High-Fidelity Generalized Method Of Cells Research Articles

Analytical and numerical evaluation of the effective properties of macro fiber composite (MFC)

The piezoelectric fibers of Macro Fiber Composite (MFC) have a rectangular cross-section with an aspect ratio of 2. This heterogeneity poses challenges for micromechanical modeling to predict the effective mechanical properties. The High-Fidelity Generalized Method of Cells (HFGMCs) is commonly used to calculate the properties of composite materials. In this paper, a Modified High-Fidelity Generalized Method of Cells (MHFGMC) is proposed to analyze MFC properties, in which the interaction between subcells is considered by defining the quadratic directional coupling term and the shear connection matrix. The accuracy of the proposed MHFGMC model is verified by using the finite element method and experimental tests. The results show that the MHFGMC can accurately predict the effective mechanical properties of MFC, and the relative error of tensile properties compared to experimental results is within 2.29%. The MHFGMC significantly improve the accuracy of the HFGMC, the relative error of its shear modulus is decreased from 105.13% to 0.71%.

Micromechanics-Based Modeling of SiC/SiC Ceramic Matrix Composites and Structures

The behavior and response of ceramic matrix composites (CMCs), in particular silicon carbide fiber reinforced silicon carbide matrix (SiC/SiC), is affected by many factors such as variation of fiber volume fraction, residual stresses resulting from processing of the composites at high temperature, random microstructures, and the presence of matrix flaws (e.g., voids, pores, cracks etc.) as well as general material nonlinearity and heterogeneity that occurs randomly in a composite. Residual stresses arising from the phase change of constituents are evaluated in this paper and it is shown that they do influence composite strength and need to be properly accounted for. Additionally, the microstructures (location of fiber centers, coating thickness etc.) of advanced CMCs are usually disordered (or random) and fiber diameter and strength typically have a distribution. They rarely resemble the ordered fiber packing (square, rectangular, or hexagonal) that is generally assumed in micromechanics-based models with periodic boundary conditions for computational expediency. These issues raise the question of how should one model such systems effectively? Can an ordered hexagonal packed repeating unit cell (RUC) accurately represent the random microstructure behavior? How many fibers need to be included to enable accurate representation? Clearly, the number of fibers within an RUC must be limited to insure a balance between accuracy and efficiency. NASA’s in-house micromechanics-based code MAC/GMC provides a framework to analyze such RUCs for the overall composite behavior and the FEAMAC computer code provides linkage of MAC/GMC to the commercial FEA code, ABAQUS. The appropriate level of discretization of the RUC as well as the analysis method employed, i.e., Generalized Method of Cells (GMC) or High Fidelity Generalized Method of Cells (HFGMC), is investigated in this paper in the context of a unidirectional as well as a cross-ply laminated CMC. Results including effective composite properties, proportional limit stress (an important design parameter) and fatigue are shown utilizing both GMC as well as HFGMC. Finally, a few multiscale analyses are performed on smooth bar test coupons as well as test coupons with features such as open-hole and double notches using FEAMAC. Best practices and guidance are provided to take these phenomena into account and keep a proper balance between fidelity (accuracy) and efficiency. Following these guidelines can account for important physics of the problem and provide significant advantages when performing large multiscale composite structural analyses. Finally, to demonstrate the multiscale analysis framework, a CMC gas turbine engine vane structure is analyzed involving a progressive damage model.

Bifurcation buckling and the effect of imperfections on the microbuckling of soft materials with periodic microstructure by the finite strain HFGMC micromechanics

Two approaches are presented for the prediction of the microbuckling of composites subjected to compressive loading and undergoing large deformations. In the first one, the finite strain high-fidelity generalized method of cells (HFGMC) micromechanical analysis is successively implemented for the prediction of the ideal bifurcation buckling stress of the composite. In the second approach which considers composites which possesses imperfections (e.g. wavy fibers), a perturbation expansion is adopted. The resulting zero and first-order problems are solved by employing the finite strain HFGMC micromechanics. Here, the ideal critical stress can be readily predicted, and from the form of the waviness growth with applied loading it is possible to estimate the actual bifurcation stress. In addition, in the particular case of a multilayered periodic composites, an exact solution is derived for the prediction of the bifurcation buckling stress. This exact solution is employed to verify the micromechanically predicted critical stresses of bi-layered composites. For fiber-reinforced composites, a comparison with available finite element solution is utilized to verify the micromechanical predictions. Applications of the offered approaches are given for periodically bi-layered composites that consist of five types of hyperelastic materials, and two types for fiber-reinforced composites. The present methods are also implemented to predict the microbuckling stresses of a porous hyperelastic material and lattice blocks whose elements consist of a hyperelastic material.

Micromechanical damping analysis of composite materials using linear viscoelasticity vs the strain energy approach

The phenomenological strain energy approach for predicting composite damping properties is widely used because of its simplicity. The composite dissipated energy is approximated as the volume average of the local (constituent) loss factor times the local strain energy. Thus, any micromechanics method (e.g., finite element or the high-fidelity generalized method of cells used herein) that can predict the local fields (and thus the strain energy) within a composite can be used to approximate the effective damping properties. In this paper, two more physics-based approaches, the correspondence principle and explicit viscoelastic modeling, are implemented within the two-scale high-fidelity generalized method of cells micromechanics theory and compared with the widely-used strain energy approach. It is shown that all three approaches provide quite similar predictions for the effective damping properties of unidirectional composites. Finally, the presented two-scale micromechanics theory is applied to predict the effective anisotropic damping properties of 2D and 3D woven composites.

A Novel Microscopic Modeling Scheme for the Shape Memory Polymer Composites with respect to the Ambient Temperature

This paper is aimed at studying the effective mechanical property of shape memory polymer composites (SMPC) reinforced with natural short fibers. To this end, a novel modeling scheme was presented. The SMPC was firstly equivalent to the composite laminates, and the natural short fibers are also subtly equivalent to the ellipsoidal inclusions distributed in the matrix materials periodically. Moreover, a represented volume element along laminate thickness can be easily chosen, and its elastic constants are accurately acquired by employing a proper microscopic mechanical model. Herein, the high-fidelity generalized method of cells, which represents a good ability in predicting the effective mechanical behaviors of composites, was used. On this basis, the classic laminate theory was improved to suitable for describing the elastic constants and failure strength the SMPC with respect to ambient temperature. Numerical results show a good consistency to the experimental data. Moreover, a higher ambient temperature tends to sharply decrease their final failure strength. It is also revealed that the presented modeling method shows a great potential in calculating the effectively mechanical property of the natural short fiber-reinforced composites.

Multiscale structural analysis of thick sandwich structures using parametric HFGMC micromechanics and isogeometric plate formulation based on refined zigzag theory

This study presents a novel multiscale modeling strategy by combining the parametric high fidelity generalized method of cells (HFGMC) micromechanical model and isogeometric analysis (IGA) based on refined zigzag theory (RZT) for thick sandwich structures. Effective behavior of sandwich face sheets at the macro scale is defined by using constituent (fiber and matrix) properties in the HFGMC model, thereby eliminating the need for time-consuming and costly experiments. At the macro level, the RZT kinematics is utilized to define zigzag distributions of in-plane displacement along thickness coordinate, which can accurately model a thick laminate or a sandwich structure as a single layer regardless of its plies. Moreover, the IGA is leveraged to discretize geometry of the structure effectively with non-uniform rational B-splines (NURBS) basis functions, i.e., free of shear locking phenomena. Various test cases of thin/thick laminates with high heterogeneity and anisotropy are analyzed to demonstrate wide-range applicability of the present approach. Results of the proposed multiscale framework provide excellent distributions of in-plane displacements and stresses along the thickness directions of the laminates as compared to the three-dimensional reference solutions. Hence, computational efficiency, high accuracy, and low cost of the present method are revealed for sandwich structures.

Analysis of bifurcation buckling and imperfections effect on the microbuckling of viscoelastic composites by HFGMC micromechanics

Two different approaches are presented for the prediction of the microbuckling of various types of viscoelastic composites under compression. In the first one, an incremental procedure in time is employed to establish, in conjunction with the Laplace’s transform and its inversion, a determinant whose first complex root indicates the occurrence of the failure stress and strain of the composite (with no imperfections) at the critical time. In the second approach, the viscoelastic composite is assumed to possess, due to faulty manufacturing, imperfections. A perturbation expansion of the field in terms of a small parameter establishes a series of problems of various order. It is shown that the solutions of the zero and first-order problems yield, in conjunction with the Laplace’s transform and its inversion, the imperfection growth with applied loading, which asymptotically approaches the bifurcation buckling stress of the viscoelastic composite. In both approaches a repeated application of the high-fidelity generalized method of cells (HFGMC) micromechanics is employed to obtain the solutions. The offered two analyses are verified and applied on bi-layered, continuous and short fiber viscoelastic composites, as well as on viscoelastic woven composites and lattice blocks. The latter two applications necessitate the employment of multiscale HFGMC micromechanical analyses since the yarns in the weaves and the elements of the lattices are themselves unidirectional viscoelastic composites.

Microbuckling of viscoplastic composites by the high-fidelity generalized method of cells micromechanics

The high-fidelity generalized method of cells (HFGMC) micromechanical analysis is employed for the prediction of microbuckling stresses of composite materials which are composed of elastic–viscoplastic constituents. The inelastic material behavior is represented by a unified viscoplasticity theory which can model non-proportional loading paths, namely it is capable of providing the effect of axial loading on the reduction of shear modulus which dominates the microbuckling. By applying an incremental procedure and in conjunction with the instantaneous stiffnesses of the viscoplastic material, an eigenvalue problem at each time step is established by the HFGMC micromechanical method, which is modified to include the buckling terms arising from the linearization of the nonlinear equations of equilibrium. These buckling terms are provided by HFGMC analysis which predicts the local stress field and the effective current tangent properties of the composite. The present approach is verified by comparison of its predictions with an exact solution that can be established in the special case of composites that consist of periodic viscoplastic layers. Applications are given for the prediction of the microbuckling stresses of bi-layered, continuously reinforced and aligned short-fiber viscoplastic composites. In addition, the microbuckling stresses of viscoplastic lattices and layered plates are presented.

Nonuniform transformation field analysis based reduced-order model of high-fidelity generalized method of cells

The accurate prediction of a composite’s nonlinear behavior using multiscale modeling techniques relies heavily on the description of microscale features and requires many internal variables, leading to high computational costs. This paper presents a reduced-order model (ROM) for composite materials based on a novel adaptation of nonuniform transformation field analysis (NTFA) in conjunction with the high-fidelity generalized method of cells (HFGMC) micromechanics theory accounting for micro constituent damage. First, the composite’s nonlinear behavior is simulated using the HFGMC technique with constituent damage represented as damage-induced inelastic strains. The nonuniformity in the inelastic strains is then approximated using a set of tensorial deformation modes. Next, a macroscopic continuum damage model is formulated based on the inelastic modes to have the same evolution function as their microscopic counterpart. The NTFA procedure is applied in a three-dimensional setting, followed by a detailed description of the localization operators necessary for linearizing the homogenized constitutive equations. Last, the ROM’s capability to capture damage effects using only a finite set of internal variables is demonstrated by comparing the predicted global and local responses through HFGMC simulations for different loading paths. The presented model is shown to decrease the computational cost of micromechanics-based unit cell homogenization significantly.

Predicting strength of a unidirectional composite containing interface crack

In this paper, both analytical Bridging Model and numerical GMC (Generalized Method of Cells) and HFGMC (High-Fidelity Generalized Method of Cells) are applied to analyze failures of a unidirectional (UD) composite with a cracked fiber and matrix interface. Interconnecting the internal stresses in the matrix with those in the fiber by a bridging tensor, the difference between Bridging Model and GMC/HFGMC is in the bridging tensors and the method of determining them. The stress concentration factors (SCFs) of the matrix in the composite before and after the cracked interface are calculated upon the different bridging tensors. Based on the true stresses resulted from the different bridging tensors and the SCFs, off-axial tensile strengths of UD composites are predicted. A total number of 9 different prediction schemes are made. The predicted results are compared with the experimental data of carbon, aramid, and flax fiber composites prepared and measured by ourselves. It is found that the predictions based on the Bridging Model equations provide overall the best agreement with the experiments.

An efficient implementation of the high-fidelity generalized method of cells for complex microstructures

The high-fidelity generalized method of cells (HFGMC) enables micromechanical analysis of heterogeneous materials with high accuracy but does so at the cost of computational efficiency. In this paper, an implementation of the triply periodic HFGMC is developed to enable high-resolution simulations of materials with complex microstructures at a significantly reduced computational cost. This paper describes efficient reformulation and develops low-cost algorithms to reduce overall computation time and memory required to analyze complex 3D microstructures. The low-cost algorithms exploit the sparsity of the data by storing and performing calculations on only the non-zero values. The Parallel Direct Sparse Solver (PARADISO) subroutine is used to execute the most computationally intensive processes in parallel on multiple cores. Simulations of two selected test cases demonstrate the validity, computational efficiency, and value of the developed implementation. The results indicate that the savings in computation time and required memory are substantial and more than 100 times in some cases. In addition, parallel processing further reduces the computation time. The efficiency achieved through this work makes the high-resolution simulation of complex microstructures using HFGMC for the prediction of accurate local stress/strain fields computationally feasible.

Finite strain parametric HFGMC micromechanics of soft tissues.

A micromechanical analysis is offered for the prediction of the global behavior of biological tissues. The analysis is based on the isotropic-hyperelastic behavior of the individual constituents (Collagen and Elastin), their volume fractions, and takes into account their detailed interactions. The present analysis predicts the instantaneous tensors from which the effective current first tangent tensor is established, thus providing the overall anisotropic constitutive behavior of the composite and the resulting field distribution in the composite. This is in contradistinction with the macroanalysis in which the composite internal energy, which involves unknown functions that depend on several strain invariants, must be proposed. The offered micromechanical analysis forms a generalization to the finite strain high-fidelity generalized method of cells (HFGMC) based on the homogenization technique for periodic composites to the parametric finite strain. This involves an arbitrary discretization of the repeating unit-cell of the periodic composites. Results are given for the response of the human abdominal aorta, which consists of three layered tissues: intima, media, and adventitia, all of which are composed out of the Collagen and Elastin. The isotropic-hyperelastic constituents (Mooney-Rivlin and Yeoh) of the composites are calibrated by utilizing available experimental data which describe the response of the tissue. Validation of the results is performed by comparison of the predicted Cauchy stress and stretches with the experimental measurements. In addition, results are given in the form of Cauchy stress and deformation gradient field distributions in the constituents of several tissues.

Analyzing the effects of interphase on the effective damping properties of aligned carbon nanotube-reinforced epoxy nanocomposites using a micromechanical approach

In this work, a micromechanical approach consisting of high-fidelity generalized method of cells (HFGMC) and Mori-Tanaka (M-T) model is proposed to calculate the damping properties of aligned carbon nanotube-epoxy nanocomposites. To determine the resultant directional specific damping coefficients, these models, by applying strain energy approach in the global system utilize each constituent’s specific damping coefficients and mechanical properties. The effects of interphase created in the contact region of the two initial phases—carbon nanotube and polymer matrix—are extensively investigated. Comparative studies show that the micromechanical results are in good agreement with experimental data. One major finding is the thickness and mechanical and damping properties of interphase significantly affect the overall specific damping coefficients of the carbon nanotube-polymer nanocomposites. It is found that by increasing the elastic modulus of the interphase, the longitudinal specific damping property continuously increases, while other components of damping, initially increase and then asymptotically decrease. The damping properties of polymer nanocomposites can be increased by increasing the interphase damping capacity. However, the rise of interphase thickness leads to a reduction of nanocomposite damping properties. Also, the influences of carbon nanotube volume fraction and radius are examined on the damping response of polymer nanocomposites.

The parametric HFGMC micromechanical model for soft UHMWPE laminated composites

The 3D parametric High-Fidelity-Generalized-Method-of-Cells (HFGMC) micromechanical model was used to predict the mechanical tensile behavior of ultra-high-molecular-weight-polyethylene (UHMWPE) cross-ply laminated composite (Dyneema®). The composite microstructure was obtained from optical microscope images and was digitized using CAD software. Two configurations of 3D representative-volume-elements (RVE)s were constructed and analyzed to demonstrate the capability of the HFGMC in predicting the effective behavior of UHMWPE laminated composites. The elastic properties of both constituents (fiber and matrix) were obtained from published experimental data and from in-house experiments. Linear and nonlinear constitutive laws were applied for the constituents for comparison. The parametric HFGMC predictions for the overall mechanical properties were compared with extensive tensile experiments. The nonlinear behavior under in-plane shear was also examined in order to evaluate the effect of the proposed RVE configurations for the laminated composite. The parametric HFGMC (PHFGMC) is formulated using the virtual work for both the global and local fields. This allows the development of a new incremental-iterative nonlinear solution scheme. Special periodicity constraints are proposed for the special, yet prevalent case, of non-periodic RVE mesh. The parametric HFGMC micromechanical model is shown to be capable to simulate the nonlinear behavior of soft composites. This model can be generalized for rate dependent constituents.

A multiscale modeling for failure predictions of fiber reinforced composite laminates

A three-dimensional multiscale damage modeling, based on the parametric High Fidelity Generalized Method of Cells (HFGMC) micromechanics and integrated with a finite-element (FE) code, is proposed to predict the mechanical response, including damage evolution, as well as to generate failure envelopes of composite laminates. A hexagonal repeating unit-cell (RUC) is used to represent explicitly the fiber and matrix phases of a lamina at the microscale level. Failure of a lamina is determined by using both strain-based and stress-based failure theories. This modeling approach is implemented as a user material subroutine (UMAT) within displacement-based Solid FE models, where each solid element is represented by one material point. When the modeled damage is propagating within the macroscale FE model, the material points are eliminated from the FE calculations by using the element deletion approach. The efficiency of the proposed multiscale modeling is demonstrated by a comparison of simulated results with measured data available in the literature.

Microscopic modeling and effective properties evaluation of glass/epoxy plain woven composites

In this study, the description of a micromechanical method to estimate the effective elastic constants of woven composites with polymer matrix is presented. To this end, woven composites are discretized to a series of unidirectional laminas. For validating the microscopic mechanical method, the calculated effective moduli of the unidirectional lamina are compared with other literatures. In addition, the fiber off-axis angle effects on longitudinal modulus, transversal modulus and Poisson’s ratio are further discussed by employing the coordinate transformation formula. It can be concluded that minimum longitudinal modulus will be acquired when the off-axis angle of 60° is considered. Combining with the classic laminate theory and 3D high-fidelity generalized method of cells, the elastic constants of E-glass/epoxy woven composites are predicted. By comparisons, it demonstrated that the proposed method is capable of mechanical modeling for plain woven composites.

A multi-scale modeling scheme for damage analysis of composite structures based on the High-Fidelity Generalized Method of Cells

This paper presents a description of a multi-scale method to investigate the failure behaviors and damage evolution of composite laminates reinforced with unidirectional fibers. The proposed approach is based on the microscopic mechanical theory and pre-processing function of the ANSYS/LS-DYNA software. At micro-scale, the High-Fidelity Generalized Method of Cells (HFGMC) is employed to establish the microscopic model, which can be used to acquire the microscopic stress distributions in the representative volume element (RVE). Moreover, a viscoplastic constitutive model is employed to describe the nonlinear behaviors of matrix materials. At macro-scale, each integration point in elements is employed to investigate the damage evolution for each lamina. In order to validate the proposed method, the numerical results of failure evolution path and stress-strain responses of the composite laminates are compared with experimental data. A good consistency between theoretical results and experimental data can be found. On this basis, the failure evolution path for each lamina is further investigated. The numerical results revealed that the crack firstly appeared in the 90° lamina. With the further increasing of external loading, the crack will accumulate along with the layer direction for −45° lamina, 45° lamina and 90° lamina.

A multiscale progressive damage analysis for laminated composite structures using the parametric HFGMC micromechanics

A nonlinear multiscale damage analysis is proposed based on the parametric High Fidelity Generalized Method of Cells (HFGMC) micromodel. Two repeating unit-cells (RUCs), square and hexagonal, for the composite microstructure are discretized into subcells which represents the fiber and matrix phases. Average traction and displacement continuities, periodicity conditions, and volumetric equilibrium are applied. The proposed modeling approach is implemented as a user material subroutine within displacement-based layered-shell FE models, where each layer can be represented with at least one RUC. The RUC through-thickness stress and stiffness contributions are eliminated in case of a failure. Two failure criteria, in conjunction with compressive strength equation, are applied to determine RUC failure. In the first, the Tsai-Wu failure criterion is used at the RUC level. The second employs the strain invariant failure theory (SIFT) within each subcell, leading to its extinction. The in-situ properties and damage parameters of the RUC are determined by calibration using axial, transverse, and shear data of unnoteched laminates. The accuracy of the proposed framework is demonstrated by a comparison with experimental results of open-hole laminates.

Solution of the Nonlinear High-Fidelity Generalized Method of Cells Micromechanics Relations via Order-Reduction Techniques

The High-Fidelity Generalized Method of Cells (HFGMC) is one technique, distinct from traditional finite-element approaches, for accurately simulating nonlinear composite material behavior. In this work, the HFGMC global system of equations for doubly periodic repeating unit cells with nonlinear constituents has been reduced in size through the novel application of a Petrov-Galerkin Proper Orthogonal Decomposition order-reduction scheme in order to improve its computational efficiency. Order-reduced models of an E-glass/Nylon 12 composite led to a 4.8–6.3x speedup in the equation assembly/solution runtime while maintaining model accuracy. This corresponded to a 21–38% reduction in total runtime. The significant difference in assembly/solution and total runtimes was attributed to the evaluation of integration point inelastic field quantities; this step was identical between the unreduced and order-reduced models. Nonetheless, order-reduced techniques offer the potential to significantly improve the computational efficiency of multiscale calculations.

Micromechanical Analysis for Two-Phase Copper-Silver Composites under Large Deformations

This study presents a homogenization based on micromechanics approach for a two-phase copper (Cu)-silver (Ag) composite undergoing finite deformations. In this approach, the high-fidelity generalized method of cells (HFGMC) is implemented for the prediction of the effective behavior of two cold-drawn Cu-Ag composites with different drawing strains and to obtain the field (deformation gradient and stress) distributions in the composite. Both metals (Cu or Ag) are rate-dependent crystal plasticity material constituents. HFGMC is applied for studying the deformation behavior of two-phase Cu-Ag composites under uniaxial compression. The micromechanical approach has been verified by comparison with experimental and finite element simulation results. Results in terms of deformation behavior and field distributions are given for two different cold-drawn composites.