An effective-medium approach to the optical properties of heterogeneous materials with nonlinear properties

Abstract

We have derived an analytic approach, based on the hypothesis of the effective-medium theory, to evaluate the effective dielectric magnitudes, including the effective third order nonlinear optical susceptibility α(χ⁽³⁾), of a heterogeneous two-component medium in which one of the components (the host matrix or the embedded particle respectively) presents nonlinear behaviour. Our formulation allows us to solve the full nonlinear problem in the whole range of concentrations without treating the nonlinear effects as a small perturbation to the linear behaviour. Therefore, it can be considered as an upper limit of the commonly used low-field nonlinear approximations for heterogeneous materials. Under certain conditions, these composites can exhibit a bistable regime that the present theory properly describes in wide ranges of the concentration, shape, wavelength, dielectric contants and intrinsic nonlinear optical susceptibility of the nonlinear component as well as in terms of the intensity of the external electric field. The present approach has been used to calculate the nonlinear optical response of Cu—Al₂O₃ nanocomposites.     

A multiscale analytical approach to evaluate osseointegration

Osseointegrated implants are frequently used in reconstructive surgery, both in the dental and orthopedic field, restoring physical function and improving the quality of life for the patients. The bone anchorage is typically evaluated at micrometer resolution, while bone tissue is a dynamic composite material composed of nanoscale collagen fibrils and apatite crystals, with defined hierarchical levels at different length scales. In order to understand the bone formation and the ultrastructure of the interfacial tissue, analytical strategies needs to be implemented enabling multiscale and multimodal analyses of the intact interface. This paper describes a sample preparation route for successive analyses allowing assessment of the different hierarchical levels of interest, going from macro to nano scale and could be implemented on single samples. Examples of resulting analyses of different techniques on one type of implant surface is given, with emphasis on correlating the length scale between the different techniques. The bone-implant interface shows an intimate contact between mineralized collagen bundles and the outermost surface of the oxide layer, while bone mineral is found in the nanoscale surface features creating a functionally graded interface. Osteocytes exhibit a direct contact with the implant surface via canaliculi that house their dendritic processes. Blood vessels are frequently found in close proximity to the implant surface either within the mineralized bone matrix or at regions of remodeling.     

A digital microfluidic approach to homogeneous enzyme assays

A digital microfluidic device was applied to a variety of enzymatic analyses. The digital approach to microfluidics manipulates samples and reagents in the form of discrete droplets, as opposed to the streams of fluid used in channel microfluidics. This approach is more easily reconfigured than a channel device, and the flexibility of these devices makes them suitable for a wide variety of applications. Alkaline phosphatase was chosen as a model enzyme and used to convert fluorescein diphosphate into fluorescein. Droplets of alkaline phosphatase and fluorescein diphosphate were merged and mixed on the device, resulting in a 140-nL, stopped-flow reaction chamber in which the fluorescent product was detected by a fluorescence plate reader. Substrate quantitation was achieved with a linear range of 2 orders of magnitude and a detection limit of approximately 7.0 x 10-20 mol. Addition of a small amount of a nonionic surfactant to the reaction buffer was shown to reduce the adsorption of enzyme to the device surface and extend the lifetime of the device without affecting the enzyme activity. Analyses of the enzyme kinetics and the effects of inhibition with inorganic phosphate were performed, and Km and kcat values of 1.35 microM and 120 s-1, respectively, agreed with those obtained in a conventional 384-well plate under the same conditions (1.85 microM and 155 s-1). A phototype device was also developed to perform multiplexed enzyme analyses. It was concluded that the digital microfluidic format is able to perform detailed and reproducible assays of substrate concentrations and enzyme activity in much smaller reaction volumes and with higher sensitivity than conventional methods.     

A differential approach to microstructure-dependent bounds for multiphase heterogeneous media

We present a new method of deriving microstructure-dependent bounds on the effective properties of general heterogeneous media. The microstructure is specified by the average Eshelby tensors. In the small contrast limit, we introduce and calculate the expansion coefficient tensors. We then show that the effective tensor satisfies a differential inequality with the initial condition given by the expansion coefficient tensors in the small contrast limit. Using the comparison theorem, we obtain rigorous bounds on the effective tensors of multiphase composites. These new bounds, taking into account the average Eshelby tensors for homogeneous problems, are much tighter than the microstructure-independent Hashin–Shtrikman bounds. Also, these bounds are applicable to non-well-ordered composites and multifunctional composites. We anticipate that this new approach will be useful for the modeling and optimal design of multiphase multifunctional composites.     

A new simulational approach to electron-photon showers in heterogeneous media

Three-dimensional Monte Carlo calculations are developed for determining the energies of electron-photon showers detected in emulsion chambers. The calculations provide better accuracy and higher precision by taking into account contributions from the Landau effect, the transition effect, etc., which are hard to solve analytically.The simulation is applicable, even in the extremely high energy region (∼ 1000 TeV), for any type of chamber design, either with wide gap or alternate mixed substances. We present here two powerful ways to save computing time without sacrificing the accuracy of numerical results. The first employs approximate formulae for the cross sections of bremsstrahlung and pair creation processes, including both the screening effect and the Landau effect. The second uses analytical formula of multiple Coulomb scattering applicable when including various electromagnetic processes, such as bremsstrahlung, pair creation and ionization loss.We compared our results with FNAL data and found remarkable agreement, both for average cascades and for fluctuation. We conclude that the position of sensitive materials inserted between heavy elements critically affects the precision of the energy determination.     

Elastic properties of RCC under flexural loading-experimental and analytical approach

In structural analysis, especially in indeterminate structures, it becomes essential to know material and geometrical properties of members. The codal provisions recommend elastic properties of concrete and steel and these are fairly accurate enough. The stress–strain curve for concrete cylinder or a cube specimen is plotted. The slope of this curve is modulus of elasticity of plain concrete. Another method of determining modulus of elasticity of concrete is by flexural test of a beam specimen. The modulus of elasticity most commonly used for concrete is secant modulus. The modulus of elasticity of steel is obtained by performing a tension test of steel bar. While performing analysis by any software for high rise building, cross area of plain concrete is taken into consideration whereas effects of reinforcement bars and concrete confined by stirrups are neglected. The aim of study is to determine elastic properties of reinforced cement concrete material. Two important stiffness properties such as AE and EI play important role in analysis of high rise RCC building idealized as plane frame. The experimental programme consists of testing of beams (model size 150 × 150 × 700 mm) with percentage of reinforcement varying from 0.54 to 1.63%. The experimental results are verified by using 3D finite element techniques. This study refers to the effect of variation of percentage of main longitudinal reinforcement and concrete grade. Effect of confinement is not considered and it appears in a separate study.     

A theoretical approach to the deformation of honeycomb based composite materials

Honeycomb sheet is already widely used as a core in aeronautical sandwich construction. An alternative application is to use it as the reinforcing element for composites in which the cells of the honeycomb are filled with various materials. This paper presents the results of a study of such a composite, in which a low modulus infill is used. The work covers, in simple terms, the elastic properties of the unfilled sheet and the composite under in-plane direct loading and out-of-plane bending. The plastic deformation characteristics under in-plane direct loading are also considered. Specimen experimental results are presented, which show that the simple analytical approach used is clearly justifiable.     

An analytical approach for calculating vacuum properties of outgassing tubing systems

A method has been investigated to calculate vacuum properties, such as flow conductance and pressure drop, for complex tubing systems, taking into account the outgassing of the tube walls. The procedure is based on the determination of the pressure drop, Δp_i, of each tube element, starting at the chamber with a given mass flow rate, q_b1. The flow conductance, G, of the system follows from the overall pressure drop, ΔpΣⁿ_i Δp_i as $\text{G\&z.dbnd;q}{}_{\mathrm{b1}}\text{/Δp}$. The results are applied to analysing a vacuum system of the First Spacelab Payload.     

A theoretical approach to the interaction between buckling and resonance instabilities

This article deals with the interaction between buckling and resonance instabilities of mechanical systems. Taking into account the effect of geometric nonlinearity in the equations of motion through the geometric stiffness matrix, the problem is reduced to a generalized eigenproblem where both the loading multiplier and the natural frequency of the system are unknown. According to this approach, all of the forms of instabilities intermediate between those of pure buckling and pure forced resonance can be investigated. Numerous examples are analyzed, including discrete mechanical systems with one to n degrees of freedom, continuous mechanical systems, such as oscillating deflected beams subjected to a compressive axial load, as well as oscillating beams subjected to lateral–torsional buckling. A general finite element procedure is also outlined, with the possibility to apply the proposed approach to any general bi- or tri-dimensional framed structure. The proposed results provide a new insight in the interpretation of coupled phenomena such as flutter instability of long-span or high-rise structures.     

An approach to micro-macro modeling of heterogeneous materials

 A micro-macro strategy suitable for modeling the mechanical response of heterogeneous materials at large deformations and non-linear history dependent material behaviour is presented. When using this micro-macro approach within the context of finite element implementation there is no need to specify the homogenized constitutive behaviour at the macroscopic integration points. Instead, this behaviour is determined through the detailed modeling of the microstructure. The performance of the method is illustrated by the simulation of pure bending of porous aluminum. The influence of the spatial distribution of heterogeneities on the overall macroscopic behaviour is discussed by comparing the results of micro-macro modeling for regular and random structures. Received 9 July 2000     

Energy absorption of graded foam subjected to blast: A theoretical approach

Energy absorption of graded foam subjected to blast is investigated, in which the high velocity crushing of foam is modeled with shock theory and rigid-perfectly-plastic-locking idealization. The characteristics of a typical blast are taken into account when determining the foam density profile. Different from the homogeneous foam, the graded foam density variation is designed largest at the loading end and smallest at the supporting end, with an exponential decay in between. It is found that, subjected to the same blast load, the total input energy, in fact the energy to be dissipated by the cladding, decreases with increasing density gradient. The final foam deformation with larger density gradient is smaller.     

A cleaner enzymatic approach for producing non-phthalate plasticiser to replace toxic-based phthalates

Clean Technologies and Environmental Policy - Dioctyl phthalate (DOP) is industrially commonly used as a polyvinyl chloride (PVC) plasticiser. As DOP does not form a chemical link with PVC, it...     

A nonholonomic approach to isoparallel problems and some applications

Isoparallel problems are a class of optimal control problems on principal fibre bundles endowed with a connection and a Riemannian metric on the base space. These problems consist of finding the shortest curve on the base among those with a given parallel transport operator. It has been shown that when the structure group of the principal bundle admits a bi-invariant metric, the normal solutions are precisely the projections of the geodesics (relative to an appropriate Riemannian metric) on the bundle. In this work we obtain a generalization of this result that holds true for any structure group, by transforming the isoparallel problem into a nonholonomic problem of a generalized type. The latter reduces to the geodesic problem if the structure group has a bi-invariant metric. We illustrate the theory with an application to the optimal control of an elastic rolling ball (the plate-ball system), relating some aspects of this problem to the dynamics of a simple pendulum. Finally, we indicate how the study of locomotion of microorganisms can benefit from this approach. This work shows how optimal control and generalized nonholonomic mechanics are related within the context of Lagrangian reduction.     

Estimating shear properties of walnut wood: a combined experimental and theoretical approach

In this study, several theoretical models to numerically estimate shear properties of orthotropic materials are introduced. These approaches are based on the combination of Hankinson’s empirically derived formula with other empirical and analytical calculations. Next to shear moduli, which are estimated from the elastic moduli and Poisson’s ratios, shear strengths are also estimated from the in-axis strengths. The models are validated by mechanical tests on walnut wood (Juglans regia L.), for which a sufficient data set can be found in literature. The Arcan test is used to estimate the shear moduli, while the shear block test is used to estimate the shear strengths. The results show that the model, which is based on a combined use of Hankinson’s formula and tensor rotation, gives the best estimation of shear moduli as evaluated by the minimum differences to the experimentally obtained results. For the shear strengths, a combination of Hankinson’s formula and Norris’ failure criterion shows the best agreement in comparison to the experimental data. The theoretical calculations may be used for a time efficient estimation of shear modulus and strength in comparison to the very time-consuming experimental estimation.     

A powerful approach to develop nitrogen-doped graphene sheets: theoretical and experimental framework

Journal of Materials Science - We have reported a novel route to develop highly conductive graphene sheets using camphor as a natural precursor followed by nitrogen doping via low-temperature...     

Axisymmetric analytical solutions for a heterogeneous multi-ferroic circular plate subjected to electric loading

In the present article, responses of an electrically loaded heterogeneous circular plate of transversely isotropic multi-ferroic materials are investigated. The axisymmetric problem is solved by adopting the direct displacement method developed by the second author. The corresponding generalized displacement functions are obtained through a step-by-step integration procedure. The coupling magneto-electro-elastic fields in the plate, which exactly satisfy the upper and lower boundary conditions and approximately meet the cylindrical boundary condition, are explicitly expressed. Numerical calculations are performed for two particular plates to investigate the influence of material heterogeneity.     

A computational homogenization approach for limit analysis of heterogeneous materials

The macroscopic strength domain and failure mode of heterogeneous or composite materials can be quantitatively determined by solving an auxiliary limit analysis problem formulated on a periodic representative volume element. In this paper, a novel numerical procedure based on kinematic theorem and homogenization theory for limit analysis of periodic composites is developed. The total velocity fields, instead of fluctuating (periodic) velocity, at microscopic level are approximated by finite elements, ensuring that the resulting optimization problem is similar to the limit analysis problem formulated for structures, except for additional constraints, which are periodic boundary conditions and averaging relations. The optimization problem is then formulated in the form of a standard second‐order cone programming problem to be solved by highly efficient solvers. Effects of loading condition, representative volume element architecture, and fiber/air void volume fraction on the macroscopic strength of perforated and fiber reinforced composites are studied in numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.