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NASA NTRS2004

Comparison of the Computational Efficiency of the Original Versus Reformulated High-Fidelity Generalized Method of Cells

The High-Fidelity Generalized Method of Cells (HFGMC) micromechanics model has recently been reformulated by Bansal and Pindera (in the context of elastic phases with perfect bonding) to maximize its computational efficiency. This reformulated version of HFGMC has now been extended to include both inelastic phases and imperfect fiber-matrix bonding. The present paper presents an overview of the HFGMC theory in both its original and reformulated forms and a comparison of the results of the two implementations. The objective is to establish the correlation between the two HFGMC formulations and document the improved efficiency offered by the reformulation. The results compare the macro and micro scale predictions of the continuous reinforcement (doubly-periodic) and discontinuous reinforcement (triply-periodic) versions of both formulations into the inelastic regime, and, in the case of the discontinuous reinforcement version, with both perfect and weak interfacial bonding. The results demonstrate that identical predictions are obtained using either the original or reformulated implementations of HFGMC aside from small numerical differences in the inelastic regime due to the different implementation schemes used for the inelastic terms present in the two formulations. Finally, a direct comparison of execution times is presented for the original formulation and reformulation code implementations. It is shown that as the discretization employed in representing the composite repeating unit cell becomes increasingly refined (requiring a larger number of sub-volumes), the reformulated implementation becomes significantly (approximately an order of magnitude at best) more computationally efficient in both the continuous reinforcement (doubly-periodic) and discontinuous reinforcement (triply-periodic) cases.

Arnold, Steven M↗

NASA NTRS2017

Order-Reduced Solution of the Nonlinear High-Fidelity Generalized Method of Cells Micromechanics Relations

The High-Fidelity Generalized Method of Cells (HFGMC) is one technique for accurately simulating nonlinear composite material behavior. The HFGMC uses a higher-order approximation for the subcell displacement field that allows for a more accurate determination of the subcell stressstrain fields at the cost of some computational efficiency. In order to reduce computational costs associated with the solution of the ensuing system of simultaneous equations, the HFGMC global system of equations for doubly-periodic repeating unit cells with nonlinear constituents was reduced in size through the use of a Petrov-Galerkin-based Proper Orthogonal Decomposition order-reduction scheme. A number of cases were presented that address the computational feasibility of using order-reduction techniques to solve solid mechanics problems involving complex microstructures.

order-reduction↗

NASA NTRS2004

Testing the Predictive Capability of the High-Fidelity Generalized Method of Cells Using an Efficient Reformulation

The High-Fidelity Generalized Method of Cells is a new micromechanics model for unidirectionally reinforced periodic multiphase materials that was developed to overcome the original model's shortcomings. The high-fidelity version predicts the local stress and strain fields with dramatically greater accuracy relative to the original model through the use of a better displacement field representation. Herein, we test the high-fidelity model's predictive capability in estimating the elastic moduli of periodic composites characterized by repeating unit cells obtained by rotation of an infinite square fiber array through an angle about the fiber axis. Such repeating unit cells may contain a few or many fibers, depending on the rotation angle. In order to analyze such multi-inclusion repeating unit cells efficiently, the high-fidelity micromechanics model's framework is reformulated using the local/global stiffness matrix approach. The excellent agreement with the corresponding results obtained from the standard transformation equations confirms the new model's predictive capability for periodic composites characterized by multi-inclusion repeating unit cells lacking planes of material symmetry. Comparison of the effective moduli and local stress fields with the corresponding results obtained from the original Generalized Method of Cells dramatically highlights the original model's shortcomings for certain classes of unidirectional composites.

Arnold, Steven M.↗

NASA NTRS2020

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 to 6.3x speedup in the equation assembly/solution runtime while maintaining model accuracy. This corresponded to a 21 to 38 percent 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.

Ricks, Trenton M.↗

NASA NTRS2012

Implementation of a Smeared Crack Band Model in a Micromechanics Framework

The smeared crack band theory is implemented within the generalized method of cells and high-fidelity generalized method of cells micromechanics models to capture progressive failure within the constituents of a composite material while retaining objectivity with respect to the size of the discretization elements used in the model. An repeating unit cell containing 13 randomly arranged fibers is modeled and subjected to a combination of transverse tension/compression and transverse shear loading. The implementation is verified against experimental data (where available), and an equivalent finite element model utilizing the same implementation of the crack band theory. To evaluate the performance of the crack band theory within a repeating unit cell that is more amenable to a multiscale implementation, a single fiber is modeled with generalized method of cells and high-fidelity generalized method of cells using a relatively coarse subcell mesh which is subjected to the same loading scenarios as the multiple fiber repeating unit cell. The generalized method of cells and high-fidelity generalized method of cells models are validated against a very refined finite element model.

Pineda, Evan J.↗

NASA NTRS2012

Progressive Failure of a Unidirectional Fiber-Reinforced Composite Using the Method of Cells: Discretization Objective Computational Results

Pineda, Evan J.↗

NASA NTRS2024

Multiscale Modeling of Thermoplastics Using Atomistic-informed Micromechanics

A multiscale repeating unit cell model of a single spherulite containing four disparate length scales was developed to predict the thermoelastic behavior of semicrystalline thermoplastic materials for composite aerospace applications. The continuum level scales were fully coupled and modeled using the generalized method of cells and the high fidelity generalized method of cells micromechanics theories. Data from molecular dynamics simulations were used as inputs for the amorphous and crystalline constituents in the multiscale continuum models. Effective Young’s modulus, shear modulus, Poisson’s ratio, coefficient of thermal expansion, and thermal conductivity were predicted for polyether ether ketone and polyether ketone ketone, showing good agreement with the available experimental data from the open literature. Moreover, it is shown that predicted properties are fairly insensitive to the fidelity of the micromechanics model used at the highest continuum scale or the assumed shape of the spherulite.

thermoplastics↗

NASA NTRSDate not supplied

Multiscale Modeling of Thermoplastics Using Atomistic-informed Micromechanics

A multiscale model was developed for predicting the thermoelastic behavior of semi-crystalline thermoplastic materials for composite aerospace applications. At the highest scale containing the semi-crystalline spherulite in an amorphous matrix, the generalized method of cells, or high fidelity method of cells, was used to perform the homogenization calculations to obtain the effective properties. Models were developed assuming a cubic, or spherical shape, for the spherulite to understand if the morphology of the spherulite affects the effective thermoelastic properties. The generalized method of cells was used to model at the repeating unit cells at the subscales of the microstructure including the lamellae stacks and granular crystal blocks. The scales are integrated using the multiscale micromechanics method in the NASA Multiscale Analysis Tool. Data from molecular dynamics simulations were used as inputs for the amorphous and crystalline constituents. Convergence studies were performed to determine the best level of discretization for the repeating unit cell at the highest scale. Effective Young’s modulus, shear modulus, Poisson’s ratio, coefficient of thermal expansion, and thermal conductivity were predicted for polyether ether ketone and polyether ketone ketone, and very good agreement between the model utilizing the cubic spherulite and the experimental data, where available, was observed for polyether ketone ketone. Normalization of the data for the bulk polyether ketone ketone, against amorphous data, improved the predictions as compared to experimental data. Overall, the high fidelity method of cells predicted a stiffer response then the generalized method of cells as the crystallinity was increased. The shape of the spherulite had a minimal effect on the predicted bulk properties of the polymers.

thermoplastics↗

NASA NTRS2008

Micromechanics-Based Progressive Failure Analysis of Composite Laminates Using Different Constituent Failure Theories

Predicting failure in a composite can be done with ply level mechanisms and/or micro level mechanisms. This paper uses the Generalized Method of Cells and High-Fidelity Generalized Method of Cells micromechanics theories, coupled with classical lamination theory, as implemented within NASA's Micromechanics Analysis Code with Generalized Method of Cells. The code is able to implement different failure theories on the level of both the fiber and the matrix constituents within a laminate. A comparison is made among maximum stress, maximum strain, Tsai-Hill, and Tsai-Wu failure theories. To verify the failure theories the Worldwide Failure Exercise (WWFE) experiments have been used. The WWFE is a comprehensive study that covers a wide range of polymer matrix composite laminates. The numerical results indicate good correlation with the experimental results for most of the composite layups, but also point to the need for more accurate resin damage progression models.

Moncada, Albert M.↗

NASA NTRS2021

Micromechanics-based Modeling of Laminated SiC/SiC Ceramic Matrix Composites

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.

Ceramic Matrix Composites↗

NASA NTRS2002

High-Fidelity Generalization Method of Cells for Inelastic Periodic Multiphase Materials

An extension of a recently-developed linear thermoelastic theory for multiphase periodic materials is presented which admits inelastic behavior of the constituent phases. The extended theory is capable of accurately estimating both the effective inelastic response of a periodic multiphase composite and the local stress and strain fields in the individual phases. The model is presently limited to materials characterized by constituent phases that are continuous in one direction, but arbitrarily distributed within the repeating unit cell which characterizes the material's periodic microstructure. The model's analytical framework is based on the homogenization technique for periodic media, but the method of solution for the local displacement and stress fields borrows concepts previously employed by the authors in constructing the higher-order theory for functionally graded materials, in contrast with the standard finite-element solution method typically used in conjunction with the homogenization technique. The present approach produces a closed-form macroscopic constitutive equation for a periodic multiphase material valid for both uniaxial and multiaxial loading. The model's predictive accuracy in generating both the effective inelastic stress-strain response and the local stress said inelastic strain fields is demonstrated by comparison with the results of an analytical inelastic solution for the axisymmetric and axial shear response of a unidirectional composite based on the concentric cylinder model, and with finite-element results for transverse loading.

Aboudi, Jacob↗

NASA NTRSDate not supplied

Comparison of Micromechanics Theories for Modeling Discontinuous Long Fiber Reinforced Composites

Two micromechanics theories, the high fidelity generalized method of cells and Carrera Unified Formulation, are used to model a repeating unit cell of a discrete long fiber composite. The models are used to predict the effective properties of a single fiber with varying fiber-to-fiber end gap lengths, as well as the local stresses in the fiber and matrix. Both models produced similar results and demonstrated that they could capture the complex local load transfer due to the shear lag effect.

discrete long fiber composites↗

NASA NTRSDate not supplied

Progressive Damage Response of 3D Woven Composites via the Multiscale Recursive Micromechanics Solution with Tailored Fidelity

Progressive failure simulations have been performed for orthogonal 3D woven composites consisting of RTM6 resin matrix and AS4 carbon fibers. The Multiscale Recursive Micromechanics approach has been used, which, while being computationally efficient, captures the primary effects of the microstructure at each considered length scale. This approach also enables use of any micromechanics theory at any length scale, and herein, the fidelity of the chosen theories across the scales has been tailored to strike a balance with computational efficiency. The Mori-Tanaka method is employed at the lowest length scale, the Generalized Method of Cells is used at intermediate scales, and the High-Fidelity Generalized Method of Cells is used at the highest woven composite repeating unit cell scale. Furthermore, two different damage models, also with different levels of fidelity and efficiency, have been used for the resin material at the lowest length scale. Results for the mechanical behavior in response to loading in various directions are compared for the two damage models and with available test data.

3D woven↗

NASA NTRS2017

Comparison of Multiscale Method of Cells-Based Models for Predicting Elastic Properties of Filament Wound C/C-SiC

Three different multiscale models, based on the method of cells (generalized and high fidelity) micromechanics models were developed and used to predict the elastic properties of C/C-SiC composites. In particular, the following multiscale modeling strategies were employed: Concurrent multiscale modeling of all phases using the generalized method of cells, synergistic (two-way coupling in space) multiscale modeling with the generalized method of cells, and hierarchical (one-way coupling in space) multiscale modeling with the high fidelity generalized method of cells. The three models are validated against data from a hierarchical multiscale finite element model in the literature for a repeating unit cell of C/C-SiC. Furthermore, the multiscale models are used in conjunction with classical lamination theory to predict the stiffness of C/C-SiC plates manufactured via a wet filament winding and liquid silicon infiltration process recently developed by the German Aerospace Institute.

multiscale modeling↗

NASA NTRS2018

Coupled Thermomechanical Micromechanics Modeling of the Influence of Thermally Grown Oxide Layer in an Environmental Barrier Coating System

Environmental Barrier Coatings (EBCs) have emerged as a promising means of protecting critical components for high temperature applications (e.g., aircraft engines). EBCs are often used to protect an underlying material (substrate) from extreme thermal/chemical environments. However, systems that utilize EBCs are susceptible to a number of failure modes including oxidation/delamination. Environmental Barrier Coatings (EBCs) have emerged as a promising means of protecting silicon based ceramic matrix composite (CMC) components for high temperature applications (e.g., aircraft engines). EBCs are often used to protect an underlying material (substrate) such as silicon carbide from extreme thermal/chemical environments. In a typical CMC/EBC system, an EBC may or may not be adhered to an underlying substrate with a bond coat (e.g., silicon). Irrespective, systems that utilize EBCs are susceptible to a number of failure modes including oxidation/delamination, recession, chemical attack and dissolution, thermo-mechanical degradation, erosion, and foreign object damage. Current work at NASA Glenn Research Center is aimed at addressing these failure modes in EBC systems and developing robust analysis tools to aid in the design process. The Higher-Order Theory for Functionally Graded Materials (HOTFGM), a precursor to the High-Fidelity Generalized Method of Cells micromechanics approach, was developed to investigate the coupled thermo-mechanical behavior of functionally graded composites and will be used herein to assess the development and growth of a low-stiffness thermally grown oxide (TGO) layer in EBC/CMC systems without a silicon bond coat. To accomplish this a sensitivity study is conducted to examine the influence of uniformly and non-uniformly grown oxide layer on the associated driving forces leading to mechanical failure (spallation) of EBC layer when subjected to isothermal loading, recession, chemical attack and dissolution, thermomechanical degradation, erosion, and foreign object damage. Current work at NASA Glenn Research Center is aimed at addressing these failure modes in EBC systems and developing robust analysis tools to aid in the design process. The Higher-Order Theory for Functionally Graded Materials (HOTFGM), a precursor to the High-Fidelity Generalized Method of Cells micromechanics approach, was developed to investigate the coupled thermo-mechanical behavior of functionally graded composites (Aboudi et al., 1999, Composites B). For example, HOTFGM was previously used (Arnold et al, 1995, NASA CP 10178, paper 34), to assess interlaminar stresses (including free edge effects) in a substrate with a thermal barrier coating (TBC). In this study, HOTFGM micromechanics analyses will be used to assess the development and growth of a low-stiffness thermally grown oxide (TGO) layer between a silicon carbide substrate and a ytterbium disilicate EBC. In order to realistically simulate TGO growth, an evolution law will be incorporated into HOTFGM. In addition, the effect of TGO roughness will be explored consistent with previous TBC work (Pindera et al., 2000. Material Science and Engineering, A284, pp. 158-175). This model represents a first step in developing a robust analysis tool that can ultimately be used to design durable EBC systems. Additional failure modes will be considered as part of a future work.

environmental barrier coating↗

NASA NTRSDate not supplied

Thermal Conductivity of 3D Woven Composite Thermal Protection System Materials via Multiscale Recursive Micromechanics

Multiscale micromechanics predictions have been made for the effective thermal conductivities and local thermal fields for a novel 3D woven composite thermal protection system material. The Multiscale Recursive Micromechanics approach, which enables micromechanics models to call other micromechanics models (or themselves recursively) to consider finer and finer length scales, has been employed. The multiscale model uses a recently developed version of the High-Fidelity Generalized Method of Cells micromechanics theory at each of three length scales. Results focus on the impact of the microstructural geometry representation at each length scale on the material’s effective thermal conductivity, along with the local thermal flux and temperature fields induced in the microstructures.

Composites↗

NASA NTRS2017

Micromechanics of Composite Materials Governed by Vector Constitutive Laws

The high-fidelity generalized method of cells micromechanics theory has been extended for the prediction of the effective property tensor and the corresponding local field distributions for composites whose constituents are governed by vector constitutive laws. As shown, the shear analogy, which can predict effective transverse properties, is not valid in the general three-dimensional case. Consequently, a general derivation is presented that is applicable to both continuously and discontinuously reinforced composites with arbitrary vector constitutive laws and periodic microstructures. Results are given for thermal and electric problems, effective properties and local field distributions, ordered and random microstructures, as well as complex geometries including woven composites. Comparisons of the theory's predictions are made to test data, numerical analysis, and classical expressions from the literature. Further, classical methods cannot provide the local field distributions in the composite, and it is demonstrated that, as the percolation threshold is approached, their predictions are increasingly unreliable. XXXX It has been observed that the bonding between the fibers and matrix in composite materials can be imperfect. In the context of thermal conductivity, such imperfect interfaces have been investigated in micromechanical models by Dunn and Taya (1993), Duan and Karihaloo (2007), Nan et al. (1997) and Hashin (2001). The present HFGMC micromechanical method, derived for perfectly bonded composite materials governed by vector constitutive laws, can be easily generalized to include the effects of weak bonding between the constituents. Such generalizations, in the context of the mechanical micromechanics problem, involve introduction of a traction-separation law at the fiber/matrix interface and have been presented by Aboudi (1987), Bednarcyk and Arnold (2002), Bednarcyk et al. (2004) and Aboudi et al. (2013) and will be addressed in the future.

micromechanics↗

DOE OSTI2021

Stochastic microstructural analysis of ceramic matrix composites using a high-fidelity multiscale framework

This paper presents a high-fidelity multiscale framework to simulate the mechanical behavior of ceramic matrix composites (CMCs), accounting for the complex micromorphology captured using detailed material characterization. First, high-resolution micrographs are obtained for the specific carbon fiber silicon-carbide-nitride matrix (C/SiNC) CMC to characterize the variability of the architectural features and manufacturing-induced defects at the microscale. An image processing algorithm is then used to precisely estimate the size and distribution of all subscale features and defects from the micrographs. The information is then used to generate a three-dimensional stochastic representative volume element (SRVE) to reconstruct microscale constituents accounting for the variability. As a result, the generated SRVEs are simulated using the high-fidelity generalized method of cells (HFGMC) micromechanics theory to investigate the effects of defects on the elastic properties of C/SiNC CMCs.

Ceramic Matrix Composites↗