Asymptotic Homogenization Method Research Articles

Effective Inertia Coefficients Prediction and Cell Size Effects in Thickness Direction of Periodic Composite Plates

This paper develops a new homogenization method for free vibration problems of periodic composite plates. In this new method, three-dimensional (3D) periodic plates are equivalent to Reissner–Mindlin plates with both effective stiffnesses and effective inertia coefficients. The effective stiffnesses for the dynamic problems are the same as those for the static problems, and they can be achieved by the equivalence principle of macro- and microscopic internal virtual work. To fully take the inertia effects into account, the effective inertia coefficients including the effective translational, translational–rotational and rotational inertias are determined by the two-scale equivalence principle of kinetic energies under three rigid modes. In addition, cell size effects in the thickness direction of composite plates are investigated by using the proposed method and the asymptotic homogenization method (AHM). Numerical experiments validate the effectiveness of the proposed equivalent method for different scale factors, and show that the rotational inertia cannot be ignored for out-of-plane deformations, especially for higher-order modes. Besides, numerical comparisons show that the cell size effects are not negligible when using the AHM to analyze the out-of-plane deformations, and three or more repeated unit cells in the thickness direction are required to ensure accuracy.

Stretching-dominated truss lattice materials: Elastic anisotropy evaluation, control, and design

In this paper, an analytical homogenization formula is derived based on the numerical representative volume element (RVE) method and asymptotic homogenization (AH) method for the effective elastic properties of truss lattice materials. Compared with the numerical homogenization method, the analytical homogenization formula can treat the geometric parameters of the truss structure, the elastic properties of the base material as well as the loading direction as independent variables without repeated modelling. Compared with the Gibson and Ashby’s meso-mechanics model, the analytical homogenization formula can not only obtain the same calculation results, but also can conveniently evaluate and control the elastic properties of the stretching-dominated compound truss lattice structure with complex truss configuration and multi-material composition. According to the anisotropy and deformation characteristics of the elementary trusses, design methodologies for stretching-dominated compound truss lattice structures with controllable anisotropy and optimal isotropic elasticity are proposed. Based on the analytical homogenization formula, the elastic isotropy conditions of the stretching-dominated compound truss lattice materials can be obtained. In the end, the designed 3D elastically isotropic truss lattice structures and 2D bi-material elastically isotropic lattice structures are manufactured and tested to verify the correctness of the isotropy conditions and calculation results obtained by the analytical homogenization formula.

Topology optimization of coated structures with layer-wise graded lattice infill for maximizing the fundamental eigenfrequency

Combining the advantages of lattice infill and coating, coated structures with lattice infill are widely used in engineering to achieve light-weight structural characteristic or certain functionalities. This paper presents a novel concurrent multiscale topology optimization (TO) framework to maximize the fundamental eigenfrequency of structures with a uniform outer coating and layer-wise graded lattice infill microstructures. The macroscale topology optimization is conducted by using the velocity field-based level set method, which inherits the implicit geometrical representation and signed distance property of the conventional level set method (smooth and clear boundaries and well-maintenance of the uniform thickness of the coating). Besides, the employment of general mathematical programming algorithms enables the velocity field-based level set method to handle multiple constraints in an easier way. At microscale, the popular density-based method is used to design layer-wise graded lattice structures. The effective material properties of microstructures are computed by using the asymptotic homogenization method, which bridges macroscale and microscale designs. With higher design flexibility, it is found that coated structures with layer-wise graded lattice infill have higher eigenfrequencies than those with periodic and uniform lattice infill microstructures. The influence of layers of lattice infill and coating thickness on the concurrent optimization of macrostructures and microstructures are carefully studied by showcasing several numerical examples, and the effectiveness of the proposed method is also confirmed, as well as the manufacturability of optimization results.

A Multiscale Study of CFRP Based on Asymptotic Homogenization with Application to Mechanical Analysis of Composite Pressure Vessels.

The application of composites is increasingly extensive due to their advanced properties while the analysis still remains complex on different scales. In this article, carbon fiber reinforced polymer (CFRP) is modeled via asymptotic homogenization employing a representative volume element (RVE) with periodic boundary conditions. A multiscale mechanical model of CFRP is established to bridge the microscopic model, mesoscopic model, and macroscopic model. According to asymptotic homogenization, the coefficients of the material constitutive equation are calculated with volume-averaged stress and strain. Using the homogenized materials properties of CFRP, the tensile experiments of composite layers with the layout of [] are carried out to validate asymptotic homogenization method. The results indicated that the asymptotic homogenization approach can be used to calculate the homogenized elastic moduli and Poisson’s ratio of the whole structure, where the numerical results are basically consistent with test data. The sequent homogenized CFRP laminate model is applied to the mechanical analysis of type III composite pressure vessels, whereby burst pressure is accurately predicted. This work might shed some light on multiscale analysis of composite pressure vessels.

Prediction of effective properties for multilayered laminated composite with delamination: A multiscale methodology proposal

In the manuscript, piezoelectric composites with non-uniform imperfect contact at the interface are studied. For this, the two-scales asymptotic homogenization method is used to determine the effective properties of said piezocomposite. First, a theoretical framework is offered to determine the effective properties of piezoelectric materials with generalized periodicity. In addition, a method to solve the local problems for the particular case of stratified media with imperfect contact conditions is described. In the second part, the above-described methodology is used to study different numerical examples and compare the results taking into account distinct contact conditions at the interfaces. Examples include wavy composites with uniform imperfect contact, laminate composite with cracks at the interface, and non-uniform imperfect contact conditions elastic and piezoelectric composites. Also, the influence on the average properties of the small geometrical parameter related to the size of the periodic cell in the homogenization is illustrated. To validate the model, some of the results are compared with the values obtained by the finite elements method (FEM).

A multiscale approach to predict the effective conductivity of a suspension using the asymptotic homogenization method

This work aims to implement the asymptotic homogenization method (AHM) to predict the effective thermal/electrical conductivity for suspensions with aligned inclusions. Exploiting the substantial separation of length scales between the macroscopic and microscopic structures, multiscale modeling using the AHM capitalizes on the perturbations of the potential field caused due to the presence of an inclusion under a macroscopic loading used to predict the effective property. The analytical formulation for the thermal/electrical conductivity problem is derived, and subsequently, the finite element formulation required to solve the unit cell problem is described. The results obtained for a cylindrical inclusion are validated against known analytical solutions for both the dilute [Mori–Tanaka (MT)] and concentrated volume fractions (ϕ) of the inclusion. This study revealed that MT estimate and AHM agree well at ϕ less than 0.4. However, in near-maximum packing fractions, the AHM results fared significantly better than MT when compared with known asymptotic forms [J. Keller, “Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders,” J. Appl. Phys. 34, 991 (1963)]. The proposed AHM method is then implemented in structures with aligned spheroidal inclusions of various aspect ratios and conductivity ratios, thus providing a more generalized approach to predict the effective thermal/electrical conductivity. The results obtained are systematically benchmarked and validated against known analytical expressions.

Investigation of concurrent multiscale topology optimization for designing lightweight macrostructure with high thermal conductivity

Lightweight and high heat conductive solid structures are playing important role in various fields of engineering. To maximize the design performance for such structures, we investigated multiscale topology optimization for excessive lightweight heat-conductive porous structures and introduced a mathematical optimization model formulation for concurrently optimizing the structures (macrostructure) and the constitutive pores (microstructure). The microscale is considered a representative volume element and designed using the asymptotic homogenization method. For each iteration, the effective heat conductivity tensor of the microstructure is evaluated during the optimization process and used as the heat conductivity of the macrostructure. Sensitivity analysis on this concurrent optimization scheme was derived to address the macro and microstructure coupling. To broaden the scope of the research applicability, three topology optimization methods, i.e., SIMP, level set and ESO are investigated, and the results are compared and discussed. The suggested formulations showed a successful application of the concurrent multiscale optimization formulations and good coupling on the macro and microscale. Also, the formulations demonstrated a strong influence between the macro and the microscale of the design problem for the topology optimization methods. Increasing the design freedom by introducing various microstructures for the macro design domain showed superior performance associated with attaining high weight reduction. In addition, the concurrent optimization scheme has enabled the microstructures to attain a good spatial layout of materials while taking into account the weight reduction constraint. The spatial arrangements of the designed microstructures have achieved conducting heat in a shorter path toward the heatsink zone of macrostructure design. This allows attaining good performance with high weight reduction. Furthermore, numerical examples of different mesh numbers were used to study mesh dependency of multiscale topology optimization. The scope of the study was broadened by the inclusion of 3D case studies. Implementing Isosurface technique to achieve high detail model was also used to attain high detailed concurrently optimized design with minimal mesh number to minimize the computational cost. The 3D optimized case was investigated experimentally.

Analytical Solution of Thermo-Mechanical Properties of Functionally Graded Materials by Asymptotic Homogenization Method.

In this work, a general mathematical model for functionally graded heterogeneous equilibrium boundary value problems is considered. A methodology to find the local problems and the effective properties of functionally graded materials (FGMs) with generalized periodicity is presented, using the asymptotic homogenization method (AHM). The present models consist of the matrix metal Mo and the reinforced phase ceramic ZrC, the constituent ratios and the property gradation profiles of which can be described by the designed volume fraction. Firstly, a new threshold segmentation method is proposed to construct the gradient structure of the FGMs, which lays the groundwork for the subsequent research on the properties of materials. Further, a study of FGMs varied along a certain direction and the influence of the varied constituents and graded structures in the behavior of heterogeneous structures are investigated by the AHM. Consequently, the closed–form formulas for the effective thermo–mechanical coupling tensors are obtained, based on the solutions of local problems of FGMs with the periodic boundary conditions. These formulas provide information for the understanding of the traditional homogenized structure, and the results also be verified the correctness by the Mori–Tanaka method and AHM numerical solution. The results show that the designed structure profiles have great influence on the effective properties of the present inhomogeneous heterogeneous models. This research will be of great reference significance for the future material optimization design.

Hierarchical heterogeneous one-dimensional problem in linear viscoelastic media

We address the homogenisation of a linear viscoelastic and hierarchical composite material in a one-dimensional (1D) framework via a three-scale asymptotic homogenisation method. We consider a family of heterogeneous problems with periodic, rapidly-oscillating and piece-wise coefficients that model a structure with two hierarchical levels of organisation. Here, we assume continuity contact conditions at the interfaces among the constituents and set a straightforward geometrical configuration in order to gain a better insight of the multiscale problem. The main goal is to provide a general overview of the procedure and validate the approach by means of a comparison between the solution of the original heterogeneous problem, the homogenised problem and the formal asymptotic solution. In addition, we show that the three-scale approach presents a clear improvement over the recursive two-scale one, and we illustrate the convergence of the solutions towards the solution of the homogenised problem when the asymptotic parameters approach.

Finite volume based asymptotic homogenization of viscoelastic unidirectional composites

The previously developed finite volume based asymptotic homogenization theory designed for unidirectional composites with finite-sized microstructures is further extended to non-ageing linear viscoelastic region via correspondence principle. The theory is a unified framework able to accommodate general loadings, both out-of-plane and in-plane. It first transforms the problems in time domain to Laplace domain via elastic–viscoelastic correspondence principle, solves the problems in transformed region, and transforms the solutions back to time domain through numerical inversion technique. In Laplace domain, the microscale problems are solved up to the third term of expansion, and the homogenized stiffness tensors of all orders are passed to the macroscale, leading to the solutions of macroscale displacements. The method’s capability is assessed through comparison with direct numerical solution, showing increasing accuracy with increasing truncation order and decreasing scale ratio, and further enhancement by reconstructing boundary layer. The robustness of theory is verified through parametric studies. The possibility of reducing the highest order of spatial derivative of homogenized displacement is also investigated. The finite volume based methodology marks the theory’s uniqueness among the existing higher order asymptotic homogenization methods, providing an efficient alternative to solve the viscoelastic composite structural problems.

Multiscale modelling of brain tissue oxygen and glucose dynamics in tortuous capillary during ischaemia-reperfusion

Brain ischaemia causes reduction of cerebral blood flow, which interrupts the transport of oxygen and glucose to brain tissue. These transport inefficiencies have also been associated with the complex cerebral microvasculature. In this study, the importance of cerebral capillary complexity and tissue interstitial porosity in oxygen and glucose transport and metabolism during brain tissue ischaemia is investigated using an asymptotic expansion homogenization method. Applying this technique produces new macroscale governing equations with associated microscale cell problems. Solving the latter on the brain tissue microstructural geometry will obtain parametric tensors, namely the conductivities of the capillary, K and interstitial space, E; and the diffusivities of a substrate in the capillary, DM,c, and in interstitial space, DM,t. From the microscale simulations, increasing the capillary tortuosity resulted in a decrement of K and DM,c, but no significant changes in E and DM,t. Meanwhile, increasing the capillary radius resulted in the decrease of all parameters except K. Then, the parametric tensors obtained are used to solve the macroscale governing equations on a one-dimensional brain model under ischaemia-reperfusion conditions. From the macroscale simulations, changing the capillary tortuosity affects the uptake and metabolism of oxygen and glucose during brain ischaemia-reperfusion. In addition, in more tortuous capillaries, oxygen and glucose are utilized rapidly at tissues proximal to the capillary during the reperfusion process, leaving only a small amount of these substrates for distal tissues. This could be associated with severe stroke outcome. It is thus proposed that the capillary tortuosity of a patient could potentially be used as an additional indicator in determining the severity of ischaemic stroke.

Time-domain asymptotic homogenization for linear-viscoelastic composites: mathematical formulae and finite element implementation

Asymptotic homogenization (AH) is a rigorous method used for modeling multiscale mechanical behavior in periodic composites. Unlike conventional AH methods, which predict the effective viscoelasticity using the elastic–viscoelastic correspondence principle, an AH method was proposed to directly calculate the effective material properties in the time domain, thereby enabling the complex transform to be omitted between the time and Laplace domains. This time-domain AH method is reformulated based on the integral form rather than the differential form of the Kelvin–Voigt viscoelastic model. Thus, multiple time-domain-based characteristic elasticity and viscosity displacement tensors are replaced by one integral form characteristic displacement tensor in the polymer matrix composite. Using mathematical formulae, a numerical implementation method was then developed by establishing location- and time-related stress loads. An in-house Python subroutine was also programmed to compute the effective viscoelastic properties from the finite element implementation results. Unidirectional and woven fabric composite homogenization results were comparable to those obtained using the conventional representative volume element method, indicating high effectiveness and accuracy. The results suggest that the proposed time-domain AH method is a powerful tool for modeling the two-scale mechanical behavior of viscoelastic composites and is compatible with the existing commercial software.

Micro–macro asymptotic approach applied to heterogeneous elastic micropolar media. Analysis of some examples

In this work, the derivation of the effective properties for heterogeneous micropolar media with periodic structure using the two-scale asymptotic homogenization method (AHM) is reported. Analytical expressions for the local problems and the effective coefficients are explicitly described. As a particular case, periodic laminated composites are also analyzed, focusing on centro-symmetric Cosserat composites with isotropic and cubic constituents. Also, closed-form formulae of the effective properties are obtained for both constituent symmetries, and numerical values are reported and discussed. The resulting composite belongs to the orthotropic symmetry under rotations of 90°about the unitary vector e3, i.e., it has eighteen effective independent properties: nine stiffness and nine torques. As a limit case, a comparison between classical and Cosserat effective elastic properties is shown for a laminated composite with isotropic constituents. Finally, the engineering moduli of centro-symmetric laminated Cosserat materials with isotropic and cubic constituents are reported, and the numerical values are analyzed.

A general boundary layer corrector for the asymptotic homogenization of elastic linear composite structures

Asymptotic homogenization method is often used in multiscale analysis of periodic structures instead of conducting a full field heterogeneous analysis, in order to achieve computational feasibility and efficiency. When completed with a relocalization process, this method may provide relevant estimates to microscale fields within the material. Nevertheless, the construction of a solution near the boundaries remains beyond the capabilities of classical relocalization schemes due to the loss of periodicity in the vicinity of the boundaries. This paper proposes a post-processing scheme in order to conduct the relocalization step within a finite element framework for periodic linear elastic composite materials. It also assesses the boundary layer effect and a new general method, effective for various boundary conditions (Dirichlet, Neumann or mixed), is proposed based on the idea of computing corrective terms as solution of auxiliary problems on the unit-cell. These terms are finally added to the usual fields obtained from the relocalization process to obtain the corrected solution near the boundaries. The efficiency, accuracy and limitation of the proposed approach are studied on various numerical examples.

MODELING OF MECHANICAL BEHAVIOUR OF A 1D, LINEAR AND MICROPERIODIC COMPOSITE, WITH FAILURE, VIA THE ASYMPTOTIC HOMOGENIZATION METHOD

In this paper the Asymptotic Homogenization Method is combined with a model of Hyperelasticity with softening (which models the mechanical failure of the material), with the purpose of estimating the mechanical behaviour of microperiodic composites in the way to reproduces better the behaviour obtained in experiments. This hybrid approach brings some satisfactory results, showing that the (hypothetical) composite studied presents effective behaviour very close to the expected from the Hyperelasticity model. That is a very important fact because can be used as a tool to predict the failure of composite materials, trough its constituent materials information. This evaluating needs more steps, to be possible much bigger and strong conclusions.

Multiscale topology optimization of coated structures with layer-wise graded lattice infill for maximum fundamental eigenfrequency

Coated structures with lattice infill have attracted great interest from academia to industry. Functionally, the outer coating can protect its inner substrate and lattice infill structures exhibit great ability to improve certain structural performance. In this research, we propose a novel concurrent multiscale topology optimization method to design structures with layer-wise graded lattice infill covered by an outer coating of uniform thickness. The fundamental eigenfrequency is selected as the design objective to be maximized. At the macroscale, the velocity field-based level set (VFLS) method is used to achieve optimal macroscale structures with clear and smooth boundaries. Besides, making use of the signed distance property, the thickness of the outer coating can be well-maintained. At the microscale, the substrate area of coated structures is uniformly (or near-uniformly) divided into several layers to have different microstructural configurations and the popular solid isotropic material with penalization (SIMP) method is used to perform microscale topology optimization. Two different scales are bridged using the asymptotic homogenization method. Numerical results show that coated structures with layer-wise graded lattice infill we design own higher fundamental eigenfrequency than those with periodic and uniform lattice infill microstructures, due to the enlargement of the design freedom.

Acoustic wave propagation in permeable lossy metamaterials

This paper investigates acoustic wave propagation in gas-saturated permeable lossy metamaterials, which have different types of resonators, namely, acoustic and elastic resonators, as building-block elements. By using the two-scale asymptotic homogenization method, the macroscopic equations that govern sound propagation in such metamaterials are established. These equations show that the metamaterials can be modeled as equivalent fluids with unconventional effective density and compressibility. Analysis of these frequency-dependent and complex-valued parameters shows that the real parts of both can take negative values within frequency bands determined by inner resonances. The upscaled theory is exemplified with the case of a permeable lossy metamaterial having a unit cell comprising two unconnected fluid networks and a solid frame. One of these fluid networks is loaded with acoustic resonators (e.g., quarter-wavelength, Helmholtz resonators), while thin elastic films are present in the other one. It is shown that the propagation of acoustic waves in permeable lossy metamaterials is determined by both classical visco-thermal dissipation and local elasto-inertial resonances. The results are expected to lead to judicious designs of acoustic materials with peculiar properties including negative phase velocity and phase constant characteristic for regressive waves, very slow phase velocity, and wide sub-wavelength bandgaps.

AN OPEN-SOURCE MATLAB IMPLEMENTATION FOR ELASTIC ANALYSES OF HETEROGENEOUS MATERIALS USING THE EXTENDED MULTISCALE FINITE ELEMENT METHOD

The extended multiscale finite element method (EMsFEM) has been more and more widely used in the multiscale analysis of the mechanical properties of heterogeneous materials and structures, since it has no assumptions of scale separation and periodicity compared to the asymptotic homogenization method and the representative volume element (RVE) method. To further facilitate the promotion and application of the EMsFEM, this paper discourses the basic ideas, principles, and implementations of this method in detail, and releases the corresponding MATLAB code for free use by researchers. On the work of this paper, scholars can carry out further research on different multiscale computation problems. In the EMsFEM, numerical basis functions play a key role as a bridge, establishing a connection between the microscopic heterogeneous properties of materials and their macroscopic equivalent mechanical quantities. In this paper, four approaches are introduced to construct the numerical shape functions and their implementation processes are also presented. Finally, some typical numerical examples are given for verifying the effectiveness of the EMsFEM. The MATLAB code can be download from the website https://github.com/hliu-whu/EMsFEM.

Verification of asymptotic homogenization method developed for periodic architected materials in strain gradient continuum

Strain gradient theory is an accurate model for capturing the size effect and localization phenomena. However, the challenge in identification of corresponding constitutive parameters limits the practical application of the theory. We present and utilize asymptotic homogenization herein. All parameters in rank four, five, and six tensors are determined with the demonstrated computational approach. Examples for epoxy carbon fiber composite, metal matrix composite, and aluminum foam illustrate the effectiveness and versatility of the proposed method. The influences of volume fraction of matrix, the stack of RVEs, and the varying unit cell lengths on the identified parameters are investigated. The homogenization computational tool is applicable to a wide class materials and makes use of open-source codes in FEniCS. We make all of the codes publicly available in order to encourage a transparent scientific exchange.

Modeling of rubber-cord layers under quasi-static loading

Modeling a pneumatic tire with a strong change in shape causes the problem of choosing an adequate model for rubber-cord plies. Generally, the classical method of asymptotic homogenization is not suitable due to physical and geometrical nonlinearity for the strain up to 15 %. The known models used for simulation the entire tire as well as rubber-cord plies are analyzed. A choice is made in favor of modeling the plies using the stress anisotropic potential, which is an anisotropic function of the strain tensor invariants. The relationship of such a constitutive law with a quasi-linear constitutive equation in terms of stress and strain differentials is indicated. A convenience of these two types of constitutive equations in terms of numerical implementation is also given. A modification of the effective properties definition for an inhomogeneous layer is explained. The difference from the standard effective moduli definition is clarified. The arrangement of the cords is supposed to be approximately periodic and all cords to be equivalent to the effective fiber. Two models of a rubber-cord plies are described under such assumption. These are a model of an orthotropic material and a model of a transversely isotropic material. Computational experiments, which make it possible to determine the material parameters of anisotropic potentials, are pointed out. Real tests with a sample of rubber ply under slow quasi-static loading were conducted. Significant hysteresis was detected. It is shown that an additive model combining a hyper elastic material with Maxwell viscoelastic model provides good accuracy in stress dependence on the strain rate. The numerical procedure developed to calculate solution to the quasi-static problem of tire deformation is described. It is implemented in home-made computer code. A numerical example on the tire simulation is given. That is so-called breaking test, in which strong deformation is achieved and the developed model is applied to.