Collapse criteria of foam cells under various loading

Recently porous materials are widely used in civil and mechanical engineering. In particular, such porous materials as metal and polymer foams have applications in lightweight structures. From mechanics point of view foams can demonstrate unusual behavior such as strain localization related to foam cells buckling under certain loads. The aim of this work is the elaboration of the model of foam material taking into account the cell collapse. We consider the cell collapse initiation during the elastic instability and its further evolution under loading. The geometrical structure of foam is generated with the use of the Voronoi algorithm. Based on stochastic distributions of cells we create various geometrical models of foams. The influence of the cell volume, wall thicknesses and material properties of the foam material on critical loads is obtained. The calculations are performed with the use of Abaqus CAD/CAE system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of porous implants with non-uniform mechanical properties distribution based on CT images

In this study, the mechanical properties (elastic modulus, yield stress, and Poisson's ratio) of rhombic dodecahedron (RD) unit cell has been studied analytically and numerically. For the analytical study, two well-known beam theories, namely Euler Bernoulli and Timoshenko, have been implemented. For validating the analytical relationships, finite element model of unit cell with repetitive boundary condition has been created. Moreover, the experimental results of recent studies have been used for validation. The results showed that the presented analytical relationships for RD lattice structure have good agreement with numerical and experimental results in all the relative densities particularly in lower relative densities. Besides, the analytical relationships based on Timoshenko theory showed closer results with numerical/experimental data. The derived analytical relationships for RD as well as the data extracted from CT scan images of a femur bone, were combined and used to create a porous femur implant model. The stress and strain distributions of the porous femur model under typical static compressive load due to human weight as well as axial rigidity of the model in the same loading conditions have been obtained and compared with the experimental results from other studies. The stress and strain distributions of the porous femur implant model based on RD unit cells, as well as its axial rigidity, showed good agreement with the results obtained for human femur.     

Wave propagation in a fluid-saturated inhomogeneous porous medium

A closed system of constitutive equations for the dynamical and geometric quantities in a fluid- saturated inhomogeneous elastic porous medium is constructed within the framework of the three-dimensional theory of elasticity. The geometrical characteristics of the wave front and of the ray in a fluid-saturated inhomogeneous medium are obtained from the Fermi's principle.     

Dynamic stability of a porous cylindrical shell

This paper is devoted to a closed cylindrical shell made of a porous-cellular material. The mechanical properties vary continuously on the thickness of a shell. The mechanical model of porosity is as described as presented by Magnucki, Stasiewicz. A shell is simply supported on edges. On the ground of assumed displacement functions the deformation of shell is defined. The displacement field of any cross section and linear geometrical and physical relationships are assumed in cylindrical coordinate system. The components of deformation and stress state were found. Using the Hamilton's principle the system of differential equations of dynamic stability is obtained. The forms of unknown functions are assumed and the system of a differential equations is reduced to a simple ordinary equation of dynamic stability of shell (Mathieu's equation). The derived equation are used for solving a problem of dynamic stability of porous-cellular shell with intensity of load directed in generators of shell. The critical loads are derived for a family of porous shells. The unstable space of family porous shells is found. The influence a coefficient of porosity on the stability regions in Figures is presented. The results obtained for porous shell are compared to a homogeneous isotropic cylindrical shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical estimation of the thermal expansion of lamellar single domains in metal/ceramic composites

A porous ceramic preform, the pore structure of which is created via a freeze-casting technique, is infiltrated with a metal melt via sqeeze-casting. The microstructure of the obtained metal/ceramic composites has lamellar domains with geometrical characteristics which are dependent on the manufacturing parameters. The aim of our study is to predict the coefficients of thermal expansion of single domains of parallel Al₂O₃ platelets embedded in Al. For this purpose, a homogenization procedure was employed for microstructure based finite element and micromechanical modeling. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A concise quaternion geometry of rotations

This communication compiles propositions concerning the spherical geometry of rotations when represented by unit quaternions. The propositions are thought to establish a two‐way correspondence between geometrical objects in the space of real unit quaternions representing rotations and geometrical objects constituted by directions in the three‐dimensional space subjected to these rotations. In this way a purely geometrical proof of the spherical Ásgeirsson's mean value theorem and a geometrical interpretation of integrals related to the spherical Radon transform of a probability density functions of unit quaternions are accomplished. Copyright © 2004 John Wiley & Sons, Ltd.     

Computational homogenization and nonlinear effects in heterogeneous fluid saturated porous media

For the two-scale modelling of deforming fluid-saturated porous structure we apply the asymptotic homogenization approach to the fluid-structure interaction problem involving linear elastic porous skeleton and the Newtonian compressible fluid. The sensitivity analysis of effective coefficients depending on the geometrical configuration is used to introduce a weakly nonlinear formulation which enables to capture influences of the deformation on the material properties. The paper is devoted to the comparison of the linear and the weakly nonlinear models (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A representation of the equations of multicomponent multiphase seepage

A mathematical model of non-isothermal multicomponent flows in a porous medium is investigated. A general case is considered when the model can be used to describe processes with an arbitrary number of components and phases. A general form of the system of mixed-type equations describing the flow, which is similar to the Godunov form for hyperbolic systems is proposed. The equations obtained are applicable to flows with gas, liquid and solid phases. The thermodynamic properties of the medium are determined solely by a single multivalued function, by changing which one can obtain models of different flows in a porous medium. A clear geometrical interpretation of the solutions of the equations is proposed. An equation for the entropy is obtained, and it is shown that in order that the model should not contradict the second law of thermodynamics, it is necessary to take into account, in the energy equation, the work of the gravity force, which is often neglected when investigating seepage.     

A Phenomenological Study of Pore-Size Dependent Thermal Conductivity of Porous Silicon

A phonon-hydrodynamics approach is used to analyze the influence of porosity and of pore size on the reduction of the thermal conductivity in porous silicon. Different geometrical arrangements of the pores have been considered. For any given value of the porosity, the theoretical results show that for increasing Knudsen number (i.e., decreasing pore size) the effective thermal conductivity decreases whatever the geometrical arrangement of the pores is.     

Slow viscous flow through a membrane built up from porous cylindrical particles with an impermeable core

This paper concerns the slow viscous flow through an aggregate of concentric clusters of porous cylindrical particles with Happel boundary condition. An aggregate of clusters of porous cylindrical particles is considered as a hydro-dynamically equivalent to solid cylindrical core with concentric porous cylindrical shell. The Brinkman equation inside the porous cylindrical shell and the Stokes equation outside the porous cylindrical shell in their stream function formulations are used. The drag force acting on each porous cylindrical particle in a cell is evaluated. In certain limiting cases, drag force converges to pre-existing analytical results, such as, the drag on a porous circular cylinder and the drag on a solid cylinder in a Happel unit cell. Representative results are then discussed and presented in graphical forms. The hydrodynamic permeability of the membrane built up from porous particles is evaluated. The variation of hydrodynamic permeability with different parameters is graphically presented. Some new results are reported for flow pattern in the porous region. Being in resemblance with the model of colloid particles with a coating of porous layers due to adsorption phenomenon, results obtained through this model can be useful to study the membrane filtration process.     

Porous-media simulation of bone-cement spreading during vertebroplasty

The reinforcement of porous vertebral cancellous bone by the injection of bone cement is a practical procedure for the stabilisation of osteoporotic compression fractures and other weakening lesions. This contribution concerns the reproduction and prediction of the resulting bone-cement distribution during the injection procedure by means of numerical simulation. A detailed micromechanical (locally single-phasic) model exhibits the drawback that all geometrical and physical transition conditions of the individual parts of the complex aggregate have to be known. Therefore, we rather proceed from a macroscopic (and multi-constituent) continuum-mechanical model based on the Theory of Porous Media. In this regard, the homogenisation of the underlying micro-structure results in a model of three constituents: these are the solid bone skeleton, which is saturated by the liquid bone marrow that may be displaced by the injected liquid bone cement. The influence of the micro-architecture of the pore space on the spreading of the bone cement is considered by a spatial diversification of the anisotropic permeability tensors, obtained through image processing techniques applied to medical imaging data (µCT). The numerical investigation of the strongly coupled problem enables the study of vertebroplasty and allows for the comparison between the simulation results and the experimentally determined bone-cement distribution that were imaged during injections. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The pyramidal configurations of N+1 bodies cannot rotate

For the (N+1)-body problem, we assume that N bodies are at the vertices of a unit regular polygon and the (N+1)st body is along the vertical line normal to the plane formed by the former N bodies. If N bodies rotate at the unit circle and the (N+1)st body oscillates along the vertical line of the plane formed by the former N bodies and passing through the geometrical center, then we prove that the (N+1)st body must locate at the geometrical center of unit regular polygon.     

多孔材料夹芯的分层复合双壁面圆柱壳体中波的传播

在经典的理论框架内,对分层的复合材料壳体——多孔材料夹芯的双壁面圆柱壳体,研究自由谐和波在其中的传播.借助于一个具有同样几何特性的展开平板,评估波通过多孔夹芯层传播时大部分有效的成分.通过有效波成分的考虑,将多孔层模拟为具有等效特性的流体.因此,模型简化为一个集满流体介质的双壁面圆柱壳体.最后,评估带宽频率中结构的传播损失,并对结果加以比较.     

Classification of Material <Emphasis Type="Italic">G</Emphasis>-Structures

A geometrical interpretation of the G-structures associated to elastic material bodies is given. In addition, characterizations of their integrability are obtained. Since the lack of integrability is a geometrical measure of the lack of homogeneity, the corresponding inhomogeneity conditions are obtained.     

A fast running numerical model based on the implementation of volume forces for prediction of pressure drop in a fin tube heat exchanger

Numerical based design of geometrical structures is common when studying systems involving heat exchangers, a central component in several fields, such as industrial, vehicle and household systems. The geometrical structure of heat exchangers is generally comprised by closely placed fins and tube bundles. The creation of a mesh grid for a geometrically compact heat exchanger will result in a dense structure, which is not feasible for personal computer usage. Hence, volume forces were created based on Direct Numerical Simulations (DNS) on a Flow Representative Volume (FRV) of a tube fin heat exchanger in an internal duct system of a heat pump tumble dryer. A relation of the volume averaged velocity and the volume averaged force was established in two different FRV models with a finite element simulation in COMSOL. This relation was subsequently used to create flow resistance coefficients based on volume averaged expressions of fluid velocity and volume forces. These flow resistance coefficients were implemented in two respective porous models, which represent the entire heat exchanger except the interior arrangements of fins and tube bundles. Hence, the computation time was reduced thanks to the absence of a dense mesh grid. Experimental results of the entire heat exchanger showed good agreement with the second porous model in terms of pressure drop and volume flow rate.     

离散二元几何分布串联系统的参数估计

本文首次给出了二元几何分布的定义及其主要性质,并针对二元几何分布串联系统给出了参数的矩估计和极大似然估计,同时通过大量Monte-Carlo模拟考察了估计的精度,文章最后通过Monte-Carlo数值例子来说明方法的运用.     

Exact solutions for the evolution of ellipsoidal inclusions in porous media

This paper derives exact mathematical solutions for the time-dependentevolution of a single ellipsoidal inclusion in a porous mediumwhen a linear straining flow is active in the far field. Thisrepresents a two-phase free boundary problem. It is shown thatthe dynamics is such that an initially ellipsoidal inclusionremains ellipsoidal under evolution. The theory of ellipsoidalharmonics is used to determine the system of ordinary differentialequations governing the geometrical parameters of the ellipsoidalinclusion.     

Deformation and Damaging of Composites with Transversally-isotropic Components under Compressive Loading

In the present paper, the problem of deformation and damage of composites with a porous isotropic matrix and transversally-isotropic unidirectional fibers under compressive loading is considered when microdamages are accumulated in the fiber. Fractured micro-volumes are modelled by a system of randomly distributed quasi spherical pores. The Shleicher-Nadai fracture criterion is used as a condition for the origin of micro-pores (micro-damage) based on the assumption of a rigid material. The limit value of the strength of the material is assumed as a stochastic function of coordinates. By using a numerical procedure, the solution of the above problem is found. The nonlinear stress-strain diagrams for a transversally-isotropic composite are obtained for the case of uniaxial compression-tension along the fibers. The nonlinearity of the deformations of the composite is caused by accumulation of micro-damages in the matrix. The influence of the physical-mechanical properties of materials, of the volume concentration, of the porosity of components, of the geometrical parameters of the structure, and of the character of the scatter of the strength in the material on the micro-damage of the material and, as a consequence, the influence on the macro-stress-macro-strain diagram is analyzed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of relief of phase reflection diffraction grating

This article describes theoretical and experimental results on phase reflection diffraction gratings. Based on Fourier optics the mathematical formulation describing diffraction dispersion of light from a relief grating of the trapezium profile is derived. We propose a method that lets us estimate the grating’s geometric parameters in a versatile modelling system. We have designed an original programme that estimates diffraction intensities and calculates diffraction efficiency. The estimated intensities are used to reconstruct the grating’s geometrical properties using our mathematical model. The precision of the method is evaluated as the deviation of obtained results from microscopy data.