Locally-exact Homogenization Theory Research Articles

Homogenized moduli and local multiphysics fields of unidirectional piezoelectric nanocomposites with energetic surfaces

The study extends the locally-exact homogenization theory (LEHT) to determine the effective parameters and localized multiphysics fields of unidirectional piezoelectric nanocomposites with energetic surfaces. The interface is simulated using the generalized Gurtin-Murdoch model for piezoelectric nanocomposites, taking into account the discontinuities between surface stresses and electrical displacement. By characterizing hexagonal and square repeating unit cells (RUC), the interactions across a fiber's or pore's interface in piezoelectric nanocomposites with energetic surfaces have been analyzed. The extended LEHT possesses the advantages of fast convergence, excellent stability, and computational efficiency due to its avoidance of mesh discretization and pointwise satisfaction of interfacial conditions. To verify the validity of the present theory, this paper also derives the Eshelby solution for piezoelectric nanocomposite with energetic surfaces. The reliability and accuracy of the present theory are demonstrated by comparing the present results from the Eshelby solution and the finite element method (FEM) with generally good agreement. On this basis, the influence of microstructural effects, such as phase volume fractions and geometrical arrangement, on the effective and local responses of piezoelectric nanocomposites with energetic surfaces are investigated. The results show that effective piezoelectric/dielectric moduli and electric displacement field for piezoelectric nanocomposites are significantly affected by the size effect, along with several other parameters such as unit cell arrays and fiber/pore volume fractions. Those results highlight the significance of neighboring pore or fiber interactions for piezoelectric nanocomposites, which are often overlooked in traditional micromechanics models and are challenging to be captured using numerical methods.

Inverse Identification and Design of Thermal Parameters of Woven Composites through a Particle Swarm Optimization Method

Designing thermal conductivity efficiently is one of the most important study fields for taking the advantages of woven composites. This paper presents an inverse method for the thermal conductivity design of woven composite materials. Based on the multi-scale structure characteristics of woven composites, a multi-scale model of inversing heat conduction coefficient of fibers is established, including a macroscale composite model, mesoscale fiber yarn model, microscale fiber and matrix model. In order to improve computational efficiency, the particle swarm optimization (PSO) algorithm and locally exact homogenization theory (LEHT) are utilized. LEHT is an efficient analytical method for heat conduction analysis. It does not require meshing and preprocessing but obtains analytical expressions of internal temperature and heat flow of materials by solving heat differential equations and combined with Fourier's formula, relevant thermal conductivity parameters can be obtained. The proposed method is based on the idea of optimum design ideology of material parameters from top to bottom. The optimized parameters of components need to be designed hierarchically, including: (1) combing theoretical model with the particle swarm optimization algorithm at the macroscale to inverse parameters of yarn; (2) combining LEHT with the particle swarm optimization algorithm at the mesoscale to inverse original fiber parameters. To identify the validation of the proposed method, the present results are compared with given definite value, which can be seen that they have a good agreement with errors less than 1%. The proposed optimization method could effectively design thermal conductivity parameters and volume fraction for all components of woven composites.

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

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

Multiscale numerical modeling for thermal behavior of plain-woven composites with interfacial and internal defects

This paper predicted thermal responses of plain-woven composites with interfacial and internal defects based on the multi-scale simulation. The effective and localized responses of plain-woven composites are simulated through “bottom-up” homogenization and “top-down” localization across several scales. Periodic representative volume elements (RVEs) are extracted at micro-scale, meso-scale and macro-scale, respectively, to represent the entire composite arrays at distinct scales. The local behavior of RVEs is predicted by either finite element method (FEM) or the locally exact homogenization theory (LEHT), the latter of which is employed to simulate the effect of imperfect interface between fiber and matrix to avoid large-scale mesh refinement in FEM, especially at the vicinity of physical interphase/ cohesive damaged interface. Based on the localized temperature/heat flux distributions, the effect constitutive relations are introduced to predict the corresponding effective thermal coefficients. The accuracy and efficiency of the proposed method is verified by validating the predicted effective coefficients against the results in literature. Several microstructural parameters, including interfacial parameters, internal porosity volume fraction, as well as fiber contents are tested on the effective response and it is found that effective thermal conductivity of plain-woven composites varies greatly when the interfacial parameter of imperfect interface lied in range of “sensitivity zone”, while the thickness of physical interphase has less significant influence on the in-plane and out-of-plane thermal conductivities. More importantly, the reinforcing/weakening schemes of fiber or damaged interface are illustrated through recovering the local thermal distribution and demonstrating the significant interfacial effect. The model established in this paper can provide theoretical guidance for the neglected imperfect interface and internal defects in plain woven composites.

The multi-scale meso-mechanics model of viscoelastic dentin

Human dentin is a hierarchical material with multi-level micro-/nano-structures, consisting of tubule, perti-tubular dentin (PTD) and intertubular dentin (ITD) as the major constituents at microscale; and the PTD and ITD are further composed of collagen and hydroxyapatite (HAp) crystals with different volume fractions at nanoscale. In most cases, the HAp is considered as elastic while the collagen as viscoelastic material. It is of great significance to study the hierarchical structure and viscoelasticity of human dentin to understand the mechanical properties of dentin for further development of restorative materials. Based on this, this paper focuses on multiscale modeling of the elastic properties and dynamic viscoelastic response of dentin and establishes a bottom-up micromechanics model from nano-to macro-scale. In order to study the nanostructural effect on the viscoelastic behavior of hierarchical structures, the homogenization theories of random platelets composites (HTRPC) and the locally-exact homogenization theory (LEHT) are introduced for the homogenization of heterogeneous materials of microstructures at different levels. The HTRPC, based on Eshelby Inclusion theory, is used to predict the effective modulus of PTD and ITD. The LEHT is a method for homogenizing multiphase dentin characterized by repeated unit cells (RUCs). The resulting predictions are in very good agreement with several experimental data from the literature. In addition, the results of nanostructrual effect on dentin show that the viscoelasticity of dentin is majorly contributed by collagen and the HAp greatly provide the strength and hardness of dentin. Furthermore, the ageing effect on dentin's viscoelasticity is considered from the proposed multiscale micromechanics model. It is demonstrated that the ageing effect is much more influential in affecting the loss moduli of dentin than the storage.

Frequency domain homogenization of effective and localized viscoelastic response of unidirectional composites with imperfect interfaces

The locally-exact homogenization theory for unidirectional periodic composites with imperfect interfaces is generalized to estimate the effective complex moduli of viscoelastic composites in the frequency domain and recover corresponding localized stress distributions. In order to avoid complex mesh discretization, the Trefftz concept is adopted to obtain the analytical expression of internal displacements and stresses of representative unit cells by directly solving elastic partial differential equations. Physical coating and linear cohesive spring models are implemented at the interface between fibers and matrix. Besides, we also prove that the two imperfect interface models are equivalent under specified viscoelastic conditions. Comparisons against simulations available in the literature are given to validate the present method where good agreement is generally obtained. The effects of phase volume fraction, coating thickness and interfacial stiffness on five independent elastic constants are effectively investigated using the present models, further the local stress distributions within unit cells are presented to illustrate the effect of imperfect interfaces on material properties. Finally, the free vibration of Euler beams composed of unidirectional composites are studied through multiscale simulations to demonstrate importance of aforementioned microstructural parameters on damping of structural vibration, offering guideline for the structural and material design of aircrafts and satellites.

Extended Locally Exact Homogenization Theory for Effective Coefficients and Localized Responses of Piezoelectric Composites

Herein, the elasticity‐based locally exact homogenization theory (LEHT) is extended to study the effective and localized responses of piezoelectric fiber composites. Based on the Trefftz concept, coupled internal solutions for electric fields and elastic displacements are developed for repeating unit cells (RUCs) that are characterized from the composite domain. The unknown coefficients of the internal fields can be determined by imposing interfacial continuity conditions between fiber and matrix and implementing the newly proposed multiphysics periodic boundary conditions. Finally, the generalized constitutive relations are established to predict the effective coefficients. The accuracy of the proposed technique is tested by validating the present simulations against the analytical Eshelby solution and asymptotic method, classical Mori−Tanaka (M−T) model, finite element (FE)‐ and finite volume (FV)‐based methods, as well as the experimental measurement in the literature with excellent agreement. What is more, several microstructural effects, such as phase volume fraction, fiber arrangement, etc., are varied to test their effects on the piezoelectric composites at both homogenized and localized levels. Based on the presented computational efficiency, the multiphysics LEHT is encapsulated into a blackbox with input/output data constructions for easy implementation by professionals or nonprofessionals.

An effective thermal conductivity and thermomechanical homogenization scheme for a multiscale Nb3Sn filaments

Abstract A comprehensive study of the multiscale homogenized thermal conductivities and thermomechanical properties is conducted towards the filament groups of European Advanced Superconductors (EAS) strand via the recently proposed Multiphysics Locally Exact Homogenization Theory (LEHT). The filament groups have a distinctive two-level hierarchical microstructure with a repeating pattern perpendicular to the axial direction of Nb3Sn filament. The Nb3Sn filaments are processed in a very high temperature between 600 and 700°C, while its operation temperature is extremely low, −269°C. Meanwhile, Nb3Sn may experience high heat flux due to low resistivity of Nb3Sn in the normal state. The intrinsic hierarchical microstructure of Nb3Sn filament groups and Multiphysics loading conditions make LEHT an ideal candidate to conduct the homogenized thermal conductivities and thermomechanical analysis. First, a comparison with a finite element analysis is conducted to validate effectiveness of Multiphysics LEHT and good agreement is obtained for the homogenized thermal conductivities and mechanical and thermal expansion properties. Then, the Multiphysics LEHT is applied to systematically investigate the effects of volume fraction and temperature on homogenized thermal conductivities and thermomechanical properties of Nb3Sn filaments at the microscale and mesoscale. Those homogenized properties provide a full picture for researchers or engineers to understand the Nb3Sn homogenized properties and will further facilitate the material design and application.

Effects of general imperfect interface/interphase on the in-plane conductivity of thermal composites

The effects of general imperfect interphase/interface models on effective and local thermal conductive responses of fiber reinforced-composites are investigated. A physical interphase model is firstly introduced and then extended to the weakly conducting (WC) and highly conducting (HC) interfaces by satisfying the discontinuity conditions of temperature and normal heat flux, respectively. To simulate different geometric arrangements of fibers, hexagonal and square repeating unit cells (RUCs) are used in this paper. Regarding simulation technique, the locally-exact homogenization theory (LEHT) is extended by considering the aforementioned interphase/interface models to re-produce the localized temperature and heat flux distributions in the unit cells and then to predict the macroscopic thermal conductivities. The present method does not require mesh discretization and pre-processing, but directly obtains the accurate analytical expression of internal domains after solving the thermal differential equations. The unknown coefficients are obtained through applying imperfect interphase/interface conditions and weak-form periodic boundary conditions. Finally, the effective Fourier's formula is established to obtain the overall thermal conductivities of composites. The LEHT is advantageous in fast convergence, numerical stability and computational efficiency, thus providing a good numerical tool in studying the influence of interface parameters on composites. The generated numerical results are in good agreement with the analytical or numerical simulations in the literature. On this basis, the effects of interfacial geometric and physical parameters on the macroscopic coefficients and local thermal responses of the composites are tested. Finally, the thickness of a physical interphase is varied to test its applicability in generating equivalent effect with WC interface and HC interface. Generally, the imperfect interfacial effects play important roles in the thermal conductive response of composite materials. In the meantime, the proposed method is of great significance for the study of thermal conductions of composite microstructures.

Global buckling and multiscale responses of fiber-reinforced composite cylindrical shells with trapezoidal corrugated cores

In order to comprehensively understand the effect of composite materials on the structural response, a novel multiscale numerical model is presented in this work to study the global buckling and localized responses of composite cylindrical shells with trapezoidal corrugated cores. Treated as hierarchical structures, the investigation of composite shells includes carbon fiber-reinforced composites at micro-scale, fiber–matrix laminates at meso-scale, as well as corrugated shells at macro-scale. The well-established locally-exact homogenization theory is firstly introduced to generate the microstructural effective coefficients, which are then transferred to the lamination theory. Finally, a simplified structural homogeneous representative volume element (RVE) is established with equivalent properties to a part of cylindrical structures with inner and outer skins embedded with corrugated cores, significantly facilitating the finite element (FE) simulations. The simplified structure is validated against the 3D full-scale analysis with good agreement for critical loads of different buckling modes. The global buckling performance of corrugated cylinders is then demonstrated by considering the effect of constituent materials, fiber volume fraction and geometric parameters. More importantly, benefited from a top-down procedure, the localized stress distributions are recovered using the three-stage multiscale model, to predict the possible crack initiations starting within the microstructures.

On the effective behavior of viscoelastic composites in three dimensions

We address the calculation of the effective properties of non-aging linear viscoelastic composite materials. This is done by solving the microscale periodic local problems obtained via the Asymptotic Homogenization Method (AHM) by means of finite element three-dimensional simulations. The work comprises the investigation of the effective creep and relaxation behavior for a variety of fiber and inclusion reinforced structures (e.g. polymeric matrix composites). As starting point, we consider the elastic-viscoelastic correspondence principle and the Laplace-Carson transform. Then, a classical asymptotic homogenization approach for composites with discontinuous material properties and perfect contact at the interface between the constituents is performed. In particular, we reach to the stress jump conditions from local problems and obtain the corresponding interface loads. Furthermore, we solve numerically the local problems in the Laplace-Carson domain, and compute the effective coefficients. Moreover, the inversion to the original temporal space is also carried out. Finally, we compare our results with those obtained from different homogenization approaches, such as the Finite-Volume Direct Averaging Micromechanics (FVDAM) and the Locally Exact Homogenization Theory (LEHT).

Generalized locally-exact homogenization theory for evaluation of electric conductivity and resistance of multiphase materials

Abstract The locally-exact homogenization theory is further extended to investigate the homogenized and localized electric behavior of unidirectional composite and porous materials. Distinct from the classical and numerical micromechanics models, the present technique is advantageous by developing exact analytical solutions of repeating unit cells (RUC) with hexagonal and rhomboid geometries that satisfy the internal governing equations and fiber/matrix interfacial continuities in a point-wise manner. A balanced variational principle is proposed to impose the periodic boundary conditions on mirror faces of an RUC, ensuring rapid convergence of homogenized and localized responses. The present simulations are validated against the generalized Eshelby solution with electric capability and the finite-volume direct averaging micromechanics, where excellent agreements are obtained. Several micromechanical parameters are then tested of their effects on the responses of composites, such as the fiber/matrix ratio and RUC geometry. The efficiency of the theory is also proved and only a few seconds are required to generate a full set of properties and concomitant local electric fields in an uncompiled MATLAB environment. Finally, the related programs may be encapsulated with an input/output (I/O) interface such that even non-professionals can execute the programs without learning the mathematical details.

The morphological effect of carbon fibers on the thermal conductive composites

The locally-exact homogenization theory (LEHT) with thermal conductive capability is developed to investigate the morphological effect of carbon/graphite fibers on the effective and localized thermal responses of periodic composites. Based on the orientations of basal planes, the material properties of carbon fibers can be transversely isotropic, radially orthotropic or circumferentially orthotropic, possibly influencing the microscopic behavior of thermal conductive composites. By taking fiber-fiber interactions into account, repeating unit cells (RUCs) of hexagonal and rectangular geometries are considered with large fiber volume fractions. The efficiency and stability of the LEHT are guaranteed by solving the complete internal eigenvalue functions, imposing point-wise continuity conditions, as well as implementing the generalized variational principle. It is demonstrated that the present theory exhibits good agreement with the independently developed Eshelby solutions and Hashin's formula. The morphological effects are also tested by generating effective coefficients over a wide range of fiber volume fractions and recovering the local thermal field concentrations within the composite microstructures. The results clearly indicate that even if the homogenized properties are not significantly affected by the morphologies of carbon fibers by satisfying the replacement scheme, the large heat-flux gradients within the orthotropic fibers could still lead to the split of carbon reinforcement.

An efficient analytical homogenization technique for mechanical-hygrothermal responses of unidirectional composites with applications to optimization and multiscale analyses

The elasticity-based Locally Exact Homogenization Theory (LEHT) is extended to study the mechanical-hygrothermal behaviors of unidirectionally-reinforced composites. Based on the framework developed previously, thermal and moisture effects are incorporated into the LEHT to study the homogenized and localized responses of heterogeneous materials, which are validated using available analytical and numerical techniques. The LEHT programs are then encapsulated as subroutines with Input/Output (I/O) interfaces, to be readily applied in different computational scenarios. In order to illustrate the efficiency of the LEHT, the theory is firstly coupled to the Particle Swarm Optimization (PSO) algorithm in order to minimize the axial thermal expansion mismatch in hexagonal and square fiber arrays by tailoring the fiber volume fraction. The LEHT is then implemented into the lamination theory to study fabrication-induced residual stresses arising during the cool-down process which introduces local laminate stresses owing to thermo-mechanical property mismatch between plies. Both of these applications illustrate the efficiency and accuracy of the LEHT in generating effective properties and local stress distributions, making the theory a golden standard in validating other analytical or numerical techniques as well as a reliable tool in composite design and practice for professionals and non-professionals alike.

Homogenized moduli and local stress fields of unidirectional nano-composites

Abstract The recently generalized locally-exact homogenization theory is further extended to enable the determination of homogenized moduli and local stress fields in unidirectional nano-composites with surface effects based on the Gurtin-Murdoch model. The excellent stability and quick convergence of the homogenization theory with the concomitant rapid execution times enables repetitive solutions of the unit cell problem with variable geometric and material parameters, thereby enabling extensive parametric studies to be conducted efficiently. The distinguishing feature of the theory is its ability to provide accurate homogenized moduli as well as accurate local stress fields for square, rectangular and hexagonal arrays of nano-inclusions or nano-porosities. The extended elasticity-based computational capability is validated by published results on homogenized moduli and stress concentrations in nanoporous aluminum obtained using classical micromechanics, elasticity-based, semi-analytical and numerical approaches. New results are generated aimed at demonstrating the effects of nanopore arrays on homogenized moduli and local stress fields in a wide range of porosity volume fractions and nanopore radii. These results highlight the importance of adjacent pore interactions either neglected or not directly taken into account in the classical micromechanics models, nor easily captured by numerical techniques.

Elasticity-based microstructural optimization: An integrated multiscale framework

An integrated multiscale computational framework is developed to identify material microstructures which minimize target field variables at the microstructural level aimed at enhancing structural performance. The computational framework is comprised of the Particle Swarm Optimization algorithm, a structural-level analytical solution to a technological problem, and the elasticity-based locally-exact homogenization theory at the microstructural level. The excellent stability and quick convergence of the homogenization theory with the concomitant rapid execution times enables repetitive solutions of the unit cell problem with variable geometric and material parameters within a multiscale framework. The unit cell solution provides accurate homogenized moduli for input into the structural-level analysis, as well as accurate local stress fields for the calculation of objective functions by the optimization algorithm. The integrated multiscale optimization capability is demonstrated in the context of the classical Kirsch problem with an underlying graded microstructure aimed at stress concentration reduction. The results reveal the limitations of stress field optimization in graded materials solely at the homogenized level, pointing to the need for repetitive micro-level stress field recovery from an accurate and efficient homogenization theory within multiscale optimization analysis.

Tailoring the moduli of composites using hollow reinforcement

Effects of hollow reinforcement on homogenized moduli and stress fields in unidirectional composites are investigated using an elasticity-based homogenization theory for periodic materials with hexagonal symmetries. The theory employs Fourier series representations for hollow fiber and matrix displacement fields in the cylindrical coordinate system which satisfy exactly the equilibrium equations and continuity conditions in the interior of the unit cell. The inseparable exterior problem requires satisfaction of periodicity conditions efficiently accomplished using previously introduced balanced variational principle. This principle ensures rapid solution convergence in the presence of thick or very thin-walled hollow fibers with relatively few harmonic terms. The solution’s analytical framework and stability facilitate parametric and optimization studies aimed at rapid identification of fiber wall-thickness impact on homogenized moduli and local fields in wide range of fiber volume fractions. This is illustrated for two material systems containing hollow glass fibers and alumina nanotubes as reinforcement, revealing new results of interest in the design of multifunctional porous materials. The results of parametric studies are made more precise upon incorporation of the locally-exact homogenization theory into the Particle Swarm Optimization algorithm, with the new computational capability employed to identify tube thickness as a function of the porosity volume fraction which minimizes the differences between target homogenized moduli and the corresponding matrix moduli. Analysis of a generalized Kirsch problem with an underlying microstructure based on targeted homogenized moduli around the cylindrical cavity demonstrates the homogenization theory’s utility in tailoring the local stress field with the aim of enhancing material toughness.

Locally-exact homogenization of viscoelastic unidirectional composites

The elasticity-based, locally-exact homogenization theory for periodic materials with hexagonal and tetragonal symmetries is extended to accommodate linearly viscoelastic phases via the correspondence principle. The theory employs Fourier series representations for fiber and matrix displacement fields in the cylindrical coordinate system that satisfy exactly equilibrium equations and continuity conditions in the interior of the unit cell. The inseparable exterior problem requires satisfaction of periodicity conditions efficiently accomplished using previously introduced balanced variational principle which ensures rapid displacement and stress field convergence in the presence of linearly viscoelastic phases with relatively few harmonic terms. The solution’s stability and efficiency, with concommitant simplicity of input data construction, facilitate rapid identification of the impact of phase viscoelasticity and array type on homogenized moduli and local fields in wide ranges of fiber volume fraction. We illustrate the theory’s utility by investigating the impact of fiber array type and matrix viscoelastic response (constant Poisson’s ratio vs constant bulk modulus) on the homogenized response and local stress fields, reporting previously undocumented differences. Specifically, we show that initially small differences between hexagonal and square arrays are magnified substantially by viscoelasticity. New results on the transmission of matrix viscoelastic features to the macroscale are also generated in support of construction of homogenized viscoelastic functions from experimental data.

On Boundary Condition Implementation Via Variational Principles in Elasticity-Based Homogenization

Convergence characteristics of the locally exact homogenization theory for periodic materials, first proposed by Drago and Pindera (2008, “A Locally-Exact Homogenization Theory for Periodic Microstructures With Isotropic Phases,” ASME J. Appl. Mech., 75(5), p. 051010) and recently generalized by Wang and Pindera (“Locally-Exact Homogenization Theory for Transversely Isotropic Unidirectional Composites,” Mech. Res. Commun. (in press); 2016, “Locally-Exact Homogenization of Unidirectional Composites With Coated or Hollow Reinforcement,” Mater. Des., 93, pp. 514–528; and 2016, “Locally Exact Homogenization of Unidirectional Composites With Cylindrically Orthotropic Fibers,” ASME J. Appl. Mech., 83(7), p. 071010), are examined vis-a-vis the manner of implementing periodic boundary conditions. The locally exact theory separates the unit cell problem into interior and exterior problems, with the separable interior problem solved exactly in cylindrical coordinates and the inseparable exterior problem tackled using a balanced variational principle. This variational principle leads to exceptionally fast and well-behaved convergence of the Fourier series coefficients in the displacement field representation of the unit cell's different phases. Herein, we compare the solution's convergence behavior based on the balanced variational principle with that based on the constrained energy-based principle originally proposed by Jirousek (1978, “Basis for Development of Large Finite Elements Locally Satisfying All Fields Equations,” Comput. Methods Appl. Mech. Eng., 14, pp. 65–92) in the context of locally exact finite-element analysis. The relevance of this comparison lies in the recently rediscovered implementation of Jirousek's constrained variational principle in the homogenization of periodic materials.

Locally-exact homogenization theory for transversely isotropic unidirectional composites

The locally-exact homogenization theory for unidirectional composites with square periodicity and isotropic phases proposed by Drago and Pindera [18] is extended to architectures with hexagonal symmetry and transversely isotropic phases. The theory employs Fourier series representation for the displacement fields in the fiber and matrix phases in the cylindrical coordinate system that satisfies exactly the equilibrium equations and continuity conditions in the unit cell's interior. The inseparable exterior problem involves satisfaction of periodicity conditions for the hexagonal unit cell geometry demonstrated herein to be readily achievable using the previously introduced balanced variational principle for square geometries. This variational principle plays a key role in the employed unit cell solution, ensuring rapid convergence of the Fourier series coefficients with relatively few harmonic terms, yielding converged homogenized moduli and local stress fields with little computational effort. The solution's stability is illustrated using the dilute case which is shown to reduce to the Eshelby solution regardless of the employed number of harmonic terms. Comparison with published results and predictions of a finite-volume based homogenization in a wide fiber volume range and different fiber/matrix modulus contrast validates the approach's accuracy, and its utility is demonstrated through rapid local stress recovery in a multi-scale application. This extension completes the development of the theory for three important classes of unidirectional reinforcement arrays, thereby providing an efficient alternative to finite-element based homogenization techniques or approximate micromechanical schemes, as well as an efficient standard against which other methods may be compared.