Finite-volume Direct Averaging Micromechanics Research Articles

An effective microscale approach for determining the anisotropy of polymer composites reinforced with randomly distributed short fibers

In this paper, an effective microscopic modeling scheme is presented to analyze mechanical properties of composites with random short fibers. To this end, the displacement-load tests of the standard samples, which are acquired by cutting a short fiber-reinforced composite plate of 650 mm × 650 mm × 2.5 mm, are firstly executed under the quasi-static tensile loads. To identify the geometric sizes of the short fibers and their distributions at microscopic scale, the advanced micro-computed tomography (micro-CT) is employed by testing a small sample of 1 cm × 2.5 mm × 2.5 mm. On this basis, a simplified microscopic model is reconstructed by the 3D parametric finite-volume direct averaging micromechanics (FVDAM) theory according to the statistic results of the micro-CT images. The proposed method is further validated by comparing the effective modulus obtained from tensile tests. The scanning electron microscopy (SEM) is also used to visualize the fracture morphology of the fibers. It is found that brittle fracture occurs in the short-fibers paralleled to the external loading.

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.

Homogenization and Localization of Ratcheting Behavior of Composite Materials and Structures with the Thermal Residual Stress Effect.

In this contribution, the ratcheting behavior and local field distribution of unidirectional metal matrix composites are investigated under cyclic loading. To that end, we extended the finite-volume direct averaging micromechanics (FVDAM) theory by incorporating the rule of nonlinear kinematic hardening. The proposed method enables efficient and accurate simulation of the ratcheting behavior of unidirectional composites. The local satisfaction of equilibrium equations of the FVDAM theory facilitates quick and rapid convergence during the plastic iterations. To verify the proposed theory, a finite-element (FE) based unit cell model is constructed with the same mesh discretization. The remarkable correlation of the transverse response and local field distribution generated by the FVDAM and FE techniques demonstrates the effectiveness and accuracy of the proposed models. The stress discontinuities along the fiber/matrix interface that are generic to the finite-element theory are absent in the FVDAM prediction. The effects of thermal residual stresses induced during the consolidation process, as well as fiber orientations, are revealed. The generated results indicate that the FVDAM is well suited for simulating the elastic-plastic ratcheting behavior of metal matrix composites, which will provide the conventional finite-element based technique with an attractive alternative.

Characterization of Interphase/Interface Parameters of Unidirectional Fibrous Composites by Optimization-Based Inverse Homogenization

Optimization-based micromechanical and inverse-homogenization models are developed to inversely calculate the interphase/interface properties of unidirectional periodic fibrous composites from prescribed effective properties or localized stress concentrations. The interphase/interface effects between fibers and the surrounding matrix are described by four different mathematical models that are reviewed in the present work. In order to guarantee the stability of the characterization process, two sophisticated micromechanical models, locally exact homogenization theory and finite-volume direct averaging micromechanics, are introduced in this work and connected to the gradient-free particle swarm optimization to search for the optimal parameters to minimize the cost/objective functions that consider the homogenized or localized responses of unidirectional composites. The accuracy and efficiency of the proposed procedure are tested by substituting the optimized parameters back into the direct micromechanical models and validating against the target functions, where good agreement is always obtained. More importantly, the numerical effects generated by those parameters are also tested on the effective properties and localized stress distributions of fibrous composites.

PSO-driven micromechanical identification of in-situ properties of fiber-reinforced composites

Within the framework of micromechanics, the newly expanded Generalized Finite-Volume Direct Averaging Micromechanics is connected to the gradient-free Particle Swarm Optimization (PSO-GFVDAM) to identify the in-situ constituent properties and residual stresses and study their effects on the effective and localized responses of fibrous composites. The present technique is advantageous by avoiding the gradient updating concept and adopting the advanced generalized finite-volume micromechanics with plasticity effects, both of which guarantee the stability and efficiency of the proposed technique. After introducing the microstructural effects from the fabrication process and chemical reaction that cannot be easily detected from the macroscopic behavior of composite materials, the PSO-GFVDAM is employed to identify the elastic constituent properties of carbon/epoxy composites, yield stress and hardening parameters of the aluminum-matrix system, as well as the residual stress state within the microstructures. The stability and accuracy of the algorithm are tested by checking the iterated particle distributions, errors generated at each step and substituting the deduced parameters back into direct analyses to predict effective responses of off-axis loaded specimens. More important to the effective response, the localized stress distributions are efficiently recovered, helping to characterize the possible damage initiation and crack propagation that usually start from the material levels.

Three-Dimensional Parametric Finite-Volume Homogenization of Periodic Materials with Multi-Scale Structural Applications

In order to satisfy the increasing computational demands of micromechanics, the Finite-Volume Direct Averaging Micromechanics (FVDAM) theory is developed in three-dimensional (3D) domain to simulate the multiphase heterogeneous materials whose microstructures are distributed periodically in the space. Parametric mapping, which endorses arbitrarily shaped and oriented hexahedral elements in the microstructure discretization, is employed in the unit cell solution. Unlike the finite-element (FE) technique, the expressions for local stiffness matrices are derived explicitly, enabling efficient global stiffness matrix assembly using an easily implementable algorithm. To demonstrate the accuracy and efficiency of the proposed theory, the homogenized moduli and localized stress distributions produced by the FE analyses are given for comparisons, where excellent agreement is always obtained for the 3D microstructures with different geometrical and material properties. Finally, a multi-scale stress analysis of functionally graded composite cylinders is conducted. This extension further increases the FVDAM’s range of applicability and opens new opportunities for pursuing other areas, providing an attractive alternative to the FE-based approaches that may be compared.

Homogenized and localized responses of coated magnetostrictive porous materials and structures

The homogenized properties and localized field distributions of coated magneto-strictive porous heterogeneous materials are extensively investigated using the extended Finite Volume Direct Averaging Micromechanics (FVDAM) with coupled mechanical-magnetic capabilities. Both square and hexagonal microstructures are employed to mimic different hollow coating arrangements. In order to validate the present technique, the corresponding finite element model is constructed in Abaqus to provide rigorous comparisons, and no visible difference is observed between two different models’ predictions. The effects of the thickness of coatings are tested on the effective and local responses, where the hollow coatings play significant roles in the variations of homogenized properties and transmissions of stress concentrations. Finally, a two-step multiscale framework is established on periodic wavy structures, wherein the layers are composed of CoFe2O4 material reinforced by BaTiO3 hollow coatings with two different volume fractions. The amplitude-to-wavelength ratio is varied to effectively study its impact on the field distributions of homogenized layers and local microstructures.

Finite-volume homogenization of elastic/viscoelastic periodic materials

In this contribution, the three-dimensional (3D) finite-volume direct averaging micromechanics (FVDAM) is reconstructed by incorporating parametric mapping capability into the theory’s analytical framework, which allows using arbitrarily shaped and oriented hexahedral subvolumes in the material microstructure discretization. The technique is then further extended to include linear viscoelastic capability to investigate the creep and relaxation behavior of polymeric matrix composites and stress field evolution with either continuous or discontinuous reinforcements. Elastic-viscoelastic correspondence principle is employed in the unit cell solution to relate the governing relations between time domain and Laplace-Carson domain. After establishing the homogenized constitutive equations and solving the boundary value problems in Laplace-Carson domain, Zakian inversion scheme is used to invert the inexplicit homogenization functions to obtain both relaxation moduli and stress distributions during a relaxation history in real space. The FVDAM theory via correspondence principle is validated extensively by comparing with Mori-Tanaka, finite-element, locally exact solutions and experimental data. Good agreement between the present theory and other methods, along with a few numerical investigations, explains not only the accuracy of the 3D parametric FVDAM but also the validity and efficiency of the correspondence principle and Zakian inversion technique.

Parametric multiphysics finite-volume theory for periodic composites with thermo-electro-elastic phases

The zeroth-order multiphysics finite-volume micromechanics has been proposed to model coupled thermo-electro-mechanical behaviors of unidirectional composites embedded with piezoelectric phases. Parametric mapping is implemented within the multiphysics finite-volume theory’s framework, facilitating modeling of multiphase piezoelectric materials with complex microstructures with relatively coarse unit cell discretization. The resulting theory admits piezoelectric materials with complete anisotropy and arbitrary poling direction and enables rapid generation of the entire set of coupled thermo-mechanical, piezoelectric properties, figures of merits, as well as the local fluctuations of fields within the composite microstructures with greater fidelity than its predecessor. The proposed method is verified extensively by comparison with the finite-element homogenization technique, which produces an excellent agreement in a wide range of volume fractions but offers much better stability and efficiency. The contrast with the rectangular theory is also presented and discussed, demonstrating the advantage and the need for the development of parametric formulation. This extension further increases the finite-volume direct averaging micromechanics theory’s range of applicability, providing an attractive standard for investigating multiphase and multiphysics problems with different microstructural architectures and scales against which other approaches may be compared.

Micromechanical modeling of plain woven polymer composites via 3D finite‐volume homogenization

In this article, an anisotropic version of 3D finite‐volume direct averaging micromechanics (FVDAM) has been developed, enabling efficient and accurate analysis of composites with woven reinforcements. The formulation is based on a local/global stiffness matrix procedure, with the equilibrium equations in conjunction with continuity of displacements and tractions imposed in an average sense. The present 3D FVDAM admits arbitrary anisotropic constituent materials, whereas the 2D FVDAM permits orthotropic subvolume materials at most. On this basis, an FVDAM‐based two‐step approach, which is expected to provide more accurate predictions than the traditional one‐step homogenization, is employed to homogenize the repeating unit cell of a plain weave. The performance of the proposed method is assessed by comparison with other analytical, numerical, and experimental data found in literature. The generated results demonstrate that the FVDAM is capable of modeling plain woven polymer composites reasonably, making FVDAM an attractive alternative to other numerical and analytical techniques for analysis of composites with periodic internal microstructures. POLYM. COMPOS., 39:3022–3032, 2018. © 2017 Society of Plastics Engineers

Micromechanical Modeling of Viscoplastic Behavior of Laminated Polymer Composites With Thermal Residual Stress Effect

In this paper, a multiscale approach has been developed for investigating the rate-dependent viscoplastic behavior of polymer matrix composites (PMCs) with thermal residual stress effect. The finite-volume direct averaging micromechanics (FVDAM), which effectively predicts nonlinear response of unidirectional fiber reinforced composites, is incorporated with improved Bodner–Partom model to describe the viscoplastic behavior of PMCs. The new micromechanical model is then implemented into the classical laminate theory, enabling efficient and accurate analysis of multidirectional PMCs. The proposed multiscale theory not only predicts effective thermomechanical viscoplastic response of PMCs but also provides local fluctuations of fields within composite microstructures. The deformation behaviors of several unidirectional and multidirectional PMCs with various fiber configurations are extensively simulated at different strain rates, which show a good agreement with the experimental data found from the literature. Influence of thermal residual stress on the viscoplastic behavior of PMCs is closely related to fiber orientation. In addition, the thermal residual stress effect cannot be neglected in order to accurately describe the rate-dependent viscoplastic behavior of PMCs.

A new and general formulation of three-dimensional finite-volume micromechanics for particulate reinforced composites with viscoplastic phases

This paper presents a fully triply periodic version of finite-volume direct averaging micromechanics (FVDAM), enabling refined analysis of multiphase composites with arbitrary internal microstructures. Explicit three-dimensional FVDAM equations including plastic effects are derived. The surface-averaged continuity of displacements and tractions, and volume-averaged equilibrium equations are imposed based on the local/global stiffness approach. Further, the viscoplastic behavior for the composites can be modeled by incorporating state-variable-based Bodner-Partom constitutive model into the three-dimensional FVDAM framework. The presented model is used to simulate uniaxial tensile and strain cycling response of SiC/Al composites with various strain rates at room and high temperatures, respectively. Simultaneously, the microscopic stress and strain fields at different loading stages are also revealed in the simulations. The theoretical results agree well with the corresponding experimental data, indicating that the proposed model can predict the rate-dependent behavior of particulate reinforced composites reasonably.

Progressive failure analysis of composite structure based on micro- and macro-mechanics models

Based on parameter design language, a program of progressive failure analysis in composite structures is proposed. In this program, the relationship between macro- and micro-mechanics is established and the macro stress distribution of the composite structure is calculated by commercial finite element software. According to the macro-stress, the damaged point is found and the micro-stress distribution of representative volume element is calculated by finite-volume direct averaging micromechanics (FVDAM). Compared with the results calculated by failure criterion based on macro-stress field (the maximum stress criteria and Hashin criteria) and micro-stress field (Huang model), it is proven that the failure analysis based on macro- and micro-mechanics model is feasible and efficient.

Failure response of fiber-epoxy unidirectional laminate under transverse tensile/compressive loading using finite-volume micromechanics

The transverse damage initiation and extension of a unidirectional laminated composite under transverse tensile/compressive loading are evaluated by means of Representative Volume Element (RVE) presented in this paper based on an advanced homogenization model called finite-volume direct averaging micromechanics (FVDAM) theory. Fiber, fiber-matrix interface and matrix phases are considered within the RVE in determining fiber-matrix interface debonding and matrix cracking. The simulated fracture patterns are shown to be in good agreement with experimental observations.

Plastic strain localization in periodic materials with wavy brick-and-mortar architectures and its effect on the homogenized response

Biomimetic or bio-inspired microstructures are increasingly being explored as a source of inspiration for material innovation. The goal of this study is to aid future design of biomimetic materials by conducting analysis of material architectures that resemble brick-and-mortar microstructures found in nacre. Specifically, this study explores the thus-far undocumented combined effects of waviness and platelet architecture on composite material ductility under unidirectional loading parallel and perpendicular to the reinforcing platelets. Model material architectures, comprised of discontinuous silicon carbide platelets suspended in aluminum matrix, that mimic nacre’s microstructure were constructed for analysis with the finite-volume direct averaging micromechanics (FVDAM) theory. The silicon carbide platelets play the role of nacre’s load-bearing calcite phase while the aluminum matrix mimics the combined effects of hierarchical load transferring mechanisms and organic protein matrix. The FVDAM simulations indicate that the introduction of waviness leads to an increase in ductility. Just as significant to material performance is the degree of relative shift between wavy rows of discontinuous hard-phase platelets. The effect of shift on ductility was found to be most significant when introduced to a degree that disrupted unit cell symmetry and when applied to configurations with low amplitude-to-wavelength ratios. The differences in the observed homogenized response are rooted in the local microstructure-controlled stress and resulting plastic strain fields that are identified in this investigation.

Cohesive Zone-Based Damage Evolution in Periodic Materials Via Finite-Volume Homogenization

The zeroth-order parametric finite-volume direct averaging micromechanics (FVDAM) theory is further extended in order to model the evolution of damage in periodic heterogeneous materials. Toward this end, displacement discontinuity functions are introduced into the formulation, which may represent cracks or traction-interfacial separation laws within a unified framework. The cohesive zone model (CZM) is then implemented to simulate progressive separation of adjacent phases or subdomains. The new capability is verified in the linear region upon comparison with an exact elasticity solution for an inclusion surrounded by a linear interface of zero thickness in an infinite matrix that obeys the same law as CZM before the onset of degradation. The extended theory's utility is then demonstrated by revisiting the classical fiber/matrix debonding phenomenon observed in SiC/Ti composites, illustrating its ability to accurately capture the mechanics of progressive interfacial degradation.

A Unified Macro- and Micromechanics Constitutive Model of Fully Coupled Fields

A unified macro- and micromechanics constitutive model of fully coupled electro-magneto-thermo-elastic multiphase functional composites is developed. The model is based on the hypothesis of periods, the theory of uniformity, and the finite-volume direct averaging micromechanics (FVDAM). By introducing quadratic displacements and electric and magnetic potentials into the constitutive model, its accuracy is improved. The efficiency of the model is also raised by adopting surface-averaged quantities to be the primary variables of the original FVDAM. A numerical example is presented, and a relation between material constants and the fiber volume fraction is obtained.

Homogenization of periodic materials with viscoelastic phases using the generalized FVDAM theory

In the last decades, new generations of advanced materials have been designed and manufactured for specific applications. The micromechanics plays an important role in the development of heterogeneous materials, enabling efficient analyses of composite materials with complex geometries, circumventing the traditional trial-and-error approach, producing substantial cost savings. The unit cell problem to the analysis of periodic heterogeneous media can be solved by the well-established 0th order version of the finite-volume theory, named finite-volume direct averaging micromechanics (FVDAM) theory. This standard version of the FVDAM theory employs an incomplete second-order displacement field within individual subvolumes of a discretized analysis domain together with a surface-averaging framework, which does not enforce displacement or traction continuity in a point-wise manner. This, in turn, produces interfacial interpenetrations and non-traction stress discontinuities, thereby demanding very refined meshes in order to produce good interfacial conformability and pointwise stress continuity between adjacent subvolumes. To overcome these shortcomings, a generalized FVDAM theory has been proposed to enable analysis of periodic heterogeneous materials in the finite-deformation domain. The generalization is based on a higher-order displacement field representation and on the definition of elasticity-based surface-averaged kinematic and static variables related through a local stiffness matrix. Herein, we specialize the generalized FVDAM theory to the infinitesimal analysis of periodic materials with viscoelastic phases, where a total or secant formulation is employed, with the viscoelastic strains evaluated incrementally using an algorithm based on the concept of state variables. The generalized or 2nd order version considerably improves interfacial conformability and pointwise traction and non-traction stress continuity between adjacent subvolumes in comparison with the 0th order version, but with a higher computational cost. Furthermore, for the same mesh discretization, these two versions provide comparable macroscopic response. Considering these features, the 0th order version is recommended to evaluate the effective elastic properties and the homogenized creep and relaxation functions, while the 2nd order version is more efficient in the evaluation of the microscopic displacement and stress fields.

Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformations—Part I: Framework

The recently constructed generalized finite-volume theory for two-dimensional linear elasticity problems on rectangular domains is further extended to make possible simulation of periodic materials with complex microstructures undergoing finite deformations. This is accomplished by embedding the generalized finite-volume theory with newly incorporated finite-deformation features into the 0th order homogenization framework, and introducing parametric mapping to enable efficient mimicking of complex microstructural details without artificial stress concentrations by stepwise approximation of curved surfaces separating adjacent phases. The higher-order displacement field representation within subvolumes of the discretized unit cell microstructure, expressed in terms of elasticity-based surface-averaged kinematic variables, substantially improves interfacial conformability and pointwise traction and nontraction stress continuity between adjacent subvolumes. These features enable application of much larger deformations in comparison with the standard finite-volume direct averaging micromechanics (FVDAM) theory developed for finite-deformation applications by minimizing interfacial interpenetrations through additional kinematic constraints. The theory is constructed in a manner which facilitates systematic specialization through reductions to lower-order versions with the 0th order corresponding to the standard FVDAM theory. Part I presents the theoretical framework. Comparison of predictions by the generalized FVDAM theory with its predecessor, analytical and finite-element results in Part II illustrates the proposed theory's superiority in applications involving very large deformations.

Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformations—Part II: Results

In Part I, a generalized finite-volume direct averaging micromechanics (FVDAM) theory was constructed for periodic materials with complex microstructures undergoing finite deformations. The generalization involves the use of a higher-order displacement field representation within individual subvolumes of a discretized analysis domain whose coefficients were expressed in terms of surface-averaged kinematic variables required to be continuous across adjacent subvolume faces. In Part II of this contribution we demonstrate that the higher-order displacement representation leads to a substantial improvement in subvolume interfacial conformability and smoother stress distributions relative to the original theory based on a quadratic displacement field representation, herein called the 0th order theory. This improvement is particularly important in the finite-deformation domain wherein large differences in adjacent subvolume face rotations may lead to the loss of mesh integrity. The advantages of the generalized theory are illustrated through examples based on a known analytical solution and finite-element results generated with a computer code that mimics the generalized theory's framework. An application of the generalized FVDAM theory involving the response of wavy multilayers confirms previously generated results with the 0th order theory that revealed microstructural effects in this class of materials which are important in bio-inspired material architectures that mimic certain biological tissues.