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The effective design of metallic metamaterials, characterized by interconnected struts or 'lattices,' hinges on the ability to predict strut and strut intersection ('node') responses. This is critical for predicting the macroscopic properties of structures comprised of thousands of struts. Computationally efficient beam descriptions, defined by strut properties like cross-sectional area, modulus, and yield stress, can significantly expedite the prediction of lattice structures and ultimately enable topology optimization. This paper provides a comprehensive examination of the properties of electron-beam melted three-dimensional printed struts and their 'nodes'—four intersecting struts. The findings elucidate the efficacy of various strategies for defining effective properties that accurately capture mechanical response. The study reveals that using a single set of effective properties can introduce inconsistencies between strut stiffness, peak load, and critical displacement. Stiffness correlates with averaged cross-sectional areas, while the peak load capacity correlates more closely with minimum inscribed cross-sectional areas. Analysis of nodes indicates that the interaction of surface defects and heterogeneity within the node strongly influences multi-strut response. Data from CT scans, EBSD scans, and nanoindentation maps highlight spatial variations comparable to the strut diameter, posing a significant challenge in defining effective homogenized properties. This study emphasizes the need for future efforts to integrate statistical property distributions with high throughput simulations to overcome the difficulty in defining a representative volume element (RVE) at the strut scale.

One of the most common approximations in the theory of composite materials is the homogenization theory. The main difficulty in its application lies in solving the cell problem, i.e., the boundary value problem on a periodically repeating element of composite material. For the case of inclusions of large size and with characteristics far superior to those of the matrix, the lubrication approach gives good results. However, the use of such an asymptotic approach is unnatural for the case when the characteristics of the matrix exceed the characteristics of the inclusion. In the presented work, the problem of conductivity for a 2D composite with rhombic voids is considered. The solution of the cell problem is carried out by two methods. First, the lubrication approach is used for this purpose. In addition, a modification of the Schwarz alternating method is proposed. This new approach has been called an "inhomogeneous cell model". Both methods made it possible to obtain analytical expressions for the effective conductivity. A comparison of the indicated approximate models is carried out, and it is shown that the obtained solutions exactly satisfy Keller's theorem. The physical foundations of the proposed inhomogeneous cell model are discussed. Its advantage in solving the considered problem is shown. [ABSTRACT FROM AUTHOR]

A variational-based homogenization method for magnetoelastic composite materials is established in a small strains framework. The existence of a non-symmetrical stress tensor motivates the elaboration of a homogenized Cosserat type magnetoelastic effective medium at the macroscale. Generic expressions of the effective magnetic and elastic properties are derived, showing the existence of couplings between the elastic and magnetic behaviors at the macrolevel. Applications of the developed homogenization methodology are done for periodic heterogeneous media prone to local bending at the scale of a few unit cells. The validation of the homogenized medium is performed by comparing its predictions versus those of fully resolved computations. The influence of the magnetic field intensity and orientation on the strength of micropolar effects is assessed. The proposed formulation opens new possibilities for the efficient design of multifunctional metamaterials via computational modelling. [ABSTRACT FROM AUTHOR]

Polymer electrolyte coated carbon fibres embedded in polymeric matrix materials represent a multifunctional material with several application scenarios. Structural batteries, thermal management materials as well as stiffness adaptive composites, made from this material, are exposed to significant joule heat, when electrical energy is transferred via the carbon fibres. This leads to a temperature increase of up to 100 K. The thermal behaviour of this composite material is characterized in this numerical study based on a RVE representation for the first time. Compared to classical fibre reinforced plastics, this material comprises a third material phase, the polymer electrolyte coating, covering each individual fibre. This material has not been evaluated for effective thermal conductivity, specific heat and thermal behaviour on the microscale before. Therefore, boundary conditions, motivated from applications, are applied and joule heating by the carbon fibres is included as heat source by an electro-thermal coupling. The resulting temperature field is discussed towards its effect on the mechanical behaviour of the material. Especially the temperature gradient is pronounced in thickness direction, leading to a temperature drop of 1 °Cmm, which needs to be included in thermal stress analysis in future thermo-mechanically coupled models. Another important emphasis is the identification of suitable homogenization and model reduction strategies in order to reduce the numerical effort spent on the thermal problem. Therefore, traditional analytical homogenization methods as well as a newly proposed “Two-Level Lewis-Nielsen” approach are discussed in comparison to virtually measured effective quantities. This extensive comparison of analytical and numerical methods is original compared to earlier works dealing with PeCCF composites. In addition, the accuracy of the new Two-Level Lewis-Nielsen method is found to fit best compared to classical methods. Finally, a first efficient and accurate 2D representation of the thermal behaviour of the PeCCF composite is shown, which reduces computational cost by up to 97%. This benefit comes with a different Temperature drop prediction in thickness direction of 1.5 °Cmm. In the context of future modelling of multifunctional PeCCF composite materials with multiphysical couplings, this deviation is acceptable with respect to the huge benefit for computational cost.

Many engineered materials display ordered or disordered microstructures. Such materials exhibit transport properties which are unmatched by their single‐phase homogeneous counterparts. These properties are obtained by the mixture of two or more phases typically characterized by a large contrast in their properties. For the development of these materials, it is critical to develop a robust computational framework in order to provide a fundamental understanding of how microstructure affects performance. This hinges on predicting their macroscopic properties, given the constitutive laws and spatial distribution of their constituents. To this end, this work presents a computational framework based on formulating periodic conduction transport problems in terms of boundary integral equations whose kernel is expressed in terms of Weierstrass zeta‐function. The components of the effective conductivity tensor are then sought in the form of power series expansions of a conductivity contrast parameter. To accelerate their convergence, these expansions are transformed into Padé approximants. Presently restricted to the case of two‐dimensional, two‐phase microstructures, this framework is shown to yield accurate results over the entire range of the contrast parameter. Representation of the kernel as a lattice sum allows the use the fast multipole method, thereby making computations significantly more efficient. [ABSTRACT FROM AUTHOR]

In this article, an investigation was carried out to verify hybrid models capabilities to predict the effective properties of heterogeneous materials. A hybrid model ANN−φ is developed by combining artificial neural networks and micromechanical modeling. The homogenization approach used in this study is mainly based on Eshelby's inclusion problem. The ANN−φ model, once trained on an Eshelby's tensors database, showed an excellent predictive capabilities of the effective mechanical behavior and local stresses in heterogeneous materials. The obtained results with ANN−φ are compared to numerical estimations which are often costly in terms of computational time. The results presented in this work show that the developed hybrid model can provide a significant computational time saving by a factor up to 2000 for 104 phases while maintaining its accuracy and reliability. [ABSTRACT FROM AUTHOR]

One of the most common approximations in the theory of composite materials is the homogenization theory. The main difficulty in its application lies in solving the cell problem, i.e., the boundary value problem on a periodically repeating element of composite material. For the case of inclusions of large size and with characteristics far superior to those of the matrix, the lubrication approach gives good results. However, the use of such an asymptotic approach is unnatural for the case when the characteristics of the matrix exceed the characteristics of the inclusion. In the presented work, the problem of conductivity for a 2D composite with rhombic voids is considered. The solution of the cell problem is carried out by two methods. First, the lubrication approach is used for this purpose. In addition, a modification of the Schwarz alternating method is proposed. This new approach has been called an “inhomogeneous cell model”. Both methods made it possible to obtain analytical expressions for the effective conductivity. A comparison of the indicated approximate models is carried out, and it is shown that the obtained solutions exactly satisfy Keller’s theorem. The physical foundations of the proposed inhomogeneous cell model are discussed. Its advantage in solving the considered problem is shown.

In this paper we present a semi-multiscale methodology, where a micrograph is split into multiple independent numerical model subdomains. The purpose of this approach is to enable a controlled reduction in model fidelity at the microscale, while providing more detailed material data for component level- or more advanced finite element models. The effective anisotropic elastic properties of each subdomain are computed using periodic boundary conditions, and are subsequently mapped back to a reduced mesh of the original micrograph. Alternatively, effective isotropic properties are generated using a semi-analytical method, based on averaged Hashin–Shtrikman bounds with fractions determined via pixel summation. The chosen discretization strategy (pixelwise or partially smoothed) is shown to introduce an uncertainty in effective properties lower than 2% for the edge-case of a finite plate containing a circular hole. The methodology is applied to a aluminium alloy micrograph. It is shown that the number of elements in the aluminium model can be reduced by 99.89 % while not deviating from the reference model effective material properties by more than 0.65 % , while also retaining some of the characteristics of the stress-field. The computational time of the semi-analytical method is shown to be several orders of magnitude lower than the numerical one. [ABSTRACT FROM AUTHOR]

This paper reviews the analytical, numerical as well as the computational homogenization multiscale modelling schemes for determining the effective material properties for heterogeneous medium at the macroscopic level. It also looked at the limitations of the analytical homogenization techniques in simulating the effective non linear heterogeneous material behaviours (for example the rapid localization of damage and so on) as well as the advancements of the computational techniques in addressing these limitations. In addition, the possible future trends for the computational technique such as the development of a fully coupled micro-macro computational scheme were also discussed. It was concluded that although, the analytical technique was quite popular and straight forward, its inability to capture rapid localization of damage limited its application and that numerical and computational schemes were able to address these limitations as they relied on the establishment of constitutive relations for the macroscopic problems in a numerical form through which the macroscopic problems were constructed and solved in a nested manner.

The gas diffusion layer (GDL) typically consists of a thin layer of carbon fiber paper, carbon cloth or nonwoven and has numerous pores. The GDL plays an important role that determines the performance of the fuel cell. It is a medium through which hydrogen and oxygen are transferred and serves as a passage through which water, generated by the electrochemical reaction, is discharged. The GDL tissue undergoes a compressive loading during the stacking process. This leads to changes in fiber content, porosity and resin content due to compressive load, which affects the mechanical, chemical and electrical properties of the GDL and ultimately determines fuel cell performance. In this study, the geometry of a GDL was modeled according to the compression ratios (10%, 20%, 30%, 40% and 50%), which simulated the compression during the stacking process and predicted the equivalent properties according to the change of GDL carbon fiber content, matrix content and pore porosity, etc. The proposed method to predict the equivalent material properties can not only consider the stacking direction of the material during stack assembling process, but can also provide a manufacturing standard for fastening compressive load for GDL. [ABSTRACT FROM AUTHOR]

A micromechanical model for short fiber-reinforced composites (SFRCs) with functionally graded interphases and a systematic prediction scheme to determine the effective properties are presented. The matrix and the fibers are regarded to be linear elastic, isotropic, and homogeneous. Fibers are assumed to be ellipsoids coated perfectly by functionally graded interphases, which is supposed to be formed chemically or physically by the constituents near the interface. First, to analyze the grading interphase effect, layer-wise concept is followed to divide the functionally graded interphases into multi-homogeneous sub-layers. Next, to take the effect of functionally graded interphases into account, a combination of multi-inclusion method and Mori–Tanaka method is applied to predict effective elastic properties of this unidirectional SFRCs with respect to the content and aspect ratio of the inclusions. By employing coordinate transformation, spatially elastic moduli are obtained. Finally, Voigt homogenization scheme is used to obtain the overall, averaged, symmetrical elastic properties of the SFRCs. Numerical examples and analyses demonstrate the applicability of the proposed method and indicate the influences of graded interphase, orientation, and aspect ratio of inclusions as well as properties and contents of the constituents on the overall properties of SFRCs. [ABSTRACT FROM AUTHOR]

[Display omitted] • The mechanical performance of thin-wall Ti-6Al-4Vadditively manufacturing metamaterials does not align with the bulk material properties. • Different area metrics need to be used for homogenization schemes to forecast the stiffness and yield stress of lattice sub-components. • The interdependence of microstructure and geometry heterogeneity is vital in identifying the weakest link for more precise predictions of mechanical response in AM structures. • Computational methods that embed statistical distributions of properties can be connected to statistic distributions of macroscopic response, which are then used to guide structural design. • Illustrative examples of results that can be used to define geometric and material property distributions in modeling approaches are presented. The effective design of metallic metamaterials, characterized by interconnected struts or 'lattices,' hinges on the ability to predict strut and strut intersection ('node') responses. This is critical for predicting the macroscopic properties of structures comprised of thousands of struts. Computationally efficient beam descriptions, defined by strut properties like cross-sectional area, modulus, and yield stress, can significantly expedite the prediction of lattice structures and ultimately enable topology optimization. This paper provides a comprehensive examination of the properties of electron-beam melted three-dimensional printed struts and their 'nodes'—four intersecting struts. The findings elucidate the efficacy of various strategies for defining effective properties that accurately capture mechanical response. The study reveals that using a single set of effective properties can introduce inconsistencies between strut stiffness, peak load, and critical displacement. Stiffness correlates with averaged cross-sectional areas, while the peak load capacity correlates more closely with minimum inscribed cross-sectional areas. Analysis of nodes indicates that the interaction of surface defects and heterogeneity within the node strongly influences multi-strut response. Data from CT scans, EBSD scans, and nanoindentation maps highlight spatial variations comparable to the strut diameter, posing a significant challenge in defining effective homogenized properties. This study emphasizes the need for future efforts to integrate statistical property distributions with high throughput simulations to overcome the difficulty in defining a representative volume element (RVE) at the strut scale. [ABSTRACT FROM AUTHOR]

Micromechanics-based homogenization has been employed extensively to predict the effective properties of technologically important composites. In this review article, we address its application to various physical phenomena, including elasticity, thermal and electrical conduction, electric, and magnetic polarization, as well as multi-physics phenomena governed by coupled equations such as piezoelectricity and thermoelectricity. Especially, for this special issue, we introduce several research works published recently from our research group that consider the anisotropy of the matrix and interfacial imperfections in obtaining various effective physical properties. We begin with a brief review of the concept of the Eshelby tensor with regard to the elasticity and mean-field homogenization of the effective stiffness tensor of a composite with a perfect interface between the matrix and inclusions. We then discuss the extension of the theory in two aspects. First, we discuss the mathematical analogy among steady-state equations describing the aforementioned physical phenomena and explain how the Eshelby tensor can be used to obtain various effective properties. Afterwards, we describe how the anisotropy of the matrix and interfacial imperfections, which exist in actual composites, can be accounted for. In the last section, we provide a summary and outlook considering future challenges.

Piezocomposite design for dedicated ultrasonic imaging applications requires precise homogenization models for predicting the electromechanical characteristics of the new material. Thus, several homogenization models have been developed. As part of this work, we applied several analytical homogenization models for piezocomposite of 2–2 and 1–3 connectivities. To validate these analytical models, a comparative study was made between various models and experimental measurements. As a result, these homogenized electromechanical properties are effectively used for the calculation and comparison of electroacoustic response for typical transducers aimed at ultrasound imaging applications. An optimal design of transducer aimed at ultrasound imaging applications is proposed as a dedicated imaging performance index, elaborated through a trade-off between sensitivity and bandwidth. [ABSTRACT FROM AUTHOR]

The objective of this paper is to critically evaluate the accuracy and efficiency of a general-purpose micromechanics approach based on the Mechanics of Structure Genome (MSG), when it is applied to the constitutive modeling of 3D structures. The Generalized Method of Cell (GMC) is chosen as a reference method during efficiency evaluation. The predictions by Three- Dimensional Finite Element Analysis (3D FEA) are chosen as benchmarks during accuracy evaluation. Composites such as a continuous fiber-reinforced composite, a particle-reinforced composite, two discontinuous fiberreinforced composites, and a woven composite are analyzed using MSG, GMC, and 3D FEA. During homogenization, MSG is found to be as accurate but much more efficient than 3D FEA, and despite high efficiency, GMC is found to sacrifice accuracy for efficiency. During dehomogenization, MSG is found to be as accurate as 3D FEA, but GMC is found not to be so accurate. The fidelity of MSG, when it is applied to the modeling of other structures (e.g., beams, plates, and shells), can be similarly evaluated. [ABSTRACT FROM AUTHOR]

A computational homogenization analysis for the simulation of porous magneto-electric composite materials is presented. These materials combine two or more ferroic states with each other enabling a coupling between magnetization and electric polarization. This magneto-electric coupling finds application in sensor technology or data storage devices. Since most single-phase multiferroics show coupling at very low temperatures beyond technically relevant applications, two-phase composites, consisting of a ferroelectric and a ferromagnetic phases, are manufactured. They generate a strain-induced magneto-electric coupling at room temperature. The performance and reliability of these materials is influenced by defects or pores, which can arise during the manufacturing process. We analyze the impact of pores on the magnitude of the magneto-electric coupling coefficient. In order to determine the effective properties of the composite, a two-scale finite element ( FE 2 ) homogenization approach is performed. It combines the macroscopic and microscopic scale by direct incorporation of the microscopic morphology. We derive the basic equations for the localization and the homogenization of the individual field variables and give an algorithmic expression for the effective tangent moduli. We discuss the influence of pores on the magneto-electric coupling in two-phase composites by analyzing numerical examples. [ABSTRACT FROM AUTHOR]

Motivated by the need to attribute a rigorous mathematical meaning to the term "metamaterial," we propose a novel approach to the homogenization of critical-contrast composites. This is based on the asymptotic analysis of the Dirichlet-to-Neumann map on the interface between different components ("stiff" and "soft") of the medium, which leads to an asymptotic approximation of eigenmodes. This allows us to see that the presence of the soft component makes the stiff one behave as a class of time-dispersive media. By an inversion of this argument, we also offer a recipe for the construction of such media with prescribed dispersive properties from periodic composites. [ABSTRACT FROM AUTHOR]

Abstract The present study introduces a methodology that allows to combine 3D printing, experimental testing, numerical and analytical modeling to create random closed-cell porous materials with statistically controlled and isotropic overall elastic properties that are extremely close to the relevant Hashin-Shtrikman bounds. In this first study, we focus our experimental and 3D printing efforts to isotropic random microstructures consisting of single-sized (i.e. monodisperse) spherical voids embedded in a homogeneous solid matrix. The 3D printed specimens are realized by use of the random sequential adsorption method. A detailed FE numerical study allows to define a cubic representative volume element (RVE) by combined periodic and kinematically uniform (i.e. average strain or affine) boundary conditions. The resulting cubic RVE is subsequently assembled to form a standard dog-bone uniaxial tension specimen, which is 3D printed by use of a photopolymeric resin material. The specimens are then tested at relatively small strains by a proper multi-step relaxation procedure to obtain the effective elastic properties of the porous specimens. [ABSTRACT FROM AUTHOR]

Hydrogel exhibits attractive mechanical properties that can be regulated to be extremely tough, strong and resilient, adhesive and fatigue-resistant, thus enabling diverse applications ranging from tissue engineering scaffolds, flexible devices, to soft machines. As a liquid-filled porous material composed of polymer networks and water, the hydrogel freezes at subzero temperatures into a new material composed of polymer matrix and ice inclusions: the frozen hydrogel displays dramatically altered mechanical properties, which can significantly affect its safety and reliability in practical applications. In this study, based upon the theory of homogenization, we predicted the effective mechanical properties (e.g. , Young's modulus, shear modulus, bulk modulus and Poisson ratio) of a frozen hydrogel with periodically distributed longitudinal ice inclusions. We firstly estimated its longitudinal Young's modulus, longitudinal Poisson ratio and plane strain bulk modulus using the self-consistent method, and then its longitudinal and transverse shear modulus using the generalized self-consistent method; further, the results were employed to calculate its transverse Young's modulus and transverse Poisson ratio. We validated the theoretical predictions against both finite element (FE) simulation and experimental measurement results, with good agreement achieved. We found that the estimated transverse Poisson ratio ranges from 0.3 to 0.53 and, at low volume fraction of ice inclusions, exhibits a value larger than 0.5 that exceeds the Poisson ratios of both the polymer matrix and the ice inclusion (typically 0.33–0.35). Compared with other homogenization methods (e.g. , the rule of mixtures, the Halpin-Tsai equations, and the Mori-Tanaka method), the present approach is more accurate in predicting the effective mechanical properties (in particular, the transverse Poisson ratio) of frozen hydrogel. Our study provides theoretical support for the practical applications of frozen liquid-saturated porous materials such as hydrogel. [ABSTRACT FROM AUTHOR]

The gas diffusion layer (GDL) typically consists of a thin layer of carbon fiber paper, carbon cloth or nonwoven and has numerous pores. The GDL plays an important role that determines the performance of the fuel cell. It is a medium through which hydrogen and oxygen are transferred and serves as a passage through which water, generated by the electrochemical reaction, is discharged. The GDL tissue undergoes a compressive loading during the stacking process. This leads to changes in fiber content, porosity and resin content due to compressive load, which affects the mechanical, chemical and electrical properties of the GDL and ultimately determines fuel cell performance. In this study, the geometry of a GDL was modeled according to the compression ratios (10%, 20%, 30%, 40% and 50%), which simulated the compression during the stacking process and predicted the equivalent properties according to the change of GDL carbon fiber content, matrix content and pore porosity, etc. The proposed method to predict the equivalent material properties can not only consider the stacking direction of the material during stack assembling process, but can also provide a manufacturing standard for fastening compressive load for GDL.

This work presents the derivation of the effective shear modulus for a heterogeneous material composed of multilayered composite spheres embedded in a linear elastic matrix. It is based on the composite spheres model known from the literature. In contrast to Herve and Zaoui (Int J Eng Sci 31:1-10, 1993), the effective shear modulus is obtained by equating the results of two models: In the first model, a heterogeneous sphere is embedded in an equivalent homogeneous material, whereas in the second model, the heterogeneous sphere is replaced by an equivalent homogeneous sphere. In the context of both, a shear stress approach and a shear deformation approach, this results in an overdetermined system of equations which is solved with the least squares method. In a numerical study, our results are compared to effective moduli and bounds from the literature. Furthermore, a convincing agreement with experimental data for glass microspheres embedded in a polyester matrix is demonstrated. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with homogenization of a double porosity fluid‐saturated periodic medium. At the mesoscopic level, dynamic behaviour of the medium is described by the Biot model featured by high contrasts in the permeability and the poroelastic coefficients. The fluid flow is governed by the Darcy flow model extended by inertia terms and by the mass conservation equation. To respect the high contrasts, some of the material properties are scaled by the small parameter which characterizes the size of the heterogeneities being subject of the asymptotic analysis. The macroscopic model of the medium is obtained using the two‐scale homogenization based on the periodic unfolding method. For this, the Laplace transformation in time is used to introduce the local autonomous problems for characteristic responses defining the effective medium properties. In comparison with the low contrast heterogeneous medium, the microflow in the double porosity gives rise to the fading memory effects involved also in the macroscopic poroviscoelastic constitutive law. The problem treated in the paper is an extension of previously studied problems with either rigid skeleton part, or deformable Biot ‘medium without high contrasts in material properties. Numerical illustrations of the homogenized effective model parameters are given. The derived two‐scale model is a convenient tool for studying wave propagation in many natural media and provides a basis for material research. [ABSTRACT FROM AUTHOR]

Because of composite materials’ inherent heterogeneity, the field of micromechanics provides essential tools for understanding and analyzing composite materials and structures. Micromechanics serves two purposes: homogenization or prediction of effective properties and dehomogenization or recovery of local fields in the original heterogeneous microstructure. Many micromechanical tools have been developed and codified, including commercially available software packages that offer micromechanical analyses as stand-alone tools or as part of an analysis chain. With the increasing number of tools available, the practitioner must determine which tool(s) provides the most value for the problem at hand given budget, time, and resource constraints. To date, simple benchmarking examples have been developed in an attempt to address this challenge. The present paper presents the benchmark cases and results from the Micromechanical Simulation Challenge hosted by the Composites Design and Manufacturing HUB. The challenge is a series of comprehensive benchmarking exercises in the field of micromechanics against which such tools can be compared. The Level I challenge problems consist of six microstructure cases, including aligned, continuous fibers in a matrix, with and without an interphase; a cross-ply laminate; spherical inclusions; a plain-weave fabric; and a short-fiber microstructure with “random” fiber orientation. In the present phase of the simulation challenge, the material constitutive relations are restricted to linear thermoelastic. Partial results from DIGIMAT-MF, ESI VPS, MAC/GMC, finite volume direct averaging method, Altair MDS, SwiftComp, and 3D finite element analysis are reported. As the challenge is intended to be ongoing, the full results are hosted and updated online at www.cdmHUB.org. [ABSTRACT FROM AUTHOR]

There is a demand for multi scale micromechanical models to disclose and analyze the effects of microstructure on macro mechanical properties of hybrid fiber reinforced concrete (HFRC). This study involved presenting a multi-level micromechanical model that involves cement paste level, concrete level, and hybrid fiber reinforced concrete level to quantitatively predict the effective isotropic and elastic properties of HFRC under ambient temperature. For the purposes of homogenization, the volume fractions of different phases at different levels are determined by means of a modified Power’s model. In the multi-level micromechanical model, hydration products of clinker, sand, coarse aggregate, and hybrid fiber are comprehensively considered. A homogenization stepping framework is presented to realize upscaling from microstructural properties to the effective elastic properties of a macrostructure for HFRC. Additionally, several substepping homogenizations are also presented to estimate the effective elastic properties of an equivalent medium with respect to the cement paste and hybrid fiber reinforced concrete. Comparisons with experimental data from extant studies are implemented level by level. Subsequently, the influences of aggregate, sand, fiber type, and hydration degree on the properties of HFRC are discussed based on a proposed multi-level micromechanical model. Finally, the mixture ratio of steel fiber and w/c are investigated with respect to the HFRC design to obtain anticipated elastic properties. [ABSTRACT FROM AUTHOR]

Our work presents an extension of a composite sphere model to temperature-dependent elastic effects accompanied by curing. Homogenization of a representative unit cell (micro-RVE) on the heterogeneous microscale which accounts for thermo-chemo-mechanical coupling with linear elasticity yields volumetric effective properties. Two conceptions are considered: Firstly, a homogeneous mixture with n -phases is assumed. Secondly, a geometric arrangement on the microscale is represented by the n -layered composite sphere model. In a numerical study for a 3-phase matrix it is demonstrated that the effective properties lie within certain bounds. [ABSTRACT FROM AUTHOR]

In this paper, an efficient image-based simulation toolkit for material characterization is presented, which is scalable to work from personal computers to workstations. The effective thermal conductivity, elasticity, and permeability are evaluated employing a computational homogenization framework based on the Finite Element Method (FEM). Two complementary open-source packages are presented: one developed in Python, which can convert digital images into voxel meshes (pyTomoviewer); the other developed in Julia, that can run numerical simulations to compute effective material properties (chpack). Also, a CUDA C version of chpack is provided (chfem_gpu). They were designed to deal with large multi-phase models, so strategies were devised to minimize their memory footprint, while avoiding a high toll on execution time. The voxel-based approach significantly simplifies the FEM meshes and allows efficient matrix-free implementations. In that sense, to handle large linear systems of equations, the element-by-element (EBE) technique is adopted, in conjunction with a low-memory implementation of the Preconditioned Conjugate Gradient (PCG) method. The code was thoroughly tested on an artificial geometry made of a square array of cylinders, for which analytical solutions exist, as well as on a real micro-tomographic reconstruction of FiberFormTM, a carbon preform commonly used in thermal protection systems. • Open-source software bundle for image-based numerical homogenization and analysis. • Developed lightweight workflow to compute effective properties of microstructures. • Verified and validated results against analytical values and experimental data. • Performed conductivity, elasticity, and permeability analyses in 500 3 voxels images. • Homogenized properties of large carbon fiber samples obtained from X-Ray μ CT. [ABSTRACT FROM AUTHOR]

The present contribution aims to study the effect the inner topology of periodic architected materials on their effective piezoelectric and flexoelectric properties. An analysis is performed to study the sensitivity of the effective flexoelectric moduli to the geometrical variables of the considered unit cells of periodic lattice materials. More specifically, hexagonal, re-entrant auxetic, and rectangular unit cells are investigated in terms of their relative piezoelectric and flexoelectric response. The base material obeys a piezoelectric behavior, and the homogenized electromechanical properties are evaluated using the extended Hill–Mandel macrohomogeneity condition articulated with variational principles in the context of periodic multiphysical homogenization. The developed modeling framework of coupled electromechanical phenomena accounts for the higher gradient effects modeled by flexoelectricity that are induced by the strong gradient of properties that exist at the microlevel of the lattice unit cell. Regular hexagonal, re-entrant hexagonal and rectangular lattice unit cells are mutually compared in terms of their flexoelectric properties. The regular hexagonal UC shows the highest piezoelectric moduli, whereas the flexoelectric moduli are found to be maximum for the re-entrant UC. The coupling between mechanical and electrical fields is attested by the non-uniform distribution of the electric potential within the three studied architected materials subjected to a uniform strain. [ABSTRACT FROM AUTHOR]

The aim of this paper is to model the effective viscoelastic properties of micro-cracked masonry based on a homogenization approach. The masonry is modeled as a mixture of a micro-cracked viscoelastic mortar matrix and isotropic elastic brick inclusions. Firstly, the effect of micro-cracks on the effective viscoelastic properties of mortar is considered at micro-scale. Secondly, brick inclusions are added in the micro-cracked mortar matrix at mesoscale to simulate effective viscoelastic properties of masonry. The modified Maxwell viscoelastic model is employed for noncracked, micro-cracked mortar matrix and micro-cracked masonry. The use of the same modified Maxwell model of the noncracked mortar to model the viscoelastic properties of the micro-cracked mortar is an approximation. This approximation is carried out in the short- and the long-term behaviors in the Laplace–Carson space. It is validated in transient situation with exact solutions obtained from the inverse Laplace–Carson transform for simple loading conditions. This technique allows avoiding the complexity of the inverse Laplace–Carson transform. The isotropic case with random orientation distribution of open micro-cracks in mortar and with spherical brick inclusions is considered. [ABSTRACT FROM AUTHOR]

In this paper we present a computational homogenization procedure for the simulation of magneto-electro-mechanically coupled boundary value problems (bvp)s on two scales. We derive the basic equations for the localization and the homogenization of the individual field variables and give an algorithmic expression for the effective tangent moduli. The resulting algorithmic two-scale transition procedure is implemented into an FE 2 -method, which allows us to compute macroscopic boundary value problems in consideration of attached microscopic representative volume elements. The challenge in the simulation of magneto-electro-mechanically coupled materials is the modeling of the complicated interactions between magnetical, electrical and mechanical quantities on both scales. A primer example for such interactions is given by magneto-electric composites. Thus, we apply the presented method to the two-scale simulation of a variety of versions of magneto-electric composites. In detail, we consider two-phase composites composed of (i) piezomagnetic and piezoelectric, (ii) piezomagnetic and non-linear electrostrictive, as well as (iii) piezomagnetic and non-linear dissipative ferroelectric phase. Based on numerical examples, we discuss aspects of the applicability, the numerical stability as well as the predictive capability of the proposed method. [ABSTRACT FROM AUTHOR]

The computation of apparent material properties for a random heterogeneous material requires the assumption of a solution field on a finite domain over which the apparent properties are to be computed. In this paper the assumed solution field is taken to be that defined by the shape functions that underpin the finite element method and it is shown that the variance of the apparent properties calculated using the shape functions to define the solution field can be expressed in terms of a variability response function (VRF) that is independent of the marginal distribution and spectral density function of the underlying random heterogeneous material property field. The variance of apparent material properties can be an important consideration in problems where the domain over which the apparent properties are computed is smaller than the representative volume element and the approach introduced here provides an efficient means of calculating that variance and performing sensitivity studies with respect to the characteristics of the material property field. The approach is illustrated using examples involving heat transfer problems and finite elements with linear and nonlinear shape functions and in one and two dimensions. Features of the VRF are described, including dependency on shape and scale of the finite element and the order of the shape functions. [ABSTRACT FROM AUTHOR]

Asymmetric piezoelectric composites exhibit coupling between their macroscopic linear momentum and electric field, a coupling that does not appear at the microscopic scale. This electromomentum coupling constitutes an additional knob to tailor the dynamic response of the medium, analogously to the Willis coupling in elastic composites. Here, we employ topology- and free material optimization approaches to maximize the electromomentum coupling of periodic piezoelectric laminates in the low frequency, long-wavelength limit. We find that the coupling can be enhanced by orders of magnitude, depending on the degrees of freedom in the optimization process. The optimal compositions that we find provide guidelines for the design of metamaterials with maximum electromomentum coupling, paving the way for their integration in wave control applications. [ABSTRACT FROM AUTHOR]

We investigate the elastic behavior of periodic hexagonal honeycombs over a wide range of relative densities and cell geometries, using both analytical and numerical approaches. Previous modeling approaches are reviewed and their limitations identified. More accurate estimates of all nine elastic constants are obtained by modifying the analysis of Gibson and Ashby (1997) to account for the nodes at the intersection of the vertical and inclined members. The effect of the nodes becomes significant at high relative densities. We then compare the new analytical equations with previous analytical models, with a numerical analysis based on a computational homogenization technique and with data for rubber honeycombs over a wide range of relative densities and cell geometries. The comparisons show that both the new analytical equations and numerical solutions give a remarkably good description of the data. The results provide new insights into understanding the mechanics of honeycombs and designing new cellular materials in the future. [ABSTRACT FROM AUTHOR]

This study investigates the influence of void microstructure on the effective elastic properties of high fiber volume fraction discontinuous fiber-reinforced composites with randomly oriented fibers. A void microstructure is characterized by using parameters such as the number of voids, void volume fraction, void Aspect Ratio and void size distribution. In order to overcome the difficulties associated with the creation of high fiber volume fraction representative volume element models containing voids, a novel automation program called “ArtiComp” is developed. This program is used to generate representative volume element models of discontinuous fiber-reinforced composites containing randomly oriented straight fibers, curved fibers and spheroidal voids. The effective properties of discontinuous fiber-reinforced composites are computed using a finite element-based micromechanics approach. For comparison, the Mori-Tanaka method is used to compute the effective properties of unidirectional discontinuous fiber-reinforced composites and then orientation averaging is performed on these properties to obtain the effective properties of discontinuous fiber-reinforced composites with randomly oriented fibers. The comparison shows that the finite element predicted properties are stiffer than the analytically predicted properties. The results indicate that the finite element predicted properties are dependent on the number of voids used to represent the void volume. It is seen that the effective properties significantly decrease when void volume fraction is increased. The void Aspect Ratio and void size distribution are found to have a negligible influence on the effective properties. Finally, it is observed that straight fiber composites with voids perform better than the corresponding curved fiber composites with voids. [ABSTRACT FROM AUTHOR]

The present study is concerned with a numerical prediction of uncertainties in the macroscopic mechanical properties of microheterogeneous materials with uncertain microstructure. As a model material, solid foams are employed. The stochastic information on the uncertainty is gained in multiple numerical homogenization analyses of small-scale testing volume elements. The local relative density, the cell size distribution, the cell geometry and the spatial orientation of the testing volume elements are assumed to form the set of the relevant stochastic variables. Selected microstructural cases are analyzed for their macroscopic material response. Based on the probability distributions for the stochastic variables defining the microstructures of the testing volume elements, the probability distributions for the mesoscopic material properties are obtained. For the numerical homogenization of the testing volume elements, an enhanced finite element technique is employed, where the components of the macroscopic deformation gradient are introduced as generalized degrees of freedom. Assuming periodic boundary conditions, the global degrees of freedom interact with the conventional displacement degrees of freedom of the discretized microstructure via special boundary coupling elements. The mesoscopic stresses are obtained in a rather efficient manner as the generalized reaction forces for the global degrees of freedom. [ABSTRACT FROM AUTHOR]

In the present work, applying the asymptotic homogenization method (AHM), the derivation of the antiplane effective properties for three-phase magneto-electro-elastic fiber unidirectional reinforced composite with parallelogram cell symmetry is reported. Closed analytical expressions for the antiplane local problems on the periodic cell and the corresponding effective coefficients are provided. Matrix and inclusions materials belong to symmetry class 6mm. Numerical results are reported and compared with the eigenfunction expansion-variational method (EEVM) and other theoretical models. Good agreements are found for these comparisons. In addition, with the herein implemented solution, it is possible to reproduce the effective properties of the reduced cases such as piezoelectric or elastic composites obtaining good agreements with previous reports. [ABSTRACT FROM AUTHOR]

Abstract: The computational cost of micromechanics for heterogeneous materials can be reduced in certain cases where symmetric boundary conditions are applicable. We derived an eighth symmetric formulation of the Generalized Method of Cells for triply periodic microstructures. During this endeavor, an inherent symmetry was discovered. This implied that all repeating unit cells may be quarter symmetric representations of other microstructures. Additionally, it was discovered that a repeating unit cell can have columns of subcells swapped with no changes to the local or global fields. We concluded that first-order micromechanics are not well suited for capturing detailed or complex microstructures; however, higher-order theories, such as High Fidelity Generalized Method of Cells, can adequately model these microstructures. [Copyright &y& Elsevier]