Asymptotic Homogenization Theory Research Articles

Unravelling Size-Dependent and Coupled Properties in Mechanical Metamaterials: A Couple-Stress Theory Perspective.

The lack of material characteristic length scale prevents classical continuum theory (CCT) from recognizing size effect. Additionally, the even-order material property tensors associated with CCT only characterize the materials' centrosymmetric behavior and overlook their intrinsic chirality and polarity. Moreover, CCT is not reducible to 2D and 1D space without adding couples and higher-order deformation gradients. Despite several generalized continuum theories proposed over the past century to overcome the limitations of CCT, the broad application of these theories in the field of mechanical metamaterials has encountered significant challenges. These obstacles primarily arise from a limited understanding of the material coefficients associated with these theories, impeding their widespread adoption. Implementing a bottom-up approach based on augmented asymptotic homogenization, a consistent and self-sufficient effective couple-stress theory for materials with microstructures in 3D, 2D, and 1D spaces is presented. Utilizing the developed models, material properties associated with axial-twist, shear-bending, bending-twist, and double curvature bending couplings in mechanical metamaterials are characterized. The accuracy of these homogenized models is investigated by comparing them with the detailed finite element models and experiments performed on 3D-printed samples. The proposed models provide a benchmark for the rational design, classification, and manufacturing of mechanical metamaterials with programmable coupleddeformation modes.

Micromechanical characterization of Carbon/PEEK thermoplastic composite material in-situ consolidated by automated fiber placement: Stiffness prediction

Despite manufacturing challenges, Automated Fiber Placement (AFP) offers a viable alternative to conventional manufacturing methods, allowing for time and cost savings. Creating a Representative Volume Element (RVE) that realistically represents long-fiber-reinforced composites with high fiber volume fraction is a challenging task in modeling their response. The present research aims to predict effective stiffness properties of in-situ-consolidated Carbon/PEEK thermoplastic composite material by considering the effect of fiber volume fraction, void content, degree of crystallinity, and interlaminar resin pocket resulting from AFP in-situ consolidation manufacturing process. In this regard, two sets of samples were manufactured by AFP in-situ consolidation and autoclave re-consolidation methods. Both of them were evaluated by micrographic study and thermoanalytical Differential Scanning Calorimetry (DSC) technique to obtain inputs required for micromechanical analysis. The 2D RVEs on a micro-scale are developed to predict the transverse elastic modulus, out-of-plane Poisson's ratio and out-of-plane shear modulus of the composite material by applying Periodic Boundary Conditions (PBCs) and using Asymptotic Homogenization Theory (AHT). Results show that AFP in-situ consolidation may lead the transverse elastic and out-of-plane shear moduli of Carbon/PEEK thermoplastic composite material to be reduced by about 10 % and 20 %, respectively, compared to autoclave re-consolidation whereas the out-of-plane Poisson's ratio remains unchanged. The findings of the present work confirm that the mechanical performance of Carbon/PEEK thermoplastic composite material could be remarkably influenced by AFP in-situ consolidation manufacturing process, particularly in the transverse direction, which must be taken into account in finite element modeling, analyses, and design of AFP-manufactured composite laminates and structures.

Enhancement effect of two-tiered heterostructure in piezoelectric filler-dispersed polymer composites

The effect of two-tiered heterostructure was verified by computational simulations and prototype experiments as a new approach for improving the performance of piezoelectric (PE) filler-dispersed polymer materials. Multiscale finite element analysis (FEA) based on asymptotic homogenization theory was applied to systematically investigate the fundamental properties of 0–3 composites in the first level and the relationship between macroscopic piezoelectricity and connectivity of the constituent phases in the second level. Simulations indicated that macroscopic piezoelectricity can be improved by controlling the PE filler content of the constituent phases in the two-tiered heterostructure, and that the connectivity of the PE phases in the direction of the applied electric field can effectively enhancing piezoelectricity. Furthermore, the two-tiered heterostructure discovered by multiscale FEA was fabricated by the melt-casting, demonstrating a piezoelectricity improvement of approximately 480 %.

Effective Lewis number and burning speed for flames propagating in small-scale spatio-temporal periodic flows

Propagation of premixed flames having thick reaction zones in rapidly-varying, small-scale, zero-mean, spatio-temporal periodic flows is considered. Techniques of large activation energy asymptotics and homogenization theory are used to determine the effective Lewis number Leeff and the effective burning speed ratio ST/SL, which are influenced by the flow through flow-enhanced diffusion. The resultant effective diffusivity matrix is, in general, neither a scalar nor a diagonal matrix and therefore induces anisotropic effects on the propagation of multi-dimensional flames. As the flow Peclet number Pe becomes large, the flow-enhanced fuel diffusion coefficient and the thermal diffusivity behave respectively like (PeLe)σ and Peσ, where Le is the Lewis number and σ≤2 is a constant which depends on the flow and the direction of flame propagation. The maximal value σ=2 is achieved for steady, unidirectional, spatially periodic shear flows, while for steady two-dimensional square vortices, we have σ=1/2. In general, the constant σ is determined by solving a linear partial differential equation. The scaling laws for the diffusion coefficients lead to corresponding scaling laws for the effective Lewis number and the effective burning speed ratio of the form Leeff≃Le1−σ and ST/SL∼(Pe/Le)σ/2. Effects of thermal expansion and volumetric heat loss on the flame are also briefly discussed. In particular, it is shown that the quenching limit is enlarged by a factor 1/Leσ for Le<1 and diminished by the same factor for Le>1, due to the flow-enhanced diffusion. The potential implications of the results to better understand turbulent combustion are discussed. A special emphasis is placed on the dependence of the flame on Le in the presence of high-intensity, small-scale flows. In particular, it is shown that this dependence is intimately linked to the flow through Taylor-dispersion like enhanced diffusion, rather than through the traditional molecular diffusion coupled with curvature effects. The flow-dependent effective Lewis number identified may also provide an explanation to the peculiar experimental observation that turbulence appears to facilitate ignition in Le>1 mixtures and to inhibit it in Le<1 mixtures. Novelty and significance statementAn original study, combining asymptotic analysis and homogenization theory, is applied to describe flame propagation in small-scale, spatio-temporal periodic flow fields. Scaling laws are derived for the effective burning speed and the effective Lewis number for high-intensity small-scale flows, which are useful to better understand the behaviour of turbulent premixed flames in the distributed reaction zone regime. The formula for the flow-dependent effective Lewis number identified herein may explain the peculiar experimental observation that turbulence appears to facilitate ignition in Le>1 mixtures and to inhibit it in Le<1 mixtures. The high-intensity small-scale flows are shown to increase the quenching limit due to volumetric heat losses in Le<1 mixtures and decrease it in Le>1 mixtures

Functionally graded structure design for magnetic field applications

This study presents topology optimization methodology of anisotropic magnetic composites, which consist of two ferromagnetic materials with low and high reluctivity values, considering the nonlinear magnetic saturation effect. Instead of employing the asymptotic homogenization theory, the representative volume element method combined with the machine learning is used to build the continuous function model and it is applied to obtain the material property according to the design variable change. Finally, the micro-scale functionally graded structure composed of two ferromagnetic materials with the macro-scale topological morphology is simultaneously designed to improve the magnetic performance of actuators. Numerical examples for symmetric and asymmetric magnetic actuator models are provided to validate the effectiveness of proposed design process. In the numerical results, optimized configurations and objective values obtained with the nonlinear magnetic composite material, which depend on the intensity of the magnetic flux density, are compared with those of the linear magnetic composite material.

Conjugate natural convection along regularly ribbed vertical surfaces: A homogenization-based study

Natural convection heat transfer from periodically ribbed vertical surfaces is targeted for upscaling, incorporating the analysis of thermal conduction through the microscale ribs. Asymptotic homogenization theory is employed, considering the steady conjugate heat transfer problem, to formulate second-order accurate effective conditions for velocity and temperature at a fictitious plane surface, beyond which the macroscale behavior of the flow is computed. This allows to avoid the numerically expensive resolution of fields within and through the microstructured corrugations. For the streamwise velocity component, the no-slip boundary condition is corrected at first order (in terms of a small parameter ϵ , ratio of microscopic to macroscopic length scales) by the classical Navier-slip condition plus a buoyancy term, while the gradient of the normal stress appears at second order together with a temperature-gradient term. The temperature at the virtual boundary deviates from the uniform value at the baseplate; the thermal slip is described via a first-order temperature-gradient term with a coefficient depending on rib geometry and thermal conductivity. Different case studies are conducted on the case of transverse square ribs, varying the density of the pattern and the rib-to-fluid thermal conductivity ratio, to provide extensive validation of the model against full feature-resolving simulations. Beyond the validation phase, a better understanding of the effects of different parameters on the heat transfer performance is pursued. The presence of ribs is found to decrease the overall heat transfer rate from the surface, and this deterioration is only partially alleviated by raising the thermal conductivity of the ribs. Increasing the number of conducting ribs on the hot surface has a complex, non-monotonic effect on the heat transfer rate, unlike the case of adiabatic ribs where the average Nusselt number decays monotonically. The performance of low-thermal-conductivity elements (e.g. wooden ribs) may considerably differ from that of perfectly adiabatic ones.

Evaluation the interface damage influences on mechanical properties of fiber-reinforced composites based on subdomain boundary element theory

In this paper, a novel microscopic method is presented to investigate the effective mechanical properties and local stress distribution of the composites with respect to some specific interfacial cracks. To this end, a subdomain boundary element method (SBEM) is combined with the asymptotic homogenization theory to calculate their effective mechanical properties. In order to accurately capture the stress distribution on the boundary and inside points (Gaussian integral points) of representative volume element (RVE), the parametric sub-cell is firstly constructed to discretize the RVE. On this basis, a node-pair decoupling method is introduced with consideration of unilateral and bilateral damage. The influence on mechanical properties of composites is revealed, which provides a solid foundation for the design and manufacture of the composite interface.

Stochastic higher-order three-scale strength prediction model for composite structures with micromechanical analysis

Stochastic multiscale modeling and analysis for the strength prediction of composite structures with complex multiscale random configurations remain a challenging problem. This is mainly due to the high-dimensional physical properties, the non-linear and non-Gaussian features, and the fact that many repeated evaluations of the corresponding stochastic multiscale model are often required. In this paper, we develop a stochastic higher-order three-scale strength prediction (SHTSP) model for composite structures, which is designed to overcome the limitations of prohibitive computation involving the microscale, the mesoscale, and the macroscale. By virtue of asymptotic homogenization theory and micromechanical analysis, the SHTSP model is established from the detailed stochastic higher-order three-scale homogenization analysis implemented with analytic solutions of typical composite structures subjected to tension, bending, and twist loads. The SHTSP model represents strength anisotropy through different strength criteria for evaluating the yield state of the different component materials of composite structures, which are induced by material interface deboning or matrix cracking in multiple scales. Moreover, these are constructed and calibrated from the high-accuracy mechanical analysis with help of two classes of mesoscopic and microscopic auxiliary cell functions, respectively. The corresponding numerical algorithm of the SHTSP model and a preprocessing multilevel strategy is designed to improve the computational efficiency. Finally, the numerical experiments in 3D cases illustrate the outstanding performance of the proposed SHTSP model, and the proposed method can significantly reduce the computational time.

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 optimization of magnetoelectric composite materials – Effect of material combination

Multiscale optimal designs that maximize the magnetoelectric (ME) effect are presented for typical ferroelectric and ferromagnetic material combinations, based on a study of novel microstructures, in an effort to achieve ME properties superior to those of the conventional structure. The asymptotic homogenization theory was employed for scale bridging and numerical solutions were obtained using the finite element method. The longitudinal homogenized ME constant was adopted as the objective function. The phase composition, arrangement, and polarization direction in the microstructure were set as design variables. For all material combinations, multiscale optimization was achieved for novel microstructures beyond existing laminated structures, with the optimum microstructure depending on the material combination. The findings of the study are expected to be useful in the development of high-performance multiferroic composite materials.

A compatible multiscale model for nanocomposites incorporating interface effect

The asymptotic homogenization theory is usually employed for analysis of composites with hierarchical microstructures, and the interface effect has to be considered for nanocomposites due to the fact that the surface-to-volume ratio of the nanofillers becomes significant. However, commonly-used two-dimensional surface theories of elasticity cannot be compatibly applied in conjunction with the asymptotic homogenization theory for analysis of nanocomposites due to traction discontinuity and the difficulty in specifying positive definiteness of elasticity tensor in terms of two-dimensional interface parameters. This study is focused on solving the incompatibility problem. A simple but useful three-dimensional interface theory is proposed, in which stress discontinuity conditions across two-dimensional interfaces are replaced by traction continuity on its two sides, and the requirement for positive definiteness of elasticity tensor can be easily specified. This three-dimensional interface theory is linked to the well-accepted two-dimensional models through energy-equivalence and kinematics of its middle surface. It is found that if the mechanical behavior of an interface can be well captured by commonly-used two-dimensional surface theories of elasticity, the three-dimensional interface can be characterized by constant elastic moduli as well as a size-independent thickness. Based on the proposed three-dimensional interface theory and the asymptotic homogenization theory, a compatible multiscale model is then developed and provides a practical way to examine the scaling law of effective moduli of typical nanocomposites. It is examined using numerical results that two-dimensional interface parameters can be easily extracted from the scaling law, and the way we link three-dimensional interfaces to two-dimensional counterparts is justified.

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.

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.

Machine learning-combined topology optimization for functionary graded composite structure design

This study presents new framework in which the representative volume element (RVE) method and machine learning (ML) model are used to construct continuous anisotropic effective material properties for simultaneous design of the overall topology configuration and local fiber material layout in functionally graded composite structures. It is an alternative to the asymptotic homogenization design method (AHDM) to obtain continuous effective material property functions. While the AHDM uses the asymptotic homogenization theory (AHT) and Legendre polynomials, the RVE method calculates anisotropic effective material properties having nonlinear behavior with respect to design variables of microstructures, and it is easier to implement than AHT given the governing equations and appropriate boundary conditions. More efficient and accurate than Legendre polynomials, ML is used to build a continuous model of the RVE results required for simultaneous design of the overall topology configuration and local fiber material layout. To show the convenience and expandability of the proposed method, a 3D RVE model is also proposed through the extension of the 2D model. The proposed method is verified through 2D and 3D numerical examples to minimize structural compliance and obtained results are compared with those from the application of AHDM.

A multiscale reduced‐order‐model strategy for transient thermoelasticity with variable microstructure

AbstractThis article deals with thermoelastic computation of heterogeneous structures containing quasi‐periodic microstructures having variable properties (geometric and/or material) using reduced‐order modeling. Such heterogeneous structure is extremely expensive to simulate using classical finite element methods, as the level of discretization required to capture the microstructural effects, is too fine. Based on the asymptotic homogenization theory, the multiscale technique explores the micro–macro behavior for thermoelasticity. Considering each integration point of the macrostructure consists of an underlying locally periodic microstructure, the overall problem is basically separated into a homogeneous problem defined over the macrostructure and a heterogeneous problem defined over each microstructure. Even though the usage of multiscale strategy helps in the reduction of numerical expense, it still deals with a full‐order finite element solution for the macroproblem and each microproblem. Using a twofold reduced‐order modeling further accentuates the cost reduction and provides a robust solution in a reduced space: (i) as an offline precomputation stage for the microstructural problem, and (ii) as an online process that can embed adaptivity for the macroscopic problem.

Topology Optimization Design of Adjustable Thermal Expansion Metamaterial Based on Independent Continuous Variables

This paper presents a topology optimization method to construct adjustable thermal expansion metamaterials. The negative thermal expansion (NTE) and extreme positive thermal expansion (EPTE) microstructures are designed, respectively. First, the effective elastic modulus and the effective thermal expansion coefficient of microstructure are derived by the multiscale asymptotic homogenization theory based on the periodic characteristics of the metamaterial. Second, a topology optimization model aiming at extreme thermal expansion coefficient is established and solved based on independent continuous topological variables, which is subjected to the effective elastic modulus and structure weight fraction. Then a cross-precision progressive optimization method is provided to save the computation time and make a smoother boundary. In addition, the specific topological feature for the initial configuration is discussed. Five holes are the necessary condition to obtain adjustable thermal expansion metamaterials for four-axisymmetric structures. Finally, a transformation strategy from NTE to EPTE and the change law of thermal expansion coefficient bounds with effective elastic modulus are studied. Numerical examples show that the NTE and EPTE can be designed in the range of effective elastic modulus from 0.01 to 0.05. The proposed method provides a reference for the novel configuration design of metamaterials.

Locally exact asymptotic homogenization of viscoelastic composites under anti-plane shear loading

The recently developed elasticity-based locally exact asymptotic homogenization theory is extended to accommodate non-ageing linearly viscoelastic composites under anti-plane shear loading. The theory is based on transforming the problem to the Laplace domain using the elastic-viscoelastic correspondence principle, solving it in the transformed domain and transforming it back to time domain using an efficient and accurate inversion technique. In the Laplace domain, the microscale unit cell interior problems are solved exactly up to the third order through Fourier series expansions, whereas the inseparable exterior periodic boundary value problems are tackled by asymptotic extensions of a balanced variational principle. Numerical examples of microscale solutions are presented and compared with the finite element method in support of the theory's accuracy. The effectiveness of the theory is assessed through a solution of a viscoelastic composite structural problem with finite-sized microstructures, illustrating good comparison with the results of a direct numerical solution. The theory is also applied to the microscale analysis of composites reinforced by viscoelastic fibers with different relaxation rates rarely reported in the literature. The analytical solution methodology in the Laplace domain is unique among the existing higher-order asymptotic homogenization methods employed in the solution of viscoelastic composite structural problems with non-vanishing microstructural scales. It serves as a gold standard for validating strictly numerical approaches.

Microstructural topology optimization of periodic beam structures based on the relaxed Saint-Venant solution

Design optimization of beam structures is a significant topic as beams are an efficient load-carrying component in engineering applications. Most of the earlier researches concentrate on the structural optimization of beams with invariant cross-section topology, and design optimization of periodic beam structures, which cross-section topology varies along the axial direction, has not been extensively investigated. This work presents a novel approach based on the relaxed Saint-Venant solution to conduct microstructural topology optimization of periodic beam structures for minimum structural compliance. One benefit of adopting Saint-Venant solution based compliance formulation is that the strain energy induced by transverse shear loading is incorporated in the structural compliance, which is not reflected in that based on first-order homogenization of the asymptotic homogenization (AH) theory, so that a more reasonable objective function is adopted for the optimization problem. In addition, a material connectivity constraint, which is constructed by restraining the ratio of the strain energy calculated from relaxed Saint-Venant solution to that obtained from first-order homogenization of the AH theory, is further proposed to prevent material separation or to strengthen material connection through the thickness direction. The detailed sensitivity analysis of the objective function and the constraints are carried out with the adjoint method, and several numerical examples are given to show the validity of the relaxed Saint-Venant solution based compliance formulation and the effectiveness of the proposed material connectivity constraint.

An Extended Multiscale Finite Element Method (EMsFEM) Analysis of Periodic Truss Metamaterials (PTMM) Designed by Asymptotic Homogenization

Asymptotic Homogenization (AH) and the Extended Multiscale Finite Element Method (EMsFEM) are both procedures that allow working on a structural macroscale that incorporates the effect of averaged microscopic heterogeneities, thus resulting in computationally efficient strategies. EMsFEM works directly on coupled finite micro and macroscales using numerically built discrete interpolation functions. Periodic Truss Metamaterials (PTMMs) are cellular materials formed by the periodic repetition of a truss-like unit cell and engineeringly tailored to show a given macroscopic response. In this work we analyze the numerical behavior of selected PTMMs that were designed for extreme Poisson ratios using AH theory. As a first issue, we study macroscopic structures made of finite unit cells and verify how close their average behavior coincides with the material properties predicted by AH. For comparison, we solve the macroscopic plane stress associate problems that employ the elastic constitutive tensor obtained by AH. The second issue is concerned with the ability of EMsFEM to reproduce the structural behavior of the full macro-micro model. We employ two versions of the EMsFEM, adopting linear (LBC) and periodic (PBC) boundary conditions to build the numerical interpolation functions. The third and most important aspect discussed in this research concerns evaluation of the EMsFEM downscaled displacement fields. We observe that according to the layout of the AH designed unit cell, to the use of LBC or PBC and, depending on the boundary conditions present in the macroscopic problem, spurious downscaled displacements might occur. Such spurious displacements are due to excessive compliance of the corresponding unit cell and can be detected when building the numerical interpolation functions. We conclude that the layout optimization of PTMM using AH must be carefully interpreted and that EMsFEM is a good tool to detect a macroscopic excessively compliant response at an early design stage.

Finite Volume-Based Asymptotic Homogenization of Periodic Materials Under In-Plane Loading

AbstractThe previously developed finite volume-based asymptotic homogenization theory (FVBAHT) for anti-plane shear loading (He, Z., and Pindera, M.-J., “Finite-Volume Based Asymptotic Homogenization Theory for Periodic Materials Under Anti-Plane Shear,” Eur. J. Mech. A Solids (in revision)) is further extended to in-plane loading of unidirectional fiber reinforced periodic structures. Like the anti-plane FVBAHT, the present extension builds upon the previously developed finite volume direct averaging micromechanics theory applicable under uniform strain fields and further accounts for strain gradients and non-vanishing microstructural scale relative to structural dimensions, albeit with multidimensional in-plane loadings incorporated. The unit cell problems at different orders of the asymptotic field expansion are solved by satisfying local equilibrium equations and displacement and traction continuity in a surface-averaged sense which is unique among the existing asymptotic homogenization schemes, leading to microfluctuation functions that yield homogenized stiffness tensors at each order for use in macroscale problems. The newly extended multiscale theory is employed in the analysis of a structural boundary-value problem under in-plane loading, illustrating pronounced boundary effects. A combination approach proposed in the literature is subsequently employed to mitigate the boundary layer effects by explicitly accounting for the microstructural details in the boundary region. This combination approach produces accurate recovery of the local fields in both regions. The extension to in-plane problem marks FVBAHT as an alternative, self-contained asymptotic homogenization tool, with documented advantages relative to current numerical techniques, for the analysis of periodic materials in the presence of strain gradients produced by three-dimensional loading regardless of microstructural scale.