On the homogenization of elastic-plastic periodic composites by the microlocal parameter approach

Summary In this paper it is shown how the nonstandard homogenization method of periodic unelastic composites leading to the microlocal parameter theories, [1], [2] can be applied to the Prandtl-Reuss elastic-plastic periodic materials with the kinematic work hardening. The problem is analysed within the theory of small elastic-plastic deformations. The cylindrical thick walled laminated tube under pressure is taken as an example of an application of the proposed approach.With 5 Figures     

A nonstandard method of modelling of thermoelastic periodic composites

The purpose of the paper is to derive from the equations of the nonlinear thermoelasticity a certain class of homogenized models of periodic composite materials. The main feature of the proposed approach is a possibility of modelling local stresses and heat fluxes with various approximations. The approach is based on the methods of nonstandard analysis. The linearized equations and an example of application to laminated materials are discussed.     

A coupled-mode theory for periodic piezoelectric composites

A coupled oscillator model to calculate the resonance spectrum of a one-dimensional piezoelectric composite plate, used in ultrasonic transducers, is proposed. Two resonant modes, one produced by the elastic wave reflection on the plate boundaries (thickness resonance) and the other by the reflection on the periodic discontinuities (lateral resonance) are considered. A Kronig-Penney model is used to calculate the lateral resonances. The thickness resonance is obtained with an effective medium model. The coupling of these two modes is described by a biquadratic equation whose solutions are the resonant frequencies of the piezoelectric composite plate. A criterion for a distribution of phases to keep the spurious lateral resonances away from the thickness resonance vicinity is obtained.     

On composites with periodic structure

The overall moduli of a composite with an isotropic elastic matrix containing periodically distributed (anisotropic) inclusions or voids, can be expressed in terms of several infinite series which only depend on the geometry of the inclusions or voids, and hence can be computed once and for all for given geometries. For solids with periodic structures these infinite series play exactly the same role as does Eshelby's tensor for a single inclusion or void in an unbounded elastic medium.For spherical and circular-cylindrical geometries, the required infinite series are calculated and the results are tabulated. These are then used to estimate the overall elastic moduli when either the overall strains or the overall stresses are prescribed, obtaining the same results. These results are compared with other estimates and with experimental data. It is found that the model of composites with periodic structure yields estimates in excellent agreement with the experimental observations.     

Micromorphic effects in a modelling of periodic multilayered elastic composites

The paper deals with the problems of modelling of multilayered periodic composites. The proposed models take into account certain micromorphic effects resulting from the periodic structure of the body. The governing equations describing motions and stresses of the nonhomogeneous elastic materials with microperiodic structure in the nonlinear as well as linear cases have been derived. As an example the time-harmonic vibrations of laminated medium consisting of alternating layers of two homogeneous, isotropic, linear-elastic materials have been discussed.     

On the modelling of dynamic behavior of periodic lattice structures

Summary. The aim of this contribution is to propose and apply a new approach to the formulation of mathematical models for the analysis of dynamic behavior of dense periodic lattice structures (space or plane trusses) of an arbitrary form. The modelling approach is carried out on two levels. First, we formulate a discrete model, represented by the system of finite difference equations with respect to the spatial coordinates. The obtained equations describe both low- and high-frequency wave propagation problems. Second, two continuum models are derived directly from the finite difference equations and represented respectively by the second- and the fourth-order PDEs with constant coefficients. These models have a physical sense provided that the considerations are restricted to the long wave propagation phenomena. The proposed approach is applied to the vibration analysis for a certain plane lattice structure. Special attention is given to the determination of the range of applicability of the continuum models.     

On finite deformation of composites with periodic microstructure

By incorporating the periodic Green function, which is obtained from the instantaneous nominal moduli of the matrix, in the method of periodic structures (MPS), and utilizing the extremum principles of Hashin and Shtrikman (1962), the overall moduli of a rate-independent elastic-plastic matrix reiforced with periodic distribution of cylindrical inclusions are obtained. The MPS, itself, is re-examined by homogenizing the quasi-static rate equilibrium equations and utilizing a periodic stress-free velocity gradient. It is shown that by evaluating the instantaneous nominal moduli of the matrix based on the velocity gradient in the matrix adjacent to the inclusion, instead of the average velocity gradient in the matrix, a lower bound to the overall moduli can be obtained. The numerical example involves analysis of the growth of circular cylindrical cavities in an isotropically hardening matrix.     

On modelling thermoelastic processes in layered composites

Summary The main aim of this contribution is to propose a 2D-mathematical model of thermoelastic layered composites. This model is based on a 3D-linearized thermoelastic continuum theory with internal kinematical and thermal constraints. A fundamental system of field equations for 3D-layered composites is reduced to a system of field equations for 2D-thermoelastic material continua, describing thermoelastic processes in composite plates and plate-like bodies.This paper has been presented at the 28^th Polish Solid Mechanics Conference, September 4–8, 1990, Kozubnik, Poland     

On the effective medium theory of subwavelength periodic structures

Abstract

The effective medium theory of one-dimensional and two-dimensional periodic structures are investigated. A method based on a Fourier decomposition of the wave propagating along the direction perpendicular to the periodic structures allows one to determine the zeroth-, first- and second-order effective indices. For one-dimensional problems, we derive closed-form expressions of the effective indices for both TE and TM polarization. Our result can be applied to arbitrary periodic structure with symmetric or non-symmetric lamellar or continuously varying index profiles. The theoretical predictions are carefully validated using rigorous coupled-wave analysis. For the two-dimensional case, only symmetric structures are discussed and the computation of the zeroth-, first-, and second-order effective indices requires the inversion of an infinite matrix which can be truncated and simply solved numerically. The EMT prediction is qualitatively validated using rigorous computation for small period-to-wavelength ratios. It is shown that for large period-to-wavelength ratios near the cutoff value, no analogy between 2-D periodic structures and homogeneous media holds for highly modulated lamellar gratings.     

On crack problems in periodic two-layered elastic composites

This paper deals with the two-dimensional static problems of the interface crack in a periodically layered space. Within the framework of the homogenized model of the linear elasticity with microlocal parameters [19, 20] the exact solutions of the considered problems are obtained. The stress singularities at the crack tips are discussed in detail from the viewpoint of the fracture theory.

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Résumé On traite du problème statique à deux dimensions d'une fissure d'interface dans un espace comportant des couches périodiques. On obtient les solutions exactes au problème considéré en recouvrant au modèle homogène d'élasticité linéaire à paramétres microlocaux. On discute dans le détail les singularités de contraintes aux extrémités de la fissure, du point de vue de la théorie de la rupture.

     

On the three-dimensional interface crack problems in periodic two-layered composites

Within the framework of the linear thermoelasticity with microlocal parameters the solutions of some three-dimensional interface crack problems in a two-layered periodic space are obtained. Effective results have been achieved through the use of similarity in governing equations of the homogenized model for the laminated body and transversely isotropic elastic solid. Some specific examples concerning interface crack problems have been considered and discussed from the point of view of fracture theory.     

On the problem of modelling for parameter identification in distributed structures

Structures are often characterized by parameters, such as mass and stiffness, that are spatially distributed. Parameter identification of distributed structures is subject to many of the difficulties involved in the modelling problem, and the choice of the model can greatly affect the results of the parameter identification process. Analogously to control spillover in the control of distributed-parameter systems, identification spillover is shown to exist as well and its effect is to degrade the parameter estimates. Moreover, as in modelling by the Rayleigh–Ritz method, it is shown that, for a Rayleigh–Ritz type identification algorithm, an inclusion principle exists in the identification of distributed-parameter systems as well, so that the identified natural frequencies approach the actual natural frequencies monotonically from above.     

Textile composites: modelling strategies

Textile materials are characterised by the distinct hierarchy of structure, which should be represented by a model of textile geometry and mechanical behaviour. In spite of a profound investigation of textile materials and a number of theoretical models existing in the textile literature for different structures, a model covering all structures typical for composite reinforcements is not available. Hence the challenge addressed in the present work is to take full advantage of the hierarchical principle of textile modelling, creating a truly integrated modelling and design tool for textile composites. It allows handling of complex textile structure computations in computer time counted by minutes instead of hours of the same non-linear, non-conservative behaviour of yarns in compression and bending. The architecture of the code implementing the model corresponds to the hierarchical structure of textile materials. The model of the textile geometry serves as a base for meso-mechanical and permeability models for composites, which provide therefore simulation tools for analysis of composite processing and properties.     

Effective elastoplastic properties of the periodic composites

The article presented deals with the homogenization of composite materials with elastoplastic constituents. The transformation field analysis (TFA) approach is presented and applied to compute the effective nonlinear behavior of multicomponent periodic composite structure. Computational implementation of the method consists in special utilization of the program ABAQUS, which makes it possible to homogenize n-component periodic composites with relatively general configuration of the periodicity cell. Numerical example of homogenization of a three-component periodic composite shows the comparison between the nonlinear behavior of a real composite and of a homogenized one in a specific boundary problem defined on its representative volume element (RVE).     

Numerical modelling of woven composites: Biaxial loading

A numerical model of a 2 × 2 twill reduced Unit Cell was developed. The elasto-plastic response of the matrix and possibility of tow/matrix debonding were modelled. The material is assumed to fail after failure of the tows is detected using maximum-stress and/or physically-based failure criteria. The support provided by the adjacent layers was accounted for considering two idealised cases of support: In-Phase (IP) and Out-of-Phase (OP); which bound the support a given layer can have within a laminate.The difference in results between IP and OP provides an estimate of the expected variation in strength due to the support. Additionally, the failure strength predictions obtained averaging the IP and OP results agree well with experimental results for uniaxial tension/compression.The in-plane biaxial failure envelope was also determined. A strong correlation was found between the loading ratio and the effect of support provided by the adjacent layers on the failure strength.     

Two-dimensional modelling of solid-fluid composites

The purpose of this paper is to apply biomimetic-designed composites to artificial structures. From the results of numeric modelling analysis in biomechanics, we have learned the bone structures optimized to lighten weight and understood that the solid-fluid composite structure of the cancellous bone at the joint part works to distribute the joint load perfectly. In this paper, the two-dimensional honeycomb structure filled with fluid was investigated by way of a simplified solid-fluid composite material model of the cancellous bone. Hybrid finite element analyses illustrated that the solid-fluid phase interaction is effective in dispersing compressive load. In-plane indentation tests were carried out and in-plane deformation distributions of the solid-fluid composite specimens were measured. Consequently, as for the solid-fluid composite specimens whose cells were filled up with glycerine, a good enough cell deformation mode was obtained.     

Analytical modelling of laminated composites

A significant number of research papers has been published on the analytical modelling of composite laminates over the past 20 years. The drive for more accurate analysis has led us to techniques which have become computationally more and more burdensome, while the engineering world continues to use simple, first-older shear deformable plate theory as its primary tool. This paper presents a unique approach to the analysis of thick laminated composites by presenting two simple finite element methods. The first uses the Predictor Corrector technique to extend the simple Mindlin type element to achieve greater accuracy, and the second develops a new Least Squares element which can approximate a C¹ continuous element. The Least Squares element has the capability to incorporate a simplified higher order basis into a piecewise continuous displacement field creating an accurate, yet computationally simple, element. These two methods have the potential to upgrade analysis methods significantly with little additional computational cost. It is hoped that this work can instigate further research into efficient modelling of composite laminates.     

Lattice modelling of the failure of particle composites

Three-phase particle composites are modelled by a beam lattice. The second, hard phase (particle) is a linear elastic material. However, to improve the representation of the lattice model, tension softening of the matrix and/or the interface between the particle and the matrix is introduced. Moreover, as each beam element in the lattice is short and deep, instead of using the Euler-Bernoulli theory, beam elements based on the Timoshenko theory are introduced to enhance accuracy. Finally, the effect of finite deformations is also investigated due to the large displacements and/or rotations likely to be involved with the evolution of the damage. Numerical results for particle composites are presented and discussed.     

Extraction of the SAW attenuation parameter in periodic reflecting gratings

In this paper, the extraction of the coupling-of-modes (COM) model attenuation parameter gamma in a finite grating is considered. We use test structures comprising identical transmitting and receiving transducers and a grating centered in the acoustic channel along the propagation direction of the surface acoustic wave (SAW). The extraction procedure proposed is based on studying the magnitude of the ratio of the reflection and transmission coefficients of the grating, R/T, obtained through time gating from the S parameter measurements of the test devices. In particular, we found that the level of the notches of R/T directly depends on the attenuation of SAW in the grating. A simple closed-form expression for the attenuation normalized to the grating length, gamma lambda0, depending on the characteristics of absolute value R/T, is given. The proposed method is applied to the measurement data for selected grating topologies to yield estimates of the attenuation.     

Overall behavior of composites with periodic multi-inhomogeneities

When applying the equivalent inclusion method (EIM) to a composite material with non-dilute distribution of reinforcement particles, due to the complex interaction between the particles, the homogenizing eigenstrain field will in general be highly nonlinear. The interaction becomes more complex, when the reinforcements are multi-phase particles, i.e., the core inhomogeneity is surrounded by many layers of coatings. In this paper, a treatment for an accurate determination of the distribution of homogenizing eigenstrain fields corresponding to composites with non-dilute periodic distribution of multi-phase reinforcement particles is given. The proposed method is applicable to problems, where the reinforcement particles have: very thick coatings; functionally graded (FG) coatings; or coatings with variable thicknesses. Strong dependence of the overall response of composites on the microstructure of their reinforcement particles is well recognized. The theory is extended to estimate the effective elastic moduli of such composites.