HOMOGENIZATION—BASED TOPOLOGY DESIGN FOR PURE TORSION OF COMPOSITE SHAFTS

In conjunction with the homogenization theory and the finite element method, the mathematical models for designing the corss-section of composite shafts by maximizing the torsion rigidity are developed in this paper. To obtain the extremal torsion rigidity, both the cross-section of the macro scale shaft and the representative microstructure of the composite material are optimized using the new models. The micro scale computational model addresses the problem of finding the periodic microstructures with extreme shear moduli. The optimal microstructure obtained with the new model and the homogenization method can be used to improve and optimize natural or artificial materials. In order to be more practical for engineering applications, cellular materials rather than ranked materials are used in the optimal process in the existence of optimal bounds for the elastic properties. Moreover, the macro scale model is proposed to optimize the cross-section of the torsional shaft based on the tailared composites. The validating optimal results show that the models are very effective in obtaining composites with extreme elastic properties, and the cross-section of the composite shaft with the extremal torsion rigidity. The project supported by the National Natural Science Foundation of China (10172078 and 10102018)     

Micro–macro analysis of shape memory alloy composites

The aim of the paper is to develop a micro–macro approach for the analysis of the mechanical behavior of composites obtained embedding long fibers of Shape Memory Alloys (SMA) into an elastic matrix. In order to determine the overall constitutive response of the SMA composites, two homogenization techniques are proposed: one is based on the self-consistent method while the other on the analysis of a periodic composite. The overall response of the SMA composites is strongly influenced by the pseudo-elastic and shape memory effects occurring in the SMA material. In particular, it is assumed that the phase transformations in the SMA are governed by the wire temperature and by the average stress tensor acting in the fiber. A possible prestrain of the fibers is taken into account in the model.Numerical applications are developed in order to analyze the thermo-mechanical behavior of the SMA composite. The results obtained by the proposed procedures are compared with the ones determined through a micromechanical analysis of a periodic composite performed using suitable finite elements.Then, in order to study the macromechanical response of structural elements made of SMA composites, a three-dimensional finite element is developed implementing at each Gauss point the overall constitutive laws of the SMA composite obtained by the proposed homogenization procedures. Some numerical applications are developed in order to assess the efficiency of the proposed micro–macro model.     

Flow of electrolytes in a porous medium

A two-scale model of ion transfer in a porous medium is obtained for one-dimensional horizontal flows under the action of a pressure gradient and an external electric field by the method of homogenization. Steady equations of electroosmotic flows in flat horizontal nano-sized slits separated by thin dielectric partitions are averaged over a small-scale variable. The resultant macroequations include Poisson’s equation for the vertical component of the electric field and Onsager’s relations between flows and forces. The total horizontal flow rate of the fluid is found to depend linearly on the pressure gradient and external electric field, and the coefficients in this linear relation are calculated with the use of microequations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 162–173, July–August, 2008.     

A version of Hill＇s lemma for Cosserat continuum

On the basis of Hill＇s lemma for classical Cauchy continuum, a version of Hill＇s lemma for micro-macro homogenization modeling of heterogeneous Cosserat continuum is presented in the flame of average-field theory. The admissible boundary conditions required to prescribe on the representative volume element for the modeling are extracted and discussed to ensure the satisfaction of Hill-Mandel energy condition and the first-order average field theory.     

Poisson–Nernst–Planck Systems for Narrow Tubular-Like Membrane Channels

We study global asymptotic behavior of Poisson–Nernst–Planck (PNP) systems for flow of two ion species through a narrow tubular-like membrane channel. As the radius of the cross-section of the three-dimensional tubular-like membrane channel approaches zero, a one-dimensional limiting PNP system is derived. This one-dimensional limiting system differs from previously studied one-dimensional PNP systems in that it encodes the defining geometry of the three-dimensional membrane channel. To justify this limiting process, we show that the global attractors of the three-dimensional PNP systems are upper semi-continuous as the radius of the channel tends to zero.     

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.     

A Two-Scale Model for Coupled Electro-Chemo-Mechanical Phenomena and Onsager’s Reciprocity Relations in Expansive Clays: I Homogenization Analysis

The macroscopic model governing coupled electro-chemo-mechanical phenomena in expansive clays is revisited within a rigorous homogenization procedure applied to the microscopic governing equations which describe the local interaction between charged clay particles and a binary monovalent aqueous electrolyte solution. The up-scaling of the microscopic electro-hydro-dynamics leads to a two-scale approach wherein the macroscopic model appears governed by a fully coupled form of Onsager’s reciprocity relations, mass conservation equations and a modified Terzaghi’s effective stress principle. In addition, the two-scale approach provides microscopic representations for the effective coefficients which are exploited herein to obtain further insight in the constitutive behavior of the electrochemical parameters and the swelling pressure. Among other effects, we show that these microscopic closure relations are mainly dictated by the spatial variability of a microscale electric potential which satisfies a local version of the Poisson–Boltzmann problem in a periodic unit cell, The proposed framework allows to address various relevant still open issues regarding the constitutive behavior of swelling systems, Among them we give particular emphasis on the analysis of the influence of the fluctuation and distortion of the electrical double layer upon the magnitude of the electrochemical coefficients and the precise local conditions for the validity of the symmetry of Onsager’s relations.     

Numerical simulation of blood flow and interstitial fluid pressure in solid tumor microcirculation based on tumor-induced angiogenesis

A coupled intravascular–transvascular–interstitial fluid flow model is developed to study the distributions of blood flow and interstitial fluid pressure in solid tumor microcirculation based on a tumor-induced microvascular network. This is generated from a 2D nine-point discrete mathematical model of tumor angiogenesis and contains two parent vessels. Blood flow through the microvascular network and interstitial fluid flow in tumor tissues are performed by the extended Poiseuille’s law and Darcy’s law, respectively, transvascular flow is described by Starling’s law; effects of the vascular permeability and the interstitial hydraulic conductivity are also considered. The simulation results predict the heterogeneous blood supply, interstitial hypertension and low convection on the inside of the tumor, which are consistent with physiological observed facts. These results may provide beneficial information for anti-angiogenesis treatment of tumor and further clinical research. The project supported by the National Natural Science Foundation of China (10372026).     

Non-localness of Excess Potentials and Boundary Value Problems of Poisson–Nernst–Planck Systems for Ionic Flow: A Case Study

Poisson–Nernst–Planck (PNP) type systems are basic primitive models for ionic flow through ion channels. Important properties of ion channels, such as current-voltage relations, permeation and selectivity, can be extracted from solutions of boundary value problems (BVP) of PNP type models. Many issues of BVP of PNP type systems with local excess potentials (including particularly classical PNP systems that treat ions as point-charges) are extensively examined analytically and numerically. On the other hand, for PNP type systems with nonlocal excess potentials, even the issue of well-posedness of BVP is poorly understood. In fact, the formulation of correct boundary conditions seems to be overlooked, even though complications of ionic behavior near the boundaries (locations of applied electrodes) have been long experienced in experiments and simulations. PNP type systems with nonlocal excess potentials can be viewed as functional differential systems and, for many approximation models of nonlocal excess potentials, as differential equations with both delays and advances. Thus PNP type systems with nonlocal excess potentials have infinite degree of freedoms and BVP with the traditional “two-point-boundary-conditions” would be severely under determined. The mathematical theory for PNP with nonlocal excess potential would be significantly different from that for PNP with local excess potentials. Taking into considerations of experimental designs of ionic flow through ion channels and in a relatively simple setting, we present a form of natural “boundary conditions” so that the corresponding BVP of PNP type systems with nonlocal excess potentials are generally well-posed. This work, at an early stage toward a better understanding of related issues, provides some insights on interpretations of experimental designs of imposing boundary conditions and for correct formulations of numerical simulations, and hopefully, will stimulate further mathematical analysis on this important issue.     

Sliding friction across the scales: Thermomechanical interactions and dissipation partitioning

A homogenization framework is developed for determining the complete macroscopic thermomechanical sliding contact response of soft interfaces with microscopic roughness. To this end, a micro–macro mechanical dissipation equality is first established which enables defining a macroscopic frictional traction. The derivation allows both contacting bodies to be deformable, thereby extending the commonly adopted setting where one of the bodies is rigid. Moreover, it forms a basis for the second step, where a novel micro–macro thermal dissipation equality is established which enables defining partitioning coefficients that are associated with the frictional dissipation as it is perceived on the macroscale. Finally, a comparison of the temperature fields from the original heterogeneous thermomechanical contact problem and an idealized homogeneous one reveals an identification of the macroscopic temperature jump. The computational implementation of the framework is carried out within an incrementally two-phase micromechanical test which delivers a well-defined macroscopic response that is not influenced by purely algorithmic choices such as the duration of sliding. Two-dimensional numerical investigations on periodic and random samples from thermo-viscoelastic boundary layers with unilateral and bilateral roughness demonstrate the temperature–velocity–pressure dependence of the macroscopic contact response.     

Novel numerical implementation of asymptotic homogenization method for periodic plate structures

The present paper develops and implements finite element formulation for the asymptotic homogenization theory for periodic composite plate and shell structures, earlier developed in  and , and thus adopts this analytical method for the analysis of periodic inhomogeneous plates and shells with more complicated periodic microstructures. It provides a benchmark test platform for evaluating various methods such as representative volume approaches to calculate effective properties. Furthermore, the new numerical implementation (Cheng et al., 2013) of asymptotic homogenization method of 2D and 3D materials with periodic microstructure is shown to be directly applicable to predict effective properties of periodic plates without any complicated mathematical derivation. The new numerical implementation is based on the rigorous mathematical foundation of the asymptotic homogenization method, and also simplicity similar to the representative volume method. It can be applied easily using commercial software as a black box. Different kinds of elements and modeling techniques available in commercial software can be used to discretize the unit cell. Several numerical examples are given to demonstrate the validity of the proposed methods.     

On the Solution of a Boltzmann System for Gas Mixtures

We study the Boltzmann equation for a mixture of two gases in one space dimension with initial condition of one gas near vacuum and the other near a Maxwellian equilibrium state. A qualitative-quantitative mathematical analysis is developed to study this mass diffusion problem based on the Green’s function of the Boltzmann equation for the single species hard sphere collision model in Liu andYu (Commun Pure Appl Math 57:1543–1608, 2004). The cross-species resonance of the mass diffusion and the diffusion-sound wave is investigated. An exponentially sharp global solution is obtained.     

On the <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript> Theory of Hydrostatic Euler Equations

In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence and uniqueness of H ^s solutions under the local Rayleigh condition. This extends Brenier’s (Nonlinearity 12(3):495–512, 1999) existence result by removing an artificial condition and proving uniqueness. In addition, we prove weak–strong uniqueness, mathematical justification of the formal derivation and stability of the hydrostatic Euler equations. These results are based on weighted H ^s a priori estimates, which come from a new type of nonlinear cancellation between velocity and vorticity.     

Effective Hamiltonian and Homogenization of Measurable Eikonal Equations

We study the homogenization problem for a class of evolutive Hamilton-Jacobi equations with measurable dependence on the state variable. We use in our analysis an adaptation of the definition of viscosity solutions to the discontinous setting, obtained through replacement, in the test inequalities, of the punctual value of the Hamiltonian by measure–theoretic weak limits.The existence of periodic solutions to the corresponding cell problem cannot be achieved, under our assumptions, through an ergodic approximation. Instead our approach is based on the introduction of an intrinsic distance defined as the infimum of some line integrals on curves joining two given points. A crucial role for this is played by the notion of transversality between curves and sets of vanishing Lebesgue measure.The asymptotic analysis is finally performed and employs a modified versions of the Evans’ perturbed test-function argument, the most relevant new fact being the use of t-partial sup-convolution and t-partial convolutions of subsolution to compensate the lack of continuity.     

On solving the problem of an ideal incompressible fluid flow around large-aspect-ratio axisymmetric bodies

Based on Babenko’s fundamental mathematical ideas, principally new (unsaturated) algorithms are developed for the numerical solution of problems of a potential axisymmetric ideal fluid flow around bodies of revolution, in particular, an ellipsoid of revolution with an aspect ratio equal to 1000. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 56–67, September–October, 2006.     

A Two-Scale Model for Coupled Electro-Chemo-Mechanical Phenomena and Onsager’s Reciprocity Relations in Expansive Clays: II Computational Validation

In Part I Moyne and Murad [Transport in Porous Media 62, (2006), 333–380] a two-scale model of coupled electro-chemo-mechanical phenomena in swelling porous media was derived by a formal asymptotic homogenization analysis. The microscopic portrait of the model consists of a two-phase system composed of an electrolyte solution and colloidal clay particles. The movement of the liquid at the microscale is ruled by the modified Stokes problem; the advection, diffusion and electro-migration of monovalent ions Na⁺ and Cl⁻ are governed by the Nernst–Planck equations and the local electric potential distribution is dictated by the Poisson problem. The microscopic governing equations in the fluid domain are coupled with the elasticity problem for the clay particles through boundary conditions on the solid–fluid interface. The up-scaling procedure led to a macroscopic model based on Onsager’s reciprocity relations coupled with a modified form of Terzaghi’s effective stress principle including an additional swelling stress component. A notable consequence of the two-scale framework are the new closure problems derived for the macroscopic electro-chemo-mechanical parameters. Such local representation bridge the gap between the macroscopic Thermodynamics of Irreversible Processes and microscopic Electro-Hydrodynamics by establishing a direct correlation between the magnitude of the effective properties and the electrical double layer potential, whose local distribution is governed by a microscale Poisson–Boltzmann equation. The purpose of this paper is to validate computationally the two-scale model and to introduce new concepts inherent to the problem considering a particular form of microstructure wherein the clay fabric is composed of parallel particles of face-to-face contact. By discretizing the local Poisson–Boltzmann equation and solving numerically the closure problems, the constitutive behavior of the diffusion coefficients of cations and anions, chemico-osmotic and electro-osmotic conductivities in Darcy’s law, Onsager’s parameters, swelling pressure, electro-chemical compressibility, surface tension, primary/secondary electroviscous effects and the reflection coefficient are computed for a range particle distances and sat concentrations.     

Generalized solitonary and periodic solutions for nonlinear partial differential equations by the Exp-function method

In this paper, the Exp-function method with the aid of the symbolic computational system Maple is used to obtain the generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics, namely, (2+1)-dimensional Konopelchenko–Dubrovsky equations, the (3+1)-dimensional Jimbo–Miwa equation, the Kadomtsev–Petviashvili (KP) equation, and the (2+1)-dimensional sine-Gordon equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

Numerical simulation of wave flows caused by a shoreside landslide

A mathematical model is developed for formation and propagation of discontinuous waves caused by sliding of a shoreside landslide into water. The model is based on the equations of a two-layer “shallow liquid” with specially introduced “dry friction” in the low layer, which allows one to describe the joint motion of the landslide and water. An explicit difference scheme approximating these equations is constructed, and it is used to develop a numerical algorithm for simulating the motion of the free boundaries of both the landslide and water (in particular, the propagation of a water wave along a dry channel, incidence of the wave on the lakeside, and flow over obstacles). Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 109–117, July–August, 1999.