Permeable Boundary Conditions Research Articles

Thermal convection in a spherical shell with melting/freezing at either or both of its boundaries

In a number of geophysical or planetological settings, including Earth’s inner core, a silicate mantle crystallizing from a magma ocean, or an ice shell surrounding a deep water ocean—a situation possibly encountered in a number of Jupiter and Saturn’s icy satellites—a convecting crystalline layer is in contact with a layer of its melt. Allowing for melting/freezing at one or both of the boundaries of the solid layer is likely to affect the pattern of convection in the layer. We study here the onset of thermal convection in a viscous spherical shell with dynamically induced melting/freezing at either or both of its boundaries. It is shown that the behavior of each interface—permeable or impermeable—depends on the value of a dimensionless number P (one for each boundary), which is the ratio of a melting/freezing timescale over a viscous relaxation timescale. A small value of P corresponds to permeable boundary conditions, while a large value of P corresponds to impermeable boundary conditions. Linear stability analysis predicts a significant effect of semi-permeable boundaries when the number P characterizing either of the boundary is small enough: allowing for melting/freezing at either of the boundary allows the emergence of larger scale convective modes. The effect is particularly drastic when the outer boundary is permeable, since the degree 1 mode remains the most unstable even in the case of thin spherical shells. In the case of a spherical shell with permeable inner and outer boundaries, the most unstable mode consists in a global translation of the solid shell, with no deformation. In the limit of a full sphere with permeable outer boundary, this corresponds to the “convective translation” mode recently proposed for Earth’s inner core. As another example of possible application, we discuss the case of thermal convection in Enceladus’ ice shell assuming the presence of a global subsurface ocean, and found that melting/freezing could have an important effect on the pattern of convection in the ice shell.

Pre-kinking analysis of a constant moving crack in a magnetoelectroelastic strip under in-plane loading

A constant moving crack in a magnetoelectroelastic strip under in-plane mechanical, electric and magnetic loading is considered for impermeable and permeable crack surface boundary conditions, respectively. Fourier transform is applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are further transformed into Fredholm integral equations of the second kind. Steady state asymptotic fields near the crack tip are obtained and the corresponding field intensity factors are defined. The exact solution for a cracked infinite magnetoelectroelastic material can be recovered if the width of the strip tends to infinity. The crack speed and the geometric size of the strip affect the singular field distribution around the crack tip and the influences of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomenon is investigated by applying the maximum hoop stress intensity factor criterion. • Constant moving crack in a magnetoelectroelastic strip is considered. • Asymptotic fields near the crack tip are obtained in an explicit form. • The geometric size of the strip affects the singular crack tip fields. • The crack kinking phenomenon is investigated.

Electromagnetic Casimir effect on the boundary of a D -dimensional cavity and the high temperature asymptotics

We consider the finite temperature Casimir stress acting on the boundary of a D ⩾ 3 dimensional cavity due to the vacuum fluctuations of electromagnetic fields. Both perfectly conducting and infinitely permeable boundary conditions are considered, and it is proved that they correspond mathematically to the relative and absolute boundary conditions. The divergence terms of the Casimir free energy are related to the heat kernel coefficients of the Laplace operator. It is shown that the Casimir stress is free of divergence if and only if D is exactly three. The high temperature asymptotics of the regularized Casimir free energy are also found to depend on the heat kernel coefficients. When D > 3, renormalization is required to remove terms of order higher than or equal to T2.

Electro-elastic fields of piezoelectric materials with an elliptic hole under uniform internal shearing forces

The existing investigations on piezoelectric materials containing an elliptic hole mainly focus on remote uniform tensile loads. In order to have a better understanding of the fracture behavior of piezoelectric materials under different loading conditions, theoretical and numerical solutions are presented for an elliptic hole in transversely isotropic piezoelectric materials subjected to uniform internal shearing forces based on the complex potential approach. By solving ten variable linear equations, the analytical solutions inside and outside the hole satisfying the permeable electric boundary conditions are obtained. Taking PZT-4 ceramic into consideration, numerical results of electro-elastic fields along the edge of the hole and axes, and the electric displacements in the hole are presented. Comparison with stresses in transverse isotropic elastic materials shows that the hoop stress at the ends of major axis in two kinds of material equals zero for the various ratios of major to minor axis lengths; If the ratio is greater than 1, the hoop stress in piezoelectric materials is smaller than that in elastic materials, and if the ratio is smaller than 1, the hoop stress in piezoelectric materials is greater than that in elastic materials; When it is a circle hole, the shearing stress in two materials along axes is the same. The distribution of electric displacement components shows that the vertical electric displacement in the hole and along axes in the material is always zero though under the permeable electric boundary condition; The horizontal and vertical electric displacement components along the edge of the hole are symmetrical and antisymmetrical about horizontal axis, respectively. The stress and electric displacement distribution tends to zero at distances far from the elliptical hole, which conforms to the conclusion usually made on the basis of Saint-Venant’s principle. Unlike the existing work, uniform shearing forces acting on the edge of the hole, and the distribution of electro-elastic fields inside and outside the elliptic hole are considered.

On Boundary Permeation Through the Walls of Blocked Cerebral Capillaries: A Non-Linear Model

In this paper, we present a mathematical model of a blockage in a cerebral capillary where the first stage of cerebrospinal fluid formation takes place. The cerebral capillary is modelled as a slightly bent cylindrical tube, and the plasma in the tube as a Newtonian fluid. The principles and methods of the “effective area” for studying the penetration of fluid into permeable walls were used to investigate the filtrate’s momentum flux. The non-linear Stokes constitutive equation for incompressible fluid is supplemented by a permeable boundary condition, to prove the existence of a unique weak solution, which depends on the viscosity and the nature of the curvature of the specialized capillary.

Implication of the lopsided growth for the viscosity of Earth's inner core

A particular mode of convection, called translation, has recently been put forward as an important mode of inner core dynamics because this mechanism is able to explain the observed east–west asymmetry of P-wave velocity and attenuation (Monnereau et al., 2010). Translation is a particular solution to Navier–Stokes equation with permeable boundary conditions, but depending on the viscosity of the solid core, modes with higher spherical harmonics degree can develop. At low viscosity, these modes can be dominant and dissipate the degree l=1 of thermal heterogeneities. Hence, a viscosity threshold may be expected below which translation cannot take place, thereby constraining the viscosity of iron at inner core conditions.Using a hybrid finite-difference spherical harmonics Navier–Stokes solver, we investigate here the interplay between translation and convection in a 3D spherical model with permeable boundary conditions. Our numerical simulations show the dominance of pure translation for viscosities of the inner core higher than 4×1018Pas. Translation is almost completely hampered by convective motions for viscosities lower than 1017Pas and the phase change becomes an almost impermeable boundary. Between these values, a well developed circulation at the harmonic degree l=1 persists, but composed of localized cold downwellings, a passive upward flow taking place on the opposite side (the melting side). Such a convective structure remains compatible with the seismic asymmetry.

Pre-curving analysis of an opening crack in a magnetoelectroelastic strip under in-plane impact loadings

An opening crack in a magnetoelectroelastic strip under in-plane mechanical, electric, and magnetic impact loadings is considered for magneto-electrically impermeable and permeable crack surface boundary conditions. Laplace and Fourier transforms are applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are expressed in terms of Fredholm integral equations of the second kind. The asymptotic fields near the crack tip are obtained in explicit form and the corresponding field intensity factors are defined. The crack curving phenomena are investigated by applying the criterion of maximum hoop stress intensity factors. Numerical results show that the hoop stress intensity factors are influenced by the electric and magnetic loadings and the geometric size ratios.

Transient thermal elastic fracture of a piezoelectric cylinder specimen

A finite piezoelectric cylinder with an embedded penny-shaped crack is investigated for a thermal shock load on the outer surface of the cylinder. The theory of linear electro-elasticity is applied to solve the transient temperature field and the associated thermal stresses and electrical displacements without crack. These thermal stresses and electrical displacements are added to the surfaces of the crack to form an electromechanical coupling and mixed mode boundary-value problem. The electrically permeable crack face boundary condition assumption is used, and the thermal stress intensity factor and electrical displacement intensity factor at the crack border are evaluated. The thermal shock resistance of the piezoelectric cylinder is evaluated for the analysis of piezoelectric material failure in practical engineering applications.

Intervallic Coiflets for Numerical Calculation of Dynamic Stress Intensity Factor

There are lots of practical problems which are related to the solution of Fredholm integral equations of the second kind. The present work proposes intervallic Coiflets for solving the equations. Illustrative problem involving dynamic stress and electric fields of a cracked piezoelectric excited by anti-plane shear wave is addressed. Permeable boundary condition has been used to obtain a pair of dual integral equations of the symmetric and antisymmetric parts which can be reduced to the solutions of two Fredholm integral equations of the second kind. The dynamic stress intensity factor is expressed in terms of the right-end values of two unknown functions in Fredholm integral equations. The two unknown functions are solved by intervallic Coiflets which have less the endpoints error. And intervallic Coiflets have low calculation cost and high accuracy due to the wavelet expansion coefficients are exactly obtained without calculating the wavelet integrations. The calculation results agree well with the existing method, which show the high accuracy of the estimation and demonstrate validity and applicability of the method.

Exact solutions for piezoelectric materials with an elliptic hole or a crack under uniform internal pressure

The existing investigations on piezoelectric materials containing an elliptic hole or a crack mainly focus on remote uniform tensile loads. In order to have a better understanding for the fracture behavior of piezoelectric materials under different loading conditions, theoretical and numerical solutions are presented for an elliptic hole or a crack in transversely isotropic piezoelectric materials subjected to uniform internal pressure and remote electro-mechanical loads. On the basis of the complex variable approach, analytical solutions of the elastic and electric fields inside and outside the defect are derived by satisfying permeable electric boundary condition at the surface of the elliptical hole. As an example of PZT-4 ceramics, numerical results of electro-elastic fields inside and outside the crack under various electric boundary conditions and electro-mechanical loads are given, and graphs of the electro-elastic fields in the vicinity of the crack tip are presented. The non-singular term is compared to the asymptotic one in the figures. It is shown that the dielectric constant of the air in the crack has no effect on the electric displacement component perpendicular to the crack, and the stresses in the piezoelectric material depend on the material properties and the mechanical loads on the crack surface and at infinity, but not on the electric loads at infinity. The figures obtained are strikingly similar to the available results. Unlike the existing work, the existence of electric fields inside an elliptic hole or a crack is considered, and the piezoelectric solid is subjected to complicated electro-mechanical loads.

Analysis of an interface crack between two dissimilar piezoelectric solids

A meshless method based on the local Petrov–Galerkin approach is proposed, to solve the interface crack problem between two dissimilar piezoelectric solids. Permeable and impermeable electrical boundary conditions are considered on the crack-faces. Quasi-static governing equations for the electrical fields and elastodynamic equations with an inertial term for the 2-D mechanical fields are considered. Nodal points are spread on the analyzed domain, and each node is surrounded by a small circle for simplicity. A Heaviside step function as the test functions is applied in the weak-form on the local subdomain. The local integral equations are derived. The spatial variations of the displacements and electric potential are approximated by the Moving Least-Squares (MLS) scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. The system of the ordinary differential equations of the second order resulting from the equations of motion is solved by the Houbolt finite-difference scheme as a time-stepping method.

Some new aspects of boundary conditions at cracks in piezoelectrics

Fracture mechanical investigations of piezoelectric materials as components of smart structures have become popular in the last 30 years. In the early years of research, boundary conditions at crack faces have been adopted from pure mechanical systems under the assumption that boundaries were traction free. From the electrostatic point of view, cracks have been assumed to be either free of charge or fully permeable. Later, limitedly permeable crack boundary conditions have become popular among the community, nevertheless still assuming traction-free crack faces. Recently, the theoretical framework has been extended to include electrostatically induced mechanical tractions in crack models yielding a significant crack closure effect. However, these models are still simple, neglecting, e.g., the piezoelectric field coupling. In this work, we present an extended model for crack surface tractions yielding some interesting effects. In particular, the orientation of the electrical field with respect to the poling axis becomes important. Furthermore, applying a collinear stress parallel to the crack faces influences the Mode-I stress intensity factor and a Mode-II shear loading couples to the Mode-I SIF.

Frequency Response of a Fractional Derivative Viscoelastic Type Lined Tunnel with Partial Sealing

Based on Biot saturated soil theory, steady state dynamic response of the system is studied in the frequency domain when the inner boundary of a fractional derivative viscoelastic type circular lined tunnel is under the axisymmetric load and fluid pressure respectively. On the basis of introducing a partial permeable boundary condition, the solutions of stress, displacement and pore pressure of the lining and saturated soil are obtained by the inner boundary of the lining and continuity conditions of the interface, besides, the stress-displacement constitutive behavior of the lining is described by fractional derivative viscoelastic constitutive model. The influence of physical parameter on the system response is investigated. It is shown that the order of fractional derivative model has a great influence on the system dynamic response, and it depends on material parameter of the lining when the inner boundary of lining is subjected to axisymmetric load. The permeability parameter of lining has significant effects on system response induced by fluid pressure.

Casimir interaction of concentric spheres at finite temperature

We consider the finite temperature Casimir effect between two concentric spheres due to the vacuum fluctuations of the electromagnetic field in the ($D+1$)-dimensional Minkowski spacetime. Different combinations of perfectly conducting and infinitely permeable boundary conditions are imposed on the spheres. The asymptotic expansions of the Casimir free energies when the dimensionless parameter $\ensuremath{\epsilon}$, the ratio of the distance between the spheres to the radius of the smaller sphere, is small are derived in both the high temperature region and the low temperature region. It is shown that the leading terms agree with those obtained using the proximity force approximation, which are of order $T{\ensuremath{\epsilon}}^{1\ensuremath{-}D}$ in the high temperature region and of order ${\ensuremath{\epsilon}}^{\ensuremath{-}D}$ in the low temperature region. Some universal structures are observed in the next two correction terms. The leading terms of the thermal corrections in the low temperature region are also derived. They are found to be finite when $\ensuremath{\epsilon}\ensuremath{\rightarrow}{0}^{+}$ and are of order ${T}^{D+1}$.

Electromagnetic Casimir piston in higher-dimensional spacetimes

We consider the Casimir effect of the electromagnetic field in a higher-dimensional spacetime of the form $M\ifmmode\times\else\texttimes\fi{}\mathcal{N}$, where $M$ is the four-dimensional Minkowski spacetime and $\mathcal{N}$ is an $n$-dimensional compact manifold. The Casimir force acting on a planar piston that can move freely inside a closed cylinder is investigated. Different combinations of perfectly conducting boundary conditions and infinitely permeable boundary conditions are imposed on the cylinder and the piston. It is verified that if the piston and the cylinder have the same boundary conditions, the piston is always going to be pulled towards the closer end of the cylinder. However, if the piston and the cylinder have different boundary conditions, the piston is always going to be pushed to the middle of the cylinder. By taking the limit where one end of the cylinder tends to infinity, one obtains the Casimir force acting between two parallel plates inside an infinitely long cylinder. The asymptotic behavior of this Casimir force in the high temperature regime and the low temperature regime are investigated for the case where the cross section of the cylinder in $M$ is large. It is found that if the separation between the plates is much smaller than the size of $\mathcal{N}$, the leading term of the Casimir force is the same as the Casimir force on a pair of large parallel plates in the ($4+n$)-dimensional Minkowski spacetime. However, if the size of $\mathcal{N}$ is much smaller than the separation between the plates, the leading term of the Casimir force is $1+h/2$ times the Casimir force on a pair of large parallel plates in the four-dimensional Minkowski spacetime, where $h$ is the first Betti number of $\mathcal{N}$. In the limit the manifold $\mathcal{N}$ vanishes, one does not obtain the Casimir force in the four-dimensional Minkowski spacetime if $h$ is nonzero. Therefore the data obtained from Casimir experiments suggest that the first Betti number of the extra dimensions should be zero.

Diffraction of antiplane shear waves by a finite crack in a piezoelectric material

AbstractThe problem of diffraction of antiplane shear waves by a crack of finite length under the permeable electric boundary conditions is investigated analytically. Using Fourier transforms the mixed boundary value problem is reduced to two pairs of dual integral equations. These dual integral equations are further reduced to a pair of Fredholm integral equations of the second kind. The iterative solutions of the Fredholm integral equations have been obtained for small values of the wave number. And analytical expressions for the dynamic stress intensity factor and electric displacement intensity factor are obtained. Finally the numerical results for dynamic stress intensity factor are displayed graphically.

Analysis of multiple axisymmetric annular cracks in a piezoelectric medium

An axisymmetric annular electric dislocation is defined. The solution of axisymmetric electric and Volterra climb and glide dislocations in an infinite transversely isotropic piezoelectric domain is obtained by means of Hankel transforms. The distributed dislocation technique is used to construct integral equations for a system of co-axial annular cracks with so-called permeable and impermeable electric boundary conditions on the crack faces where the domain is under axisymmetric electromechanical loading. These equations are solved numerically to obtain dislocation densities on the crack surfaces. The dislocation densities are employed to determine field intensity factors for a system of interacting annular and/or penny-shaped cracks.

Comparison of transmissive permeable and reflective impermeable interfaces between electrode and electrolyte

This article described the basic concepts of the permeable boundary (PB) and impermeable boundary (IPB) conditions between electrode and electrolyte that are essential in studying diffusion and migration of ions through the electrode for electrochemical devices. The transmission line models (TLMs) were introduced to explain the boundary conditions at the electrode/electrolyte interfaces. The impedance data were simulated based upon the TLMs for PB and IPB conditions, giving attention to the different behaviors of low-frequency impedance. In addition, this article explained that the electrodes used for fuel cells and batteries can be classified according to the PB and IPB conditions.

Identification of aquifer-recharge zones and sources in an urban development area (Delhi, India), by correlating isotopic tracers with hydrological features

Recharge zones and sources in an urban setup (NCT of Delhi, India) were identified using environmental isotopes (2H, 3H, 18O); they were then correlated with hydrogeological conditions. The isotopic results showed that groundwater is being recharged by surface water during the dry season, while recharge associated with local precipitation becomes prominent during the monsoon. The effect of source-water evaporation and altitude on the isotopic characteristics of groundwater was clearly noted. A gradual increase in groundwater age, i.e. decrease in tritium content, while moving away from the river/canals/drains, suggests a degree of mixing of old-aged groundwater with relatively young recharging water. Further, to substantiate the findings of isotopic investigations, surface recharge conditions were differentiated into potential pervious (recharge prone) and impervious (recharge resistant) surfaces through mapping of potential recharge areas based on soil type and water-table depth, to depict a three-dimensional illustration of hydrogeologically mediated recharge zones of the area. The hydrogeological evidence thus obtained about the spatial distribution of permeable zones, slope and boundary conditions, aptly substantiates the isotopic findings. The study seeks its impact by correlation of the isotopic findings with the regional groundwater flow regime which has been altered by the urban development.

Electrostatic Tractions at Crack Faces Taking into Account Full Piezoelectric Field Coupling

Cracks in piezoelectric solids have been subject to fracture mechanical investigations for more than 30 years. In the early years of research, boundary conditions at crack faces have been adopted from pure mechanical systems assuming traction free boundaries. From the electrostatic point of view, cracks have been assumed to be either free of surface charge or fully permeable. Later, limited permeable crack boundary conditions have become popular among the community, however still assuming traction-free crack faces. Recently, the theoretical framework has been extended including electrostatically induced mechanical tractions into crack models yielding a significant crack closure effect. However, these models are still simple neglecting e.g. piezoelectric field coupling. As one consequence, the tractions do not depend on the direction of the electric field with respect to the direction of material polarization. Here, an extended model is presented, entirely accounting for the piezoelectric coupling effect. In this case, the crack closure effect depends on the electric field direction. Additionally, the model yields a new shear effect.