Asymptotic Homogenization Method Research Articles

Morphology influence on the acoustics of permeo-elastic media

This paper deals with the acoustics of permeo-elastic media. These media consist of a rigid porous skeleton on which flexible elastic films are fixed and interact with the fluid saturating the pores. They can correspond to foams, the thin membranes playing the role of films and the rigid structure being formed by the struts. The acoustics of permeo-elastic media can deviate from that of conventional porous media due to the fluid/film interaction at the pore scale. This introduces an elastic component in the energy that interacts with viscous and inertial effects. The actual behavior is established by the two-scale asymptotic homogenization method. It is shown that the film effects depend strongly on the pores and the film arrangement. Different wave propagation regimes are identified depending on the nature of the local flow. Consider- ing (i) fully connected, i.e., no pores are closed by the films, (ii) unconnected, i.e., all pores are closed, and (iii) partially connected, i.e., some pores are closed by the films, the study re- veals under which conditions the fluid-film interaction significantly influences the macroscopic acoustic behavior. Depending on the morphology, the permeability or compressibility (or both) can exhibit singularities or anomalous properties due to inner resonance or anti-resonance.

Coupled micro-macroscopic modeling of layered composites with finite deformations

A coupled micro- and macroscopic modeling of an incompressible layered composite material under finite deformations is performed based on the asymptotic homogenization method and universal semi-linear models. The deformation diagrams of the periodicity cell are calculated. A method for searching for effective transversally isotropic properties of LCM based on the obtained diagrams is considered. Cylindrical bending of an LCM plate is simulated using the model of a transversally isotropic medium. The microstresses in the periodicity cell are calculated based on the homogenized stresses.

Modeling the edge effect in composites based on asymptotic homogenization method

The purpose of this work is to develop a numerical algorithm for solving the problem of calculating the stress-strain state of structures made of composite materials, taking into account edge effects. The solution of the problem is carried out by modifying the asymptotic averaging method by taking into account an additional solution of the boundary layer type, which quickly fades away from the boundary. This solution makes a significant contribution to the neighborhood of the composite surface itself. A numerical example of calculating an element of a composite structure with holes is given, which showed the feasibility of the proposed method. The calculations are compared with a three-dimensional finite element solution in the ANSYS software package.

An Asymptotic-Homogenization Explicit Time-Domain Method for Random Multiscale Vibration Analysis of Porous Material Structures

Porous material structures have been widely used in civil engineering, mechanical engineering, aerospace engineering and other fields due to their high specific strength and specific stiffness. The stochastic response analysis of porous material structures under random excitations deserves more attention. The multiscale differential governing equations of porous material structures are derived based on the multiscale asymptotic homogenization method (AHM), and the macroscale and microscale explicit time-domain expressions of structural responses are further established. On this basis, the statistical moments of dynamic responses of porous material structures under non-stationary random excitations can be achieved with the explicit time-domain method (ETDM). The purposed method combines the advantages of AHM for high-efficient explicit formulation of macroscale and microscale dynamic responses of porous material structures and the advantages of ETDM for fast analysis of non-stationary random vibration problems. A numerical example is presented to validate the computational accuracy and efficiency of the present approach for non-stationary random vibration analysis of porous material structures.

Elasticity-based locally-exact homogenization theory for three-phase composites considering the morphological effect of carbon fibers

Elasticity-based locally-exact homogenization theory (LEHT) is further extended in this paper to investigate the mechanical behaviors of three-phase composites with orthotropic interlayer as well as circumferentially orthotropic (CO) and radially orthotropic (RO) carbon fibers that are determined by their morphological basal plane. Taking advantage of the Fourier series representations for displacements of each phase, in-plane and out-of-plane exact solutions are obtained for orthotropic media through rigorous mathematical derivations and the internal eigenvalue functions are directly solved within rectangular and hexagonal representative unit cells (RUCs) subjected to axial normal, transverse loading and axial shear loading, respectively. The balanced variational principle is applied to enforce periodical boundary conditions which ensures rapid convergence of displacement fields. The solutions of LEHT are verified with far-field Eshelby problems with good agreement and the morphological effects of fiber are investigated by comparing CO/RO properties and equivalent transversely isotropic (TI) parameters obtained from replacement scheme. Homogenized elastic moduli of RUCs with orthotropic interlayer are predicted through homogenization constitutive relations and compared to special extension of the composite cylinders approach and numerical results based on the asymptotic expansion homogenization method to demonstrate excellent correlation. Finally, the effects of orthotropic interlayer have been demonstrated and local stress distributions are obtained from a “top-to-down” procedure, illustrating constituents’ interactions from microscope perspective and identifying the possible crack initiations/propagations within microstructures.

Modeling of effective elastic-plastic properties of layered composites with a periodic structure in the framework of the anisotropic flow theory

The article is devoted to the development of a method for constructing theoretical strain diagrams. The method is based on the use of a model of effective constitutive relations for approximating the deformation diagrams of layered composites obtained using the asymptotic averaging method. To find the elastic constants of the model of a transversally isotropic composite, the method of minimizing the deviation of the approximation deformation diagrams from the diagrams obtained by the asymptotic homogenization (AH) method is used for a series of standard problems of deformation at small deformations. Minimization problems were solved using the Hooke-Jeeves method. The results of numerical simulation by the proposed method for layered composites are presented, which showed good approximation accuracy, which is achieved due to the proposed method for separating the coupled problems of micro- and macroscopic deformation.

Structural design of patient-specific vascular ring stents

The expediency of replacing the corrugated ring of a vascular stent with a constant corrugation pitch by a corrugated ring of variable pitch is being investigated. To describe the change in pitch of corrugation, a special function is introduced. The equilibrium and deformation equations were analyzed using the asymptotic homogenization method. The dependence of the radial stiffness of a corrugated ring with a variable pitch of corrugation on the constructive parameters is obtained. The effectiveness of the design under consideration is evaluated by the area of the lumen provided by the vascular stent. Compared to stents with regularly corrugated rings, this area is larger which improves the ring stent efficiency.

Higher-order asymptotic homogenization for piezoelectric composites

In this work, we developed a higher-order asymptotic homogenization method (AHM) for efficient multiscale analysis of piezoelectric composite structures. Instead of solving the composite structural problems directly with all heterogeneities considered, AHM first solves the microscale problem at each order of expansion to obtain the corresponding microfluctuation functions and effective properties; the latter is then passed to the macroscale to solve coupling governing differential equations, with the macroscale physical quantities passed back to the microscale and combined with microfluctuation functions to recover the local response. Governing differential equations are formulated and solved in Cartesian coordinates, which is much beneficial for understanding the multiscale-multiphysics behavior of flat piezocomposite structures such as Micro Fiber Composite transducers. Two numerical examples are solved using AHM, i.e., a 1D piezocomposite rod and a 2D piezocomposite plate, and the results are compared with direct numerical solution (DNS) to validate the accuracy and efficiency of the proposed method. The developed multiscale analysis algorithm largely saves the computational cost compared with DNS when a large number of inclusions are included in piezocomposites, thus making it an efficient tool for solving multiphysics composite structural problems with piezoelectricity.

Using asymptotic homogenization to determine effective thermo-viscoelastic properties of fibrous composites with interphase layer

In this paper, we apply the method of asymptotic homogenization in parametric space in order to determine thermo-viscoelastic properties of materials composed of cylindrical fibers surrounding by an intermediate layer separating them from viscoelastic matrix. We use the Maxwell model of viscoelasticity. The volumetric fraction and physical properties of the layer can vary. There is a possibility of a contrast between the properties of fibers, the layer, and the matrix. Our approach allows us to treat temperature as a parameter and study the response of mechanical characteristics on the change in temperature. We perform the Fourier transform of governing equations and then implement asymptotic homogenization of the resulting equations in parametric space. Microscopic boundary value problem is defined on a periodic cell consisting of a fiber surrounded by a layer and embedded in the matrix. Macroscopic equations are derived for the complex Young’s and shear moduli. Effective viscoelastic properties of the material are computed for different values of the frequency of harmonic oscillations and for various temperatures. This paper discusses how effective storage and loss Young’s and shear moduli depend on volumetric fraction and mechanical properties of the material of the layer. The obtained results can provide information on how concentration, structure and properties of inclusions can affect the effective storage and loss Young’s and shear moduli.

Multiscale structural analysis for plain weave composite plates using asymptotic homogenization method

Asymptotic homogenization method (AHM) is used to perform multiscale structural analysis for plain weave composite plates (PWCPs). First, AHM is used to obtain the effective elastic properties of yarns. Then, the effective plate properties of PWCPs are predicted using AHM. Finally, an equivalent plate model is established to perform the macroscale structure analysis, and recover the mesoscale fields. The accuracy and computational efficiency of the multiscale structural analysis are verified by comparing the results with those of direct numerical simulation. Furthermore, the influences of the geometrical parameters and laying angle of yarns on the equivalent plate stiffness are investigated.

Mechanism exploration and effective analysis method of shear effect of helically wound structures

Helically wound structures are widely used in practical engineering due to its excellent mechanical property, such as the steel rope, the reinforced armor layer of the marine flexible pipe/cables. The shear stiffness plays an important role in exactly predicting the mechanical response of the helically wound structure, especially for the short structure. There has been no general methodology to directly calculate the shear effect of this type of structure because of the geometrical complexity. This paper introduces and modifies a novel implementation of asymptotic homogenization method so-called NIAH to effectively calculate the shear property of the helically wound structure. This modification of NIAH is derived based on the strain energy equivalence of the macroscopic structure and the microscopic structure so-called the microscopic unit cell, which can be used to quickly calculate the effective stiffness and effective stress. In this paper, the effective stiffness is only discussed. The mechanical mechanism of the shear effect of helically wound structures is firstly explained, and then taking into account the shear effect, a quickly effective analysis method of the mechanical response for the helically wound structure is proposed. The efficient and accurate finite-element model of the unit cell which is used in the numeric implementation of the effective analysis method, is determined through the sensitivity analysis of meshing methods and periodic boundary conditions. Considering the practical application, the implementation of this method is validated for helically wound structures with equal-scale subcomponents and non-equal-scale subcomponents. The influence of the slenderness ratio on the shear effect is also explored in this work. This study provides a meaningful reference for the loading analysis and structural design of helically wound structures. • A method for calculating shear property of helically wound structure is proposed. • Mechanical mechanism of shear effect of helically wound structure is revealed. • The influence of the slenderness ratio of structure on shear effect is studied. • An accurate and efficient model of unit cell is determined by sensitivity analysis. • Shear property of equal-scale and non-equal-scale structures is discussed.

Two-Scale Asymptotic Homogenization Method for Composite Kirchhoff Plates with in-Plane Periodicity

This paper develops a two-scale asymptotic homogenization method for periodic composite Kirchhoff plates. In this method, a three-dimensional (3D) periodic plate problem is simplified as a Kirchhoff plate problem, which is governed by a fourth-order uniformly elliptic partial differential equation (PDE) with periodically oscillating coefficients. Then, a two-scale solution in an asymptotic expansion form is presented for the PDE, and it is found that the first-order perturbed displacement in the asymptotic solution is zero. Additionally, periodic boundary and normalization constraint conditions are proposed to determine the unique solution to unit cell problems. Moreover, standard finite element formulations for calculating the perturbed displacements are derived from the principle of virtual work. Physical interpretations of the influence functions are presented by analyzing the properties of self-balanced quasi-load vectors used for solving the influence functions. Numerical comparisons show that the present method is physically acceptable and highly accurate.

Calculation of the Single-Walled Carbon Nanotubes' Elastic Modulus by Using the Asymptotic Homogenization Method

This article uses the asymptotic homogenization method to investigate the mechanical characteristics of single-walled carbon nanotubes. The asymptotic homogenization method has been used to derive the equations for the homogenized elastic properties matrix. Then, MATLAB is used to model nanotubes and implement the finite element simulation. Two types of chiralities, including armchair and zigzag, are taken into account in this regard. Young's modulus and shear modulus have been estimated for both armchair and zigzag nanotubes in accordance with the relationships between the derived coefficients and molecular mechanical characteristics. Investigations have been done into how diameter and orientation affect the mechanical characteristics of carbon nanotubes. The research's findings are corroborated and generally consistent with those found in other articles.

On gas transport modes in matrix-fracture systems with arbitrary multiscale configurations

Tight shale reservoirs exhibit high heterogeneity and strong anisotropy in multiscale pore/fracture networks, with highly variable properties. The local equilibrium or non-equilibrium states vary spatially and are strongly controlled by the gas transport modes at each scale. A fundamental understanding of the coupling effects of gas flow in heterogeneous porous media with arbitrary scale ratios is crucial but not yet available. Here, we systematically and theoretically study the gas transport modes and gas flow velocity in multiscale matrix-fracture systems using the asymptotic homogenization method. A series of exact scaling laws for the gas velocity in heterogeneous porous media with arbitrary multiscale configurations are established, and the local equilibrium/non-equilibrium effects at each scale are analyzed in detail. It is shown that the gas transport modes between two adjacent porous media can be classified into four distinct types governed by two characteristic time scales (rather than two types as commonly reported). We demonstrate an ultrahigh pressure gradient in a thin depressurized zone in the matrix that can reach 103∼105 times the macroscopic pressure gradient, greatly increasing gas flow rates by three to five orders of magnitude. The hydraulically-created fractures not only provide preferential flow pathways, but more importantly, they increase the gas velocity in the matrix (which does not contain any fractures) by several orders of magnitude. The work also sheds light on the discrepancy between the observed high gas production and the experimentally measured low permeability in drilled cores.

Multiscale Analysis of Bi-Layer Lattice-Filled Sandwich Structure Based on NIAH Method.

Bi-layer lattice-filled sandwich structures have good application prospects for multi-physics problems; however, high-precision numerical analysis methods are lacking. Recently, the newly proposed asymptotic homogenization method called the novel numerical implementation of asymptotic homogenization (NIAH) was further developed based on the Mindlin plate theory, which is a potential method for overcoming the above limitation. This study investigates the feasibility of this method for Bi-layer lattice-filled sandwich structures. The obtained results are compared to those from homogenization methods developed based on the Kirchhoff theory, and accordingly, the influence of the shear effect on the accuracy of the structural responses of the considered structures is studied. Subsequently, the impacts of the size effect, macrostructure type, and lattice type are also considered. The analysis results showed that, for most cases, the NIAH method can yield high-precision results for Bi-layer lattice-filled sandwich structures. When the number of lattice cells is insufficient or different layers of the lattice have excessive differences in their stiffness, the accuracy of the results obtained using the NIAH method is degraded.

Isogeometric analysis-based buckling optimization framework for grid-stiffened shells using asymptotic homogenization method and Rayleigh–Ritz method

Isogeometric analysis-based buckling optimization framework for grid-stiffened shells using asymptotic homogenization method and Rayleigh–Ritz method

Morphology influence on the acoustic properties of permeo-elastic media

This paper deals with the acoustics of permeo-elastic media. These media comprise a rigid porous skeleton with attached elastic films that interact with the fluid that saturates the pores. The pore-scale fluid/film interaction introduces elastic energy that interacts with viscous dissipation and inertial effects. The corresponding non-conventional flow regimes are studied by the two-scale asymptotic homogenisation method by successively considering permeo-elastic media with (i) fully connected pores, (ii) strictly unconnected pores (i.e. pores closed by films), (iii) partially connected pores, and (iv) mixed morphologies. This analysis allows to precisely describe how the morphology both influences, quantitatively and qualitatively, the interaction of the elastic and kinetic energies of the films with the viscous and kinetic energies of the fluid, and modifies the effective properties. It is shown that, in specific frequency ranges, the film–fluid system behaves as an equivalent visco-elastic fluid and that local film–fluid resonances can be observed. Numerical examples illustrate the main features in the different cases. The presented synthetic approach enables the design of materials that exhibit unconventional conductivity, unconventional compressibility, or both, in different or the same frequency range, by means of tuning their morphological and/or mechanical parameters.

Integrated wavelet-learning method for macroscopic mechanical properties prediction of concrete composites with hierarchical random configurations

In this research, a novel integrated wavelet-learning approach with high-accuracy and efficient prediction capability is developed to predict the effective elastic properties of concrete composites with hierarchical random configurations and heterogeneous elasticity in multiple spatial scales. The significant contributions of this work include: (1) The random structural heterogeneities and material uncertainties of concrete composites in meso-scale and micro-scale are modeled on the basis of modified authors’ previous researches and Weibull probabilistic model separately. (2) Innovative stochastic three-scale asymptotic homogenization method (STAHM), background grid technique and padding strategy are presented to thoroughly acquire the doubly random geometry and material features of concrete composites with hierarchical random configurations for constructing concrete material databases. (3) The wavelet transform is utilized to preprocess the high-dimensional data features of the material databases, and the wavelet coefficients are employed as the novel input neurons for efficiently establishing more plain wavelet-neural networks that possess increasing capacity and resisting noise for approximating mappings of high complexity and nonlinearity. The numerical experiments and achieved results demonstrate that the integrated wavelet-learning approach is robust for the high-accuracy and efficient prediction of effective elastic properties of concrete materials with random heterogeneity patterns in multiple spatial scales.

Closed-form expressions for the effective properties of piezoelastic composites reinforced with cylindrical fibers by Maxwell scheme

In this work, the Maxwell scheme is implemented to characterize the effective properties of piezoelastic fiber composites reinforced by unidirectional cylindrical fibers distributed along the x3-axis. The piezoelastic composites are assumed to have transversely isotropic constituent materials. Analytical expressions for the effective properties (elastic, piezoelectric, and dielectric) are reported. The Mori–Tanaka approach is also developed for piezoelastic composites and it is shown that their analytical results coincide with the Maxwell scheme and the effective field method. Numerical analysis and comparisons with the two-scale asymptotic homogenization method and the semi-analytical finite element method are shown and discussed. As a summary, the novelty value of this work is that the Maxwell and Mori–Tanaka schemes have been extended to heterogeneous piezoelectric structures. Closed formulas are derived for the effective elastic, piezoelectric, and dielectric moduli. A numerical implementation of the effective moduli is carried out and the overall properties are validated with other methods.

Multiscale embedded models to determine effective mechanical properties of composite materials: Asymptotic Homogenization Method combined to Finite Element Method

A reliable way to determine the effective properties of heterogeneous media plays an important role in engineering design. As an experimental test to obtain effective properties may be tedious and expensive to perform, and as computational approaches, when all heterogeneities of the media are considered during the simulations, might become impractical due to computational effort, homogenization procedures are a good alternative to estimate the effective properties of the media, mainly considering damage and manufacturing defects in the material. Among several homogenization methods, the Asymptotic Homogenization Method (AHM), a well-established mathematically based method, is considered for deriving the relations for the effective properties of the media, and the Finite Element Method (FEM) is used to solve the equilibrium relations. In this work, a fiber-reinforced composite is considered, with three different regions depicted: intact, damaged fibers, and regions with resin/void pockets. The proposed AHM-FEM approach is used to simulate a tensile test in all scenarios with standard and multiscale embedded Unit Cells used to obtain the effective elastic modulus on the fiber direction. Discussions regarding computational efficiency are carried out, as well as numerical comparisons among the types of Unit Cells used to model the media. In a second scenario, the complete fourth-order elasticity tensor of the media is investigated, considering the fibers as both isotropic and transversely isotropic. The results for the transversely isotropic fibers approach literature data, while when isotropic fibers are considered, poor results are achieved. Finally, the approach is extrapolated to obtain the mechanical properties of metamaterials, defined here as a composite reinforced by two different kinds of fibers.