On the complex potentials of the linear thermoelasticity with microlocal parameters

Summary The linear theory of thermoelasticity with microlocal parameters was proposed lately in [1], [2], [3] as a certain alternative approach to the modelling of microperiodic composites. In this paper we adopt the complex variable method for two-dimensional problems of the periodically layered thermoelastic composites treated within the framework of the linear thermoelasticity with microlocal parameters. The presented complex potentials reduce two-dimensional static problems of the microperiodically layered thermoelastic composites to the boundary values problems for analytical functions. As an example illustrating this method the plane-strain problem of stress and temperature distribution in the microperiodic two-layered half-space subjected to general loading and temperature on the boundary is considered.With 3 Figures     

Asymptotic solution of mode-III crack in damaged softening materials

In this paper, a constitutive model of elasticity coupled with damage suggested by Lemaitre et al. [1] is used. The macroscopic stress-strain response of the model includes two stages: strain hardening and strain softening. The basic equation is derived for the anti-plane shear problem. Several lowest order asymptotic solutions are obtained, and assembled for the crack-tip fields.     

Multiscale finite element method for a locally nonperiodic heterogeneous medium

A generalization of the mathematical homogenization theory to account for locally nonperiodic solutions is presented. Such nonperiodicity may arise either due to the rapidly varying microstructure (e.g.: graded materials, microcracks) or because the macroscopic solution is not smooth and may have significant variation within a microstructure. In the portion of the problem domain where the material is formed by a spatial repetition of the base cell and the macroscopic solution is smooth, a double scale asymptotic expansion and solution periodicity are assumed, and consequently, mathematical homogenization theory is employed to uncouple the microscopic problem from the global solution. For the rest of the problem domain it is assumed that the periodic solution does not exist (cutouts, cracks, free edges in composites, etc.) and the approximation space is decomposed into macroscopic and microscopic fields. Compatibility between the two regions is explicitly enforced. The proposed method is applied to resolve the structure of the microscopic fields in the single ply composite plates with a centered hole and with a centered crack and in the [0/90] _s laminated plate. Numerical results are compared to the reference solution, an engineering global-local approach, and the direct extraction from the mathematical homogenization method.     

Two‐scale computational modelling of water flow in unsaturated soils containing irregular‐shaped inclusions

The focus of this paper is two‐dimensional computational modelling of water flow in unsaturated soils consisting of weakly conductive disconnected inclusions embedded in a highly conductive connected matrix. When the inclusions are small, a two‐scale Richards’ equation‐based model has been proposed in the literature taking the form of an equation with effective parameters governing the macroscopic flow coupled with a microscopic equation, defined at each point in the macroscopic domain, governing the flow in the inclusions. This paper is devoted to a number of advances in the numerical implementation of this model. Namely, by treating the micro‐scale as a two‐dimensional problem, our solution approach based on a control volume finite element method can be applied to irregular inclusion geometries, and, if necessary, modified to account for additional phenomena (e.g. imposing the macroscopic gradient on the micro‐scale via a linear approximation of the macroscopic variable along the microscopic boundary). This is achieved with the help of an exponential integrator for advancing the solution in time. This time integration method completely avoids generation of the Jacobian matrix of the system and hence eases the computation when solving the two‐scale model in a completely coupled manner. Numerical simulations are presented for a two‐dimensional infiltration problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Mathematical modelling of the interface crack. A new improved numerical method

A pre-stressed fiber elastic composite containing an anti-plane interface crack is considered. The boundary conditions of our problem are leading to a homogeneous and a non-homogeneous problem. Using the theory of Cauchy’s integral ([1], [2], [3] and [4]), Guz’s representation ([1] and [3]) and the numerical analysis we determine the incremental fields in the vicinity of the crack tips.In [5] we presented a numerical method for calculating the complex integrals which appear in the non-homogeneous complex differential equation for the complex potential. In the present paper we shall improve the method, providing an appropriate treatment of the singularities. We shall deliver a very good formula for calculating numerically the complex integrals employing Plemelj’s fomulas and Cauchy’s integral. A numerical example regarding the propagation of an anti-plane interface crack in glass-epoxy is studied.     

Computational study of plasma-solid interaction in mixtures of oxygen with rare gases

The computer experiment describing the interaction of electronegative plasma with immersed substrates is presented and results of modelling are discussed. The main attention is devoted to the sheath region and to the influence of plasma parameters on its formation. The used computational method is the self-consistent particle modelling. The input parameters for modelling were derived both by the measurement in mixtures of oxygen with rare gases and by the simulation of oxygen plasma by the macroscopic kinetic approach.     

Mass transport with enzyme reactions

Summary A mathematical approach is presented for modelling the reactions catalyzed by enzymes attached to the inner surface of a tube through which the substrate solution passes. In a classical paper of Kobayashi and Laidler [4] a theoretical treatment of this problem was considered but the resulting nonlinear equation appeared to be intractable, except for the simple cases. In the present paper, the central integro-differential equation is reduced to an integral Volterra equation. The analysis is extended to different transport situations and to reactions with classical Michaelis-Menten kinetics and with different types of inhibition. Explicit power series representations of the solutions are established for several reaction velocity models. The consistency of the mathematical model used is established by proving the existence and the uniqueness of the continuous solution of the basic integral equation.     

Velocity inversion for acoustic waves with wavelet operator

Seismic exploration is an important method in geophysical engineering. In this paper, with three-dimensional wavelet operator, a new inverse and imaging method is presented to the seismic inversion problem. Compared with the singular-function method in [1], [2] and [3], the method in this paper can suppress the noise in real-world data and can provide a formula to estimate the error produced by the band-limited nature of the data. As a result, the location of the interior surface in the earth can be detected more precisely.     

Problems of identification of mechanical characteristics of viscoelastic composites

Summary The problem of modelling and identification of viscoelastic properties of composites as asphalt concrete, based on experimental investigations is the subject of this paper. Only viscoelastic properties of such materials are taken into consideration and usefulness of linear constitutive relations in differential form is examined. The identification problem is formulated in terms of sensitivity analysis and optimization theory. An approach to select the best from the considered class of material models is proposed and the values of associated material parameters are determined by minimization of objective function representing a measure of a difference between theoretically calculated and experimentally obtained response of a sample to sinusoidal loading. Obtained this way, results are compared with those calculated using the method based on the approach proposed by Bland and Lee [1] for calculation material parameters of Burgers model. It has been shown that differential constitutive relations with constant coefficients have very restricted application in the considered case because they are in force in narrow domains of excitation frequency. The application of models with frequency-dependent coefficients is one of the possible solutions of this problem [2]. Another approach can also be proposed based on the application of fractional calculus in viscoelasticity, see Bagley and Torvik [3].     

Modeling macroscopic extended continua with the aid of numerical homogenization schemes

Even in the range of small elastic deformations the behavior of foams is not well described by only two elastic constants. Usually the manufacturers give values of the material parameters depending on the loading conditions. This problem is investigated on a microscopic scale by a simple beam model and on the macroscopic scale by an extended continuum model. It has been found that this approach shows the size effect [J. Mater. Sci. 18 (1983) 2572] that cannot be described within the framework of the standard continuum mechanical setting. The existence of the size effect within this model can be explained by independent rotations which do not scale with the displacement field.While macroscopic material parameters are generally unknown for foams the macroscopic properties are derived from the microscale where the parameters are assumed to be known. After evaluation of the microscopic constitutive equations, which are also considered to be known, the quantities are mappped back to the macroscale by a homogenization procedure. This approach is known from literature as FE² model, see e.g. [V. Kouznetsova, Computational homogenization for the multi-scale analysis of multi-phase materials, PhD-thesis, Technical University of Eindhoven, 2002], [Int. J. Numer. Meth. Eng., 54 (2002) 1235] or [Arch. Appl. Mech., 72 (2002) 300]. It is shown that the Cosserat continuum and the FE² model are able to describe the same effects qualitatively.     

Introduction of damage evolution in a scale transition approach for highly-filled particulate composites

The present paper deals with a non-conventional scale transition for modelling the behaviour of highly-filled particulate composites, starting from a methodology initially proposed by Christoffersen [Christoffersen J. Bonded granulates. J Mech Phys Solids 1983;31:55–83] and recently extended by Nadot et al. [Nadot C, Dragon A, Trumel H, Fanget A. Damage modelling framework for viscoelastic particulate composites via a scale transition approach. J Theor Appl Mech 2006;44(3):553–83] in presence of damage. The model thus obtained is here completed with several ingredients allowing to describe damage evolution and in particular a defect nucleation criterion as well as a closure criterion. These criteria are formulated in terms of displacement, and so as to ensure continuity in terms of macroscopic stress. They are finally introduced in an iterative numerical solving procedure which allows to follow damage evolution as a discrete sequence of interfacial debonding including also eventual closure of defects.     

A load-cycling technique for R-curve behaviour: application to a low cement refractory

The data presented below for a low cement refractory shows that the material has strong R-curve behaviour for certain specimen sizes. The superposition method proposed by Sakai and Bradt [1] was coupled with the effective crack model developed by Karihaloo and Nallathambi [2] and used to investigate this R-curve behaviour. The technique that was developed involves load cycling on one specimen to evaluate K _IC values with crack extension, and was shown to give favourable results for this material.     

Niobium/alumina bicrystal interface fracture: A theoretical interlink between local adhesion capacity and macroscopic fracture energies

In the presented work, an effort has been put to clear up the theoretical interlink between local adhesion capacity and macroscopic fracture energies by bridging different length scales, such as nano-, meso-, and macro-scale. Crystal plasticity theory along with a cohesive modelling approach has been used during this work. The influence of different cohesive law parameters (cohesive strength, work of adhesion) on the macroscopic fracture energies for three different orientations of niobium/alumina bicrystal specimens has been presented. It is found that cohesive strength has a stronger effect on macroscopic fracture energies as compared to work of adhesion. In the last part a generalized correlation among macroscopic fracture energy, cohesive strength, work of adhesion and yield stress is derived. The presented results can provide a great help to experimentalists in order to design better metal/ceramic interfaces.     

Bridging the length-scale gap—short fibre composite material as an example

A sequential modelling approach consisting of passing information across length scales is presented to simulate macroscopic behavior of composite materials. The modeling procedure utilizes a proper flow of information from molecular scale to macroscopic scale including material characteristics at different length scales. Both molecular dynamics and analytical/numerical methods were used in the multiscale analysis together with some experimental observations obtained from Raman microspectroscopy and X-ray microtomography. The multiscale procedure is systematically applied to short glass fibre polypropylene composite material.     

Some asymptotic results in the theory of wave-energy devices

Summary Evans [2] has shown how the solution of the radiation problem for water-waves gives all the necessary parameters for determining the efficiency of a wave-energy device. We consider how this radiation problem can be solved in certain limits for several types of device. First, a moving-cylinder device which is many wavelengths below the surface is examined, using Leppington and Siew's [11] method. We then calculate the efficiency of a Cockerell raft device which is either many wavelengths, or only a small fraction of a wavelength, long. Finally, the problem of a thin, symmetric gas-filled bag in a wave field is solved and the first-order solution given for the efficiency.     

Elastoplastic analysis with adaptive boundary element method

The purpose of this paper is to examine the capabilities provided by an iterative – adaptive mesh redesign method developed for the Boundary Integral Equation (BIE) method for elastoplastic analysis. In previous papers [1, 2] the fundamentals of the method where presented in elasticity and elastoplasticity. It was shown there that the adaptive procedure converges fast and is specially developed to reveal the ability of the BIE to provide high accuracy with very economic deployment of quadratic boundary and interior elements. In this paper the results of method applied on particular problems in elastoplasticity are compared against well known experimental solutions. The ability of the adaptive scheme to converge fast is demonstrated, but more important finding is the fact that the progressive refinement of the discretisation gives more insight into the physical problem.     

Local heating effect on film flow structure

Studies of fundamental regularities governing film flow regimes are of interest for wide range of practical problems appearing in projecting and optimization of technological plants in energetic, chemical industry and other branches of industry, including space technologies. The present work is devoted to theoretical study and numerical modeling of processes in film flow of fluid on inclined surface with local heat source. Experimental researches carried out at the Institute of Thermophysics SB RAS [1] show that the effect of thermocapillarity under certain conditions can significantly influence the regime of film flow. Forming of “roller” of fluid is observed in the experiments in the area with high gradient of film surface temperature. If the temperature (or surface tension) gradient exceeds certain critical level then the periodical 3-D flow structure appears. The main quantity of fluid is gathered in periodical streams (or “fingers”). Between the streams the thickness of film decreases significantly [2]. The authors’ previous theoretical results described 2-D regime of locally heated film flow [3, 4, 5]. Those results allow us to state the following hypothesis: 2-D flow structure becomes unstable and 3-D perturbations grow as the local arrest of liquid is achieved due to thermocapillary effect (in the frame of reference moving with the heat source) [6, 7]. The results of linear stability analysis and numerical modelling are presented.     

Detection of multiple cracks using frequency measurements

A method for detection of multiple open cracks in a slender Euler-Bernoulli beams is presented based on frequency measurements. The method is based on the approach given by Hu and Liang [J. Franklin Inst. 330 (5) (1993) 841], transverse vibration modelling through transfer matrix method and representation of a crack by rotational spring. The beam is virtually divided into a number of segments, which can be decided by the analyst, and each of them is considered to be associated with a damage parameter. The procedure gives a linear relationship explicitly between the changes in natural frequencies of the beam and the damage parameters. These parameters are determined from the knowledge of changes in the natural frequencies. After obtaining them, each is treated in turn to exactly pinpoint the crack location in the segment and determine its size. The forward, or natural frequency determination, problems are examined in the passing. The method is approximate, but it can handle segmented beams, any boundary conditions, intermediate spring or rigid supports, etc. It eliminates the need for any symbolic computation which is envisaged by Hu and Liang [J. Franklin Inst. 330 (5) (1993) 841] to obtain mode shapes of the corresponding uncracked beams. The proposed method gives a clear insight into the whole analysis. Case studies (numerical) are presented to demonstrate the method effectiveness for two simultaneous cracks of size 10% and more of section depth. The differences between the actual and predicted crack locations and sizes are less than 10% and 15% respectively. The numbers of segments into which the beam is virtually divided limits the maximum number of cracks that can be handled. The difference in the forward problem is less than 5%.     

冲击载荷作用下大挠度圆板的弹塑性动力分析

本文利用已知弹性解分析阶跃载荷作用下大挠度回板的弹塑性动力响应，提出了本征函数法。这是对文献［３，４］的进一步发展。此外在文献［３，４］中，解决大挠度问题时，采用基于卡门方程的混合法，但在本文中，采用了位移法，它在进行分析计算时更为有利。