Finite-volume Direct Averaging Micromechanics Research Articles

Multiscale modelling for fatigue crack propagation of notched laminates using the UMAP clustering algorithm

In this study, a novel multiscale fatigue-damage model was developed based on the parametric finite-volume direct-averaging micromechanics theory (FVDAM), the uniform manifold approximation and projection (UMAP) algorithm, and the extended finite element method (XFEM) to accurately portray the progressive propagation of fatigue cracks in notched laminates. The UMAP clustering technique was integrated with fatigue damage evolution for the first time, enabling the construction of a reduced-order unit cell with fatigue information. Compared with the conventional FVDAM unit cell, the data was reduced to 0.4% of its original size, significantly accelerating fatigue damage calculations. The reduced-order unit cell was then incorporated into XFEM, functions as the fatigue damage criterion, transmitting fatigue damage information to meso- and macro-scales, and enables fatigue crack simulation in notched laminates. To further accelerate the calculation, a cycle-jump scheme was integrated, resulting in computational time being 10 times shorter compared to cycle-by-cycle simulation while maintaining accuracy. To validate the effectiveness of the proposed model, experiments of notched [±60]7s glass-fibre/epoxy laminates under three different fatigue loads were conducted. The simulation results of all three loads were within a scatter band of factor two, which was a good accuracy in fatigue, showing the effectiveness of the proposed model.

Finite-volume micromechanics-based multiscale analysis of composite structural model accounting for elastoplastic and ductile damage mechanisms

A hierarchical homogenization framework is proposed for nonlinear elastic-plastic and damage simulation of a unidirectional composite laminate. Proper scale transition rules between the macro- and the micro-scale variables were introduced, wherein the finite-element and finite-volume methods are employed for predicting the structural and material point response, respectively. Specifically, at the macroscopic scale, the ABAQUS/CAE structural analysis provides macroscopic strains for the lower-scale unit cell simulation (top-down) at each load increment. Reversely, at the microscopic scale, the unit cell problem is modelled using extended parametric finite-volume direct averaging micromechanics (FVDAM) through a Meta-UMAT subroutine that links the global finite element solver and the constitutive laws (UMAT subroutines). Homogenization of stresses and effective stiffness tensor obtained from the unit cell predictions is passed back to the ABAQUS/CAE for the higher-scale finite-element structural model computation (bottom-up). The stresses and strains at the global and local scales can be tracked during the incremental loading on the macroscopic structures through continuous homogenization and localization. The predictive capabilities of the proposed technique are verified vis-à-vis the experimental response of the Boron/Aluminum (B/Al) laminate obtained from off-axis tensile and shear loading available in the literature.

A multiscale interfacial cyclic debonding model for fibre-reinforced composites using micromechanics and molecular dynamics

In this study, a novel multiscale model based on the finite-volume direct averaging micromechanics (FVDAM) theory and molecular dynamics (MD) was developed to predict the interfacial cyclic debonding behaviour of composites. At the microscale, a solid interface with accumulated damage was incorporated using FVDAM, which enabled the simulation of both localised and homogenised interfacial damage responses under cyclic loading; a significant reduction in strength was observed after 10 loading cycles, implying the interfacial damage accumulation. At the atomic-scale, an interface model was built and subjected to cyclic loadings using MD simulation; the stress peak after 5 cycles was approximately half of the initial value, which provides damage parameters for upper-scale calculations and reveals the fundamental mechanism of interfacial cyclic debonding. The experimental data of unidirectional SCS-6/Ti-15-3 composites under cyclic loading were adopted to verify the proposed model. Furthermore, the influence of thermal residual stress and fibre orientation was investigated, which offers valuable insights for composite design and manufacturing.

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.

Continuum damage mechanics-based finite-volume homogenization of unidirectional elastoplastic fiber-reinforced composites

The Finite-Volume Direct Averaging Micromechanics (FVDAM) method has been further extended to predict nonlinear behaviors of unidirectional boron/aluminum (B/Al) composites, induced by plastic deformation and ductile damage for the first time. The latter is considered by gradual stiffness degradation via the continuum damage mechanics (CDM) model. The convergence of homogenized and local response generated by the new FVDAM damage framework has been demonstrated by considering the hexagonal and square repeating unit cells (RUC) with different mesh discretizations under transverse tensile, transverse shear and axial shear loading paths. Then, the reliability of the developed approach to predict the nonlinear response of B/Al composites has been verified extensively vis-à-vis the experimental response available in the literature, under monotonic loading path with various off-axis angles. To accurately capture the experimental response, an RUC with actual microstructure characterized by a random fiber distribution is constructed and utilized in the numerical simulations. In addition, the developed model is also applied to predict the cyclic response of B/Al composites with different off-axis angles in further support of the theory’s potential to capture the macroscopic behaviors of B/Al composites under complex loading conditions.

A new multi-scale framework for analyzing the thermo-mechanical coupling mechanism of the composite laminates

The present article investigates the failure mechanism of the composite laminates with respect to a coupled thermo-mechanical effect. To this end, a novel coupled thermo-mechanical modeling scheme is presented. To accurately capture the nonlinear deformation at macro- and micro-scale affected by the temperature field, the finite volume method is employed to compute the thermal stress, which is introduced into the representative volume element to evaluate the whole microscopic stress field and the macroscopic mechanical behaviors on basis of the finite volume direct averaging micromechanics. In addition, a visco-plastic constitutive model with full consideration of the temperature effect is implanted into the presented multi-scale model to investigate their nonlinear responses. The thermo-mechanical coupling test scheme is subtly designed, and the experimental data shows a high consistency with the theoretical results. On this basis, the microscopic morphology of the fracture surface is further characterized. It is revealed that an elevate temperature improve the matrix toughness to some extent.

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.

Multiscale modeling of unidirectional composites with interfacial debonding using molecular dynamics and micromechanics

In this study, a new multiscale micromechanical model based on the finite-volume direct averaging micromechanics (FVDAM) theory and molecular dynamics (MD) is constructed. It describes the interfacial debonding phenomena of unidirectional composites, whereby a solid fracture interface is incorporated with the FVDAM for the first time. To better elucidate the debonding behavior of the solid interface, a six-degrees-of-freedom (6-DoF) cohesive zone model (CZM) is proposed using MD simulation. The damage matrix Dc(k) in the kth subvolume of the interface is obtained using the 6-DoF CZM, and its explicit form is described herein to facilitate its implementation in commercial software packages. To verify the effectiveness of the model, both experimental data and numerical simulations were adopted for comparison. The experimental data of the unidirectional SiCf/Ti6Al4V composite were compared with the simulation results of FVDAM, and a good agreement was established. A finite element method-based unit cell model is constructed using ABAQUS for numerical validation, and homogenized responses and local fields are compared. The excellent correlations prove the effectiveness of the proposed model. Furthermore, the effects of thermal residual stress and fiber orientations are revealed, which may provide some useful information for the design and manufacture of composites.

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.

Deep long short-term memory neural network for accelerated elastoplastic analysis of heterogeneous materials: An integrated data-driven surrogate approach

In this work, an integrated data-driven surrogate approach based on the finite-volume direct averaging micromechanics (FVDAM) and the long short-term memory (LSTM) neural network is explored to predict the elastoplastic response of composite materials. In particular, the FVDAM is first applied to generate the uniaxial and cyclic response of unidirectional composites with various off-axis orientations. Next, a two-layered neural network is trained to associate the applied strains to the corresponding stresses, which is subsequently evaluated using the separate, hold-out testing dataset. The LSTM-estimated stress–strain responses coincide with the FVDAM reference results for all the loading cases. The advantage of the LSTM to naturally capture the history-dependent stress–strain behavior over the fully connected neural network is presented, with the percentage prediction errors of the former approach an order of magnitude lower than the latter. Moreover, the robustness of the LSTM surrogate model is examined by analyzing the training data with white noise. The proposed framework offers a viable alternative for the determination of the history-dependent response of composites directly from data analysis without the need to understand the underlying deformation mechanism in the techniques of homogenization, as well as provides a foundation for efficient multiscale analysis of composite materials and structures.

Electromechanical response of multilayered piezoelectric BaTiO3/PZT-7A composites with wavy architecture

Electromechanical laminated composites with piezoelectric phases are increasingly being explored as multifunctional materials providing energy conversion between electric and mechanical energies. The current work explores thus-far undocumented combined microstructural effects of amplitude-to-wavelength ratio, volume fraction, poling direction of piezoelectric phases on both the homogenized properties and localized stress/electric field distributions in multilayered configurations under fully coupled electro-mechanical loading. In particular, the Multiphysics Finite-Volume Direct Averaging Micromechanics (FVDAM) and its counterpart, an in-house micromechanical multiphysics finite-element model, are utilized to investigate the homogenized and localized responses of wavy multilayered piezoelectric BaTiO3/PZT-7A architectures. These two methods generate highly agreeable results. Moreover, we critically examine the convergence of the finite-volume and finite element-based approaches via the Average Stress Theorem and Average Electric Displacement Theorem. The comparison shows the finite volume-based approach possesses a better numerical convergence. This study illustrates the FVDAM’s ability toward the analysis and design of engineered multilayered piezoelectric materials with wavy architecture.

Finite Volume-Based Asymptotic Homogenization of Periodic Materials Under In-Plane Loading

AbstractThe previously developed finite volume-based asymptotic homogenization theory (FVBAHT) for anti-plane shear loading (He, Z., and Pindera, M.-J., “Finite-Volume Based Asymptotic Homogenization Theory for Periodic Materials Under Anti-Plane Shear,” Eur. J. Mech. A Solids (in revision)) is further extended to in-plane loading of unidirectional fiber reinforced periodic structures. Like the anti-plane FVBAHT, the present extension builds upon the previously developed finite volume direct averaging micromechanics theory applicable under uniform strain fields and further accounts for strain gradients and non-vanishing microstructural scale relative to structural dimensions, albeit with multidimensional in-plane loadings incorporated. The unit cell problems at different orders of the asymptotic field expansion are solved by satisfying local equilibrium equations and displacement and traction continuity in a surface-averaged sense which is unique among the existing asymptotic homogenization schemes, leading to microfluctuation functions that yield homogenized stiffness tensors at each order for use in macroscale problems. The newly extended multiscale theory is employed in the analysis of a structural boundary-value problem under in-plane loading, illustrating pronounced boundary effects. A combination approach proposed in the literature is subsequently employed to mitigate the boundary layer effects by explicitly accounting for the microstructural details in the boundary region. This combination approach produces accurate recovery of the local fields in both regions. The extension to in-plane problem marks FVBAHT as an alternative, self-contained asymptotic homogenization tool, with documented advantages relative to current numerical techniques, for the analysis of periodic materials in the presence of strain gradients produced by three-dimensional loading regardless of microstructural scale.

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.

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).

Homogenization and localization of unidirectional fiber-reinforced composites with evolving damage by FVDAM and FEM approaches: A critical assessment

The recently-developed finite-volume direct averaging micromechanics (FVDAM) with progressive damage simulation capability modeled using the cohesive zone model is critically and fully assessed vis-à-vis Abaqus, a widely-adopted commercial finite-element code with cohesive element for the first time. The evolving debonding along the fiber/matrix interface of a unidirectional metal matrix composite is simulated using comparable bilinear traction-displacement separation laws in the two computational approaches. Differences between the two approaches are highlighted, including shortcomings of the Abaqus-based finite-element analysis of evolving damage. These shortcomings include the need of optimizing the compressive stiffness of the interface in order to: avoid material interpenetration; predict correct homogenized response; prevent numerical instabilities. These problems are not present in our version of the finite-volume homogenization approach since the traction-separation relations in the affected normal direction are directly eliminated when the interface is under compression. Nonetheless, comparison of the homogenized response and localized stress fields generated by the finite-element and finite-volume techniques demonstrates good agreement between the two approaches provided that a suitable compression factor is chosen in Abaqus.

Fully-coupled electro-magneto-elastic behavior of unidirectional multiphased composites via finite-volume homogenization

The effective and localized electro-magneto-elastic behavior of periodic unidirectional composites is investigated in this work. Instead of adopting the classical micromechanics models or variational principle-based finite-element (FE) techniques, the finite-volume direct averaging micromechanics (FVDAM) is extended to fulfill this task by incorporating fully-coupled electro-magneto-elastic constitutive relations. Consistent with the mathematical homogenization theories, the mechanical displacements and electric/magnetic potentials are partitioned using two-scale expansion involving the macroscopic and fluctuating contributions. The generalized local stiffness matrices are constructed explicitly by relating the surface-averaged tractions, electric displacements, and magnetic inductions to the surface-averaged mechanical displacements and electric/magnetic potentials, followed which the continuity and periodic boundary conditions are applied. The homogenized coefficients and localized stress, electric/magnetic field distributions are validated extensively against the exact generalized Eshelby solution and an in-house FE program, as well as the experimental measurements, where perfect agreements are observed for all cases. The efficiency and convergence of the proposed multiphysics FVDAM (MFVDAM) are tested by comparing the execution time and localized fields as a function of mesh discretization, with the multiphysics FE (MFEM) results as references. Besides, the MFVDAM is encapsulated into the particle swarm optimization algorithm to deduce the optimal fiber volume fraction at which maximum magnetoelectric coupling effect may occur in the composite system with piezomagnetic ceramics reinforced with piezoelectric unidirectional fibers.

Deep learning in heterogeneous materials: Targeting the thermo-mechanical response of unidirectional composites

In this communication, a multi-task deep learning-driven homogenization scheme is proposed for predicting the effective thermomechanical response of unidirectional composites consisting of a random array of inhomogeneity. Toward this end, 40 000 repeating unit cells (RUCs) comprising an arbitrary number of locally irregular inclusions are generated over a wide range of fiber volume fractions. The finite-volume direct averaging micromechanics is then employed to evaluate the homogenized thermo-mechanical moduli of each RUC. Subsequently, a two-dimensional deep convolution neural network (CNN) is constructed as a surrogate model to extract the statistical correlations between the RUC geometrical information and the corresponding homogenized response. The RUC images together with their homogenized moduli are divided into two datasets in a ratio of 9:1 with the former part used for training the CNN model and the latter part used for verification. The results presented in this contribution demonstrate that the deep CNN predictions exhibit remarkable correlations with the theoretical values generated by the finite-volume micromechanics, with a maximum relative prediction error of less than 8%, providing good support for the data-based homogenization approach.

Evolution of interfacial debonding of a unidirectional graphite/polyimide composite under off-axis loading

In this communication, the inelastic behavior of a unidirectional graphite/polyimide composite, whose constituent phases are both elastic and brittle, is characterized based on the hypothesis of shear-dominated fiber/matrix interfacial degradation as the primary cause of the observed nonlinearity. To accommodate a combined thermo-mechanical multiaxial loading in the off-axis specimens, the finite-volume direct averaging micromechanics (FVDAM) with damage evolution capability is established within a unified framework that uses discontinuity functions in conjunction with the bilinear cohesive-zone model. Meanwhile, the stand-alone FVDAM homogenization approach is combined with the particle swarm optimization algorithm to identify consistent in-situ fiber and matrix properties. The accuracy and efficiency of the present model with deduced properties are validated by comparing the simulated stress-strain responses against the experimental data in the literature with various fiber orientations and good agreements are obtained for all cases. Concomitant local stress distributions at different load steps are examined to demonstrate the stress transfer mechanism from the region of the damage interface to the surrounding matrix. For the first time, this study reveals that the off-axis dependent nonlinearity in this material system comprised of elastic fibers and brittle, linearly elastic matrix may be accurately captured using a damage evolution model rather than plasticity, viscoelasticity or viscoplasticity approaches typically employed for the matrix phase.