Yield criterion for porous media with spherical voids

The yield criterion of a porous material satisfying Gurson criterion conditions is studied here. We use the twofold limit analysis approach applied to a representative volume element of the porous media and recent optimization codes. Both upper and lower bounds are very close, and they give quasi-exact solutions. As a result, the Gurson approach is slightly improved and, for the first time, validated by a rigorous, full 3D static approach.     

Numerical limit analysis bounds for ductile porous media with oblate voids

The paper is devoted to a numerical limit analysis of a hollow spheroidal model with a von Mises solid matrix. To this purpose, existing kinematic and static 3D-FEM codes for the case of spherical cavities have been modified and improved to account for the model of a spheroidal cavity confocal with the external spheroidal boundary. The optimized conic programming formulations and the resulting codes appear to be very efficient. This framework is then applied to the derivation of numerical upper and lower anisotropic bounds in the case of an oblate void. The numerical results obtained from a series of tests are presented and allow to assess the accuracy of closed-form expressions of the macroscopic criteria proposed by [Gologanu et al., 1994] and [Gologanu et al., 1997] for porous media with oblate voids.     

Bounds for the non-local effective properties of random media

The paper deals with a random medium subjected to a static scalar field with inhomogeneous mean values. Then, effective linear material parameters show dispersion, i.e. they depend on the “wave vector” k of the mean field. The variational methods of P.H. Dederichs and R. Zeller (1973) are generalized to derive upper and lower bounds for scalar effective material parameters as functions of k. In the limit k → 0 (homogeneous mean fields), bounds of the Hashin-Shtrikman type are reproduced. For k → ∞, the bounds coincide with the exact result. In the general case, a two-point moment of the stochastic material parameter is involved. Especially, composites with cell structure and binary mixtures are considered. Detailed calculations are carried out for effective dielectricity, relating mean electric displacement to the mean electric field (which is mathematically equivalent to electrical and thermal conductivities and other scalar parameters), of a binary system composed of nearly spherical grains of equal size.     

Natural convection in porous media above a near horizontal uniform heat flux surface

The boundary layer in free convection above a uniformly heated semi-infinite flat plate, which is inclined at a small angle to the horizontal in porous media is discussed. For positive inclinations of the plate, series solutions, one valid near the leading edge and the other at large distances from it, are obtained. When the inclination is negative, a series solution valid near the leading edge is again obtained. A step-by-step numerical technique, based on a scheme by Keller, is used to complete the solution in the region where neither series is adequate. For the negative inclinations of the plate, the boundary layer separates and a region of reverse flow develops.Hier wird die Grenzschicht bei freier Konvektion in einem porösem Medium oberhalb einer gleichmäßig beheizten, halbunendlichen flachen Platte, die mit einem kleinen Winkel gegen die Horizontale geneigt ist, untersucht. Für positive Neigungen der Platte sind zwei Reihenlösungen, eine gültig für den Anlaufbereich und die andere für einen großen Abstand davon, emittelt worden. Eine gültige Reihenlösung für eine negative Neigung in der Nähe des Anlaufbereiches ist ebenfalls bestimmt worden.Es ist ein schrittweises numerisches Verfahren, das auf der Methode von Keller basiert, benützt worden, um das Ergebnis im Bereich, in welchem keine Reihenansätze existieren, zu vervollständigen. Für negative Neigungen der Platte zerteilt sich die Grenzschicht und es entwickelt sich ein Bereich von Rückströmung.     

Bounds for the non-local effective properties of random media—II

The paper deals with a random medium subjected to a static field with inhomogeneous mean values. Then, effective linear material parameters show dispersion, i.e. they depend on the ‘wave vector’ k of the mean field. Starting from a variational method previously developed by the authors, upper and lower bounds for k-dependent scalar effective parameters are derived in terms of two-point and three-point correlation functions of the stochastic material parameters. Taking into consideration the three-point correlation function gives a substantial improvement of the generalized Hashin-Shtrikman bounds obtained previously. In particular, composites with cell structure and arbitrary binary systems are considered. In order to illustrate the general results, numerical evaluations are carried out for effective permittivity of a binary cell material composed of nearly spherical grains of equal size.     

Variational principles and estimates for the rigidities of bodies with voids

Variational principles and estimates (including two-sided estimates) are obtained for the rigidities of bodies containing periodic systems of pores (voids). The problem is examined on the basis of the asymptotic averaging method. Siberian State Academy of Telecommunications and Informatics, Novosibirsk 630009. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 148–154, July–August, 1998.     

Plastic potential and yield function of porous materials with aligned and randomly oriented spheroidal voids

Based on an energy approach, the plastic potential and yield function of a porous material containing either aligned or randomly oriented spheroidal voids are developed at a given porosity and pore shape. The theory is applicable to both elastically compressible and incompressible matrix and, it is proved that, in the incompressible case, the theory with spherical and aligned spheroidal voids also coincides with Ponte Castaneda's bounds of the Hashin-Shtrikman and Willis types, respectively. Comparison is also made between the present theory and those of Gurson and Tvergaard, with a result giving strong overall support of this new development. For the influence of pore shape, the yield function and therefore the stress-strain curve of the isotropic porous material are found to be stiffest when the voids are spherical, and those associated with other pore shapes all fall below these values, the weakest one being caused by the disc-shaped voids. The transversely isotropic nature of the yield function and stress-strain curves of a porous material containing aligned pores are also demonstrated as a function of porosity and pore shape, and it is further substantiated with a comparison with an exact, local analysis when the void shape becomes cylindrical.     

Theory of nonequilibrium effects associated with the flow of nonuniform fluids in porous media

It is of interest to study nonequilibrium effects associated with the motion of nonuniform fluids in a porous medium because they may have a considerable significance both for an understanding of the physics of flow in porous medium as well as for applications, primarily in the exploitation of oil and gas deposits. The aim of the present paper is to make a theoretical analysis of some aspects of the nonequilibrium state associated with the length of the process of establishment of capillary equilibrium, and also to consider possible consequences of these aspects.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 52–57, May–June, 1980.     

Effect of a nonuniform distribution of voids on the plastic response of voided materials: a computational and statistical analysis

This study investigates the overall and local response of porous media composed of a perfectly plastic matrix weakened by stress-free voids. Attention is focused on the specific role played by porosity fluctuations inside a representative volume element. To this end, numerical simulations using the Fast Fourier Transform (FFT) are performed on different classes of microstructure corresponding to different spatial distributions of voids. Three types of microstructures are investigated: random microstructures with no void clustering, microstructures with a connected cluster of voids and microstructures with disconnected void clusters. These numerical simulations show that the porosity fluctuations can have a strong effect on the overall yield surface of porous materials. Random microstructures without clusters and microstructures with a connected cluster are the hardest and the softest configurations, respectively, whereas microstructures with disconnected clusters lead to intermediate responses. At a more local scale, the salient feature of the fields is the tendency for the strain fields to concentrate in specific bands. Finally, an image analysis tool is proposed for the statistical characterization of the porosity distribution. It relies on the distribution of the ‘distance function’, the width of which increases when clusters are present. An additional connectedness analysis allows us to discriminate between clustered microstructures.     

Free convection from a vertical plate with nonuniform surface heat flux and embedded in a porous medium

An analysis is presented for the calculation of heat transfer due to free convective flow along a vertical plate embedded in a porous medium with an arbitrarily varying surface heat flux. By applying the appropriate coordinate transformations and the Merk series, the governing energy equation is expressed as a set of ordinary differential equations. Numerical solutions are presented for these equations which represent universal functions and several computational examples are provided.     

Wave propagation at the boundary surface of elastic and initially stressed viscothermoelastic diffusion with voids media

The present investigation is concerned with the wave propagation at the boundary surface of elastic half-space and initially stressed viscothermoelastic diffusion with voids half-space. The longitudinal and transverse waves are incident obliquely at the plane interface between uniform elastic half-space and initially stressed viscothermoelastic diffusion with voids half-space. It is found that the amplitude ratios of various reflected and transmitted waves are functions of angle of incidence, frequency of incident wave and are influenced by the initial stress, diffusion, voids, elastic and viscoelastic properties of media. The expressions of amplitude ratios and energy ratios are obtained in closed form and computed numerically for a specific model. The variations of energy ratios with angle of incidence are shown graphically. The conservation of energy at the interface is verified.     

The stokes problem for a porous particle with radially nonuniform porosity

The flow past a nonuniform porous spherical particle immersed in a uniform steady-state stream is studied in the Stokes approximation. For a power-law radial dependence of the particle permeability coefficient, an analytical solution for the velocity and pressure fields outside and inside the particle is obtained. Volgograd, Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 179–184, January–February, 2000.     

Size- and shape-dependent effective conductivity of porous media with spheroidal gas-filled inclusions

Porous media containing gas-filled inclusions embedded in a solid phase constitute an important class of natural or artificial materials of both theoretical and practical interest. In these materials, thermal conductivity is one of the most important properties. In a variety of situations of practical interest, when the characteristic size of gas-filled inclusions is comparable with the mean free path of gas molecules and when the slip flow regime is considered, the behavior of gas near solid surfaces cannot be described by classical thermal conductivity equations. In fact, the boundary conditions at the solid surfaces must be modified by considering that the temperature and normal heat flux simultaneously suffer a discontinuity. The first purpose of the present work is to develop an efficient and accurate micromechanical model capable of estimating the effective conductivity of porous materials while taking into account the discontinuities of the temperature and normal heat flux across solid surfaces and the non-spherical form of gas-filled inclusions. The second purpose of the present work is to study the dependencies of the effective conductivity on the size and shape of gas-filled inclusions. By applying the micromechanical model based on the differential scheme and by using the solution results obtained for auxiliary dilute problem accounting for modified boundary conditions on surface solids, the closed-form expression for the effective conductivity is obtained. Numerical results are provided to illustrate the dependence of the effective conductivity on the size and shape of gas-filled inclusions in the case of randomly oriented inclusions.     

Bounds and estimates for the effect of strain gradients upon the effective plastic properties of an isotropic two-phase composite

Predictions are made for the size effect on strength of a random, isotropic two-phase composite. Each phase is treated as an isotropic, elastic-plastic solid, with a response described by a modified deformation theory version of the Fleck-Hutchinson strain gradient plasticity formulation (Fleck and Hutchinson, J. Mech. Phys. Solids 49 (2001) 2245). The essential feature of the new theory is that the plastic strain tensor is treated as a primary unknown on the same footing as the displacement. Minimum principles for the energy and for the complementary energy are stated for a composite, and these lead directly to elementary bounds analogous to those of Reuss and Voigt. For the case of a linear hardening solid, Hashin-Shtrikman bounds and self-consistent estimates are derived. A non-linear variational principle is constructed by generalising that of Ponte Castañeda (J. Mech. Phys. Solids 40 (1992) 1757). The minimum principle is used to derive an upper bound, a lower estimate and a self-consistent estimate for the overall plastic response of a statistically homogeneous and isotropic strain gradient composite. Sample numerical calculations are performed to explore the dependence of the macroscopic uniaxial response upon the size scale of the microstructure, and upon the relative volume fraction of the two phases.     

An experimental study of steam injection into a uniform water flow through porous media

An experimental study of steam injection into a porous media was carried out in a 2-dimensional plane porous channel. The steam was injected into a uniform downward water flow in a vertically aligned porous channel. The steam-water interface was carefully observed to understand the underlying physics. Two steam injection rate bounds were found for a given water flow rate and water subcooling. The upper bound is the steam flow rate at which the steam zone grows without limit and the lower bound is the steam flow rate at which a steam zone is just initiated. The bounds were determined experimentally for a porous channel with different permeabilities and thermal conductivities. For large particle size, chaotic oscillation of steam water interface was observed. The oscillation is believed to enhance heat and momentum transfer mechanisms. The steam zone size and shape were measured to evaluate heat transfer characteristics. The average Nusselt number is presented in terms of steam and water Reynolds numbers and the Stefan number.     

Bounds for the effective properties of heterogeneous plates

This paper presents new bounds for heterogeneous plates which are similar to the well-known Hashin–Shtrikman bounds, but take into account plate boundary conditions. The Hashin–Shtrikman variational principle is used with a self-adjoint Green-operator with traction-free boundary conditions proposed by the authors. This variational formulation enables to derive lower and upper bounds for the effective in-plane and out-of-plane elastic properties of the plate. Two applications of the general theory are considered: first, in-plane invariant polarization fields are used to recover the “first-order” bounds proposed by Kolpakov [Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneous plate. J. Meth. Phys. Solids 47, 2075–2092] for general heterogeneous plates; next, “second-order bounds” for n-phase plates whose constituents are statistically homogeneous in the in-plane directions are obtained. The results related to a two-phase material made of elastic isotropic materials are shown. The “second-order” bounds for the plate elastic properties are compared with the plate properties of homogeneous plates made of materials having an elasticity tensor computed from “second-order” Hashin–Shtrikman bounds in an infinite domain.     

Similarity properties with the flow of supersonic uniform and nonuniform flows of gas around a sphere

The article proposes an approximation formula for the velocity gradient at the critical point for ellipsoids of revolution, suitable for both an equilibrium and a perfect gas. It is shown that an approximate solution to the problem of the flow of a stream arriving from a supersonic source around a sphere can be obtained, having the solution of the problem of uniform flow around a sphere.     

Bounds and self-consistent estimates for the overall properties of anisotropic composites

Bounds of Hashin-Shtrikman type and self-consistent estimates for the overall properties of composites, which may be anisotropic, are developed. Bodies containing aligned ellipsoidal inclusions are considered particularly, generalizing previously known results. The overall thermal conductivity of a body containing aligned spheroidal inclusions is discussed as an example including, as limiting cases, bodies containing highly-conducting aligned needles and bodies containing aligned pennyshaped cracks.