Asymptotic Homogenization Theory Research Articles

Multi-level modeling of viscoelastic microcracked masonry

• We propose a multi-level model to predict the orthotropic behavior of cracked masonry • This model relies on the coupling between Griffith’s theory and homogenization methods • Overall estimates of Cecchi & Taliercio model are softer than the Cecchi & Tralli’s ones • The Modified Maxwell model is relevant for the mortar’s creep at short and long terms • Both short and long terms, constant and time-dependent crack densities were investigated This paper provides multi-level modeling of microcracked viscoelastic masonries generally present in historical masonries or refractory linings. It is an extension of the Cecchi and Tralli (2012) or Cecchi & Barbieri (2008) and Cecchi & Taliercio (2013) works for the typical Burgers and Modified Maxwell models followed by both undamaged and microcracked masonries. For the sake of simplicity and in order to provide rigorous analytical global estimates, only the mortar is assumed to be viscoelastic and microcracked. Bricks are assumed to be undamaged, elastic or quasi-rigid. The distribution of microcracks is assumed to be isotropic. The effective behavior of the viscoelastic microcracked masonry is provided by two steps. The first one relies on the coupling between the Griffith’s brittle fracture theory and linear mean-field homogenization scheme in order to account for the effect of microcracks on the macroscopic deformation of the mortar and establishes a linear relation between apparent macroscopic stress and strain. This step allows to easily and fast determine the effective creep function of the microcracked mortar without recourse to ’complex’ or heavy numerical inversion of the Laplace–Carson transform. The second step is based on the coupling between asymptotic analysis and homogenization theory applied for a periodic masonry. The proposed models provide analytical solutions - explicit functions of the crack density parameter - for the effective orthotropic behavior of a microcracked viscoelastic periodic masonry cell. This study proves that the Cecchi & Tralli’s and Cecchi & Taliercio’s extension estimates are close and that the later are softer. Such overall properties are used to perform finite element computations on a compressed masonry panel as a first application. These models allow then the prediction of mostly stressed and deformed areas in microcracked masonry structures. This study demonstrates that modeling a mortar (at its undamaged or microcracked state) with this Burgers formulation is only suitable for a masonry with too high values of the Maxwell’s relaxation time otherwise it yields to vanishing effective properties with the increase of time and crack density leading thus to a premature collapse of the masonry. On the other hand, the Modified Maxwell model permits the masonry to preserve a certain resistance for every range value of Maxwell’s relaxation time. These conclusions are valid for masonry either with elastic or quasi-rigid undamaged bricks.

Multiscale simulation of domain switching behavior in polycrystalline ferroelectric materials

This paper presents a multiscale nonlinear finite element simulation aimed at revealing the relationship between the macroscopic hysteresis response and the microscopic crystal morphology of polycrystalline ferroelectric materials. We utilized an incremental form of constitutive law to consider changes in material properties caused by domain switching, and employed the asymptotic homogenization theory based on the perturbation method for scale-bridging of the macro- and microstructures. The proposed multiscale simulation was applied to two different ferroelectric materials, viz., barium titanate and lead titanate, both of which have a perovskite-type tetragonal crystal structure. The correlation between the macroscopic response under an external electric load and the microstructural change in crystal orientations caused by 90° and 180° domain switching was investigated. We also discussed the influence of the initial crystal orientation distribution and the difference between the two ferroelectric materials.

Asymptotic homogenization of three-dimensional thermoelectric composites

Thermoelectric composites are promising for high efficiency energy conversion between thermal flows and electric conduction, though their effective behaviors remain poorly understood due to nonlinear thermoelectric coupling. In this paper, we develop an asymptotic homogenization theory to analyze the effective behavior of three-dimensional (3D) thermoelectric composites, built on the observation that the equations governing microscopic field fluctuations in the composite are actually linear instead of nonlinear after separation of length scales. A set of solutions similar to Green's function method are used to construct the unit cell problem, and appropriate interfacial continuity conditions and boundary conditions are derived. The homogenized governing equations are then developed for thermoelectric composites, and they are further reduced for a special case wherein the heat flow and electric conduction in the composite remains one-dimensional (1D) at macroscopic scale, even though the composite itself is 3D in general. The general homogenization theory is implemented using finite element method, and a key constant in the constructed solutions is determined using the reformulated eigenvalue problem. The algorithm is validated, and is applied for a number of case studies for the effective behavior of thermoelectric composites.

Homogenization modeling of domain switching in ferroelectric materials

We presented a multiscale nonlinear finite element simulation to analyze domain switching behaviors in ferroelectric materials. We utilized an incremental form of fundamental constitutive law to consider changes in the material properties caused by domain switching. A multiscale nonlinear problem was formulated by employing the asymptotic homogenization theory based on the perturbation method and implemented using finite element analysis. The developed simulation was applied to barium titanate with a Perovskite-type tetragonal crystal structure. The 90° and 180° domain switching behaviors of a single crystal were computed for verification. The nonlinear behaviors of a bulk polycrystal with virtual microstructure were analyzed as a case study. The variation of the crystal orientation distribution in the polycrystalline microstructure was analyzed to reveal its influence on macroscopic hysteresis and butterfly curves.

Novel numerical implementation of asymptotic homogenization method for periodic plate structures

The present paper develops and implements finite element formulation for the asymptotic homogenization theory for periodic composite plate and shell structures, earlier developed in Kalamkarov (1987) and Kalamkarov (1992), and thus adopts this analytical method for the analysis of periodic inhomogeneous plates and shells with more complicated periodic microstructures. It provides a benchmark test platform for evaluating various methods such as representative volume approaches to calculate effective properties. Furthermore, the new numerical implementation (Cheng et al., 2013) of asymptotic homogenization method of 2D and 3D materials with periodic microstructure is shown to be directly applicable to predict effective properties of periodic plates without any complicated mathematical derivation. The new numerical implementation is based on the rigorous mathematical foundation of the asymptotic homogenization method, and also simplicity similar to the representative volume method. It can be applied easily using commercial software as a black box. Different kinds of elements and modeling techniques available in commercial software can be used to discretize the unit cell. Several numerical examples are given to demonstrate the validity of the proposed methods.

Two-step homogenization simulation for multidomain and multigrain structures in piezoelectric materials

In this paper, a two-step scale-up procedure based on asymptotic homogenization theory is proposed for hierarchical structures consisting of multigrains and multidomains in piezoelectric materials. Intragranular domains are modeled as a microstructure during the first-step homogenization. Then, in the second-step homogenization, an aggregate of randomly oriented grains is modeled by applying the first-step homogenized material properties of multidomains to every grain. A dual-domain structure consisting of positive and negative directional domains with a 180° orientation gap is computed for case study analysis. The three-dimensional electron backscatter diffraction-measured microstructure is employed for the multigrain structure. The effect of the domain configuration on the macroscopic homogenized material properties of polycrystalline piezoelectric materials is investigated through the two-step homogenization process. In particular, the material property changes caused by the piezoelectric effect, which cannot be estimated by the mixture law, are discussed for multigrain and multidomain structures.

Nonlinear asymptotic homogenization and the effective behavior of layered thermoelectric composites

Thermoelectric materials are promising in converting heat directly into electricity, and composite materials are often used to enhance thermoelectric figure of merit. In this work, we develop a nonlinear asymptotic homogenization theory to analyze the effective behavior of layered thermoelectric composite with coupled transport of electricity and heat. The nonlinearly coupled thermoelectric transport equations are homogenized using asymptotic analysis, from which the macroscopic field distributions are derived with local fluctuation averaged out, and overall thermoelectric conversion efficiency is established using an idealized thermoelectric module. It is discovered that the thermoelectric field distributions in the composite are different from those in a homogeneous material, and they are difficult to be fitted by homogeneous solution. Furthermore, it is noted that while the effective thermoelectric properties of the composite can be defined through a set of equivalency principle, these effective properties depend on specific boundary conditions, resulting in effective figure of merit that is not correlated with thermoelectric conversion efficiency directly. The analysis thus sheds considerable insight into the effective behavior of thermoelectric composites for their design and optimization.

Fatigue design of a mechanically biocompatible lattice for a proof-of-concept femoral stem

A methodology is proposed to design a spatially periodic microarchitectured material for a two-dimensional femoral implant under walking gait conditions. The material is composed of a graded lattice with controlled property distribution that minimizes concurrently bone resorption and interface failure. The periodic microstructure of the material is designed for fatigue fracture caused by cyclic loadings on the hip joint as a result of walking. The bulk material of the lattice is Ti6AL4V and its microstructure is assumed free of defects. The Soderberg diagram is used for the fatigue design under multiaxial loadings. Two cell topologies, square and Kagome, are chosen to obtain optimized property gradients for a two-dimensional implant. Asymptotic homogenization (AH) theory is used to address the multiscale mechanics of the implant as well as to capture the stress and strain distribution at both the macro and the microscale. The microstress distribution found with AH is also compared with that obtained from a detailed finite element analysis. For the maximum value of the von Mises stress, we observe a deviation of 18.6% in unit cells close to the implant boundary, where the AH assumption of spatial periodicity of the fluctuating fields ceases to hold.In the second part of the paper, the metrics of bone resorption and interface shear stress are used to benchmark the graded cellular implant with existing prostheses made of fully dense titanium implant. The results show that the amount of initial postoperative bone loss for square and Kagome lattice implants decreases, respectively, by 53.8% and 58%. In addition, the maximum shear interface failure at the distal end is significantly reduced by about 79%.A set of proof-of-concepts of planar implants have been fabricated via Electron Beam Melting (EBM) to demonstrate the manufacturability of Ti6AL4V into graded lattices with alternative cell size. Optical microscopy has been used to measure the morphological parameters of the cellular microstructure, including cell wall thickness and pore size, and compared them with the nominal values. No sign of fracture or incomplete cell walls was observed, an assessment that shows the satisfactory metallurgical bond of cell walls and the structural integrity of the implants.

Electron backscatter diffraction crystal morphology analysis and multiscale simulation of piezoelectric materials

A computational approach based on electron backscatter diffraction (EBSD) measurement was proposed to estimate the effects of crystal morphology on the overall response of polycrystalline piezoelectric ceramics. EBSD-measured crystal orientations of a polycrystalline piezoelectric ceramic, barium titanate, were applied to a multiscale fi nite element simulation based on asymptotic homogenization theory. First, the orientation dependence of material properties, such as elastic compliance constants, dielectric and piezoelectric strain constants, was discussed for a single-domain crystal of tetragonal perovskite structure. The computation indicated that piezoelectric strain constants are more sensitive to crystal orientation compared with other properties. Then the single-crystalline material properties were introduced into multidirectionally oriented grains in the polycrystalline microstructure, the multiscale fi nite element analysis between macrostructure and EBSD-measured microstructure was performed. In this paper emphasis was placed on the diminution of microstructure. The authors discussed about the adverse effect on each component of macrostructural homogenized material properties, which is useful for micromechanics approaches.

OS0817 均質化法に基づくマルチフェロイック複合材料のマルチスケール有限要素解析

Multiferroic composite materials, which consist of ferroelectric and ferromagnetic phases, enable us to control magnetic permeability by electric field and electric permeability by magnetic field. In order to estimate the performance and reliability of these materials, it is necessary to understand the correlation between microstructural crystal morphologies and macroscopic response. In this study we employed the asymptotic homogenization theory for scale bridging between macrostructure and microstructure. The objective of this study is to develop a multiscale finite element method to design crystal morphologies of microstructure for upgrading macroscopic homogenized material properties. As a computational example, the effect of volume fraction on homogenized material properties was investigated for a composite material of barium titanate and cobalt ferrite. The development method was verified by comparison with Mori-Tanaka theory.

Fatigue design of lattice materials via computational mechanics: Application to lattices with smooth transitions in cell geometry

A numerical method based on asymptotic homogenization theory is presented for the design of lattice materials against fatigue failure. The method is applied to study the effect of unit cell shape on the fatigue strength of hexagonal and square lattices. Cell shapes with regular and optimized geometry are examined. A unit cell is considered to possess a regular shape if the geometric primitives defining its inner boundaries are joined with an arc fillet, whose radius is 1% of the cell length. An optimized cell shape, on the other hand, is obtained by minimizing the curvature of its interior borders, which are conceived as continuous in curvature to smooth stress localization.For a given multi-axial cyclic loading, failure surfaces of metallic hexagonal and square lattices are provided along with their modified Goodman diagrams to assess the effect of mean and alternating stresses on the fatigue strength. In good agreement with the experimental data available in literature, the numerical results show that the shape of the unit cell has a major impact on the fatigue performance of the lattice material. For example under fully reversible tension, the fatigue strength of optimized square lattices of 10% relative density is shown to be 54% higher than that of their conventional counterparts with regular cell geometry.

A New Approach to Calculate the Mud Density for Wellbore Stability Using the Asymptotic Homogenization Theory

This study addresses mud density prediction as part of wellbore stability analyses considering effective rock properties and using friction and elastic theories to determine the deviator stress at failure. The effective rock properties for different sandstone porosities were obtained through the theory of asymptotic homogenization, and the wall vicinity stresses at the borehole wall were estimated considering friction and elastic theories. The use of the method of asymptotic homogenization (MHA) allows obtaining effective rock mechanical properties varying in porosity and saturation conditions.

High order asymptotics for wave propagation across thin periodic interfaces

This work deals with the scattering of acoustic waves by a thin ring that contains many regularly-spaced heterogeneities. We provide and justify a complete description of the solution with respect to the period and the thickness of the heterogeneities. Our approach mixes matched asymptotic expansions and homogenization theory.

Mechanical characterization and numerical simulation of polyether–ether–ketone (PEEK) cranial implants

Cranial implants have experienced a significant evolution in the last decade in different aspects such as materials, method of fixation, and the structure. In addition, patient-specific cranial implants have recently been started to be developed. To achieve this objective, efficient mechanical characterization and numerical modeling of the implant are required to guarantee its functionality on each patient as well as to facilitate further developments. In this work, mechanical characterization and numerical models have been performed for patient-specific Polyaryletherketone (PEEK) scaffold cranial implants. Mechanical characterization has been performed at the scaffold and the whole implant levels under displacement control tests. Two different implant designs for the same patient but with different scaffold structure were experimentally characterized, and finite element models of the implants were developed within the framework of linear elasticity. Two types of finite element models were developed: a detailed finite element model with the actual scaffold geometry, and a solid shell-like model with effective material properties. These effective material properties were obtained by means of the Asymptotic Expansion Homogenization (AEH) theory which accounts for the periodicity of the underlying structure of the material. Experimental results showed a linear response of the material and the implant up to failure, therefore supporting the use of linear elastic models for simulation. Numerical models showed excellent agreement with experiments regarding load–displacement response. Models also showed a very consistent behavior with regard to the location and the value of the maximum principal stress in the implant when subjected to the maximum load of the experiments. The two numerical models were compared. The homogenized model gave results that were very close to those obtained with the detailed model, while reducing the number of degrees of freedom by 90%, and therefore the overall computational burden. The results showed that the models are able to reproduce experimental results conducted on actual implants, offering a valid alternative to be used in the design of customized cranial implants with a scaffold structure.

The Asymptotic Homogenization Method of the Settlement Calculation of the Cement-Soil Pile Composite Foundation

The finite element calculation model of the cement-soil pile composite foundation based on asymptotic homogenization theory has been built, under the state of three-dimensional stress, the equivalent elastic parameter of the composite elastic foundation model has been calculated, and depending on which, the settlement quantity has been calculated. Analysis through comparison with the testing result shows that it is feasible to use the homogenization method for calculating the settlement quantity of the composite elastic foundation, which sets up a theoretical basis for further analysis of homogenization method about composite foundation.

Multiple spatial and temporal scales method for numerical simulation of non-Fourier heat conduction problems: Multidimensional case

A multiple spatial and temporal scales method is developed to numerically simulate the phenomenon of non-Fourier heat conduction for periodic heterogeneous materials in multi-dimensions by high-order asymptotic homogenization theory. Amplified spatial and reduced temporal scales are introduced respectively to better account for the fluctuations of the temperature field due to material heterogeneity and non-local effect of the homogenized solution. In the previous work by Zhang et al. [25], a one-dimensional case has been addressed, and the aim of the present manuscript is to extend one-dimensional solution to multidimensional case. A multidimensional high-order non-local model of non-Fourier heat conduction is derived. The relationships of homogenized heat conduction coefficients for different orders are determined and a nested finite element solution procedure is outlined for the homogenized coefficients. The validity and effectiveness of the model is demonstrated by illustrating the two-dimensional numerical examples.

THE RESEARCH PROGRESS ON MULTISCALE METHOD FOR THE MECHANICAL ANALYSIS OF COMPOSITES

Multi-scale analysismethod is a new method which considers the characters of thespatial and temporal cross-scales and cross-levels, and also therelevant scales' coupling. This method uses the overallhomogeneous material to replace the original non-homogeneousmaterials, and can also meet the requirement of strain energy ofthe two systems being completely or approximately the same. Thismethod is a very important technique for solving complex problemsin engineering practice and materials science fields. In recentyears, with the development of micromechanics and the advancementof asymptotic homogenization theory, the multi-scale analysismethod has been gradually applied to the analysis of themechanical properties of fiber-reinforced composites. This paperbriefly presents the classification of the multi-scale analysismethods and their corresponding ranges of application, withemphasis laid on its main method-homogenization theory. Then itsresearch developments in the field of mechanical properties, suchas elasticity, visco-elasticity, plasticity, failure degradation,thermodynamics and others, are given in detail. Finally, theprospect of multi-scale analysis of fiber-reinforced compositesare given.

Performance Analysis on Equivalent Elasticity of 3D 4-directional Braided Composites

Based on the small parameter asymptotic homogenization theory, this article adopts the technology of digitized cell-based finite element method (DCB-FEM) to establish the digital cell-based mechanics analysis model of three-dimensional 4-directional braided composite material, analyzes and calculates the equivalent elasticity performance of 3D braided composite material using homogenization method, and the results show a good agreement with the experimental data.

A mathematical model for bone tissue regeneration inside a specific type of scaffold

Bone tissue regeneration using scaffolds is receiving an increasing interest in orthopedic surgery and tissue engineering applications. In this study, we present the geometrical characterization of a specific family of scaffolds based on a face cubic centered (FCC) arrangement of empty pores leading to analytical formulae of porosity and specific surface. The effective behavior of those scaffolds, in terms of mechanical properties and permeability, is evaluated through the asymptotic homogenization theory applied to a representative volume element identified with the unit cell FCC. Bone growth into the scaffold is estimated by means of a phenomenological model that considers a macroscopic effective stress as the mechanical stimulus that regulates bone formation. Cell migration within the scaffold is modeled as a diffusion process based on Fick's law which allows us to estimate the cell invasion into the scaffold microstructure. The proposed model considers that bone growth velocity is proportional to the concentration of cells and regulated by the mechanical stimulus. This model allows us to explore what happens within the scaffold, the surrounding bone and their interaction. The mathematical model has been numerically implemented and qualitatively compared with previous experimental results found in the literature for a scaffold implanted in the femoral condyle of a rabbit. Specifically, the model predicts around 19 and 23% of bone regeneration for non-grafted and grafted scaffolds, respectively, both with an initial porosity of 76%.

Effective wave propagation in a prestressed nonlinear elastic composite bar

The problem of determining the effective incremental response of nonlinearly elastic composite materials given some initial pre-stress, is of interest in numerous application areas. In particular the case when small amplitude elastic waves pass through a pre-stressed inhomogeneous structure is of great importance. Of specific interest is how the initial finite deformation affects the microstructure and thus the subsequent response of the structure. Modelling this effect is in general extremely difficult. In this article we consider the simplest problem of this type where the material is a one dimensional composite bar consisting of two distinct phases, periodically distributed. Neglecting lateral contractions, the initial deformation is thus piecewise homogeneous and we can therefore determine the incremental behaviour semi-analytically, given the constitutive behaviour (strain energy function) of the phases in question. We apply asymptotic homogenization theory in the deformed configuration in order to find the effective response of the deformed material in the low frequency limit where the wavelength of the propagating waves is much longer than the characteristic lengthscale of the microstructure. We close by considering the arbitrary frequency case and illustrate how the initial deformation affects the location of stop and pass bands of the material. Work is underway to confirm these results experimentally.