Stability of frictional slipping at an anisotropic/isotropic interface

The stability of steady, quasi-static slip at a planar interface between an anisotropic elastic solid and an isotropic elastic solid is studied. The paper begins with an analysis of anti-plane sliding at an orthotropic/isotropic interface. Friction at the interface is assumed to follow a rate- and state-dependent law. The stability to spatial perturbations of the form exp(ikx₁), where k is the wavenumber and x₁ is the coordinate along which the interface is studied. An expression is derived for the critical wavenumber ∣k∣_cr above which there is stability. In-plane sliding at an anisotropic/isotropic interface is subsequently studied. In this case, slip couples with normal stress changes and a constitutive law for dynamic normal stress changes is adopted. Again a formula for ∣k∣_cr is derived and the results are specialized to the case of an orthotropic/isotropic interface. Numerical plots of the dependence of ∣k∣_cr on the orientation of the orthotropic solid, as well as on the material parameters are provided.     

Correcting Anisotropic Gradient Transport of k

The gradient transport model for k is extended to classes of turbulent flows for which the gradient transport hypothesis is relevant but the anisotropy of the Reynolds stress, to which the eddy diffusivity is proportional, is large and variable. In highly anisotropic turbulence the standard isotropic model used in engineering practice is fundamentally wrong and the conventional anisotropic approximation inadequate. The work is motivated by the important observations that the eddy diffusivity coefficient for a standard gradient transport model for various transported quantities is a factor of 3–10 times larger in highly anisotropic turbulence than that used in standard engineering models. While the conventional anisotropic eddy diffusivity approximation appears adequate for material conserved scalars it is inadequate for k. The problem is solved by addressing the anisotropy of the turbulent transport of k at the level of the underlying third order tensor. It is shown that, unlike the traditional transport models for k, the orientation of the anisotropy with respect to the direction of the gradient plays a crucial role not accounted for in conventional models used in engineering calculations. The new anisotropic eddy diffusivity tensor is quadratic in the anisotropy (the traditional model is linear in the anisotropy). It is shown that the new more rigorous anisotropic eddy diffusivity varies 300% more than the standard model comparing the isotropic limit to the value for the two-dimensional limit. The two-dimensional limit is important in strongly stably stratified flows, in pressure gradient or shock driven flows and in rotating flows. Using the simple shear and the homogeneous non-equilibrium Rayleigh Taylor turbulence the new anisotropic diffusivity tensor is validated in inhomogeneous Rayleigh Taylor turbulence at early and late times.     

An improved anisotropy-resolving subgrid-scale model for flows in laminar–turbulent transition region

Some types of mixed subgrid-scale (SGS) models combining an isotropic eddy-viscosity model and a scale-similarity model can be used to effectively improve the accuracy of large eddy simulation (LES) in predicting wall turbulence. Abe (2013) has recently proposed a stabilized mixed model that maintains its computational stability through a unique procedure that prevents the energy transfer between the grid-scale (GS) and SGS components induced by the scale-similarity term. At the same time, since this model can successfully predict the anisotropy of the SGS stress, the predictive performance, particularly at coarse grid resolutions, is remarkably improved in comparison with other mixed models. However, since the stabilized anisotropy-resolving SGS model includes a transport equation of the SGS turbulence energy, k_SGS, containing a production term proportional to the square root of k_SGS, its applicability to flows with both laminar and turbulent regions is not so high. This is because such a production term causes k_SGS to self-reproduce. Consequently, the laminar–turbulent transition region predicted by this model depends on the inflow or initial condition of k_SGS. To resolve these issues, in the present study, the mixed-timescale (MTS) SGS model proposed by Inagaki et al. (2005) is introduced into the stabilized mixed model as the isotropic eddy-viscosity part and the production term in the k_SGS transport equation. In the MTS model, the SGS turbulence energy, k_es, estimated by filtering the instantaneous flow field is used. Since the k_es approaches zero by itself in the laminar flow region, the self-reproduction property brought about by using the conventional k_SGS transport equation model is eliminated in this modified model. Therefore, this modification is expected to enhance the applicability of the model to flows with both laminar and turbulent regions. The model performance is tested in plane channel flows with different Reynolds numbers and in a backward-facing step flow. The results demonstrate that the proposed model successfully predicts a parabolic velocity profile under laminar flow conditions and reduces the dependence on the grid resolution to the same degree as the unmodified model by Abe (2013) for turbulent flow conditions. Moreover, it is shown that the present model is effective at transitional Reynolds numbers. Furthermore, the present model successfully provides accurate results for the backward-facing step flow with various grid resolutions. Thus, the proposed model is considered to be a refined anisotropy-resolving SGS model applicable to laminar, transitional, and turbulent flows.     

Heat transfer through partitioned enclosures

A closed-form model for the computation of heat transfer rates through the rectangular-partitioned enclosures is investigated. The rectangular-partitioned enclosures may contain solid or gas with or without a constant and uniformly distributed heat generation. The conduction in the enclosures is considered as two-dimensional, whereas one-dimensional heat transfer through the fin-type partition is assumed. Dimensionless heat flux plots are parametrically studied by varying the aspect ratio (L/H) of the enclosure, the ratio of thermal conductivities of the enclosure to the fin-type partition (k _a /k _f ), the Biot number (β _a =h _a L/k _a ), and the reduced partition thickness (t ^*/L). It is demonstrated through an example problem that there is a large error in using one-dimensional analysis, particularly at lower values of k _a /k _f , and β _a .     

DEM investigation on the effect of sample preparation on the shear behavior of granular soil

The effect of initial fabric anisotropy produced by sample preparation on the shear behavior of granular soil is investigated by performing discrete element method (DEM) simulations of fourteen biaxial tests in drained conditions. Numerical test specimens are prepared by three means: gravitational deposition, multi-layer compression, and isotropic compression, such that different initial inherent soil fabrics are created. The DEM simulation results show that initial fabric anisotropy exerts a considerable effect on the shear behavior of granular soil, and that the peak stress ratio and peak dilatancy increase with an increase in the fabric index a_n that is estimated from the contact orientations. The stress–dilatancy relationship is found to be independent of the initial fabric anisotropy. The anisotropy related to the contact orientation and contact normal force accounts for the main contribution to the mobilized friction angle. Also, the occurrence of contractive shear response in an initial shearing stage is accompanied by the most intense particle rearrangement and microstructural reorganization, regardless of the sample preparation method. Furthermore, the uniqueness of the critical state line in e–log p′ and q–p′ plots is observed, suggesting that the influence of initial fabric anisotropy is erased at large shear strains.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.     

Sharp vorticity gradients in two-dimensional turbulence and the energy spectrum

Formation of sharp vorticity gradients in two-dimensional (2D) hydrodynamic turbulence and their influence on the turbulent spectra are considered. The analog of the vortex line representation as a transformation to the curvilinear system of coordinates moving together with the di-vorticity lines is developed and compressibility of this mapping appears as the main reason for the formation of the sharp vorticity gradients at high Reynolds numbers. In the case of strong anisotropy the sharp vorticity gradients can generate spectra which fall off as k ^?3 at large k, which appear to take the same form as the Kraichnan spectrum for the enstrophy cascade. For turbulence with weak anisotropy the k dependence of the spectrum due to the sharp gradients coincides with the Saffman spectrum: E(k) ~ k ^?4. Numerical investigations of decaying turbulence reveal exponential growth of di-vorticity with a spatial distributed along straight lines. Thus, indicating strong anisotropy and accordingly the spectrum is close to the k ^?3-spectrum.     

Eulerian–Lagrangian bridge for the energy and dissipation spectra in isotropic turbulence

We study, numerically and analytically, the relationship between the Eulerian spectrum of kinetic energy, E _E(k, t), in isotropic turbulence and the corresponding Lagrangian frequency energy spectrum, E _L(ω, t), for which we derive an evolution equation. Our DNS results show that not only E _L(ω, t) but also the Lagrangian frequency spectrum of the dissipation rate ${\varepsilon_{\rm L} (\omega, t)}$ has its maximum at low frequencies (about the turnover frequency of energy-containing eddies) and decays exponentially at large frequencies ω (about a half of the Kolmogorov microscale frequency) for both stationary and decaying isotropic turbulence. Our main analytical result is the derivation of equations that bridge the Eulerian and Lagrangian spectra and allow the determination of the Lagrangian spectrum, E _L (ω) for a given Eulerian spectrum, E _E (k), as well as the Lagrangian dissipation, ${\varepsilon_{\rm L}(\omega)}$ , for a given Eulerian counterpart, ${\varepsilon_{\rm E} (k)=2\nu k^2 E_{\rm E}(k)}$ . These equations were derived from the Navier–Stokes equations in the sweeping-free coordinate system (intermediate between the Eulerian and Lagrangian frameworks) which eliminates the effect of the kinematic sweeping of the small eddies by the larger eddies. We show that both analytical relationships between E _L (ω) and E _E (k) and between ${\varepsilon_{\rm L} (\omega)}$ and ${\varepsilon_{\rm E} (k)}$ are in very good quantitative agreement with our DNS results and explain how ${\varepsilon_{\rm L} (\omega, t)}$ has its maximum at low frequencies and decays exponentially at large frequencies.     

Extending the bounds of ‘steady’ RANS closures: Toward an instability-sensitive Reynolds stress model

The incapability of the conventional Unsteady RANS (Reynolds–Averaged Navier Stokes) models to adequately capture turbulence unsteadiness presents the prime motivation of the present work, which focuses on formulating an instability-sensitive, eddy-resolving turbulence model on the Second-Moment Closure level. The model scheme adopted, functioning as a ‘sub-scale’ model in the Unsteady RANS framework, represents a differential near-wall Reynolds stress model formulated in conjunction with the scale-supplying equation governing the homogeneous part of the inverse turbulent time scale ω_h (ω_h = ɛ_h/k). The latter equation was straightforwardly obtained from the model equation describing the dynamics of the homogeneous part of the total viscous dissipation rate ɛ, defined as ɛ_h = ɛ − 0.5ν∂²k/(∂x_j∂x_j) (Jakirlic and Hanjalic, 2002), by applying the derivation rules to the expression for ω_h. The model capability to account for vortex length and time scales variability was enabled through an additional term in the corresponding length-scale determining equation, providing a selective enhancement of its production, pertinent particularly to the highly unsteady separated shear layer region, modeled in terms of the von Karman length scale (comprising the second derivative of the velocity field) in line with the SAS (Scale-Adaptive Simulation) proposal (Menter and Egorov, 2010). The present model formulation, termed as SRANS model (Sensitized RANS), does not comprise any parameter depending explicitly on grid spacing. The predictive capabilities of the newly proposed length-scale determining model equation, solved in conjunction with Jakirlic and Hanjalic’s (2002) Reynolds stress model equation, are presently demonstrated by computing the flow configurations of increasing complexity featured by boundary layer separation from sharp-edged and continuous curved surfaces: backward-facing step flow, flow over a wall-mounted fence, flow over smoothly contoured periodically arranged hills and flow in a 3-D diffuser. The model performances are also assessed in capturing the natural decay of the homogeneous isotropic turbulence and the near-wall Reynolds stress anisotropy in a plane channel. In most cases considered the fluctuating velocity field was obtained starting from steady RANS results.     

Dynamic anti-plane sliding of dissimilar anisotropic linear elastic solids

The stability of steady, dynamic, anti-plane slipping at a planar interface between two dissimilar anisotropic linear elastic solids is studied. The solids are assumed to possess a plane of symmetry normal to the slip direction, so that in-plane displacements and normal stress changes on the slip plane do not occur. Friction at the interface is assumed to follow a rate and state-dependent law with velocity weakening behavior in the steady state. The stability to spatial perturbations of the form exp(ikx₁), where k is the wavenumber and x₁ is the coordinate along the interface is studied. The critical wavenumber magnitude, |k|_cr, above which there is stability and the corresponding phase velocity, c, of the neutrally stable mode are obtained from the stability analysis. Numerical plots showing the dependence of |k|_cr and c on the unperturbed sliding velocity, V_o, are provided for various bi-material combinations of practical interest.     

Wave propagation in an inhomogeneous transversely isotropic material obeying the generalized power law model

The analytical solutions for body-wave velocity in a continuously inhomogeneous transversely isotropic material, in which Young’s moduli (E, E′), shear modulus (G′), and material density (ρ) change according to the generalized power law model, (a+b z) ^c , are set down. The remaining elastic constants of transversely isotropic media, ν, and ν′ are assumed to be constants throughout the depth. The planes of transversely isotropy are selected to be parallel to the horizontal surface. The generalized Hooke’s law, strain-displacement relationships, and equilibrium equations are integrated to constitute the governing equations. In these equations, utilizing the displacement components as fundamental variables, the solutions of three quasi-wave velocities (V _SV , V _P ,?V _SH ) are generated for the present inhomogeneous transversely isotropic materials. The proposed solutions are compared with those of Daley and Hron (Bull Seismol Soc Am 67:661–675, (1977)), and Levin (Geophysics 44:918–936, (1979)) when the inhomogeneity parameter c?=?0. The agreement between the present results and previously published ones is excellent. In addition, the parametric study results reveal that the magnitudes of wave velocity are remarkably affected by (1) the inhomogeneity parameters (a, b, c); (2) the type and degree of material anisotropy (E/E′, ν/ν′, G/G′); (3) the phase angle (θ); and (4) the depth of the medium (z). Consequently, it is imperative to consider the effects of inhomogeneity when investigating wave propagation in transversely isotropic media.     

Piezoelastic study on singularities interacting with circular and straight interfaces

The piezoelastic investigation for a circular inclusion embedded in a sandwich has been carried out. Each medium of the composite is assumed to be transversely isotropic with hexagonal symmetry, which has an isotropic basal plane of x₁x₂-plane and a poling direction of x₃-axis. The electromechanical loadings considered in this paper include a point force and a point charge located in the middle layer of the sandwich. An efficient procedure is established by combining the analytical continuation method and alternating technique to derive the general forms of the piezoelastic fields in terms of the corresponding problem. Numerical results are provided for a number of particular examples to study the influence of material combinations, geometry, and loading condition on both the mechanical and electric response.     

One parameter model for creep in a whisker reinforced anisotropic rotating disc of Al–SiCw composite

Steady state creep in a rotating disc of anisotropic aluminum silicon carbide whisker composite has been studied in the present study. The creep behavior is described by Norton's power law. Stress and strain rate distributions for anisotropic discs have been calculated and compared with those obtained for isotropic disc. It is concluded that the radial strain rate which always remained compressive for the isotropic composite ($\alpha =1.0$) and anisotropic disc ($\alpha =1.3$), becomes tensile in the middle region of the disc when the anisotropy parameter $\alpha =0.7$. Also if α is reduced from 1.3 to 0.7 the variation of tensile strain rate in the tangential direction remains similar but the magnitude reduces by five orders of magnitude. The study revealed that anisotropy introduces significant change in the strain rates although its effect on the resulting stress distribution may be relatively small.     

Extrema of Young’s modulus for cubic and transversely isotropic solids

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young’s modulus E(n), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young’s modulus on direction n are given as well.     

On relations for general two-dimensional ideal plasticity problems under the full plasticity condition

Relations for two-dimensional ideal plasticity problems under the full plasticity condition are determined with material anisotropy, inhomogeneity, and compressibility properties taken into account. These properties are determined by the direction cosines of the principal stress, the coordinates of points in space, and the mean stress.For the yield strength we take a function of the form k = k(σ, n ₁, n ₂, n ₃, x, y, z). The desired relations are determined for the general plane ideal plasticity problem. The relations thus obtained are generalized to the cases of axisymmetric and spherical plasticity problems.     

Acoustics of two-component porous materials with anisotropic tortuosity

The paper is devoted to the analysis of monochromatic waves in two-component poroelastic materials described by a Biot-like model whose stress–strain relations are isotropic but the permeability is anisotropic. This anisotropy is induced by the anisotropy of the tortuosity which is given by a second order symmetric tensor. This is a new feature of the model while in earlier papers only isotropic permeabilities were considered. We show that this new model describes four modes of propagation. For our special choice of orientation of the direction of propagation these are two pseudo longitudinal modes P1 and P2, one pseudo transversal mode S2 and one transversal mode S1. The latter becomes also pseudo transversal in the general case of anisotropy. We analyze the speeds of propagation and the attenuation of these waves as well as the polarization properties in dependence on the orientation of the principal directions of the tortuosity. We indicate the practical importance of different shear (transversal) modes of propagation in a possible new nondestructive test of geophysical materials.     

On Pressurized Curvilinearly Orthotropic Circular Disk, Cylinder and Sphere Made of Radially Nonuniform Material

A unified analysis is presented for the elastic response of a pressurized cylindrically anisotropic hollow disk under assumed conditions of plane stress, or a hollow cylinder under plane strain conditions, and a spherically anisotropic hollow sphere, made of material which is nonuniform in the radial direction according to the power law relationship. The solution for a cylinder under generalized plane strain is also presented. Two parameters play a prominent role in the analysis: the material nonuniformity parameter m, and the parameter ?? which accounts for the combined effects of material anisotropy, represented by the specified parameters (??, ??, ??), and material nonuniformity, represented by the parameter m. The radial and circumferential stresses are the linear combinations of two power functions of the radial coordinate, whose exponents (n ₁ and n ₂) depend on the parameters m and ??. New light is added to the stress amplification and shielding under combined effects of curvilinear anisotropy and radial nonuniformity. Different loading combinations are considered, including the equal pressure at both boundaries, and the uniform pressure at the inner or the outer boundary. While the stress state for the equal pressure loading is uniform in the case of isotropic uniform material (m=0, ??=1), and for one particular radially nonuniform and anisotropic material, it is strongly nonuniform for a general anisotropic or nonuniform material. If the aspect ratio of the inner and outer radii decreases (small hole in a large disk/cylinder or sphere), the magnitude of the circumferential stress at the inner radius increases for n ₁>0 (stress amplification), and decreases for n ₁<0 (stress shielding). Both can be achieved by various combinations of the material parameters m, ??, ??, and ??. While the stress amplification in the case of a pressurized external boundary occurs readily, it occurs only exceptionally in the case of a pressurized internal boundary. The effects of material parameters on the displacement response are also analyzed. The approximate character of the plane stress solution of a pressurized thin disk is discussed and the results are compared with those obtained by numerical solution of the exact three-dimensional disk model.     

A modified Lin equation for the energy balance in isotropic turbulence

At sufficiently large Reynolds numbers, turbulence is expected to exhibit scale-invariance in an intermediate (“inertial”) range of wavenumbers, as shown by power law behavior of the energy spectrum and also by a constant rate of energy transfer through wavenumber. However, there is an apparent contradiction between the definition of the energy flux (i.e., the integral of the transfer spectrum) and the observed behavior of the transfer spectrum itself. This is because the transfer spectrum T(k) is invariably found to have a zero-crossing at a single point (at k = k_*), implying that the corresponding energy flux cannot have an extended plateau but must instead have a maximum value at k = k_*. This behavior was formulated as a paradox and resolved by the introduction of filtered/partitioned transfer spectra, which exploited the symmetries of the triadic interactions (J. Phys. A: Math. Theor., 2008). In this paper we consider the more general implications of that procedure for the spectral energy balance equation, also known as the Lin equation. It is argued that the resulting modified Lin equations (and their corresponding Navier–Stokes equations) offer a new starting point for both numerical and theoretical methods, which may lead to a better understanding of the underlying energy transfer processes in turbulence. In particular the filtered partitioned transfer spectra could provide a basis for a hybrid approach to the statistical closure problem, with the different spectra being tackled using different methods.     

Advective Transport in Heterogeneous Formations: The Impact of Spatial Anisotropy on the Breakthrough Curve

Water flow and solute transport take place in formations of spatially variable conductivity K. The logconductivity Y?= ln K is modeled as a random stationary space function, of normal univariate pdf (of mean In K _G and variance ${\sigma_{Y}^{2}}$ ) and of axisymmetric autocorrelation of integral scales I _h,I _v (anisotropy ratio f?=?I _v/I _h?<?1). The head gradient and the velocity are uniform in the mean, parallel to bedding, and of constant and given as J and U, respectively. Transport is ruled by advection, which typically overwhelms pore scale dispersion in the breakthrough curve (BTC) determination. In the present study we analyze the impact of anisotropy f on the BTC of a passive solute, which is related to the mass flux??? (t, x) at a control plane at x. While a considerable body of literature dealt with weakly heterogeneous formations ( ${\sigma _{Y}^{2} <1 }$ ), the present study addresses the case of ${\sigma _{Y}^{2} >1 }$ , which is of interest for many aquifers and is more difficult to solve either numerically or by approximations. We approach the three dimensional problem by modeling the structure as an ensemble of densely packed oblate spheroids of semi-major and semi-minor axis R and f R, respectively, and independent lognormal K, submerged in a matrix of uniform conductivity K _ef, the effective conductivity of the ensemble. The detailed numerical simulations of transport show that the BTC is insensitive to the value of the anisotropy ratio f, i.e.,??? (t, x) I _h/U depends only on ${\sigma _{Y}^{2}}$ (except for small differences in the tail). This important result implies that transport, as quantified by BTCs or spatial longitudinal mass distributions, can be modeled accurately by the much simpler solutions developed in the past for isotropic media, like e.g., the semi-analytical self-consistent approximation.