Asymptotic Homogenization Method Research Articles

Optimization design of assembly sequence for the proton exchange membrane fuel cell stacks with dimensional errors

Proton exchange membrane fuel cell (PEMFC) stacks are assembled by numerous bipolar plates (BPPs) and membrane electrode assemblies (MEAs). Component manufacturing errors are inevitably caused during the production process and brought into the stack, affecting the mechanical consistency of every cell. Aiming to improve the uniformity of the assembled stack effectively, this paper establishes a novel assembly deviation model based on the asymptotic homogenization (AH) method and proposes an assembly sequence optimization approach for the stack with dimensional errors using the genetic algorithm. The results reveal that dimensional errors significantly affect the equivalent mechanical properties of the stack. Reasonable matching of components with different types of dimensional errors is an effective method to avoid the cumulative negative impacts on the mechanical behavior of the stack. Statistically, the optimization of the assembly sequence can improve contact pressure uniformity in a 10-cell stack by more than 9.00%, while reducing the maximum contact pressure by 15.55%. This methodology is also applicable to improving the assembly uniformity of other stacked structures.

Acoustic properties of porous metamaterials featured by acoustic streaming, homogenization based modelling

We consider secondary effects induced by acoustic waves in viscous barotropic fluids saturating pores in elastic, or rigid scaffolds with elastically suspended particles. The asymptotic homogenization method is applied to derive models of the acoustic wave propagation and of the related acoustic streaming (AS) as the secondary effect. This phenomenon originating due to the nonlinearity (divergence of the Reynolds stress) in the Navier Stokes equation is retained using the perturbation analysis. This yields the first and the second order sub-problem enabling to linearize the model governing fluid dynamics. The local AS in the microstructure produces forces acting on the suspended particles, transforming the pore geometry, hence, modulating the acoustic properties. As the consequence, the permeability involved in both the subproblems is modified due to the slowly oscillating particles. The sensitivity analysis can be employed to introduce the homogenized coefficients of the model depending on the particle motion.

Acoustics of wet porous media with evaporation/condensation

This paper investigates acoustic wave propagation in wet rigid-frame porous media accounting for evaporation and condensation. At equilibrium, the solid walls are covered by a thin water film, and water vapor in the air is at its temperature-dependent saturation pressure. Small acoustic perturbations cause water to vaporize or condense, which together with the reversibility of the phase change, lead to a linear problem where the usual local poro-acoustics physics is enriched with the (i) Clapeyron relation linking liquid-wall temperature, vapor pressure, and latent heat of vaporization, (ii) latent heat transfer in the solid frame, (iii) diffusion equation for water vapor in air, and (iv) water vapor's equation of state. The equilibrium temperature highly influences the vapor concentration and the physical parameters of saturated moist air. Using the two-scale asymptotic homogenization method, it is shown that the dynamic permeability is determined similarly to classical porous media, while the effective compressibility is modified by evaporation/condensation and the equilibrium temperature. This modification is influenced by vapor mass and heat flows associated with phase changes through a local fully coupled heat transfer and water vapor diffusion problem, with specific boundary conditions at the gas–water interface. The analysis identifies dimensionless parameters and characteristic frequencies defining the upscaled model's features. Depending on equilibrium temperature, the theory qualitatively and quantitatively determines the characteristics of acoustic waves propagating through the media. The results are illustrated and discussed with analytically developed models.

Development of a numerical and analytical methodology for analyzing hybrid laminates with multi-oriented piezoelectric and structural layers

Piezoelectric materials can generate an electrical response under mechanical stress, functioning as sensors. Conversely, applying an electrical field to these materials enables precise motion control, making them effective as actuators. Using a proposed methodology, this work focuses on evaluating the effective properties of hybrid multi-oriented composite laminates consisting of structural and piezoelectric layers driven by monoclinic constitutive equations. Square unit cell model was used to calculate all coefficients of the material tensor. The finite element (FE) homogenization method and periodic boundary conditions implemented by node-to-node constraint equations are used to study a representative volume element (RVE) modeled as three-layer unit cell in the mesoscale. The pre-processing, FE analysis, and post-processing are conducted in the FE package ABAQUSⓇ through Python scripts. The computational approach yields results that align closely with the analytical effective coefficients derived from the Asymptotic Homogenization Method (AHM), demonstrating the robustness and accuracy of the proposed methodology.

Electro-mechanical coupling isogeometric analysis of static characteristics in piezoelectric composite materials based on asymptotic homogenization method

Accurate mechanical analysis is essential for reliable utilization of piezoelectric composite materials (PCMs). Isogeometric analysis (IGA) of PCMs (termed PCMIGA) based on the asymptotic homogenization method (AHM) is presented in this study and employed to investigate the static mechanical characteristics of PCMs. PCMIGA provides accurate curve representation and shorter preprocessing time, and thus demonstrates both precision and efficiency. First, AHM is utilized to calculate the effective parameters of PCMs at different volume fractions. Next, these effective parameters are combined with the basic equations and boundary conditions of PCMs to derive equations of PCMIGA based on AHM. Finally, the results from several numerical examples are compared with the reference solution to validate the convergence and precision. PCMIGA is proven to be a reliable and accurate method for analyzing the mechanical properties of PCMs.

Two-scale asymptotic homogenization analysis of piezoelectric composite materials in generalized curvilinear coordinates

This paper presents a comprehensive study on applying the two-scale asymptotic homogenization method to derive the effective properties of piezoelectric composite materials characterized by generalized periodicity in curvilinear coordinates. The methodology involves formulating the homogenization problem in a general curvilinear framework suitable for materials with complex geometric configurations. By systematically deriving the homogenized equations and effective coefficients, the paper provides a robust theoretical foundation for understanding the macroscopic behavior of piezoelectric composites under various loading conditions. The model’s versatility is demonstrated through its application to specific curvilinear coordinate systems, including the helix coordinate system and cylindrical coordinates with wavy geometry. These examples highlight the potential of the proposed approach in analyzing and designing piezoelectric materials with intricate geometries. The state-of-the-art numerical homogenization methods have also validated the proposed homogenization method, ensuring its accuracy and robustness.

Multiscale characterization of effective mechanical properties of graphene-chitosan composite aerogels

This work reports on numerical characterizations of effective mechanical properties associated with graphene-polymer composite aerogels produced using an environmentally friendly freeze-drying process. To this purpose, a multiscale approach was implemented, in which geometrical configurations were constructed based on the results of experimental characterizations. A homogenization procedure based on molecular mechanics, the Milton method, and the asymptotic homogenization method was applied. In the asymptotic homogenization method, cell problems were formulated and solved within a representative volume element using the finite element method. After validating the numerical model through experimental results from compression tests, a parametric study on the influence of microstructural parameters of the materials, such as dispersion state, aspect ratio, volume fraction of nanoinclusions, and material porosity, on the macroscopic mechanical behavior of the composite aerogels was conducted. The simulation results provided a deeper understanding of the mechanisms that enhanced the mechanical properties of such aerogels by adding various graphene derivatives, allowing for adjustments in the elaboration process to obtain materials with improved mechanical properties.

Modal Analysis of a Multi-Supported Beam: Macroscopic Models and Boundary Conditions

This paper deals with the long-wavelength behaviour of a Euler beam periodically supported by co-located rotation and compression springs. An asymptotic homogenization method is applied to derive the several macroscopic models according to the stiffness contrasts between the elastic supports and the beam. Effective models of differential order two or four are obtained, which can be merged into a single unified model whose dispersion relations at long and medium wavelengths fit those derived by Floquet-Bloch. Moreover, the essential role of rotation supports is clearly evidenced. A mixed “discrete/continuous” approach to the boundary conditions is proposed, which allows the boundary conditions actually applied at the local scale to be expressed in terms of Robin-type boundary conditions on macroscopic variables. This approach can be applied to both dominant-order and higher-order models. The modal analysis performed with these boundary conditions and the homogenised models gives results in good agreement with a full finite element calculation, with great economy of numerical resources.

Analysis of micropolar elastic multi-laminated composite and its application to bioceramic materials for bone reconstruction.

The asymptotic homogenization method is applied to characterize the effective behaviour of periodic multi-laminated micropolar elastic heterogeneous composites under perfect contact conditions. The local problem formulations and the analytical expressions for the effective stiffness and torque coefficients are derived for the centrosymmetric case. One of the main findings in this work is the analysis of the rotations effect of the layers' constitutive properties on the mechanical response of bi-laminated composites. The effects of microstructure and interfacial interactions on the composite's mechanical behaviour are captured through the independent effective moduli. Comparisons with the classical elastic case show the approach validation. Some numerical examples are shown. Furthermore, considering the micropolar media's prevalence in bio-inspired systems, the model's applicability is evaluated for reconstructing bone fractures using multi-laminated biocomposites. An important finding in this bio-inspired simulation is related to the analysis of a periodic bi-laminated micropolar composite whose isotropic constituents are a bioceramic material and a compact bone. This artificial bio-inspired material should integrate with host tissue to support cell growth and be stable and compatible. These characteristics are crucial in the enhancement of the fractured bone.

Corrigendum to “Study on equivalent mechanical properties of U-shaped bellows based on novel implementation of asymptotic homogenization method” [Reg Stud Mar. Struct. 103622, March 2024]

Corrigendum to “Study on equivalent mechanical properties of U-shaped bellows based on novel implementation of asymptotic homogenization method” [Reg Stud Mar. Struct. 103622, March 2024]

A Constitutive Law Modeling the Mixed Hardening Behavior of Particle-Reinforced Composites

Abstract A homogenized elasto-plastic constitutive law (including both constitutive equations and inversed constitutive equations) is proposed in an incremental form for the particle-reinforced composites based on the flow theory of plasticity and the asymptotic homogenization method. The constitutive law can be used to predict the mixed hardening behavior of particle-reinforced composites under arbitrary loading conditions if the uniaxial tension test curve of matrix materials is known. It is found that the constitutive law of particle-reinforced composites is similar in form to the law of matrix materials. There is a simple proportional relationship between the yield stress, the plastic modulus, and the deviatoric back stress of particle-reinforced composites and the corresponding parameters of matrix materials, which is equal to the ratio of the shear modulus of composites to the shear modulus of matrix materials. The tangent modulus of particle-reinforced composites can be calculated using a simple arithmetic formula according to the tangent modulus of matrix materials. A numerical algorithm is suggested.

Multiscale modeling of gas-induced fracturing in anisotropic clayey rocks

In the context of repositories for nuclear waste, understanding the behavior of gas migration through clayey rocks with inherent anisotropy is crucial for assessing the safety of geological disposal facilities. The primary mechanism for gas breakthrough is the opening of micro-fractures due to high gas pressure. This occurs at gas pressures lower than the combined strength of the rock and its minimum principal stress under external loading conditions. To investigate the mechanism of microscale mode-I ruptures, it is essential to incorporate a multiscale approach that includes subcritical microcracks in the modeling framework. In this contribution, we derive the model from microstructures that contain periodically distributed microcracks within a porous material. The damage evolution law is coupled with the macroscopic poroelastic system by employing the asymptotic homogenization method and considering the inherent hydro-mechanical (HM) anisotropy at the microscale. The resulting permeability change induced by fracture opening is implicitly integrated into the gas flow equation. Verification examples are presented to validate the developed model step by step. An analysis of local macroscopic response is undertaken to underscore the influence of factors such as strain rate, initial damage, and applied stress, on the gas migration process. Numerical examples of direct tension tests are used to demonstrate the model's efficacy in describing localized failure characteristics. Finally, the simulation results for preferential gas flow reveal the robustness of the two-scale model in explicitly depicting gas-induced fracturing in anisotropic clayey rocks. The model successfully captures the common behaviors observed in laboratory experiments, such as a sudden drop in gas injection pressure, rapid build-up of downstream gas pressure, and steady-state gas flow following gas breakthrough.

Size Effect of Yielding of Particle-Reinforced Composites

Abstract This paper illustrates how particle size affects the initial yield stress of particle-reinforced composites. A formulation in a closed form is presented to demonstrate the size effect of yielding of the composites. This paper also demonstrates that there is an upper bound and a lower bound for the size-dependent yield stress with the change of particle size. This means that decreasing particle size increases its yield stress up to an upper bound. Similarly, increasing particle size decrease its yield stress up to a lower bound. In this paper the asymptotic homogenization method is used in framework of the Cosserat elasticity. A virtual “unreinforced matrix” is introduced as a reference configuration. As a numerical example, the size effect of yielding of SiCp/Al is predicted.

Study on equivalent mechanical properties of U-shaped bellows based on novel implementation of asymptotic homogenization method

As typical metal thin-walled structure, bellows are used widely in various engineering fields, especially in storage and transportation of floating liquefied natural gas (FLNG) system on the sea. While the shapes of the bellows are relatively complex with the characteristics of geometric nonlinearity in the structure, it is quite challenging to calculate basic mechanical properties of the bellows accurately through theoretical analysis. Concurrently, both numerical simulation and experiments also require high computational and economic cost. Given the typical one-dimensional periodicity of the structure of U-shaped bellows, novel implementation of asymptotic homogenization (NIAH) method was secondary developed in finite element software and unit-cell model of the whole structure with periodic boundary conditions was established, realizing equivalent analysis of the overall mechanical properties of the bellows accurately and efficiently. By comparing the NIAH equivalent results with the fine finite element model results, it was found that the relative error was within 3.00 % and the calculation cost was reduced by 40 times. Compared with the experimental results, the error of NIAH equivalent results was also less than 6.00 %, which verified the accuracy and high efficiency of the NIAH equivalent method. Furthermore, the influence of unit-cell model with different structural sizes on the prediction accuracy of the NIAH equivalent stiffness results of the bellows was also discussed. This study provides a new effective method for the design and analysis of the structure of bellows.

Prediction of Thermo-Mechanical Properties of 8-Harness Satin-Woven C/C Composites by Asymptotic Homogenization.

The elasticity matrix and the coefficients of thermal expansion (CTEs) of 8-harness satin-woven (8HS) carbon-fiber-reinforced carbon matrix (C/C) composites at high temperatures were obtained by the asymptotic homogenization method (AHM) and finite element method (FEM). By analyzing the microstructure of the 8HS C/C composites, a representative volume element (RVE) model considering a braided structure was established. The effects of the temperature and component volume fraction on the elasticity matrix and CTEs of the composites were investigated. The sensitivity of model parameters, including the size of RVE model and mesh sensitivity, were studied. The optimal calculation model was employed. In addition, the effects of the 4HS methods and 8HS methods on the elastic constants of the composites were compared. The temperature and variation in the carbon fiber volume fraction were found to have a significant impact on the elasticity matrix and CTEs of composite materials. At the same volume fraction of carbon fibers, some elastic coefficients of the 4HS composite material were slightly lower than those of 8HS composite material. This research affords a computational strategy for the accurate prediction of the themo-mechanical properties of satin-woven C/C composites.

Apparent properties of porous support structure with imperfections in metal additive manufacturing

In the laser powder bed fusion (L-PBF) additive manufacturing (AM) of metals, support structure is necessary for overhang shaped components. This support is porous with periodic 2D lattice microstructure. The purpose of this paper is to predict numerically the apparent properties of porous support made by titanium alloy, Ti-6Al-4V. One of the problems involved in L-PBF AM is the geometrical imperfections, which are also seen in the microstructure of the support. Micro-CT imaging revealed the variation in strut width and the disconnections. They were parameterized and statistically measured for two cases with different design parameter that determines the density of the 2D lattice microstructure. Based on the statistically measured data, three probabilistic microstructure models were generated per one design parameter. The asymptotic homogenization method was applied to calculate the apparent Young’s moduli, shear moduli and thermal conductivities assuming orthotropic material model. The calculated results were compared with those based on the CAD data without imperfections. Finally, the apparent properties were correlated with the design parameter so that they can be used in the process simulation in the design phase before manufacturing.

Cross-Scale Reliability Analysis Framework for LNG Storage Tanks Considering Concrete Material Uncertainty

The reliability of liquefied natural gas (LNG) storage tanks is an important factor that must be considered in their structural design. Concrete is a core component of LNG storage tanks, and the geometric uncertainty of concrete aggregate material has a significant impact on their reliability. However, owing to the significant size difference between the concrete aggregate compared to the LNG storage tank, structural analysis using an accurate finite element model that includes all the geometric characteristics of the aggregate incurs significant analytical costs. In particular, for reliability analysis requiring a large number of samples, the computational costs incurred by finite element models are infeasible. Therefore, a dual acceleration strategy based on the asymptotic homogenization method and surrogate model technology is proposed to improve the efficiency of LNG storage tank reliability analysis. In the cross-scale analysis of a LNG storage tank based on asymptotic homogenization, order reduction of the LNG storage tank analysis model was realized. Based on this, a surrogate model construction method with the aggregate fraction and mass moment as inputs was proposed to further accelerate the reliability analysis of LNG storage tanks. Subsequently, a Monte Carlo method was used to perform a reliability analysis of the LNG storage tank considering the uncertainty of the concrete aggregate geometry and distribution under the action of liquid weight and wind load. The analysis showed that the wind load has a significant influence on the safety of the design of the roof of a LNG storage tank. The directionality of the wind load has a significant impact on the distribution of the sample point response for reliability analysis and the failure mode of the LNG storage tank. Owing to the directionality of the wind load, the response distributions of the maximum displacement and maximum stress of LNG were more concentrated, and the reliability of the LNG storage tank decreased after considering the wind load. In particular, the stress reliability of the tank decreased by 5.86%. When only the liquid load was considered, the maximum displacement and stress exhibited asynchronous failure, and the two almost never occurred simultaneously. When the wind load was considered, the failure mode of the LNG storage tank was dominated by the maximum stress. Moreover, the numerical example also demonstrated that the degree of freedom involved in structural analysis, as well as the time of structural analysis can be significantly reduced. So, the proposed cross-scale analysis framework can significantly improve the efficiency of reliability analysis. The conclusions established in this study provide theoretical and methodological guidance for the reliable design of LNG storage tanks.

Auxetic Properties of the Stiffest Elastic Bodies as a Result of Topology Optimization and Microstructures Recovery Based on Homogenization Method

Herein, several semianalytical formulae for the optimal distribution of elastic moduli in 2D or 3D domains occupied by nonhomogeneous, least compliant bodies are presented and three different methods for recovering microstructures predicted by the isotropic material design method are proposed, which is a stress‐based variant of topology optimization for isotropic bodies. In all five presented variants of the topology optimization of elastic bodies, the numerical implementation of these formulae requires only the use of any algorithm of searching for the minimum of functional in multidimensions without any constraints, even box constraints on design parameters. For each variant considered, the formulae defining this functional are defined analytically. All results concern the design of composite structures that maximize their stiffness (equivalently minimize their compliance) while satisfying a certain isoperimetric condition introducing the upper limit of the available material resource. The article pays special attention to the fact that the stiffest, heterogeneous isotropic elastic material has almost always auxetic properties. Based on the asymptotic homogenization method and adopted bending‐free lattice model, an algorithm for the numerical recovering of nonhomogeneous isotropic microstructure in the entire domain occupied by the plate is proposed, with a fairly easy‐to‐implement ability to model an auxetic behavior.

Homogenization of composite materials reinforced with unidirectional fibres with complex curvilinear cross section: a virtual element approach

<p>The paper presents an augmented curvilinear virtual element method to determine homogenized in-plane shear material moduli of long-fibre reinforced composites in the framework of asymptotic homogenization method. The new virtual element combine an exact representation of the curvilinear computational geometry for complex fibre cross section shapes through an innovative two-dimensional cubature suite for NURBS-like polygonal domains. A selection of representative numerical tests supports the accuracy and efficiency of the proposed approach for both doubly periodic and random fibre arrangement with matrix domain.</p>

Influence of temperature on the elastic properties of graphene and graphene-like nanosheets based on the asymptotic homogenization method

In this paper, the effective elastic properties of hexagonal monolayers such as graphene, boron nitride, silicon carbide, and aluminum nitride are evaluated using the asymptotic homogenization method taking into account temperature changes. The effective properties are determined analytically by considering temperature, force constants, bond lengths, and thickness, using atomic interactions within the context of the unit cell problem. The relationship between environmental temperature and the coefficient of thermal expansion is established by considering changes in the values ​​of bond stretching, bond angle bending, and torsion resistance constants at different temperatures from 0 to 1800 K. The results provided by the proposed homogeneous model are consistent with the valid findings of other researchers in the literature. It is found that these different hexagonal nanosheets exhibit isotropic temperature-dependent properties and their Young's and shear modulus decrease almost linearly as the temperature rises, considering a constant coefficient of thermal expansion. However, Poisson's ratio is found to be independent of temperature. Furthermore, the temperature-dependent elastic properties of these nanosheets become nonlinear when the variation of the coefficient of thermal expansion with temperature is taken into account. The Young and shear moduli of the nanosheets increase as the coefficient of thermal expansion decreases and vice versa. The highest elastic modulus values are obtained when the thermal expansion is close to its minimum threshold. However, it is recognized that the elastic modulus of graphene and boron nitride nanosheets is higher than that of silicon carbide and aluminum nitride at any temperature. As an alternative to lengthy computer modeling or rigorous testing, the homogeneous models provided can be used to simulate nanosheets.