Finite-volume Direct Averaging Micromechanics Theory Research Articles

Tailoring superelastic and transformation-induced plastic response of shape memory multilayers with wavy architectures

The finite-volume direct averaging micromechanics (FVDAM) has been extended to simulate the superelastic and plastic response of shape memory composites. The homogenized stress–strain response and local field variables generated with FVDAM were first validated vis-à-vis the exact solution under axisymmetric loading conditions. Furthermore, an in-house finite-element code was developed to assess the FVDAM theory’s accuracy for simulating wavy multilayers. Parametric studies were conducted to quantify the effect of microstructural parameters on the nonlinear response of multilayers. The common mechanism is responsible for the differences in the nonlinear stress–strain response of multilayers is rooted in the effective stress differences that alter the phase transformation.

Reinforced masonry homogenization by the finite-volume direct averaging micromechanics—FVDAM

The theory of finite-volume direct averaging micromechanics (FVDAM) has recently been used to analyze composite materials to obtain their effective properties. This technique has been continuously improved and applied to different problems and is promising in the heterogeneous materials research area with excellent results compared to other numerical approaches. As it is a relatively new technique, many applications have not been performed yet. This work applies the FVDAM for the first time to the numerical study of reinforced masonry, which presents complex behavior, often requiring structural reinforcement despite being an old material. Thus, the FVDAM theory is compared with results obtained by mechanics of structure genome (MSG), which has already been applied to the homogenization of unreinforced and FRP-reinforced masonry and by the finite element method (FEM). Lastly, FVDAM is applied to the solution of some examples involving different types of masonry unit cells, including the investigation of the effects of structural reinforcement, volume fraction, unit cell geometry, and the influence of block properties on effective mechanical properties. The results of such examples are compared with numerical methods such as MSG and FEM, demonstrating the good performance of FVDAM against the methods used in the comparison.

A UMAP-based clustering method for multi-scale damage analysis of laminates

In this study, a novel multi-scale damage model based on the parametric finite-volume direct averaging micromechanics theory (parametric FVDAM), extended finite element method (XFEM), and a clustering algorithm, is built, to describe the progressive damage evolution of laminates, where a uniform manifold approximation and projection (UMAP)-based clustering method is specially designed and incorporated with the FVDAM and XFEM for the first time. The subvolumes in the unit cell generated by FVDAM with similar mechanical responses were included in the same cluster using the UMAP-based clustering method. The semi-analytical framework of FVDAM allows the efficient generation of unit cells. The built-in advantage in the visualization of the proposed UMAP-based clustering method allows researchers in the field of mechanics to have a clear and intuitive understanding of the clustering results. The clustered unit cell is then conjugated to the XFEM, which allows the damage information to be transmitted between different scales. Three damage mechanisms of laminates under a three-dimensional (3D) stress state are considered, which include matrix damage, fiber breakage, and delamination. Furthermore, a shear nonlinear matrix Ds for a 3D element is derived and introduced in the model, which enables the description of the unique elastic nonlinear behavior of laminates. The model is implemented using UDMGINI and USDFLD subroutines in ABAQUS, which not only allows the calculation of the load–displacement curve, but also the simulation of clear and smooth crack propagation paths. To verify the effectiveness of the model, the experiments of [±60]2s and [90]8 notched glass-fiber/epoxy laminates under tension were performed by digital image correlation (DIC) monitoring, and the results compared were in good agreement with the simulations, thus proving the effectiveness of the proposed model. The code of the UMAP-based clustering method developed in this study can be downloaded from https://github.com/Danhui-Yang/MicroMechanicsUmapClustering.

A tangent finite-volume direct averaging micromechanics framework for elastoplastic porous materials: Theory and validation

In this communication, we present a reformulated finite-volume direct averaging micromechanics (FVDAM) framework for periodic multiphase materials with elastic-plastic phases. The original elastoplastic version is formulated using the secant stiffness matrix approach, which requires a computationally intensive solution procedure of the governing differential equations. For the first time, the reformulation makes use of the tangent plasticity matrix approach that incorporates linearities to the unit cell boundary value problem. The tangential FVDAM theory is implemented using a quasi-Newton-Raphson strategy that quantifies errors in the evaluation of surface-averaged stresses due to the linearization, hence allowing large load increments. The tangential formulation is vigorously and fully assessed vis-à-vis the secant approach for porous unit cells on several aspects, including the convergence of the algorithmic implementation with mesh and step sizes, solution accuracy, and computation times. The reformulation simplifies the solution to the governing differential equations relative to its predecessor, albeit at the cost of more computational efforts for the reforming and reinverting of the global tangent stiffness matrix (or the so-called tangent operator). Additionally, advantages of the FVDAM theory relative to the classical finite-element homogenization are highlighted. The tangent FVDAM is employed to demonstrate the plasticity-triggered architectural effects in the response of periodic porous materials under different loading modes. The common mechanisms responsible for differences in the elastic-plastic response are attributed to the effective and hydrostatic stress alteration in the matrix phase. The current work paves the ground for the incorporation of FVDAM into readily-available commercial computational tools through the user material subroutines that are based on tangent stiffness approaches, which will produce a paradigm shift for FVDAM to solve the challenging multiscale structural problem.

Micromechanical modeling of cyclic elasto-viscoplastic behavior of unidirectional metal matrix composites under elevated temperature

In this paper, a new format is proposed for rate-dependent cyclic behavior of unidirectional metal matrix composites, which combines the cyclic elasto-viscoplastic constitutive model with the parametric finite-volume direct averaging micromechanics theory (FVDAM). The cyclic constitutive model provides the capability of simulating materials rate-dependency for the FVDAM theory, and the FVDAM theory’s ability of calculating responses of composites makes it a suitable solution for the title problem. To verify the effectiveness of the proposed model, both experimental data and numerical simulations are adopted for comparison. The experimental data of unidirectional SCS-6/Timetal21S was compared with the simulation results of FVDAM, with a good agreement. A finite-element method (FEM) based unit cell model is constructed using ABAQUS; and both homogenized responses and local fields are compared. The excellent correlations prove the effectiveness of the proposed model. The effect of fiber orientations is also revealed. The generated results indicate that the FVDAM is well-suited for simulating the cyclic elasto-viscoplastic behavior of unidirectional metal matrix composites under elevated temperature.

Finite volume based asymptotic homogenization theory for periodic materials under anti-plane shear

A finite volume based approach is employed in the solution of unit cell problems at different orders of the asymptotic field expansion to construct a homogenization theory for anti-plane shear loading of unidirectional fiber-reinforced periodic structures. This new construction complements and further extends our recent contribution to asymptotic homogenization based on locally-exact elasticity unit cell solutions, He and Pindera (2020), to unit cells with multiple inclusions of arbitrary shapes. The present approach builds upon the previously developed finite-volume direct averaging micromechanics theory applicable under uniform strain fields, and extends it to account for strain gradients and non-vanishing microstructural scale relative to structural dimensions. The unit cell problems at different orders of the asymptotic field expansion are solved by satisfying local equilibrium equations in each subvolume of the discretized microstructure in a surface-averaged sense. This facilitates construction of local stiffness matrices at the subvolume level and subsequent assembly into the global stiffness matrix for the unit cell response under uniform and gradient strain fields. Comparison of the calculated microfluctuation functions and associated stress fields under uniform and gradient strain fields with those reported in the literature verifies the finite volume asymptotic solutions. The new theory's ability to accurately recover local fields is further illustrated through comparison with the direct numerical solution of a periodic structure with varying number of inclusions under gradient loading. The proposed homogenization approach is an efficient and accurate alternative to current numerical techniques for the analysis of periodic materials experiencing strain gradients regardless of microstructural scale, inclusion shape and number, demonstrated by analyzing arrays characterized by elliptical and square inclusions and multi-inclusion unit cells.

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.

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.

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.

The High-Fidelity Generalized Method of Cells with arbitrary cell geometry and its relationship to the Parametric Finite-Volume Micromechanics

This contribution is motivated by the recent IJSS article titled “Formulation of the High-Fidelity Generalized Method of Cells with Arbitrary Cell Geometry for Refined Micromechanics and Damage in Composites” by Haj-Ali and Aboudi. In the present discussion, we critically compare these authors’ approach with the Parametric Finite-Volume Direct Averaging Micromechanics (FVDAM) theory developed in a sequence of papers dating to 2004. We show that the extension proposed by Haj-Ali and Aboudi follows the framework of the parametric FVDAM theory, and not the original HFGMC model as claimed. Just as importantly, we demonstrate that the proposed extension is fundamentally flawed as it violates accepted principles which every mechanics theory must satisfy, namely correct reducibility and coordinate frame indifference. Direct comparison of predictions generated by the parametric FVDAM theory, the proposed extension called HFGMC with arbitrary cell geometry and finite-element method demonstrates no need for the extension carried out in the specific manner proposed by Haj-Ali and Aboudi, which in fact produces inferior results.

Homogenization of elastic–plastic periodic materials by FVDAM and FEM approaches – An assessment

The standard computational approach for the homogenization of elastic–plastic response of periodic materials is the finite-element method. Amongst the emerging alternatives, the finite-volume direct averaging micromechanics (FVDAM) theory has shown promise. Herein, we present a systematic comparison of the predictive capabilities of both approaches in the context of two technologically important material systems, each of which exhibits interesting plasticity-driven local and global responses in its own right, which have not been well-documented. This comparison is conducted for the first time on the same footing using an in-house developed finite-element code which closely mimics the homogenization framework, displacement field approximation and unit cell discretization employed by the parametric FVDAM theory. A new stress measure is introduced which highlights the fundamental differences in the variational-based and direct averaging-based solution approaches employed by the two methods. The comparison provides firm support for the parametric FVDAM theory’s ability to capture highly localized plasticity effects in different classes of heterogeneous materials, which lead to interesting and technologically important phenomena at the homogenized scale.

Micromechanics of spatially uniform heterogeneous media: A critical review and emerging approaches

Outside of the classical microstructural detail-free estimates of effective moduli, micromechanical analyses of macroscopically uniform heterogeneous media may be grouped into two categories based on different geometric representations of material microstructure. Analysis of periodic materials is based on the repeating unit cell (RUC) concept and the associated periodic boundary conditions. This contrasts with analysis of statistically homogeneous materials based on the representative volume element (RVE) concept and the associated homogeneous boundary conditions. In this paper, using the above classification framework we provide a critical review of the various micromechanical approaches that had evolved along different paths, and outline recent emerging trends. We begin with the basic framework for the solution of micromechanics problems independent of microstructural representation, and then clarify the often confused RVE and RUC concepts. Next, we describe classical models, including the available RVE-based models, and critically examine their limitations. This is followed by discussion of models based on the concept of microstructural periodicity. In the final part, two recent unit cell-based models, which continue to evolve, are outlined. First, a homogenization technique called finite-volume direct averaging micromechanics theory is presented as a viable and easily implemented alternative to the mainstream finite-element based asymptotic homogenization of unit cells. The recent incorporation of parametric mapping into this approach has made it competitive with the finite-element method. Then, the latest work based on locally-exact solutions of unit cell problems is described. In this approach, the interior unit cell problem is solved exactly using the elasticity approach. The exterior problem is tackled with a new variational principle that successfully overcomes the non-separable nature of the overall unit cell problem.

Parametric finite-volume micromechanics of periodic materials with elastoplastic phases

The recently incorporated parametric mapping capability into the finite-volume direct averaging micromechanics (FVDAM) theory has produced a paradigm shift in the theory’s development. The use of quadrilateral subvolumes made possible by the mapping facilitates efficient modeling of microstructures with arbitrarily shaped heterogeneities, and eliminates artificial stress concentrations produced by the rectangular subvolumes employed in the standard version. Herein, the parametric FVDAM theory is extended to the inelastic domain by implementing additional formulation required to accommodate plastic and thermal loading. Two different approaches of implementing plasticity have been investigated. The first approach is based on the treatment employed in previous versions of the theory wherein plastic strain fields are represented by a series expansion in Legendre polynomials. The second approach is based on direct surface-averaging of plastic strains calculated at a number of collocation points along the quadrilateral subvolumes’ surfaces, and offers substantial simplification in the parametric finite-volume theory’s elastic–plastic framework. Moreover, substantial reductions in execution times without loss of accuracy are realized due to the elimination of redundant plastic strain calculations in the subvolumes’ interiors employed in the evaluation of the Legendre polynomial coefficients. Numerical studies demonstrate the advantages of the parametric FVDAM theory relative to the standard version, together with new results that highlight its modeling capabilities vis-a-vis an emerging class of periodic lamellar materials with wavy microstructures and the thus-far undocumented architectural effects amplified by plasticity.

Thermo-elastic moduli of periodic multilayers with wavy architectures

The recently developed parametric finite-volume direct averaging micromechanics theory for periodic materials is employed to investigate the effective moduli and thermal expansion coefficients of lamellar composites with wavy architectures. In the parametric version, a reference square subvolume is mapped onto a quadrilateral subvolume in the actual discretized microstructure to accurately capture the in situ microstructural details. The mapping is used to construct local stiffness matrices of quadrilateral subvolumes which are employed in the local/global stiffness matrix solution strategy for the unit cell problem within a homogenization framework. Complete set of homogenized moduli and thermal expansion coefficients of multilayers comprised of alternating soft and hard laminae with two types of waviness is generated for the first time as a function of the volume content of the hard phase for two amplitude-to-wavelength ratios. The observed changes in the homogenized mechanical and thermal properties relative to the reference flat-layer configuration depend on the wavy microstructure orientation and become greater with increasing amplitude-to-wavelength ratios. Examination of local stress fields explains the differences observed in the homogenized moduli of multilayers with sinusoidal and corrugated waveforms for the two amplitude-to-wavelength ratios.

Parametric Finite-Volume Micromechanics of Uniaxial Continuously-Reinforced Periodic Materials With Elastic Phases

The finite-volume direct averaging micromechanics (FVDAM) theory for periodic heterogeneous materials is extended by incorporating parametric mapping into the theory’s analytical framework. The parametric mapping enables modeling of heterogeneous microstructures using quadrilateral subvolume discretization, in contrast with the standard version based on rectangular subdomains. Thus arbitrarily shaped inclusions or porosities can be efficiently rendered without the artificially induced stress concentrations at fiber/matrix interfaces caused by staircase approximations of curved boundaries. Relatively coarse unit cell discretizations yield effective moduli with comparable accuracy of the finite-element method. The local stress fields require greater, but not exceedingly fine, unit cell refinement to generate results comparable with exact elasticity solutions. The FVDAM theory’s parametric formulation produces a paradigm shift in the continuing evolution of this approach, enabling high-resolution simulation of local fields with much greater efficiency and confidence than the standard theory.