Mean-field Homogenization Scheme Research Articles

Mean field homogenization schemes using averages of stress, strain and strain energy density and applications

The average of stress (AS) and average of strain (AN) were often used in the development of conventional mean-field homogenization (MFH) schemes, but their correlation with the average of strain energy density (AE) has rarely been discussed. In this work, we examined the correlations of AE with AS and AN, and found that AE can be reduced to the product of AS or AN under uniform boundary stress or linear surface displacement associated with a uniform strain field. By introducing a reference matrix and using any two among AE, AS and AN, three MFH schemes were developed: NS (using AN and AS); NE (using AN and AE); and SE (using AS and AE). Following the concept of the Mori-Tanaka's (MT) method, three MFH schemes obtained above were reduced to MT-NS, MT-NE and MT-SE, respectively. Effective elastic properties of composites consisting of isotropic elastic matrices and spherical isotropic elastic inclusions were predicted using these schemes and compared with experimental results. The comparison revealed that MT-SE and MT-NS can reasonably replicate the experimental results in the case of moderate particle volume fractions and ratios between the Young's moduli of matrix and the reinforcement particles, demonstrating the validity of the MFH schemes. This work is significant because it not only clarifies the correlation of AE with AN and AS, but also provides more options for the development of MFH schemes, which can enrich the theory and methods for the analysis of the effective properties of composites.

Modeling strain-induced martensitic transformation in austenitic stainless steels subjected to cryogenic temperatures using incremental mean-field homogenization schemes

Austenitic stainless steels are commonly used as structural materials in high-field superconducting magnet systems because they retain high strength, ductility, and toughness at very low temperatures, and they are paramagnetic or antiferromagnetic under the Néel temperature in their fully austenitic state. However, they are susceptible to strain-induced martensitic transformation, especially at cryogenic temperatures, which modifies the material properties, induces volume changes and additional strain hardening, and leads to ferromagnetic behavior. Thus, accurate predictions of the structural performance of these materials at very low temperatures are of great interest for the conception and design of these cryo-magnetic systems. In this paper, we propose an adequate constitutive model for the evolving bi-phase material—austenite and martensite—based on a Hill-type incremental formulation. Two different versions of the model are proposed based on the linear mean-field homogenization scheme: Mori-Tanaka and Self-Consistent. Moreover, a rate-independent nonlinear mixed kinematic-isotropic hardening law is used for each phase, and the martensitic transformation is described by the nonlinear kinetic law proposed by Olson and Cohen (1975). The constitutive model is implemented in ABAQUS/Standard through a UMAT user subroutine, for which a return mapping algorithm based on the implicit backward Euler integration scheme is used and a closed-form expression of the consistent Jacobian tensor is provided. The Mori-Tanaka and Self-Consistent approaches are evaluated in terms of their ability to describe the mechanical behavior of the bi-phase aggregate by comparing the predictions of the homogenization schemes with unit-cell finite element calculations with an explicit description of the martensite inclusions and the austenite matrix. The comparison is carried out for different stress states with controlled triaxiality and Lode parameter under monotonic and cycling loading, paying special attention to the evolution of the mechanical fields in each phase. The unit-cell calculations are performed for both constant and evolving martensite volume fractions. In addition, numerical simulations of tensile tests on samples subjected to different initial temperatures are carried out for the transforming bi-phase material and the results are compared to experimental data for AISI 304L and AISI 316LN steels.

Numerical homogenization of poly-crystalline silicon wafer based photovoltaic modules including pre-cracks

The photovoltaic (PV) modules containing multiple polycrystalline silicon solar cells (PSSCs) are one of the most common devices for solar energy production. PSSCs are finding different applications such as on buildings and on solar-driven airplanes, hence, their mechanical analyses are required. During the lifespan of PSSCs, the performance degrades, which is predominantly associated with the cracking of the crystalline silicon wafer (PSW). Moreover, because of the multiple materials involved and the complicated microstructure of the energy-producing component, polycrystalline silicon wafer (PSW), modeling becomes computationally expensive. Therefore, model reduction techniques, such as homogenization, are needed. The mechanical response of polycrystalline silicon solar cells (PSSCs) is investigated in this work. The crystalline patterns of the PSW were generated using a Voronoi-tessellation scheme. A mean-field homogenization scheme using the finite element (FE) method was employed to predict the homogenized response of the PSSCs. The response of the PSSC with heterogeneous and homogeneous modeling was compared. The stiffness degradation due to the existing microcracks of the PSW was investigated. It is evident from these investigations that the homogenized FE solution provides an accurate and computationally efficient representation of the progressive failure in solar cells. The approach presented here offers a basis to further enhance the knowledge of failure analysis in PV modules and quantify the subsequent degradation of their energy producing capacity.

Microstructure-Based Modelling of Flow and Fracture Behavior of Tailored Microstructures of Ductibor® 1000-AS Steel

Emerging grades of press-hardening steels such as Ductibor® 1000-AS are now commercially available for use within tailor-welded blanks (TWBs) to enhance ductility and energy absorption in hot-stamped automotive structural components. This study examines the constitutive (hardening) response and fracture limits of Ductibor® 1000-AS as functions of the as-quenched microstructure after hot stamping. Three different microstructures consisting of bainite and martensite were obtained by hot stamping with die temperatures of 25 °C, 350 °C, and 450 °C. Mechanical characterization was performed to determine the hardening curves and plane-stress fracture loci for the different quench conditions (cooling rates). Uniaxial-tension and shear tests were conducted to experimentally capture the hardening response to large strain levels. Shear, conical hole-expansion, plane-strain notch tension, and Nakazima tests were carried out to evaluate the stress-state dependence of fracture. A mean-field homogenization (MFH) scheme was applied to model the constitutive and fracture behavior of the mixed-phase microstructures. A dislocation-based hardening model was adopted for the individual phases, which accounts for material chemistry, inter-phase carbon partitioning, and dislocation evolution. The per-phase fracture modelling was executed using a phenomenological damage index based upon the stress state within each phase. The results revealed that the 25 °C hot-stamped material condition with a fully martensite microstructure exhibited the highest level of strength and the lowest degree of ductility. As bainite was formed in the final microstructure by quenching at higher die temperatures, the strength decreased, while the ductility increased. The predicted constitutive and fracture responses in the hot-stamped microstructures were in line with the measured data. Accordingly, the established numerical strategy was extended to predict the mechanical behavior of Ductibor® 1000-AS for a broad range of intermediate as-quenched microstructures.

A binary-medium-based constitutive model for geological materials based on the statistical meso-breakage concept and mean-field homogenization

This paper is devoted to determining the inelastic responses of geological materials by the binary-medium-based constitutive model. The geomaterials are conceptualized as two-phase composites that are composed of the bonded medium of elastic–brittle nature and the frictional medium of elastoplastic nature. The evolution of meso-structure upon loading is considered by the progressive transformation of the bonded medium into the frictional medium. By introducing the statistical meso-breakage concept, the breakage ratio indicating the degree of such transformation is determined by statistical approaches. The interactions among inclusions (the bonded medium) and matrix (the frictional medium) and the influence of the varying breakage ratio are considered with the help of the double-inclusion model within the framework of the mean-field homogenization scheme. An application is presented for the structured/bonding geomaterials based on the proposed model. The comparisons between computed results and experimental data of three kinds of structured soils show that the proposed model can reproduce the typical behaviors of cemented/structured geomaterials.

A micromechanical analysis of strain concentration tensor for elastoplastic medium containing aligned and misaligned pores

In this paper, making use of Eshelby tensor for the elastoplastic medium, will be presented an approach for the strain concentration tensor for materials that present spherical or spheroidal pores. The pores are embedded in a homogeneous isotropic matrix which presents an elastoplastic behavior, governed by the von Mises yield criterion and isotropic hardening. In this context, it will also be analyzed the behavior of a porous material through the influence of the extraction of the isotropic part of the anisotropic operator. The effective properties of the material in each incremental step are obtained through the micromechanical Mori-Tanaka's model. Through elastoplastic strain concentration tensor for aligned or misaligned pores, it was observed that the pore direction represents a greater influence on the elastoplastic behavior of the material studied than aspect ratio (length divided by pore diameter). This approach has the advantage of being easily incorporated into conventional mean-field homogenization (MFH) schemes for the case where inclusion stiffness is neglected and presents different geometries or directions in 3D space.

Improved incrementally affine homogenization method for viscoelastic-viscoplastic composites based on an adaptive scheme

We proposed an adaptive incrementally affine method, a micromechanics-based mean-field homogenization scheme for viscoelastic-viscoplastic particle-reinforced composites, which is applicable for predicting the mechanical response under complex loading conditions. The formulation is based on an incrementally affine scheme using algorithmic tangent operators while adaptively adjusting the mean strain of each constituent at every step of the loading process to reflect the changes in tangent operators and the shape of the reinforcing particles. We found that its prediction beyond the viscoelastic regime can be further improved by enforcing consistency between the accumulated strain states of each phase and the concentration tensors, excluding accumulated affine strain and stress. It was inevitable that the plastic deformation of the composite was initiated earlier than what was predicted by the mean-field theory owing to the local stress concentration near the particles. Hence, we proposed a yield reduction method that enforces the earlier initiation of plastic deformation in the matrix phase when obtaining an effective mechanical response. We observed that the predictions of the adaptive incrementally affine scheme adjusted with the yield reduction agreed well with various numerical simulations of particle-reinforced composites considering viscoelastic, elastic-viscoplastic, and viscoelastic-viscoplastic matrices under uniaxial, cyclic, and biaxial loadings.

An improved mean-field homogenization model for the three-dimensional elastic properties of masonry

Accurate assessment of the overall mechanical behavior of masonry, composed of bricks and mortar joints, remains challenging due to its inhomogeneous and orthotropic nature. In this study, the feasibility of various mean-field homogenization schemes for the three-dimensional orthotropic elastic properties of masonry is comprehensively investigated. Three kinds of masonry patterns are considered, including the stack bonded pattern, the running bonded pattern and the double-leaf Flemish bonded pattern that has received limited attention so far. Special attention is paid to the homogenization schemes which have not been applied to the masonry case, such as Lielens’ interpolative double inclusions (D-I) and the interaction direct derivative (IDD) schemes. After a comparison between the well-known mean-field homogenization schemes, an improved micro-mechanical model is proposed by combining the advantages of the IDD and D-I models. The validation of the proposed model is conducted through a comparison against experimental data from literature and numerical results obtained via finite element analyses (FEA). The results show that the proposed model can accurately evaluate the orthotropic elastic properties of the three masonry typologies for a wide range of stiffness ratios between brick and mortar, ranging from 1 to 1000. The proposed model also shows better performance than the classical schemes especially when the stiffness ratios between brick and mortar are higher than 10, which is of major importance for the application of mean-field homogenization based multiscale methods to the nonlinear analysis of masonry. Furthermore, the presented homogenization method can be of interest for other anisotropic materials, e.g., laminate materials.

Effective thermo-viscoelastic behavior of short fiber reinforced thermo-rheologically simple polymers: An application to high temperature fiber reinforced additive manufacturing

This paper presents a procedure for the estimation of the effective thermo-viscoelastic behavior in fiber-reinforced polymer filaments used in high temperature fiber-reinforced additive manufacturing (HT-FRAM). The filament is an amorphous polymer matrix (PEI) reinforced with elastic short glass fibers treated as a single polymer composite (SPC) holding the assumption of thermo-rheologically simple matrix. Effective thermo-viscoelastic behavior is obtained by implementing mean-field homogenization schemes through the extension of the correspondence principle to continuous variations of temperature by using the time–temperature superposition principle and the internal time technique. The state of the fibers in the composite is described through the use of probability distribution functions. Explicit forms of the effective properties are obtained from an identification step, ensuring the same mathematical structure as the matrix behavior. The benchmark simulations are predictions of residual stress resulting from the cooling of the representative elementary volumes (REVs) characterizing the composite filament. The computation of the averaged stress in the benchmarking examples is achieved by solving numerically the stress–strain problem via the internal variables’ framework. Reference solutions are obtained from Fast Fourier Transform based full-field homogenization simulations. A comparative analysis is performed, showing the reliability of the proposed homogenization procedure to predict residual stress against extensive computations of the macroscopic behavior of a given microstructure.

An incremental‐secant mean‐field homogenization model enhanced with a non‐associated pressure‐dependent plasticity model

AbstractThis article introduces a, possibly damage‐enhanced, pressure‐dependent‐based incremental‐secant mean‐field homogenization (MFH) scheme for two‐phase composites. The incremental‐secant formulation consists on a fictitious unloading of the material up to a stress‐free state, in which a residual stress is attained in its phases. Then the secant method is performed in order to compute the mean stress fields of each phase. One of the main advantages of this method is the natural isotropicity of the secant tensors that allows defining the linear‐comparison‐composite (LCC). In this work, we show that this isotropic nature is preserved for a non‐associated pressure dependent plastic flow, making possible the direct definition of the LCC. This model is thus able to represent the physics of real polymeric composites. The MFH scheme is then verified by testing its prediction capabilities in several cases, including cyclic and nonproportional loading involving perfectly elastic phases, elasto‐plastic and damage‐enhanced elasto‐plastic phases in random representative volume elements (RVE) of uni‐directional (UD) composites and of composites reinforced with spherical inclusions.

On the dependence of orientation averaging mean field homogenization on planar fourth-order fiber orientation tensors

A comprehensive study on the influence of planar fourth-order fiber orientation tensors on effective linear elastic stiffnesses predicted by orientation averaging mean field homogenization is given. Fiber orientation states of sheet molding compound (SMC) are identified to be in most cases approximately planar. In the planar case, all possible fourth-order fiber orientation tensors are given by a minimal invariant set of structurally differing planar fourth-order fiber orientation tensors. This set defines a three-dimensional body and forms the basis for a comprehensive study on the influence of a fiber orientation distribution in terms of a fourth-order tensor on homogenized stiffnesses. The methodology of this study is the main contribution of this work and can be adopted to analyze the orientation dependence of any quantity which is a function of a planar fourth-order fiber orientation tensor. At specific points inside the set of planar fiber orientation tensors, effective stiffnesses are calculated with selected mean field homogenization schemes. These schemes are based on orientation averaging of transversely isotropic elasticity tensors following Advani and Tucker (1987), which is explicitly recast as linear invariant composition in the fiber orientation tensors of second and fourth order of Kanatani third kind. A maximum entropy reconstruction of a fiber orientation distribution function based on leading fiber orientation tensors, enables a new numerical formulation of the Advani and Tucker average for the special planar case. Polar plots of Young’s modulus and generalized bulk modulus obtained by selected homogenization schemes are arranged on two-dimensional slices within the body of admissible fiber orientation tensors, visualizing the influence of the orientation tensor on the stiffness tensor. The orientation-dependence of the generalized bulk modulus differs significantly between selected homogenizations. Restrictions on the effective anisotropic material response caused by orthotropy of closure approximations are discussed.

Micromechanical schemes for Stokes to Darcy homogenization of permeability based on generalized Brinkman inhomogeneity problems

Mean field homogenization schemes are formulated for the Stokes to Darcy upscaling of the permeability on the basis of generalized inhomogeneity problems in which flow is described by Brinkman equations. The average velocity and drag force concentrations for flow in a potentially composite inclusion are characterized in terms of a permeability contribution tensor and an equivalent permeability, which allow for the direct transposition to Stokes to Darcy upscaling of most homogenization schemes available in elasticity (self-consistent, differential, Mori–Tanaka, Maxwell, …). The unified framework extends existing effective medium and cell model permeability estimates. Its flexibility is illustrated on a panel of microstructures of porous media: granular or fibrous materials, materials with spanning cylindrical or crack-like pores, double porosity materials with disconnected or connected meso-porosity and compared with existing or newly produced full field simulation results.

Residual stresses in deep-drawn cups made of duplex stainless steel X2CrNiN23-4

Residual stress development in deep drawing processes is investigated based on cylindrical cups made of duplex stainless steel sheet. Using a two-scale approach combining finite element modelling with a mean field homogenization scheme the macro residual stresses as well as the phase-specific micro residual stresses regarding the phases ferrite and austenite are calculated for steel X2CrNiN23‑4 for various drawing depths. The simulation approach allows for the numerical efficient prediction of the macro and phase-specific micro residual stress in every integration point of the entire component. The simulation results are validated by means of X‑ray diffraction residual stress analysis applied to a deep-drawn cup manufactured using corresponding process parameters. The results clearly indicate that the fast simulation approach is well suited for the numerical prediction of residual stresses induced by deep drawing for the two-phase duplex steel; the numerical results are in good agreement with the experimental data. Regarding the investigated process, a significant influence of the drawing depth, in particular on the evolution of the residual stress distribution in drawing direction, is observed. Considering the appropriate phase-specific strain hardening, the two-scale approach is also well suited for the prediction of phase specific residual stresses on the component level.

On mean field homogenization schemes for short fiber reinforced composites: Unified formulation, application and benchmark

This paper revisits the topic of mean field homogenization for short fiber reinforced composite materials. A short glass fiber reinforced thermoplastic polyamide 6.6 with a fiber mass fraction of 35 % is used as an example material for which microstructural analyses and experimental tests were performed. We cover a set of common models: the Self-Consistent (SC), Mori–Tanaka (MT), Ponte Castañeda-Willis (PCW), Interaction Direct Derivative (IDD) and Two-Step (TS) schemes. We first recast them into a unified formulation that permits a thorough theoretical comparison, including the topics of loss of major symmetry as well as connections between the models. We are able to show that the MT, PCW and IDD schemes can be expressed in a surprisingly similar form. This extends to the equations for prediction of effective stiffness and compliance tensors as well as the strain and stress localization tensors. Secondly, we address the resolution of the material microstructure within mean field homogenization schemes, comparing classical and more recent and efficient methods. Last, we benchmark the different mean field schemes and modeling approaches, including comparisons to numerical homogenization by Fast Fourier Transformation (FFT) and experimental data from tensile tests. These show that the MT and TS schemes are both capable of accurately modeling the composite material for complex orientations and homogeneous fiber phases. For possibly inhomogeneous fiber phases only the TS scheme yields both accurate and physically reasonable results. Other models such as the IDD or PCW are of great theoretical importance, but cannot be generally applied for the given material class.

A mean-field homogenization approach to predict fracture in as-quenched microstructures of Ductibor® 500-AS steel: characterization and modelling

Ductibor® 500-AS is a hot-stamping steel that has been designed to introduce local ductile regions within hot-stamped tailor-welded blanks (TWBs) used for energy-absorbing sections of automotive structures. This study investigates the flow and fracture behavior of a range of the microstructures of this alloy, corresponding to mostly ferritic (~96% ferrite plus 4% martensite), intermediate ferritic + martensitic (~57% ferrite plus 43% martensite), and mostly martensitic (~10% ferrite plus 90% martensite), obtained by applying various quench rates after austenitization. A series of mechanical tests in various stress states, ranging from simple shear to biaxial tension, was performed to characterize fracture for the developed microstructures. A mean-field homogenization (MFH) technique was then used to predict the flow and fracture response as a function of the microstructure. An MFH scheme designated as the interpolative Samadian-Butcher-Worswick 1 (INSBW1) within the first-order secant-based linearization method was considered to predict the macroscopic hardening behavior of the ferritic-martensitic microstructures. In the MFH modelling, the flow response of the ferritic and martensitic phases was modelled based on a dislocation-based hardening model, taking into account the chemical composition, carbon partitioning between phases, and dislocation generation and annihilation. Damage accumulation and the onset of fracture were predicted using a phenomenological damage indicator defined within each constituent phase given their calculated fracture loci and instantaneous stress and strain states. The experimental results showed that the mostly-martensitic microstructure had the lowest ductility and highest strength, and ductility and strength increased and decreased, respectively, as the martensite volume fraction in the microstructure decreased. The predicted flow curves and fracture strains as well as the fracture-limit curves for all of the microstructures agreed well with the measured data and interpolated modified Mohr-Coulomb (MMC) fracture loci.

Experimental study and micromechanical modelling of the effective elastic properties of Fe–TiB2 composites

The aim of this paper is to investigate the effective properties of Fe–TiB2 composites obtained after hot or cold rolling. The elastic moduli of both hot and cold rolled composites are measured experimentally using several methods. Microstructure analyses based on SEM observations are performed to characterize the distribution of particles and cracks, and are then used to generate 3D representative microstructures using the RSA method. This allows the numerical determination of the overall elastic behavior of Fe–TiB2 composites using full-field FFT-based simulations. In addition, Young’s moduli of the hot rolled Fe–TiB2 composites are also determined analytically using the mean-field homogenization scheme of Mori–Tanaka. The elastic properties determined experimentally, analytically and numerically are in a good agreement. Overall, a significant improvement of the specific stiffness in comparison to standard steels is achieved irrespective of the processing conditions.

Model reduction by mean-field homogenization in viscoelastic composites. III. Dual theory

The mean-field homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2013 Int. J. Plasticity 42, 1–13 ( doi:10.1016/j.ijplas.2012.09.005 )) is revisited from the standpoint recently adopted in a companion paper (Idiart MI et al. 2020 Proc. R. Soc. A 20200407 ( doi:10.1098/rspa.2020.0407 )). It is shown that the scheme generates a reduced-order approximation wherein the microscopic kinetics of the composite are described in terms of a finite set of macroscopic forces identified with the phase averages and intraphase covariances of the various microscopic force fields, which can be evaluated by mean-field homogenization techniques. The approximation exhibits a two-potential structure with a convex complementary energy density but a non-convex force potential. The consequential properties of the approximation are exposed and their implications are discussed. The exposition is supplemented by proofs of equivalence between the present scheme and other candidate schemes proposed in the literature for composites with elementary local rheologies of Maxwellian type.

Effective viscoelastic behavior of polymer composites with regular periodic microstructures

Polymer matrix exhibit linear viscoelasticity during processing of composites. The viscoelastic homogenization problem in the time-domain play a vital role in the virtual process simulations. In this paper, full-field simulations using finite element (FE) are carried out for different regular periodic microstructures of unidirectional (UD) fiber reinforced polymer (FRP) (i.e., short, and long fiber) composite and particulate composites. The incremental variational based mean-field homogenization (MFH) as proposed by Lahellec and Suquet (Lahellec and Suquet, 2007a) is used here for comparing with the full-field solutions. Two different elastic bounds-based homogenization methods are coupled with this scheme. Implementation is validated with benchmark problems of particulate composites. These are then compared with the solutions of different particulate microstructures (face centered cubic (FCC), body centered cubic (BCC) and simple cubic (SC)). Mean-field response is closer to FCC arrangement. Additionally, FCC and BCC arrangement indicated nearly the same response with different local field statistics. Relaxation in the effective modulus of UD FRP composites obtained from the MFH compared well with the full-field solution as a function of direction and time for hexagonal arrangement of long fiber and staggered arrangement of ellipsoidal short fibers. The effect of different aspect ratio, volume fraction and contrast in properties on the macroscopic response is additionally investigated. Glass and carbon fiber with an isotropic approximation is used for studying the effect of contrast. Double inclusion MFH scheme coupled with the incremental variational approach indicated better comparisons with the full-field solution of UD FRP composites.

Model reduction by mean-field homogenization in viscoelastic composites. II. Application to rigidly reinforced solids.

The mean-field homogenization scheme proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44, 507-529 (doi:10.1016/j.ijsolstr.2006.04.038)) and revisited in a companion paper (Idiart et al. 2020 Proc. R. Soc. A 20200407 (doi:10.1098/rspa.2020.0407)) is applied to random mixtures of a viscoelastic solid phase and a rigid phase. Two classes of mixtures with different microstructural arrangements are considered. In the first class the rigid phase is dispersed within the continuous viscoelastic phase in such a way that the elastic moduli of the mixture are given exactly by the Hashin-Shtrikman formalism. In the second class, both phases are intertwined in such a way that the elastic moduli of the mixture are given exactly by the Self-Consistent formalism. Results are reported for specimens subject to various complex deformation programmes. The scheme is found to improve on earlier approximations of common use and even recover exact results under several circumstances. However, it can also generate highly inaccurate predictions as a result of the loss of convexity of the free-energy density. An auspicious procedure to partially circumvent this issue is advanced.

A Mori–Tanaka Based Micromechanical Model for Predicting the Effective Electroelastic Properties of Orthotropic Piezoelectric Composites with Spherical Inclusions

Piezoelectric nanocomposites consisting of an orthotropic piezo-active polymer matrix and with piezo-ceramic nanoparticles as reinforcement are potential candidates as materials for structural components of the next-generation energy-harvesting micro-devices that are compatible with flexible electronics. Prediction of effective electroelastic properties of such piezoelectric composites is an essential step towards design and development of such energy-harvesting micro/nano-structures. This paper proposes a micromechanical model for determination of effective elastic, piezoelectric and dielectric properties of unidirectional piezoelectric polymer composites having spherical type inclusions embedded in an orthotropic matrix. The model is based on Mori–Tanaka’s mean field homogenization scheme and involves the determination of piezoelectric Eshelby tensors. As an example, the effective electromechanical properties of PVDF (polyvinylidene fluoride)/PZT 7A (lead zirconate titanate) composite were determined using the micromechanical model and the results were validated with a finite element model and with experimental data available in literature. The results demonstrate that the micromechanics model developed here successfully predicts the effective electroelastic properties of piezoelectric orthotropic composite at low volume fractions of the reinforcement.