Reflection from a semi-infinite stack of layers using homogenization

A canonical scattering problem is that of a plane wave incident upon a periodic layered medium. Our aim here is to replace the periodic medium by a homogenized counterpart and then to investigate whether this captures the reflection and transmission behaviour accurately at potentially high frequencies.We develop a model based upon high frequency homogenization and compare the reflection coefficients and full fields with the exact solution. For some material properties it is shown that the asymptotic behaviour of the dispersion curves are locally linear near critical frequencies and that low frequency behaviour is replicated at these critical, high, frequencies. The homogenization approach accurately replaces the periodic medium and the precise manner in which this is achieved then opens the way to future numerical implementation of this technique to scattering problems.     

周期性结构热动力时间-空间多尺度分析

研究一种时间-空间多尺度渐近均匀化分析方法，模拟不同的极端热和动力载荷下微尺度多 相周期性结构中热动力响应问题，并建立一个广义的波动函数场控制方程描述热动力响应. 通过引入一个放大空间尺度和两个缩小时间尺度，在不同时间尺度上获得由空间非均匀性引 起的波动效应和非局部效应. 根据高阶均匀化理论在空间和时间上进行均匀化，获得高阶非 局部函数场波动方程. 并进一步用C0连续修正了高阶非局部函数场波动方程的有限元近 似解，使问题的求解避免了对有限元离散的C1连续性要求. 并与经典的空间均匀化方法 相比较，指出了经典的空间均匀化方法的局限性，进一步以一维非傅立叶热传导和热动力问 题为例，讨论了各种情况下方法的正确性与有效性.     

Homogenization in non-linear dynamics due to frictional contact

This work is devoted to a study of the classical homogenization process and its influence on the behavior of a composite under non-linear dynamic loading due to contact and friction. First, the general problem of convergence of numerical models subjected to dynamic contact with friction loading is addressed. The use of a regularized friction law allows obtaining good convergence of such models. This study shows that for a dynamic contact with friction loading, the classical homogenization process, coupled with an homogenization of the frictional contact, enables replacing the entire heterogeneous model by a homogenized one. The dynamic part of the frictional contact must be homogenized by modifying the dynamic parameter of the friction law. Modification of the dynamic parameter of the friction law is function of the type and regime of instability. A calculation of a homogenized friction coefficient is presented in view to homogenizing the static part of the frictional contact when the friction coefficient is not constant over the contact surface. Finally matrix and heterogeneities stresses in the heterogeneous models are identified by using the relocalization process and a frictional contact dynamic analysis of a homogeneous model.     

Phononic layered composites for stress-wave attenuation

The aim is to design a layered metamaterial with high attenuation coefficient and high in-plane stiffness-to-density ratio using homogenization to calculate and optimize the dynamic effective stiffness and mass density of layered periodic composites (phononic layers) over a broad frequency band. This is achieved by: (1) minimizing the frequency range of the first pass band, (2) maximizing the frequency range of the stop band, and (3) creating local resonance over the second pass band. To verify the theoretical calculation, laboratory samples were fabricated and their attenuation coefficient were measured and compared with the theoretical results. It is observed that over 4–20 kHz frequency range the attenuation per unit length in the optimally designed composite can exceed 500 dB/m; which increases with increasing frequency. A dynamic Ashby chart, depicting attenuation coefficient vs. in-plane stiffness-to-density ratio, is presented for various engineering materials and is compared with the fabricated metamaterial to show the significance of our design. This method can be used in variety of applications for stress wave management, e.g., in addition to match the impedance of the resulting composite to that of its surrounding medium to minimize (or essentially eliminate) stress wave reflection.     

Multiscale homogenization of n-component composites with semi-elliptical random interface defects

Effective elastic characteristics of periodic multicomponent composite materials with random interface defects are studied in the paper. The defects are assumed to be semi-elliptical and lying with major semi axes along the interfaces, where minor and major semi-axes as well as the defects number are given as input random variables. The homogenization approach has a multiscale character—some algebraic approximation is used first to calculate effective elastic parameters of the interphase including all defects located at the same interface. Equations for interphase random elastic parameters are obtained using MAPLE symbolic mathematics in conjunction with probabilistic generalized perturbation method. A different homogenization method is applied at the micro scale, where the cell problem is solved numerically using the Finite Element Method (FEM) program. Since the composites considered exhibit random variations of both elastic properties and the interface defects, the overall homogenized characteristics must be obtained as random quantities, which is realized on the micro scale by the Monte-Carlo simulation. The proposed interface defects model obeys the porosity effects resulting from the nature of some matrices in engineering composites as well as the interface cracks appearing as a result of composites ageing during static or fatigue fracture.     

Higher order approximate dynamic models for layered composites

Based on a higher order dynamic approximate theory developed in the present study for anisotropic elastic plates, two dynamic models, discrete and continuum models (DM and CM), are proposed for layered composites. Of the two models, CM is more important, which is established in the study of periodic layered composites using smoothing operations. CM has the properties: it contains inherently the interface and Floquet conditions and facilitates the analysis of the composite, in particular, when the number of laminae in the composite is large; it contains all kinds of deformation modes of the layered composite; its validity range for frequencies and wave numbers may be enlarged by increasing, respectively, the orders of the theory and interface conditions. CM is assessed by comparing its prediction with the exact for the spectra of harmonic waves propagating in various directions of a two-phase periodic layered composite, as well as, for transient dynamic response of a composite slab induced by waves propagating perpendicular and parallel to layering. A good comparison is observed in the results and it is found that the model predicts very well the periodic structure of spectra with passing and stopping bands for harmonic waves propagating perpendicular to layering. In view of the results, the physical significance of Floquet wave number is also discussed in the study.     

Effective longitudinal shear moduli of periodic fibre-reinforced composites with radially-graded fibres

This paper presents a closed-form expression for the homogenized longitudinal shear moduli of a linear elastic composite material reinforced by long, parallel, radially-graded circular fibres with a periodic arrangement. An imperfect linear elastic fibre-matrix interface is allowed. The asymptotic homogenization method is adopted, and the relevant cell problem is addressed. Periodicity is enforced by resorting to the theory of Weierstrass elliptic functions. The equilibrium equation in the fibre domain is solved in closed form by applying the theory of hypergeometric functions, for new wide classes of grading profiles defined in terms of special functions. The effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions. A parametric analysis investigating the influence of microstructural and material features on the effective moduli is presented. The feasibility of mitigating the shear stress concentration in the composite by tuning the fibre grading profile is shown.     

Homogenization of acoustic waves in strongly heterogeneous porous structures

We consider acoustic waves in fluid-saturated periodic media with dual porosity. At the mesoscopic level, the fluid motion is governed by the Darcy flow model extended by inertia terms and by the mass conservation equation. In this study, assuming the porous skeleton is rigid, the aim is to distinguish the effects of the strong heterogeneity in the permeability coefficients. Using the asymptotic homogenization method we derive macroscopic equations and obtain the dispersion relationship for harmonic waves. The double porosity gives rise to an extra homogenized coefficient of dynamic compressibility which is not obtained in the upscaled single porosity model. Both the single and double porosity models are compared using an example illustrating wave propagation in layered media.     

Quantitative Stochastic Homogenization of Elliptic Equations in Nondivergence Form

We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge–Ampère equation.     

Sensitivity and randomness in homogenization of periodic fiber-reinforced composites via the response function method

The main issue this paper addresses is the derivation and implementation of a general homogenization method, including the simultaneous determination of sensitivity gradients and probabilistic moments of the effective elasticity tensor. This is possible with an application of the perturbation method based on Taylor expansion and with the effective modules method. The computational procedure is implemented using plane strain analysis carried out with the finite element method (program MCCEFF) and the symbolic computations system MAPLE. The sensitivity gradients and probabilistic moments are commonly determined on the basis of partial derivatives for the homogenized elasticity tensor, calculated using the response function method with respect to some composite parameters. They are subjected separately to a normalization procedure (in deterministic analysis) and the relevant algebraic combinations (for the stochastic case). This enriched homogenization procedure is tested on a periodic fiber-reinforced two component composite, where the material parameters are taken as design variables and then, the input random quantities. The results of computational analysis are compared against the results of the central finite difference approach in the case of sensitivity gradients determination as well as the direct Monte-Carlo simulation approach. This numerical methodology may be further applied not only in the context of the homogenization method, but also to extend various discrete computational techniques, such as Boundary/Finite element and finite difference together with various meshless methods.     

Semi-analytical method for computing effective properties in elastic composite under imperfect contact

A parallel fiber-reinforced periodic elastic composite is considered with transversely iso-tropic constituents. Fibers with circular cross section are distributed with the same periodicity along the two perpendicular directions to the fiber orientation, i.e., the periodic cell of the composite is square. The composite exhibits imperfect contact, in particular, spring type at the interface between the fiber and matrix is modeled. Effective properties of this composite for in-plane and anti-plane local problems are calculated by means of a semi-analytic method, i.e. the differential equations that described the local problems obtained by asymptotic homogenization method are solved using the finite element method. Numerical computations are implemented and comparisons with exact solutions are presented.     

Second-gradient homogenized model for wave propagation in heterogeneous periodic media

The paper is focused on a homogenization procedure for the analysis of wave propagation in materials with periodic microstructure. By a reformulation of the variational-asymptotic homogenization technique recently proposed by Bacigalupo and Gambarotta (2012a), a second-gradient continuum model is derived, which provides a sufficiently accurate approximation of the lowest (acoustic) branch of the dispersion curves obtained by the Floquet–Bloch theory and may be a useful tool for the wave propagation analysis in bounded domains. The multi-scale kinematics is described through micro-fluctuation functions of the displacement field, which are derived by the solution of a recurrent sequence of cell BVPs and obtained as the superposition of a static and dynamic contribution. The latters are proportional to the even powers of the phase velocity and consequently the micro-fluctuation functions also depend on the direction of propagation. Therefore, both the higher order elastic moduli and the inertial terms result to depend by the dynamic correctors. This approach is applied to the study of wave propagation in layered bi-materials with orthotropic phases, having an axis of orthotropy parallel to the direction of layering, in which case, the overall elastic and inertial constants can be determined analytically. The reliability of the proposed procedure is analysed by comparing the obtained dispersion functions with those derived by the Floquet–Bloch theory.     

Piezocomposite with reciprocal polarization of oriented ellipsoidal inclusions and the matrix

We consider statistically homogeneous two-phase random piezoactive structures with deterministic properties of inclusions and the matrix and with random mutual location of inclusions. We present the solution of a coupled stochastic boundary value problem of electroelasticity for the representative domain of a matrix piezocomposite with a random structure in the generalized singular approximation of the method of periodic components; the singular approximation is based on taking into account only the singular component of the second derivative of the Green function for the comparison media. We obtain an analytic solution for the tensor of effective properties of the piezocomposite in terms of the solution for the tensors of effective properties of a composite with an ideal periodic structure or with the “statistical mixture” structure and with the periodicity coefficient calculated for a given random structure with its specific characteristics taken into account. The effective properties of composites with auxiliary structures (periodic and “statistical mixture”) are also determined in the generalized singular approximation by varying the properties of the comparisonmedium. We perform numerical computations and analyze the effective properties of a quasiperiodic piezocomposite with reciprocal polarization of oriented ellipsoidal inclusions and the matrix, the layered structures with reciprocal polarization of the layers [1] of a polymer piezoelectric PVF, and find their unique properties such as a significant increase in the Young modulus along the normal to the layers and in dielectric permittivities, the appearance of negative values of the Poisson ratio under extension along the normal, and an increase in the absolute values of the basic piezomoduli.     

Micro-stress prediction in composite laminates with high stress gradients

The objective of this research is to develop a macroscopic theory, which can provide the connection between macro-mechanics and micro-mechanics in characterizing the micro-stress of composite laminates in regions of high macroscopic stress gradients. The micro-polar theory, a class of higher-order elasticity theory, of composite laminate mechanics is implemented in a well-known Pipes–Pagano free edge boundary problem. The micro-polar homogenization method to determine the micro-polar anisotropic effective elastic moduli is presented. A displacement-based finite element method based on micro-polar theory in anisotropic solids is developed in analyzing composite laminates. The effects of fiber volume fraction and cell size on the normal stress along the artificial interface resulting from ply homogenization of the composite laminate are also investigated. The stress response based on micro-polar theory is compared with those deduced from the micro-mechanics and classical elasticity theory. Special attention of the investigation focuses on the stress fields near the free edge where the high macro-stress gradient occurs. The normal stresses along the artificial interface and especially, the micro-stress along the fiber/matrix interface on the critical cell near the free edge where the high macro-stress gradient detected are the focus of this investigation. These micro-stresses are expected to dominate the failure initiation process in composite laminate. A micro-stress recovery scheme based on micro-polar analysis for the prediction of interface micro-stresses in the critical cell near the free edge is found to be in very good agreement with “exact” micro-stress solutions. It is demonstrated that the micro-polar theory is able to capture the micro-stress accurately from the homogenized solutions.     

Effective longitudinal shear moduli of periodic fibre-reinforced composites with functionally-graded fibre coatings

This paper presents a homogenization method for unidirectional periodic composite materials reinforced by circular fibres with functionally graded coating layers. The asymptotic homogenization method is adopted, and the relevant cell problem is addressed. Periodicity is enforced by resorting to the theory of Weierstrass elliptic functions. The equilibrium equation in the coating domain is solved in closed form by applying the theory of hypergeometric functions, for different choices of grading profiles. The effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions. The influence of microgeometry and grading parameters on the shear stress concentration at the coating/matrix interface is addressed, aimed at the composite optimization in regards to fatigue and debonding phenomena.     

A computational homogenization approach for the yield design of periodic thin plates. Part II : Upper bound yield design calculation of the homogenized structure

In the first part of this work (Bleyer and de Buhan, 2014), the determination of the macroscopic strength criterion of periodic thin plates has been addressed by means of the yield design homogenization theory and its associated numerical procedures. The present paper aims at using such numerically computed homogenized strength criteria in order to evaluate limit load estimates of global plate structures. The yield line method being a common kinematic approach for the yield design of plates, which enables to obtain upper bound estimates quite efficiently, it is first shown that its extension to the case of complex strength criteria as those calculated from the homogenization method, necessitates the computation of a function depending on one single parameter. A simple analytical example on a reinforced rectangular plate illustrates the simplicity of the method. The case of numerical yield line method being also rapidly mentioned, a more refined finite element-based upper bound approach is also proposed, taking dissipation through curvature as well as angular jumps into account. In this case, an approximation procedure is proposed to treat the curvature term, based upon an algorithm approximating the original macroscopic strength criterion by a convex hull of ellipsoids. Numerical examples are presented to assess the efficiency of the different methods.     

基于渐近均匀化方法的准周期梁结构等效刚度数值实现方法研究

相比周期梁结构,准周期梁结构沿轴向梯度变化,具有更大的设计自由度,能够获得更好的结构性能。由于其非均质性,一般将其均匀化为具有等效性质的均质梁结构,但现有工作很少涉及准周期梁结构等效性质的计算。本文针对由周期梁结构映射而成的准周期梁结构,通过引入雅可比矩阵,基于渐近均匀化方法推导的单胞方程及其等效性质计算列式,并建立了其单胞方程及等效刚度的有限元求解列式。该方法可以处理沿轴向变形的任意微单胞构型,数值算例验证了其正确性和有效性。     

Reiterated homogenization of a laminate with imperfect contact: gain-enhancement of effective properties

A family of one-dimensional (1D) elliptic boundary-value problems with periodic and rapidly-oscillating piecewise-smooth coefficients is considered. The coefficients depend on the local or fast variables corresponding to two different structural scales. A finite number of imperfect contact conditions are analyzed at each of the scales. The reiterated homogenization method (RHM) is used to construct a formal asymptotic solution. The homogenized problem, the local problems, and the corresponding effective coefficients are obtained. A variational formulation is derived to obtain an estimate to prove the proximity between the solutions of the original problem and the homogenized problem. Numerical computations are used to illustrate both the convergence of the solutions and the gain of the effective properties of a three-scale heterogeneous 1D laminate with respect to their two-scale counterparts. The theoretical and practical ideas exposed here could be used to mathematically model multidimensional problems involving multiscale composite materials with imperfect contact at the interfaces.     

A homogenized Reissner–Mindlin model for orthotropic periodic plates: Application to brickwork panels

This paper describes a new procedure for the homogenization of orthotropic 3D periodic plates. The theory of Caillerie [Caillerie, D., 1984. Thin elastic and periodic plates. Math. Method Appl. Sci., 6, 159–191.] – which leads to a homogeneous Love–Kirchhoff model – is extended in order to take into account the shear effects for thick plates. A homogenized Reissner–Mindlin plate model is proposed. Hence, the determination of the shear constants requires the resolution of an auxiliary 3D boundary value problem on the unit cell that generates the periodic plate. This homogenization procedure is then applied to periodic brickwork panels.A Love–Kirchhoff plate model for linear elastic periodic brickwork has been already proposed by Cecchi and Sab [Cecchi, A., Sab, K., 2002b. Out-of-plane model for heterogeneous periodic materials: the case of masonry. Eur. J. Mech. A-Solids 21, 249–268 ; Cecchi, A., Sab, K., 2006. Corrigendum to A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork [Int. J. Solids Struct., vol. 41/9–10, pp. 2259–2276], Int. J. Solids Struct., vol. 43/2, pp. 390–392.]. The identification of a Reissner–Mindlin homogenized plate model for infinitely rigid blocks connected by elastic interfaces (the mortar thin joints) has been also developed by the authors Cecchi and Sab [Cecchi A., Sab K., 2004. A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork. Int. J. Solids Struct. 41/9–10, 2259–2276.]. In that case, the identification between the 3D block discrete model and the 2D plate model is based on an identification at the order 1 in the rigid body displacement and at the order 0 in the rigid body rotation.In the present paper, the new identification procedure is implemented taking into account the shear effect when the blocks are deformable bodies. It is proved that the proposed procedure is consistent with the one already used by the authors for rigid blocks. Besides, an analytical approximation for the homogenized shear constants is derived. A finite elements model is then used to evaluate the exact shear homogenized constants and to compare them with the approximated one. Excellent agreement is found. Finally, a structural experimentation is carried out in the case of masonry panel under cylindrical bending conditions. Here, the full 3D finite elements heterogeneous model is compared to the corresponding 2D Reissner–Mindlin and Love–Kirchhoff plate models so as to study the discrepancy between these three models as a function of the length-to-thickness ratio (slenderness) of the panel. It is shown that the proposed Reissner–Mindlin model best fits with the finite elements model.