Asymptotic Homogenization Research Articles

Fiber contribution on elastic properties of cellulose composites: A multiscale numerical study

AbstractMultiscale finite element analysis based on the asymptotic homogenization theory was carried out for short fiber reinforced plastics (FRP), and the effects of fiber morphologies on mechanical properties were systematically organized. Cellulose microfibers were used as the reinforcing fibers and polypropylene was used as the matrix. In order to quantify the enhancement effect of intrinsic fibers on the mechanical properties, a new indicator based on strain energy, contribution proportion of fiber (CPf) was proposed. In this paper, as the first demonstration of introducing the CPf, unidirectionally oriented short FRP was focused and elastic properties were analyzed for fundamental fiber morphologies. In terms of the fluctuation rate based on the elastic modulus of the matrix, when the volume content of fiber is 3.0 vol%, the elastic modulus of composites varied by 106%, 105%, 97.6%, and 35.1%, respectively, depending on the fiber orientation, fiber aspect ratio, fiber/matrix interface and fiber‐to‐fiber distance. Based on the CPf, fiber morphologies could be divided into two. The first factors are fiber orientation angle and fiber content, which determine the rate of change in mechanical properties with respect to the CPf. The second factors are fiber aspect ratio, fiber/matrix interface, and fiber‐to‐fiber distance, which affect the mechanical properties according to the rate of change determined by the first factors. The complicated influence of the fiber morphologies on the mechanical properties could be unified by introducing the CPf. These computations show that CPf is effective for systematic analysis and design of fiber morphologies.

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.

Modeling of the mechanical properties of fused deposition modeling (FDM) printed fiber reinforced thermoplastic composites by asymptotic homogenization

Fused deposition modeling (FDM) is a low-cost additive manufacturing method with moderate tolerances and high design flexibility. Ample studies are being undertaken for modeling the mechanical properties of FDM by using the Finite Element Method (FEM). The process technique of FDM results in anisotropic inner structures that are affected by the chosen manufacturing parameters. Moreover, composite filaments, such as fiber-reinforced polymers, have anisotropy even in filament form before FDM printing. These anisotropic effects are needed to be examined and incorporated for an adequate model. In order to speed up the design stage, we aim to prepare a practical method for simulating the mechanical properties of FDM-printed fiber-reinforced polymer composites. In this work, we computed the homogenized material properties for various fiber lengths, fiber volume percentages, and fiber orientations by asymptotic homogenization at the microscale. Then, mesoscale simulations are carried out through FEM simulations by incorporating the influences of process parameters. In this way, we demonstrate the effect of various micro- and mesostructural features on the homogenized properties step by step.

Band Gaps in Metamaterial Plates: Asymptotic Homogenization and Bloch-Floquet Approaches

In this work, we study the transversal vibration of thin periodic elastic plates through asymptotic homogenization. In particular, we consider soft inclusions and rigid inclusions with soft coatings embedded in a stiff matrix. The method provides a general expression for the dynamic surface density of the plate, which we compute analytically for circular inclusions or numerically for two-way ribbed plates. Through asymptotic homogenization, we find that band gaps related to in-plane propagating transversal waves occur for frequency intervals in which the effective surface density is negative. The same result is obtained via an asymptotic analysis of the Bloch-Floquet problem on a unit cell, showing the equivalence of the two approaches. Finally, we validate the method by comparing in several examples the predicted band gaps with those obtained from numerical Bloch-Floquet analyses on the real unit cell.

Constitutive Correlations for Mass Transport in Fibrous Media Based on Asymptotic Homogenization

Constitutive Correlations for Mass Transport in Fibrous Media Based on Asymptotic Homogenization

Multiscale optimal design for transverse magnetoelectric effect of ceramic composite

Magnetic sensors, angle sensors, energy harvesters, etc. have been developed using the transverse magnetoelectric(ME) constant, but in general, the transverse ME constant is one order of magnitude smaller than the longitudinal ME constant. Therefore, it is necessary to design and develop a heterostructure that is as effective as the longitudinal ME constant. In this study, multiscale optimal design was performed to find microscopic heterostructures that maximize the macroscopic ME effect. Asymptotic homogenization theory was employed in the scale coupling, and the microheterostructure was searched for by using the content ratio, arrangement, and polarization orientation of the constituent materials as design variables. As a result, in the case of barium titanium(BTO) for ferroelectric phase and cobalt ferrite(CFO) for ferromagnetic phase, the transverse ME constant a31 was 75.9 nNS/VC, which was a dramatic improvement over the conventional stacking structure.

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.

Homogenization of Surface Energy and Elasticity for Highly Rough Surfaces

Abstract Surface energy plays a central role in several phenomena pertaining to nearly all aspects of materials science. This includes phenomena such as self-assembly, catalysis, fracture, void growth, and microstructural evolution among others. In particular, due to the large surface-to-volume ratio, the impact of surface energy on the physical response of nanostructures is nothing short of dramatic. How does the roughness of a surface renormalize the surface energy and associated quantities such as surface stress and surface elasticity? In this work, we attempt to address this question by using a multi-scale asymptotic homogenization approach. In particular, the novelty of our work is that we consider highly rough surfaces, reminiscent of experimental observations, as opposed to gentle roughness that is often treated by using a perturbation approach. We find that softening of a rough surface is significantly underestimated by conventional approaches. In addition, our approach naturally permits the consideration of bending resistance of a surface, consistent with the Steigmann–Ogden theory, in sharp contrast to the surfaces in the Gurtin–Murdoch surface elasticity theory that do not offer flexural resistance.

Mechanical analysis of heterogeneous materials with higher-order parameters

Even though heterogeneous porous materials are widely used in a variety of engineering and scientific fields, such as aerospace, energy-storage technology, and bio-engineering, the relationship between effective material properties of porous materials and their underlying morphology is still not fully understood. To contribute to this knowledge gap, this paper adopts a higher-order asymptotic homogenization method to numerically investigate the effect of complex micropore morphology on the effective mechanical properties of a porous system. Specifically, we use the second-order scheme that is an extension of the first-order computational homogenization framework, where a generalized continuum enables us to introduce length scale into the material constitutive law and capture both pore size and pore distribution. Through several numerical case studies with different combinations of porosity, pore shapes, and distributions, we systematically studied the relationship between the underlying morphology and effective mechanical properties. The results highlight the necessity of higher-order homogenization in understanding the mechanical properties and reveal that higher-order parameters are required to capture the role of realistic pore morphologies on effective mechanical properties. Furthermore, for specific pore shapes, higher-order parameters exhibit dominant influence over the first-order continuum.

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.

Analysis of unbalanced laminate composites with imperfect interphase: Effective properties via asymptotic homogenization method

This work addresses the Asymptotic Homogenization Method (AHM) to find all the non-zero independent constants of the fourth-order elasticity tensor of a theoretically infinite periodically laminated composite. The concept of Unit Cell describes the domain, comprised of two orthotropic composite plies separated by an isotropic interphase. A general case with an unbalanced composite is considered. Thus, the coupled components of the tensor are expected. Both analytical and numerical solutions are derived. In addition, an interphase degradation model is proposed to evaluate its effect on the effective properties of the media. Two different stacking sequences are considered with five degrees of interphase imperfection each. The results show good agreement between the analytical and numerical solutions. In addition, it is clear that the more imperfect the interphase is, the more affected the effective properties of the media are, especially those dependent on the stacking direction.

Multi-Objective Collaborative Flexibility Optimization Design of the Strengthening Layer of a Flexible Riser

Abstract Marine flexible risers are widely used in ocean oil and gas extraction, and need to withstand environment loads (wave and current) and the large offset of the floater. Therefore, the flexible riser is subjected to tension, bending, and torsion loads, which are mainly resisted by the key strengthening layer. Small bending stiffness of a cross section of the strengthening layer with larger tension and torsion stiffness are required to be compliant with the ocean environment. The traditional design of the key strengthening layer is partially rigid with larger cross-sectional stiffnesses. Therefore, the innovative configurations of the strengthening layer are imperative to make sure that the flexible riser is reliable and safe during the installation and operation. The strengthening layer of the flexible riser is treated as the cylindrical shell composed of periodic unit-cell beam structures, which is a hypothetical model. The optimization design is conducted through the novel implementation of the asymptotic homogenization (NIAH) method. The multi-objective collaborative flexibility optimization formulation of cylindrical shell structure is proposed, considering the ratio of cross-sectional tensile torsion stiffness to bending stiffness of the strengthening layer as the objectives. The optimal configuration results, the helically wound structures, are obtained, which are the alternative strengthening components of flexible risers. Finally, the optimal structures are compared with the commonly used marine flexible riser, which gives a great verification of the methodology feasibility and explains why the strengthening layer is designed as the type of helically wound structure.

Applications of the asymptotic homogenization to materials with periodic and non‐periodic micro structures

AbstractThe first part of the paper presents the application of the asymptotic homogenization for determining the effective elastic and elasto‐plastic properties and stress concentrations in the B4C/2024Al composite using 3D x‐ray images of the internal structure. For comparison, calculations were performed for a dispersed composite with an aluminum matrix, in which boron carbide inclusions were simulated by ellipsoids. The elasto‐plastic behavior of the real structure of the B4C/2024Al composite was investigated in computational experiments on uniform compression, uniaxial tension, and a combination of pure shear and uniform compression. The calculation results were compared with experimental data. The second study concerns the application of asymptotic homogenization to metamaterials. Metamaterials may have unusual behaviour such as tension–bending coupling or tension–torsion coupling. Most metamaterials have a periodic structure in Cartesian coordinates. We studied tension‐compression‐shear coupling provided by the metamaterial using second order asymptotic representation.

Thermo-mechanical analysis of wood through an asymptotic homogenisation approach

In the last decades the use of wood as a construction material has been steadily increasing. Among the main reasons behind this, are its renewable resource nature and its low environmental footprint. In this context, one of the main challenges faced by engineers during the design process is the knowledge and characterization of wood’s thermo-mechanical properties. This is related to the large morphological variations present at the microstructural level, that lead to a wide scatter of the macroscopic properties. To circumvent this issue, in this work a multiscale modelling strategy based on asymptotic homogenisation is proposed. The model is based on the hierarchical nature of wood and incorporates the three material scales generally identified in soft woods: (i) the microfibril scale, (ii) the wood cell scale, and (iii) the growth ring scale. The effective thermo-mechanical macroscopic properties are obtained by sequentially applying the homogenisation procedure from the microfibril scale all the way up to the macroscopic scale. The model is employed here to investigate the thermo-mechanical response of radiata pine grown in Chile. To determine values of the microstructural parameters that yield macroscopic properties consistent with those observed experimentally, a parameter identification strategy is proposed. The latter considers four elements: an existing experimental database on timber boards density and bending tests, the multiscale model, a timber board bending test finite element model and a genetic algorithm for the optimization procedure. With the resulting microstructural parameters the model is then used to estimate the effective elastic, thermal, and thermo-mechanical properties of radiata pine wood. When compared with measured experimental data and typical experimental values found in the literature, the numerical estimates demonstrate the model predicting capabilities.

Higher order asymptotic homogenization for dynamical problems

In general, asymptotic homogenization methods (AHMs) are based on the hypothesis of perfect scale separation. In practice, this is not always the case. The problem arises of improving the solution in such a way that it becomes applicable if the inhomogeneity parameter is not small. Our study focuses on the higher order asymptotic homogenization for dynamical problems. Systems with continuous and piecewise continuous parameters, discrete systems, and also continuous systems with discrete elements are considered. Both low-frequency and high-frequency vibrations are analyzed. For low-frequency vibrations, several approximations of the AHM are constructed. The influence of the boundary conditions and the system parameters is investigated. The accuracy of the resulting approximation is assessed in numerical simulations.

Asymptotic homogenization of metamaterials elastic plates

The asymptotic homogenization technique is applied to evaluate the effective properties of thin plates with periodic heterogeneity. The effect of shear deformation in the homogenization process is evidenced and the role of cell slenderness, besides that of the plate, is clarified by several numerical analyses.

Asymptotic homogenization approach for anisotropic micropolar modeling of periodic Cauchy materials

A micropolar-based asymptotic homogenization approach for the analysis of composite materials with periodic microstructure is proposed. The upscaling relations, conceived to determine the macro-descriptors (macro displacement and the micropolar rotation fields) as a function of the micro displacement field, are consistently derived in the asymptotic framework. In particular, the micropolar rotation field is expressed in terms of the microscopical infinitesimal rotation tensor and perturbation functions. The micro displacement field is, in turn, obtained by choosing a third order approximation of the asymptotic expansion, in which the macroscopic fields are expressed as a third order polynomial expansion. It follows that the macro descriptors are directly related to both perturbation functions and micropolar two-dimensional deformation modes. Furthermore, a properly conceived energy equivalence between the macroscopic point and a microscopic representative portion of the periodic composite material is introduced to derive the consistent overall micropolar constitutive tensors. It is pointed out that these constitutive tensors are not affected by the choice of the periodic cell. Moreover, in the case of vanishing microstructure the internal-length-scale-dependent constitutive tensors tend to zero, as expected. Finally, the capabilities of the proposed approach are shown through some illustrative examples.