Parametric frequency analysis of oscillatory behavior of mouse Zona Pellucida spherical net model: cases of successful and of unsuccessful fertilization

The current paper proposes a model for describing mechanical phenomena that occur during the process of mammal fertilization when spermatozoa impact the surface of Zona Pellucida. Zona pellucida (ZP) is a dynamical 3D matrix that surrounds the mammalian oocyte. In the process of fertilization, sperm cell has to penetrate this structure. To describe impact of sperm cells with velocities that are effective and those that are ineffective relative to the oscillatory behavior of ZP, the discreet continuum model in the form of spherical net model is used. Resultant trajectories of knot mass particles dynamics of mouse ZP spherical net model in the form of generalized Lussajous curves are presented. Using generalized Lussajous curves, parametric frequency analysis of oscillatory behavior of knot material particles in the mouse ZP spherical net model is conducted. The influence of impact angles of sperm cells on corresponding knot mass particle trajectory is discussed. Favorable and unfavorable trajectories of knot mass particle motions are discussed in the context of successful fertilization. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The influence of porosity on the elastoplastic behavior of high performance cast alloys

Based on geometrical data of porous components obtained by computer tomography, a micromechanical unit cell model is proposed to predict the influence of porosity on mechanical properties of high performance cast alloys. Simulations of uniaxial loaded periodic unit cells are compared to experimental data of tensile tests on porous materials. In this way a relationship between some significant geometrical data of the pores and the macroscopic material parameters is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

ORDER EXTENSIONS AS ADJOINT FUNCTORS

Abstract

A standard extension (resp. standard completion) is a function Z assigning to each poset P a (closure) system ZP of subsets such that x ? y iff x belongs to every Z ε ZP with y ε Z. A poset P is Z -complete if each Z ε 2P has a join in P. A map f: P → P′ is Z—continuous if f^?1 [Z′] ε ZP for all Z′ ε ZP′, and a Z—morphism if, in addition, for all Z ε ZP there is a least Z′ ε ZP′ with f[Z] ? Z′. The standard extension Z is compositive if every map f: P → P′ with {x ε P: f(x) ? y′} ε ZP for all y′ ε P′ is Z -continuous. We show that any compositive standard extension Z is the object part of a reflector from IP_Z, the category of posets and Z -morphisms, to IR_Z, the category of Z -complete posets and residuated maps. In case of a standard completion Z, every Z -continuous map is a Z -morphism, and IR2 is simply the category of complete lattices and join—preserving maps. Defining in a suitable way so-called Z -embeddings and morphisms between them, we obtain for arbitrary standard extensions Z an adjunction between IP_Z and the category of Z -embeddings. Many related adjunctions, equivalences and dualities are studied and compared with each other. Suitable specializations of the function 2 provide a broad spectrum of old and new applications.     

Variational Homogenization of Micro-Electro-Elasticity

Understanding of micromechanical mechanisms in functional materials with electro-mechanical coupling is a highly demanding area of simulation technology and increasing interest has been shown in the last decades. Smart materials are characterized by microstructural properties, which can be changed by external stress and electric field stimuli, and hence find use as the active components in sensors and actuators. In this context, a key challenge is to combine models for microscopic electric domain evolution with variational principles of homogenization. We outline a variational-based micro-electro-elastic model for the micro-structural evolution of electric domains in ferroelectric ceramics. The micro-to-macro transition is performed on the basis of variational principles, extending purely mechanical formulations to coupled electro-mechanics. We focus on an electro-mechanical Boltzmann continuum on the macro-scale with mechanical displacement and electric potential as primary variables. The material model on the micro-scale is described by a gradient-extended continuum formulation taking into account the polarization vector field and its gradient, see Landis [1] and Schrade et al. [2] for conceptually similar approaches. A crucial aspect of the proposed homogenization analysis is the derivation of appropriate boundary conditions on the surface of the representative volume element. In this work we derive stiff Dirichlet-type, soft Neumann-type, and periodic boundary constraints starting from a generalized Hill-Mandel macrohomogeneity condition. Furthermore, we propose techniques to incorporate these boundary conditions in the variational principles of homogenization by means of Lagrange multiplier methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite-element modeling of the particle clustering effect in a powder-metallurgy-processed ceramic-particle-reinforced metal matrix composite on its mechanical properties

A new numerical method is proposed to predict the effect of particle clustering on grain boundaries in a ceramic- particle-reinforced metal matrix composite on its mechanical properties, and micromechanical finite-element simulation of stress–strain responses in composites with random and clustered arrangements of ceramic particles are carried out. A particular material modeled and analyzed is a TiC-particle-reinforced Al matrix composite processed by powder metallurgy. A representative volume element of a composite microstructure with 5 vol.% TiC is reconstructed based on the tetrakaidecahedral grain boundary structure by using a modified random sequential adsorption. The model proposed in this study accurately represents the stress concentrations and particle-particle interactions during deformation of the powder-metallurgy-processed composite. A comparison with the random-arrangement model shows that the present numerical approach is more accurate in simulating the behavior of the composite material.     

Magneto-Sensitive Elastomers: An Experimental Point of View

Magneto-sensitive elastomers (MSEs) are smart materials changing their shape and mechanical properties in the presence of a magnetic field. Focussing on model systems, silicone based MSEs are prepared by a multi-step mixing process and characterised using a rotational rheometer (plate-plate). Data obtained by relaxation tests is used to set-up a material model coupling the theories of viscoelasticity and magnetoelasticity. The behaviour of MSEs in quasi-static and dynamic mechanical shear experiments can be successfully predicted by the analytical model using parameters received by fitting the transient experiments. The model is validated for small shear deformations (γ = 0.02) and low magnetic fields (𝔹 = 0.2 T). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Microscopic model for a cell-seeded material

The aim of the present paper is to account for the growth of fiber which is observed in a cell-seeded material stimulated in a bioreactor. For this purpose, the change of mass is considered in the balance laws, and the deformation energy is assumed to be a function of varying mass and the Helmholtz free-energy. Fiber growth at the microscopic level causes a macroscopic change of the material's mechanical properties. The study is a first approach towards a micromechanical model accounting for remodelling in cartilage replacement materials. In so doing, constitutive equations for renewable soft tissues are proposed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of plain-weave composites

The paper analyzes the mechanical properties of plain-weave composites by employing different micromechanical models. Their in-plane and flexural properties are estimated using analytical approaches, including the rule of mixtures and one-dimensional composite beam and two-dimensional mosaic models. The expressions of effective material properties are obtained for one-, two-, and three-ply woven composites. The results obtained are in agreement with published data.     

The thermally stressed state of a thickwalled hollow composite cylinder

We propose an approach to the determination of the temperature, stress, and displacement, fields in laminated hollow composite cylinders. The cylinders are studied in three-dimensional formulation; the material of each lamina has the properties of thermal and elastic orthotropy. We study the influence of the heat flow passing through the end surfaces and nonuniform heating of the cylinder to its thermally stressed state. We exhibit peculiarities in the stress and displacement distributions caused both by anisotropy of the mechanical and thermophysical properties and by the nature of the thermal actions applied. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 14–18     

Granular materials: Parameter identification and modelling of localisation problems

The prediction of landsliding requires an exact knowledge of the mechanical behaviour of granular materials. This kind of materials, e. g., sand, have a very complex deformation behaviour, which depend on the stress state and on the loading history. In this work, the deformation behaviour of the solid skeleton is characterised via homogeneous triaxial tests on dry sand specimens. Additionally, an appropriate elasto-plastic material law to describe the solid skeleton in the frame of Theory of Porous Media (TPM) is used, which is implemented in the FE tool PANDAS. Furthermore, a single-surface yield criterion with isotropic hardening, which limits the elastic domain, and a non-associated plastic flow are employed. The determination of the material parameters of the linear elasticity law as well as the single-surface yield criterion are based on test data of triaxial experiments. The material parameters are identified using a derivative-based optimisation method (donlp2), which is coupled with PANDAS. Finally, a simulation of a benchmark test is presented to show shear band localisation effects, where the material behaviour is described by a triphasic porous media model based on the TPM, where the constituents are a deformable solid skeleton and two pore fluids, water and air. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A contribution to the modeling of metal foams on the mesoscale

Due to their useful properties in lightweight construction and due to their excellent behavior in energy absorption for example in crash mechanics, metal foams became an interesting, often utilized and investigated material. For the determination of the mechanical properties of foams without the help of expensive experiments, a way for computing these properties is searched. The problem in doing so is that foams can be composed out of randomly distributed edges and faces with varying thickness and of other inhomogeneities on the mesoscale like imperfections. The goal in this paper is, to investigate the influence of these irregularities on the mechanical, linear elastic properties of a metal foam on the macroscale and to determine the size of a representative volume element, for which the irregularities on the mesoscale do not have a great influence on the linear elastic properties. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micromechanical modeling of twinning

There exist material models that incorporate mechanical twinning in a homogenized sense, or consider specific aspects, like grain refinement or texture evolution. However, since the RVE-technique became a standard method, it is possible to obtain more detailed predictions based on micromechanical models. In this work, an approach based on a nonconvex elastic potential and the corresponding results of FE calculations are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Representative Volume in the Micromechanics of Particulate Composites

A rectangle filled with closely packed spheres of random size and properties is considered as a micromechanical model of a two-phase particulate composite. A numerical simulation is used to determine the effective mechanical properties of the assembly and their scatter as a function of the number of spheres. It is shown that, in a system with relatively small number of particles (up to 300), the scatter of Young's modulus decreases with the system size. However, the rate of the scatter decrease becomes smaller with growing size of the system, so that the convergence to zero most likely takes place at infinity.     

Linking macroscopic deformation processes to microstructure evolution using an FE-FFT-based micro-macro transition and non-conserved phase-fields

The purpose of this work is the multiscale FE-FFT-based prediction of macroscopic material behavior, micromechanical fields and bulk microstructure evolution in polycrystalline materials subjected to macroscopic mechanical loading. The macroscopic boundary value problem (BVP) is solved using implicit finite element (FE) methods. In each macroscopic integration point, the microscopic BVP is embedded, the solution of which is found employing fast Fourier transform (FFT), fixed-point and Green's function methods. The mean material response is determined by the stress-strain relation at the micro scale or rather the volume average of the micromechanical fields. The evolution of the microstructure is modeled by means of non-conserved phase-fields. As an example, the proposed methodology is applied to the modeling of stress-induced martensitic phase transformations in metal alloys. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constitutive modeling of fiber-reinforced aerogels

Aerogels are open-cell highly porous solids having exclusive properties such as lowest bulk density, lowest thermal conductivity and lowest acoustic velocity. These properties make them useful over a large spectrum of applications. However, their brittle and hydrophilic natures result in a road block which is overcome by polymer or fiber reinforcement. In this study, we are focussing on fiber-reinforced aerogels which retain the exclusive properties of aerogels mentioned above while giving them a good load bearing strength at the same time. These fiber-reinforced aerogels show many complex phenomena under loading such as, a strong nonlinearity, cyclic stress softening, and permanent set. However, there have been very few studies on fiber-reinforced aerogels oriented towards their mechanical characterization. To the best of our knowledge, there also is no micromechanical model that can capture the constitutive response of these fiber-reinforced aerogels. In this contribution, we propose a micro-mechanically motivated constitutive model of fiber reinforced aerogels. The prime source of elasticity and damage in these aerogels is the effect of bending and breakage of fibers in the material network. Accordingly, we consider non-linear bending of fibers supported by an elastic foundation of particles. Based on this concept, the strain energy function is analytically derived for a single fiber in a particular direction. The strain energy of the whole network is then obtained by integration over an unit sphere. The model shows good agreement with experimental data. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation analysis of the HIV‐1 within host model

In this paper, a bifurcation solution's analysis is proposed for an HIV‐1 within the host model around its chronic equilibrium point, this is carried out based on Lyapunov–Schmidt approach. It is shown that the coefficient b, which represents the healthy CD4⁺ T‐cells growth rate, is a bifurcation parameter; this means that the rate of multiplication of healthy cells can have serious effects on the qualitative dynamical properties and structural stability of the infection evolution dynamics. Copyright © 2015 John Wiley & Sons, Ltd.     

Structure,properties, and tests of carbonreinforced plastics

The effect has been examined of various factors on the realization of mean values of mechanical properties in monofilaments in composites; the necessity has been pointed out of taking account of specific characteristics of the structure and behavior of reinforced polymers based on carbon fibers during determination of strain and tensile properties of these materials.     

Regularity of solutions to a reaction–diffusion equation on the sphere: the Legendre series approach

In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two‐dimensional sphere close to its poles. This equation is the counterpart of the well‐studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C¹ smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.     

RESEARCH IN MATHEMATICS EDUCATION: SOME ISSUES AND SOME EMERGING INFLUENCES

In this article, I introduce a typology of forms of algebraic thinking. In the first part, I argue that the form and generality of algebraic thinking are characterised by the mathematical problem at hand and the embodied and other semiotic resources that are mobilised to tackle the problem in analytic ways. My claim is based not only on semiotic considerations but also on new theories of cognition that stress the fundamental role of the context, the body and the senses in the way in which we come to know. In the second part, I present some concrete examples from a longitudinal classroom research study through which the typology of forms of algebraic thinking is illustrated.     

Cellular articular cartilage replacement tissues: Modeling and analysis

Long-term studies reveal that mechanical stimulation causes growth and remodeling phenomena within biological tissues. The main aim of this research is to fully understand and control these phenomena. For accomplishing that, two steps are considered: first, we determine a suitable numerical model based on different approaches by a comparative study using experimental validations, and second, investigate the mechanical properties of the tissue specimens after a remodeling process. We start with the first step by choosing a convenient model that mimics the biotissue for running the numerical simulations in the second step. There are different models available that determine the mechanical properties of soft replacement tissues seeded with human chondrocytes in modern medical applications. It is our objective to achieve a common methodology of theory and experiments that allows the determination of the mechanical properties of the native material. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)