials and will enrich the research content of fracture mechanics of multi-field coupling materials.

Renewal equationsfor single-particle diffusion through a semipermeable interface.

Diffusion through semipermeable interfaces has a wide range of applications, ranging from molecular transport through biological membranes to reverse osmosis for water purification using artificial membranes. At the single-particle level, one-dimensional diffusion through a barrier with constant permeability κ_{0} can be modeled in terms of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflected BMs that are restricted to either the left or right of the barrier. Each round is killed (absorbed) at the barrier when its Brownian local time exceeds an exponential random variable parameterized by κ_{0}. A new round is then immediately started in either direction with equal probability. It has recently been shown that the probability density for snapping out BM satisfies a renewal equationthat relates the full density to the probability densities of partially reflected BM on either side of the barrier. Moreover, generalized versions of the renewal equationcan be constructed that incorporate non-Markovian, encounter-based models of absorption. In this paper we extend the renewal theory of snapping out BM to single-particle diffusion in bounded domains and higher spatial dimensions. In each case we show how the solution of the renewal equationsatisfies the classical diffusion equationwith a permeable boundary condition at the interface. That is, the probability flux across the interface is continuous and proportional to the difference in densities on either side of the interface. We also consider an example of an asymmetric interface in which the directional switching after each absorption event is biased. Finally, we show how to incorporate an encounter-based model of absorption for single-particle diffusion through a spherically symmetric interface. We find that, even when the same non-Markovian model of absorption applies on either side of the interface, the resulting permeability is an asymmetric time-dependent function with memory. Moreover, the permeability functions tend to be heavy tailed.

Analysis of mode III interface fracture for hole-initiated cracks in magnetic-electro-elastic bimaterials

The analytical solution of two magnetic-electro-elastic materials with hole-initiated cracks are investigated under anti-plane shearing. The modeling of the cracks is achieved by applying a pair of negative shear stresses in the region where the cracks occur. During this investigation, the fundamental solution of the displacement field is expressed by Green’s function method. The expressions of the magneto-electric-elastic wave fields around the interface can be obtained by using the wave function expansion method and Sommerfeld radiation conditions. Then, combined with permeable electromagnetic boundary conditions, the crack problem is reduced to a system of the first kind of Fredholm’s integral equations, from which the dynamic stress intensity factor (DSIF) at the crack tip is obtained. Finally, various examples are provided to illustrate the effects of the geometrical parameters, material properties, wave number, and incident angle on the DSIF. Compared with previous traditional numerical and analytical methods, the methods proposed in this article are more applicable in practical engineering and theoretical analysis.

Energy release rate for cracks in hydrogels undergoing finite deformations

The rupture of hydrogels and swellable elastomers involves large deformations, and there exists a large literature devoted to their experimental characterization including methods for measuring and enhancing fracture toughness. Analytical investigations of the fracture of hydrogels have recognized the importance of large deformations and the contributions of liquid flow, but they have largely been restricted to plane-strain formulations that ignore through-thickness effects. In this paper, boundary–initial value problems for cracked specimens are solved in plane-strain and three dimensions for both permeable and impermeable boundary conditions and various rates of loading. The transient stress, strain and chemical potential fields near the crack tip/front are found to be notably different than the asymptotic solutions of linear poroelasticity and large deformation plane-strain formulations. The energy release rate is computed using a poroelastic path-independent integral ( J ∗ ), and generally that also is a function of the liquid flow. It is shown for moderately thin three-dimensional specimens that liquid flow is largely in the out-of-plane direction under permeable boundaries and largely in-plane for impermeable boundaries; thus, liquid flow makes larger contributions to the energy release in the latter. In agreement with experiments, the energy release rate tends to be larger at higher loading rates due to the contributions of liquid flow. Finally, criteria for crack growth based on the critical stretch ahead of the crack are adopted to predict the critical energy release rate as a function of solid volume fraction, and the possibility of a non-monotone dependence of energy release on solid volume fraction is uncovered. The methods presented in this paper can be utilized to analyze a wide variety of problems in the rupture of hydrogels including applications to soft tissues and fibrous gels.

Wave-induced thermal flux and scattering of P waves in a medium with aligned circular cracks

High temperature affects the seismic properties of cracked and faulted reservoirs and can be an indicator for their detection. To this purpose, the authors study the wave-induced thermal flux (WITF) and develop two exact solutions for the scattering of compressional waves by a circular crack filled with a compressible fluid, in which the approach is based on thermally permeable and impermeable boundary conditions. The authors obtained the phase velocity and attenuation as a function of frequency, which found that there are two loss mechanisms, i.e., thermoelastic dissipation at low frequencies and elastic scattering at high frequencies. Basically, when the crack size is comparable to the thermal and elastic wavelengths, there are substantial dispersion and attenuation (anelasticity) in the WITF and scattering frequency ranges, respectively. This means that the spatial inhomogeneity scale for inducing WITF is much smaller than that of scattering and the two mechanisms can be discriminated. The dependence of the compressional-wave velocity and attenuation on the compressibility and thermal expansion of the crack-filling fluid are different depending on the thermal diffusion rates at the crack interface. The anelasticity is much higher in the fully permeable case. This model has the potential to evaluate thermoelastic properties and heterogeneity at different scales from seismic responses.

Correction to: Fractional derivative modelling for rheological consolidation of multilayered soil under time-dependent loadings and continuous permeable boundary conditions

Correction to: Fractional derivative modelling for rheological consolidation of multilayered soil under time-dependent loadings and continuous permeable boundary conditions

Entropy and stability analysis on blood flow with nanoparticles through a stenosed artery having permeable walls.

In this research, the electro-osmotic effects are highlighted for a blood-based hybrid nanofluid flow across an artery infected with multiple stenosis. The artery has permeable walls together with slip boundary effects. The slip and permeable boundary conditions model the more realistic blood flow problems. The governing equations of the problem are converted into non-dimensional form by introducing adequate dimensionless variables and acquired the exact solutions. The detailed study of heat transfer is given by Joule heating and viscous dissipation effects. The disorder of fluid flow is investigated by the mathematical study of entropy generation. Analytically attained solutions are examined graphically for both symmetric and non-symmetric shapes of stenosis. Streamlines are analyzed for varying values of flow rate Q and electro-osmotic parameter m. The flow velocity has smallest values on the axis of channel and gets higher value near the boundary walls. The temperature profile delineates opposite behavior to the velocity, and it is parabolic in nature. The velocity reduces towards the non-uniform stenosis except for electroosmotic parameter m. The temperature has larger magnitude in the case of anti-symmetric stenosis. Moreover, the stability of velocity solution is also analyzed.

General analytical solutions for the equal‐strain consolidation of prefabricated vertical drain foundation in unsaturated soils under time‐dependent loading

AbstractExternal disturbances were scarcely considered in the consolidation of unsaturated vertical drain foundation, in which the ideally permeable or impermeable boundary conditions were always adopted to investigated the consolidation patterns. In view of this, by taking the flows along the vertical and radial orientations simultaneously, this study proposes the general analytical solutions for the axisymmetric consolidation of prefabricated vertical drain foundations in unsaturated soils which consider drain resistance, smear effect, and time‐dependent loading under the equal‐strain assumption. Through theories related to consolidation in unsaturated soils, the governing equations of air and water phases in the matrix form are proposed. Subsequently, Fourier series expansion, matrix analysis and decoupling method are used to acquire the analytical solutions. The separate expressions of smear coefficient and drain resistance factors, which can represent the potential influence of the external boundaries more obviously, are included in the solutions. Degradation validity method is used to verify the credibility of the current solutions, and the vertical flow is found to promote the consolidation process under external conditions. Finally, the solutions obtained in this paper can provide guidance for more complicated consolidation problems in unsaturated soils.

Fractional derivative modelling for rheological consolidation of multilayered soil under time-dependent loadings and continuous permeable boundary conditions

Fractional derivative modelling for rheological consolidation of multilayered soil under time-dependent loadings and continuous permeable boundary conditions

An approach for permeable boundary conditions in SPH

Smoothed Particle Hydrodynamics (SPH) has intrinsic advantages over the Finite Volume (FV) method for predicting multiphase flows, since the need for fluid phase interface reconstructions is avoided due to the natural advection of those interfaces based on the Lagrangian description of the control volumes. However, permeable boundary conditions are less straightforward to handle with SPH, even though domain inlets and outlets are needed in most engineering problems. In this work we propose a new framework for permeable boundary conditions for the Weakly Compressible SPH based on ghost particles. The major characteristics of our approach are the application of Navier Stokes Characteristic Boundary Conditions (NSCBC) and a particle mass generation and removal procedure at the boundary. This approach enables the application of non-reflective boundary conditions and redundantizes the need for advecting ghost particles from an external buffer zone to the internal domain. Additionally, the particle mass variation can exactly be balanced with the mass flux defined at the boundary. Furthermore, we are proposing a novel approach to set the ghost particle state based on an extrapolation of the state of particles located in the internal domain and on the domain boundary. We validate the proposed framework for permeable boundaries with one- and two-dimensional test cases. The major findings are as follows. Due to the particle mass generation and removal at the domain boundary, spurious oscillations are introduced due to a non-optimal particle distribution. Those oscillations are reduced in our work to an acceptable level (1×10−5 to 1×10−4 relative to the values of the fluid dynamics variables) using existing stabilization schemes as well as the aforementioned novel methods for setting ghost particle states and particle masses at the permeable boundaries. We observe a very good agreement between the solutions obtained with the presented approach and analytical as well as numerical solutions obtained with the FV method. The proposed approach for setting the ghost particle states is indispensable for achieving a perfect balance between particle mass variation and mass flux across the permeable interface as well as the steady state and the temporal accuracy of the solution. Additionally, the pressure reflectivity at the permeable boundary is found to be in a very good agreement with the theory of the NSCBC method. Furthermore, for a SPH-FV coupled simulation we could demonstrate an excellent agreement compared to the FV method as well as the capability of our approach to handle significant flow unsteadiness and gradients at the permeable coupling interfaces without the introduction of spurious flow behavior.

Surface effects on a mode-III reinforced nano-elliptical hole embedded in one-dimensional hexagonal piezoelectric quasicrystals

To effectively reduce the field concentration around a hole or crack, an anti-plane shear problem of a nano-elliptical hole or a nano-crack pasting a reinforcement layer in a one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) is investigated subject to remotely mechanical and electrical loadings. The surface effect and dielectric characteristics inside the hole are considered for actuality. By utilizing the technique of conformal mapping and the complex variable method, the phonon stresses, phason stresses, and electric displacements in the matrix and reinforcement layer are exactly derived under both electrically permeable and impermeable boundary conditions. Three size-dependent field intensity factors near the nano-crack tip are further obtained when the nano-elliptical hole is reduced to the nano-crack. Numerical examples are illustrated to show the effects of material properties of the surface layer and reinforced layer, the aspect ratio of the hole, and the thickness of the reinforcing layer on the field concentration of the nano-elliptical hole and the field intensity factors near the nano-crack tip. The results indicate that the properties of the surface layer and reinforcement layer and the electrical boundary conditions have great effects on the field concentration of the nano-hole and nano-crack, which are useful for optimizing and designing the microdevices by PQC nanocomposites in engineering practice.

Semianalytical Solution for One-Dimensional Consolidation in a Multilayered Unsaturated Soil System with Exponentially Time-Growing Permeable Boundary

AbstractThis paper proposes new semianalytical solutions to one-dimensional consolidation of unsaturated soils with a multilayered system under the partially permeable boundary condition, in which ...

Poroelastic theory of consolidation for a two-layer system with an upper unsaturated soil and a lower saturated soil under fully permeable boundary conditions

Soils are naturally present in the form of stratified layers, each being distinguishable from adjacent layers by a distinctive set of hydraulic and elasticity properties. Lo et al. (2020) have recently presented a theoretical model of poroelasticity that provides a detailed description of simultaneous solid framework deformation and immiscible fluid flow in a two-layer soil system that is made up of upper unsaturated and lower saturated zones caused by a vertical constant surface load under semi-permeable boundary drainage conditions. Due to the existence of different moisture contents (partial and full saturations) in the upper and lower layers, excess pore fluid pressure does not exhibit a symmetric distribution with respect to depth. Therefore, an exact, analytical solution for fully permeable boundary drainage conditions cannot be determined readily by applying scaling transformations and spatial translations from our recent result even though the coupled model equations possess an invariance feature.In this paper, a well-defined boundary-value problem based on the decoupled model equations of poroelasticity generalized for a two-fluid system is rigorously formulated for a double-layer (unsaturated-saturated) soil profile that allows water drainage at both top and bottom surfaces. Using Laplace time transformation, we derive the closed-form, exact-analytical solution accounting for the intricate interaction between deformable solid matrix and compressible interstitial fluids in such a two-layer system. The solution takes a more complicated form than that for the semi-permeable case. Boundary drainage conditions indeed affect the development of excess pore water and air pressures along with the time evolution of total settlement. We show that when the bottom boundary of the saturated layer overlain by the unsaturated layer with a lower hydraulic conductivity is impervious, the curve describing the relationship between the depth and excess water pressure at the interface has a negative slope, whereas the slope is positive when the boundary is pervious. The total settlement in the lower saturated layer would thus achieve an equilibrium state faster.

Antiplane Fracture Problem of Three Nanocracks Emanating from an Electrically Permeable Hexagonal Nanohole in One-Dimensional Hexagonal Piezoelectric Quasicrystals

Based on the Gurtin-Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping, the electrically permeable boundary condition with surface effect is established, and the antiplane fracture problem of three nanocracks emanating from a hexagonal nanohole in one-dimensional hexagonal piezoelectric quasicrystals with surface effect is studied. The exact solutions of the stress intensity factor of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate are obtained under the two electrically permeable and the electrically impermeable boundary conditions. The numerical examples show the influence of surface effect on the stress intensity factors of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate under the two boundary conditions. It can be seen that the surface effect leads to the coupling of the phonon field, phason field, and electric field, and with the decrease of cavity size, the influence of surface effect is more obvious.

Local thermal non-equilibrium effects on thermal pressurization in saturated porous media considering thermo-osmosis and thermal-filtration

This paper presents new fully-coupled analytical thermo-poroelastic solutions, introducing four new components to the traditional Local Thermal Non-Equilibrium (LTNE) theory: thermo-osmosis; thermal-filtration; heat sinks due to thermal expansion of the pore fluid and the solid phase; and vertical confinement. Thermo-osmosis, thermal-filtration, and fluid dilation are found to have very different effects in case of LTNE versus local thermal equilibrium (LTE), resulting in a fundamentally different thermo-hydraulic-mechanical (THM) response. The aforementioned non-traditional thermal processes cause generation of notably higher pore pressures in the porous medium that cannot be otherwise predicted using current solutions. Fluid temperatures are more sensitive to these non-traditional thermal processes, compared to the solid phase temperatures. Thermo-osmosis is found to have more significant effects under LTNE compared to LTE, and to be the dominant mechanism affecting solid temperatures, pore pressures and stresses. Thermal-filtration and heat sink due to fluid thermal expansion are the dominant mechanisms affecting fluid temperatures. The vertical confinement and formation anisotropy is shown to notably impact the THM response of a porous layer to heat source or sink. The impact of the aforementioned four novel components are found to be enhanced under an impermeable hydraulic boundary compared to a permeable boundary condition.