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Efficient and Validated Framework for Probabilistic Progressive Failure Analysis of Composite Laminates
Abstract
In this paper, a finite element-based framework is presented to model the probabilistic progressive failure of fiber-reinforced composite laminates with high fidelity and efficiency. The framework is based on the semidiscrete modeling approach that can be seen as a good compromise between continuum and discrete methods. The enhanced semidiscrete damage model (ESD2M) tool set comprises a smart meshing strategy with failure mode separation, a new version of the enhanced Schapery theory with a novel generalized mixed-mode law, and a novel probabilistic modeling strategy. These three joined components make the model efficient in capturing failure modes such as matrix cracks, fiber tensile failure, and delamination, as well as their interactions with high fidelity, while taking material nonuniformities into account. The model capabilities are demonstrated using single-edge notched tensile cross-ply laminates as an example. The ESD2M was not only capable of capturing the complex damage progression but also provided insights and explanations for some of the failure events observed in the laboratory. The presented framework efficiently integrates failure mode predictions with probabilistic modeling and enables Monte Carlo simulations to predict the ultimate failure strength with good accuracy, as well as its scatter.
... In this paper, we apply the approach to the laminate/ structural level, hence an Integrated Computational Materials and Structures Engineering (ICMSE). In a finite element (FE) framework, a curing simulation is performed to calculate the residual stress field in a laminate and a subsequent progressive failure analysis -using the enhanced semi-discrete damage modeling (eSD2M) framework [3] -is performed to predict the ultimate failure response after cure, as illustrated in the flowchart in Fig. 2. There are a few models to characterize this process and to calculate the stresses. ...
... The equations are given in Eqn. (1)(2)(3). In the present work, the cure kinetics equation is solved using a fourth order Runge Kutta scheme. ...
... Fig. 6 shows the predicted matrix crack damage with and without incorporating residual stresses. As clearly demonstrated, the model with residual stresses captures the extent of matrix cracks much better and resembles the damage extent observed in the experiment [3]. ...
The failure modes and deformation response of composite laminates are influenced by multiple factors such as stacking sequence, lamina and interlayer properties, and geometry. One of the critical failure modes in laminated composites is delamination which may lead to premature load drops. This paper investigates the effects of ply stacking and interlayer toughening through a comprehensive numerical study. The numerical framework is the latest iteration of the enhanced semi-discrete damage model (eSD2M) advanced by the authors. The present numerical framework captured the delamination type failure and pull-out failure in open-hole tensile (OHT) laminates, considering both sublaminate-level and ply-level scaled laminates. Experimental and numerical results confirmed that blocked plies can lead to premature specimen failure due to delamination. However, interlayer toughening can delay delamination and improve the structural integrity, even for ply-scaled laminates. On the other hand, interlayer toughening may have adverse effects on sublaminate-level scaled specimens. Ply-level scaled laminates may even outperform them when interlayers are slightly toughened.
Foreign object impact on composite materials continues to be an active field due to its importance in the design of load bearing composite aerostructures. The problem has been studied by many through the decades. Extensive experimental studies have been performed to characterize the impact damage and failure mechanisms. Leaders in aerospace industry are pushing for reliable, robust and efficient computational methods for predicting impact response of composite structures. Experimental and numerical investigations on the impact response of fiber reinforced polymer matrix composite (FRPC) laminates are presented. A detailed face-on and edge-on impact experimental study is presented. A novel method for conducting coupon-level edge-on impact experiments is introduced. The research is focused on impact energy levels that are in the vicinity of the barely visible impact damage (BVID) limit of the material system. A detailed post-impact damage study is presented where non-destructive inspection (NDI) methods such as ultrasound scanning and computed tomography (CT) are used. Detailed fractography studies are presented for further investigation of the through-the-thickness damage due to the impact event. Following the impact study, specimens are subjected to compression after impact (CAI) to establish the effect of BVID on the compressive strength after impact (CSAI). A modified combined loading compression (CLC) test method is proposed for compression testing following an edge-on impact. Experimental work on the rate sensitivity of the mode I and mode II inter-laminar fracture toughness is also investigated. An improved wedge-insert fracture (WIF) method for conducting mode I inter-laminar fracture at elevated loading rates is introduced. Based on the experimental results, a computational modeling approach for capturing face-on impact and CAI is developed. The model is then extended to edge-on impact and CAI. Enhanced Schapery Theory (EST) is utilized for modeling the full field damage and failure present in a unidirectional (UD) lamina within a laminate. Schapery Theory (ST) is a thermodynamically based work potential material model which captures the pre-peak softening due to matrix micro-cracking such as hackling, micro fissures, etc. The Crack Band (CB) method is utilized to capture macroscopic matrix and fiber failure modes such as ply splitting and fiber rupture. Discrete Cohesive Zone Method (DCZM) elements are implemented for capturing inter-laminar delaminations, using discrete nodal traction-separation governed interactions. The model is verified against the impact experimental results and the associated CAI procedures. The model results are in good agreement with experimental findings. The model proved capable of predicting the representative experimental failure modes.
Stiffness degradation and progressive failure of composite laminates are complex processes involving evolution and multi-mode interactions among fiber fractures, intra-ply matrix cracks and inter-ply delaminations. This paper presents a novel finite element model capable of explicitly treating such discrete failures in laminates of random layup. Matching of nodes is guaranteed at potential crack bifurcations to ensure correct displacement jumps near crack tips and explicit load transfer among cracks. The model is entirely geometry-based (no mesh prerequisite) with distinct segments assembled together using surface-based tie constraints, and thus requires no element partitioning or enrichment. Several numerical examples are included to demonstrate the model’s ability to generate results that are in qualitative and quantitative agreement with experimental observations on both damage evolution and tensile strength of specimens. The present model is believed unique in realizing simultaneous and accurate coupling of all three types of failures in laminates having arbitrary ply angles and layup.
A new method is introduced in part I of this two-part paper [1] to model the progressive failure of composite laminates. In this second part, the method is applied to unnotched (UNT) and open hole tensile (OHT) specimens. With semi-discrete meshing features and a randomized material distribution the proposed model is capable of capturing both ultimate failure and damage progression. The increase of crack density and splitting as well as fiber failure localization are demonstrated on the basis of the cross-ply tensile specimen. Similar mechanisms, including free edge cracking, are predicted in the more complex (50/40/10) layup. With a simple technique, the meshes are efficiently generated for unnotched and notched specimens using Abaqus CAE/ Python scripting. As a result, the model is able to capture the discrete cracking near the hole as well as stress concentrations using a locally refined mesh region. By unifying features of both smeared and discrete methods, the presented semi-discrete damage model (SD2M) is seen to provide a pathway to model a variety of failure mechanisms and their interactions fairly accurately and efficiently, leading to very good predictions of ultimate strengths.
A novel semi-discrete damage model (SD2M) is proposed for the progressive failure analysis (PFA) of composite structures. The presented method utilizes a separation of fiber and matrix failure modes and introduces a material strength distribution to thin strips of matrix-splitting elements in order to capture the progression of transverse crack density. With the proposed meshing strategy, the discreteness is greatly enhanced, while the model remains efficient and simple by using solely continuum elements. The SD2M approach aims to unify desirable features of both smeared and discrete methods and utilizes a practical and simple mesh generation procedure. In addition, a novel mixed-mode law is proposed to model the damage evolution of matrix cracks. The proposed mixed-mode formulation ensures smooth and simultaneous vanishing of traction components and is suitable for generic softening laws. Applications of SD2M to several examples are presented in part II [1] of this two part paper.
The change in the critical strain energy release rate as damage evolves, known as the R-curve, is of crucial importance to the understanding of fracture behaviour. The examination of damage evolution ahead of the crack tip in order to determine accurately the crack increment is key for the determination of the R-curve. Conventional in-situ methods such as optical measurements only examine the specimen surfaces. X-ray Computed Tomography (CT) offers satisfactory image quality, but conventional CT scanning requires the removal of the specimens from the test machine. If no dye penetrant is used, the specimens can be reloaded , but some important information will be missing such as the early load drops corresponding to damage initiation. If dye penetrant is used, the specimens can no longer be re-tested. In contrast, in-situ CT scanning can capture the detailed damage states ply-by-ply while the specimen is loaded and diminish the need for multiple specimens. In-situ characterization of trans-laminar fracture toughness of composites using CT has not been attempted in the past. It has been proven successful in this research, and shown that a partial R-curve can be constructed with a single Extended Single-Edge-notch Tension (ESET) specimen.
Validation of a progressive damage finite element analysis model using the CompDam material model was performed on a large multi-stringer panel subjected to compressive loading. The panel had a Teflon insert embedded between the skin panel and stiffener to cause an ultimate failure mode of stiffener separation. The compressive loading of the panel caused the skin between the stiffeners to locally buckle before any damage began to occur. The post-buckled behavior of the panel was the driving mechanism to the onset of damage and the damage growth that led to skin-stiffener separation. An integrated global-local modeling approach was used to validate several aspects the overall behavior of the panel from start to failure. The global region captured the pre-buckled stiffness of the test panel within 10% and the buckling load of the skin panel at the critical location within 2%. The skin’s five half-wave buckled mode shape was accurately predicted by the model. The integrated local model captured the load corresponding to the onset of damage within 5% of the average test data. Comparisons are made for key aspects of the damage morphology, such as a growth pattern that included a significant matrix split in the first ply of the skin and a migration of the delamination from the skin-stiffener interface to the skin’s ply1-ply2 interface. The ultimate failure mode was shown to be an unstable growth of delamination damage under the stiffeners which led to a peak load of the analysis that was approximately 5% above the average peak load of the testing. The validated global-local modeling approach used on the multi-stringer post-buckled panel with a Teflon insert used a methodology and lessons learned using smaller specimens applied to a subcomponent that captured several aspects of a modern aircraft design.
The validation of progressive damage analysis methods requires experimental data that captures the initiation of damage, its evolution with continued loading, and the morphology of damage throughout its evolution. In support of the NASA Advanced Composites Project, several validation specimens of increasing complexity have been tested and characterized in order to provide the data required to assess the capabilities of several progressive damage analysis methods for fiber-reinforced composite materials. The analytical capabilities of interest for this validation program include networks of interacting matrix cracks and delaminations in post-buckled stiffened structures. A series of single stringer compression specimens was tested and characterized at NASA Langley Research Center. The experimental data from these tests are compared with progressive damage finite element analysis results. Test/analysis comparisons are made in terms of load/displacement histories, damage morphology, and damage size versus load metrics.
Composite structures must endure a great variety of multi-axial stress states during their lifespan while guaranteeing their structural integrity and functional performance. Understanding the fatigue behavior of these materials, especially in the presence of notches that are ubiquitous in structural design, lies at the hearth of this study which presents a comprehensive investigation of the fracturing behavior of notched quasi-isotropic [+45/90/−45/0] s and cross-ply [0/90] 2s laminates under multi-axial quasi-static and fatigue loading. The investigation of the S-N curves and stiffness degradation, and the analysis of the damage mechanisms via micro-computed tomography clarified the effects of the multi-axiality ratio and the notch configuration. Furthermore, it allowed to conclude that damage progression under fatigue loading can be substantially different compared to the quasi-static case. Future efforts in the formulation of efficient fatigue models will need to account for the transition in damaging behavior in the context of the type of applied load, the evolution of the local multi-axiality ratio, the structure size and geometry, and stacking sequence. By providing important data for model calibration and validation, this study represents a first step towards this important goal.
Toughened composite materials, where the toughening agents are micro-particles dispersed within the interlayer resin rich regions, provide enhanced delamination resistance. This delay in delamination promotes intralaminar micro-cracking and macro-cracking, thus providing superior toughness. In this paper, a thorough study of damage accumulation was conducted to identify and quantify failure mechanisms in the +/-45° laminate axial tensile test. Progression of damage accumulation was captured with digital image correlation techniques, in-situ inspection with high-resolution cameras and interrupted tests at different loading stages. Detailed cross-section microscopy of specimens were used to observe progression of crack formation, density of cracks formed and crack branching at the toughened interface.
Apart from major cracks and delamination, extensive and diffused short matrix micro-cracks are generally observed in composite laminates. In principle, discrete crack models (DCM) offer greatest fidelity to the mechanics of crack growth, including interactions between different failure modes, but it is impractical to model numerous cracks explicitly beyond coupon-sized laminates. Smeared crack models (SCM), on the other hand, treat diffused damage by effective material degradation. However, homogenization of cracks does not model strong discontinuities and delamination-matrix crack interactions. An adaptive discrete-smeared crack model is proposed here to preserve fidelity to the fracturing process while maintaining efficiency and simplicity in modelling distributed damage. DCM is used to model individual matrix cracks at the initial stages of propagation. Subsequently, adaptive transition from DCM to coupled DCM/SCM is performed. Critical cracks are explicitly retained, while non-critical ones are modelled by dispersed damage. The method is demonstrated with off-axis and open-hole tension tests.
An accurate characterization of intra- and inter-laminar damage is essential for the prediction of the damage onset and evolution in composite laminates. This paper presents a discrete crack-informed 3D continuum damage mechanics (CDM) model and its application for the simulation of delamination migration in tape laminates. The proposed approach aims at capturing the initiation and propagation of discrete matrix cracks and their interactions with interface delaminations without introducing a physical crack along with its explicit front tracking. The 3D Hashin failure criteria are employed first to predict the damage initiation of composite materials under an arbitrary stress state. A crack orientation-dependent damage evolution model is developed to dictate the composite stiffness degradation associated with a matrix crack plane. The matrix crack plane is parallel to the fiber direction and its orientation is determined by the maximum principal stress in the transverse plane. The corresponding matrix crack-informed stiffness is degraded based on the orientation of the crack plane. To alleviate mesh dependency within a continuum damage modeling framework, an energy-driven failure mechanism is adopted for characterizing damage evolution of matrix cracking and fiber breakage. By coupling the discrete crack informed model for intra-laminar damage prediction and the cohesive model for interface
delamination within the same CDM framework, their synergistic interaction can be captured during the progressive failure analysis of composite laminates. The predictive capability of the enhanced modeling strategy is examined through its application of a delamination migration problem with the presence of the delamination
initiation and migration via matrix cracking. The matrix cracks and delamination growths are captured for different loading positions without a pre-assumption on the failure mode. Good agreement is achieved by comparing the predictions with the experimental data from a published NASA report.
The R-curve for Mode I trans-laminar fracture energy in quasi-isotropic IM7/8552 carbon/epoxy laminates is here deduced numerically from a virtual Over-height Compact Tension (OCT) test. A High-fidelity Finite Element Method (Hi-FEM) using the explicit Finite Element (FE) software LS-Dyna was adopted. Cohesive interface elements and a Weibull fibre failure criterion were used to predict failure. The input parameters for the Hi-FEM were measured from independent characterisation tests. OCT specimens were tested to verify the Hi-FEM results with good agreement. The R-curve effect is postulated to be caused by the growth of the height of the Fracture Process Zone (FPZ) with crack length. Hi-FEM can be used to better understand Mode I trans-laminar fracture toughness tests and generate fracture properties such as damage heights and R-curves for future structural scale models.
Continuum Damage Mechanics (CDM) based progressive damage and failure analysis (PDFA) methods have demonstrated success in a variety of finite element analysis (FEA) implementations. However, the technical maturity of CDM codes has not yet been proven for the full design space of composite materials in aerospace applications. CDM-based approaches represent the presence of damage by changing the local material stiffness definitions and without updating the original mesh or element integration schemes. Without discretely representing cracks and their paths through the mesh, damage in models with CDM-based materials is often distributed in a region of partially damaged elements ahead of stress concentrations. Having a series of discrete matrix cracks represented by a softened region may affect predictions of damage propagation and, thus, structural failure. This issue can be mitigated by restricting matrix damage development to discrete, fiber-aligned rows of elements; hence CDM-based matrix cracks can be implemented to be more representative of discrete matrix cracks. This paper evaluates the effect of restricting CDM matrix crack development to discrete, fiber-aligned rows where the spacing of these rows is controlled by a user-defined crack spacing parameter. Initially, the effect of incrementally increasing matrix crack spacing in a unidirectional center notch coupon is evaluated. Then, the lessons learned from the center notch specimen are applied to open-hole compression finite element models. Results are compared to test data, and the limitations, successes, and potential of the matrix crack spacing approach are discussed.
This article presents a non-probabilistic fuzzy based multi-scale uncertainty propagation framework for studying the dynamic and stability characteristics of composite laminates with spatially varying system properties. Most of the studies concerning the uncertainty quantification of composites rely on probabilistic analyses, where the prerequisite is to have the statistical distribution of stochastic input parameters. In many engineering problems, these statistical distributions remain unavailable due to the restriction of performing large number of experiments. In such situations, a fuzzy-based approach could be appropriate to characterize the effect of uncertainty. A novel concept of fuzzy representative volume element (FRVE) is developed here for accounting the spatially varying non-probabilistic source-uncertainties at the input level. Such approach of uncertainty modelling is physically more relevant than the prevalent way of modelling non-probabilistic uncertainty without considering the ply-level spatial variability. An efficient radial basis function based stochastic algorithm coupled with the fuzzy finite element model of composites is developed for the multi-scale uncertainty propagation involving multi-synchronous triggering parameters. The concept of a fuzzy factor of safety (FFoS) is discussed in this paper for evaluation of safety factor in the non-probabilistic regime. The results reveal that the present physically relevant approach of modelling fuzzy uncertainty considering ply-level spatial variability obtains significantly lower fuzzy bounds of the global responses compared to the conventional approach of non-probabilistic modelling neglecting the spatially varying attributes.
This article presents a probabilistic framework to characterize the dynamic and stability parameters of composite laminates with spatially varying micro and macro-mechanical system properties. A novel approach of stochastic representative volume element (SRVE) is developed in the context of two dimensional plate-like structures for accounting the correlated spatially varying properties. The physically relevant random field based uncertainty modelling approach with spatial correlation is adopted in this paper on the basis of Karhunen-Loève expansion. An efficient coupled HDMR and DMORPH based stochastic algorithm is developed for composite laminates to quantify the probabilistic characteristics in global responses. Convergence of the algorithm for probabilistic dynamics and stability analysis of the structure is verified and validated with respect to direct Monte Carlo simulation (MCS) based on finite element method. The significance of considering higher buckling modes in a stochastic analysis is highlighted. Sensitivity analysis is performed to ascertain the relative importance of different macromechanical and micromechanical properties. The importance of incorporating source-uncertainty in spatially varying micromechanical material properties is demonstrated numerically. The results reveal that stochasticity (/ system irregularity) in material and structural attributes influences the system performance significantly depending on the type of analysis and the adopted uncertainty modelling approach, affirming the necessity to consider different forms of source-uncertainties during the analysis to ensure adequate safety, sustainability and robustness of the structure.
A reliable virtual testing framework for unidirectionally laminated composites is presented that allows the prediction of failure loads and modes of general in-plane coupons with great realism. This is a toolset based on finite element analysis that relies on a cohesive-frictional constitutive formulation coupled with the kinematics of penalty-based contact surfaces, on sophisticated three-dimensional continuum damage models, and overall on a modelling approach based on mesh structuring and crack-band erosion to capture the appropriate crack paths in unidirectional fibre reinforced plies. An extensive and rigorous validation of the overall approach is presented, demonstrating that the virtual testing laboratory is robust and can be reliably used in for composite materials screening, design and certification.
The initial fracture propagation within a full-scale stiffened quasi-isotropic composite panel and coupons with stringer feet under tensile loads was investigated. The specimens were made from Non-Crimp Fabric through Vacuum assisted Resin Transfer Moulding. The failure loads of all configurations were successfully related using the same value of trans-laminar fracture energy. The method involved independent tests of scaled-down Over-height Compact Tension specimens and the Virtual Crack Closure Technique. It was found to be crucial to include the fully developed damage process zone in the crack length and to interpret the results carefully in order to identify the failure loads consistently.
Static and dynamic analysis of the fracture tests of fiber composites in hydraulically servo-controlled testing machines currently in use shows that their grips are much too soft and light for observing the postpeak softening. Based on static and dynamic analysis of the test setup, far stiffer and heavier grips are proposed. Tests of compact-tension fracture specimens of woven carbon-epoxy laminates prove this theoretical conclusion. Sufficiently stiff grips allow observation of a stable postpeak, even under load-point displacement control. Dynamic stability analysis further indicates that stable postpeak can be observed under CMOD control provided that a large mass is rigidly attached to the current soft grips. The fracture energy deduced from the area under the measured complete load-deflection curve with stable postpeak agrees closely with the fracture energy deduced from the size effect tests of the same composite. Previous suspicions of dynamic snapback in the testing of composites are dispelled. So is the previous view that fracture mechanics was inapplicable to the fiber-polymer composites.
This paper presents the modelling of tensile failure of composites using novel enriched elements defined based on the floating node method. An enriched ply element is developed, such that a matrix crack can be modelled explicitly within its domain. An enriched cohesive element is developed to incorporate the boundaries of matrix cracks on the interface, such that the local stress concentrations on the interface can be captured. The edge status variable approach allows the automatic propagation of a large number of matrix cracks in the mesh. A laminate element is formed, such that a fixed, planar mesh can be used for laminates of arbitrary layups. The application examples demonstrate that the proposed method is capable of predicting several challenging scenarios of composites tensile failure, such as the large number matrix cracks, grip-to-grip longitudinal splits, widespread delamination, explosive splitting and distributed fibre breaking in the 0 plies, etc. The complete failure process of ply-blocked composite laminates, up to the final breaking of the loosened 0° strips, are here firstly reproduced by modelling.
Design of large composite structures requires understanding the scaling of their mechanical properties, an aspect often overlooked in the literature on composites. This contribution analyzes, experimentally and numerically, the intra-laminar size effect of textile composite structures. Test results of geometrically similar Single Edge Notched specimens made of 8 layers of 0 degree epoxy/carbon twill 2 by 2 laminates are reported. Results show that the nominal strength decreases with increasing specimen size and that the experimental data can be fitted well by Bazant's size effect law, allowing an accurate identification of the intra-laminar fracture energy of the material. The importance of an accurate estimation of Gf in situations where intra-laminar fracturing is the main energy dissipation mechanism is clarified by studying numerically its effect on crashworthiness of composite tubes. Simulations demonstrate that, for the analyzed geometry, a decrease of the fracture energy to 50% of the measured value corresponds to an almost 42% decrease in plateau crushing load. Further, assuming a vertical stress drop after the peak, a typical assumption of strength-based constitutive laws implemented in most commercial Finite Element codes, results in an strength underestimation of the order of 70%. The main conclusion of this study is that measuring accurately fracture energy and modeling correctly the fracturing behavior of textile composites, including their quasi-brittleness, is key. This can be accomplished neither by strength- or strain-based approaches, which neglect size effect, nor by LEFM which does not account for the finiteness of the Fracture Process Zone.
This paper presents an extension of the semi-discrete damage model (SD2M) (Nguyen and Waas, 2020 [1,2]), implemented using the finite element (FE) method, to capture the interaction of intra-laminar matrix cracks and inter-laminar failure (delamination migration). In the enhanced semi-discrete damage model (eSD2M), the plies are modeled with continuum finite elements and the interlayers are represented with cohesive elements. A meshing strategy is presented to create compatible meshes for interlayers in order to capture a proper load transfer between the laminae and the interfaces. The material pre-peak non-linearity of the composite is modeled with Schapery theory (ST), the post-peak strain softening is captured with the crack band (CB) method including a mixed-mode law for matrix failure. For the cohesive behavior of the interfaces, a novel mixed-mode traction-separation law is proposed. The model is validated on the basis of a ply-scaled quasi-isotropic open-hole tensile (OHT) laminate that shows a delamination-dominated failure. The eSD2M model is able to capture major matrix cracks at the hole including 0° splitting as well as distributed matrix cracks and free edge cracking. The present model is capable of predicting the delamination migration mechanism with an accurate peak load and the total failure due to fiber tensile rupture in a realistic manner.
Failure in a fiber-reinforced polymer composite is a complex event, with the possibility of intra-laminar and inter-laminar modes interacting. These modes may initiate, propagate and accumulate for prescribed loading conditions, stacking sequences and geometries of the structure. In this paper, the failure progression in a cross-ply tensile specimen with a single-edge notch is examined through experiments that provide detailed observations using digital image correlation. A numerical model based on the semi-discrete methodology presented by the authors earlier is developed further to analyze and understand the fracture mechanisms and interactions between the different failure modes, namely transverse cracking/ splitting, delamination and fiber breakage. The present modeling strategy is able to reproduce the experimental results accurately regarding the damage progression, final failure and overall response.
Keywords: Fiber composites, Progressive failure, Notched laminates, Cross-ply, Digital image correlation, Virtual testing, Semi-discrete method
The NASA Multiscale Analysis Tool (NASMAT) software is examined and discussed as a platform for multiscale modeling of composites for engineering applications. NASMAT performs analysis of materials with any arbitrary number of length scales through the use of recursive procedures and data structures and also can function as an effective material model within structural finite element models. The software’s efficiency is studied by varying the size of the structural finite element model and the local micromechanics model, while also varying the number of CPUs used to execute simulations in a high-performance computing environment. The code is then applied to a massively multiscale fractal problem with up to 150 hierarchical length scales. Finally, two practical progressive damage problems are examined involving a 3D woven polymer matrix composite material and a ceramic matrix composite gas turbine engine vane structure with an embedded micromechanics model.
Progressive damage finite element (FE) analysis methods based on continuum damage mechanics (CDM) use mesh regularization algorithms to ensure that fracture energy dissipation predictions are independent of problem discretiza-tion. Mesh regularization algorithms require some geometric input related to the discretization. When using crack band theory for mesh regularization, a characteristic element length is used to approximate the width of the region affected by the continuum crack, i.e., the crack band width. Inaccuracy in representing the crack band width significantly affects predictions in terms of fracture energy dissipation. For square elements misaligned by 45 , using a typical line length across an element rather than the crack band width overestimates dissipated fracture energy by 41%. Not accounting for element aspect ratio underestimates dissipated fracture energy by 29% and 50% for ratios of two and four, respectively. Herein, methods for calculating characteristic element lengths in fiber-reinforced materials are presented that account for meshes being misaligned with respect to material directions, element aspect ratio, and element skew. The limits of applicability of different crack band width approximations are explored through numerical crack growth studies and center notch tension FE analyses for different discretizations. Results are compared in terms of fracture energy dissipation to linear elastic fracture mechanics. Analyses with the proposed characteristic element lengths predict consistent fracture energy dissipation with various meshes. The proposed methods and the included studies on potential error in fracture energy dissipation provide analysts the basis to better understand error in CDM model predictions associated with simplified FE model preprocessing.
Results of a comprehensive experimental and computational low velocity impact (LVI) damage study is reported for T800s/3900-2B laminated composites with three stacking sequences. The stacking sequences studied are [0/45/0/90/0/-45/0/45/0/-45]s, [45/0/-45/90]3s, and [45/-45/0/45/-45/90/45/-45/45/-45]s, which are [50/40/10], [25/50/25] and [10/80/10] where each number indicates the percentage of zero, 45 and 90 plies. For each layup, three impact energy levels have been examined. After the LVI tests, the induced damage has been characterized with ultrasound C-scanning and high-resolution micro computed tomography (micro-CT). Computational predictions have been conducted using the enhanced Schapery Theory (EST) model. The numerical results are accurate in terms of the impact load responses and induced damage. The physics of the LVI damage of the samples with various stacking sequences and as a function of impact energy has been resolved using the EST model. The damage predicted by the EST model and the revealed damage mechanisms contribute to understanding the physics of the LVI behavior of laminated composites.
In situ X-ray synchrotron radiation computed tomography (SRCT) of carbon fiber composite laminates reveals the first-ever qualitative and quantitative comparisons of 3D progressive damage effects introduced by two mechanical enhancement technologies: aligned nanoscale fiber interlaminar reinforcement and thin-ply layers. The technologies were studied individually and in combination, using aerospace-grade unidirectional prepreg standard-thickness prepreg (‘std-ply’) and thin-ply composite laminates. The relatively weak interlaminar regions of the laminates were reinforced with high densities of aligned carbon nanotubes (A-CNTs) in a hierarchical architecture termed ‘nanostitching’. Quasi-isotropic double edge-notched tension (DENT) laminates were tested and simultaneously 3D-imaged via SRCT at various load steps, revealing a progressive 3D network of damage micro-mechanisms that were segmented according to modality and extent. For load steps of 0%, 70%, 80%, and 90% of baseline ultimate tensile strength (UTS), intralaminar matrix cracking and fiber/matrix interfacial debonding are found to be the dominant damage mechanisms, common to all laminate types. For both std-ply and thin-ply, nanostitched laminates had qualitatively and quantitatively similar matrix damage modality and extent compared to the baseline laminates through 90% UTS, including relatively few delaminations, despite an ∼9% increase in std-ply nanostitched UTS over the std-ply baseline. Complementary finite element-based modeling of damage predicts greater delamination extent in std-ply vs. thin-ply laminates that manifests only between 90% and 100% UTS, offering an explanation for the observed positive nanostitch effect in the std-ply, which is known to be more susceptible to delamination formation and growth than the thin-ply laminates. Thin-ply, with and without nanostitch, intrinsically suppresses matrix damage, as expected from past work and evidenced here by 6.5X less overall matrix damage surface area vs. std-ply baseline laminates averaged over all load steps. These findings contribute new insights from high-resolution experimental mapping of composite damage states that can guide and inform mechanical enhancement approaches and improved damage models.
Using a novel semi-discrete method that is presented in [1, 2] and a mesh-enhanced delamination model [3], numerical analyses of open hole tensile (OHT) specimens have been performed. The novel method combines high predictive fidelity due to discrete meshing features and high computational efficiency by using continuum damage and failure modeling using enhanced Schapery theory and the crack band method. The proposed model is capable of accurately predicting the failure progression and tensile strength of laminates with different layups ("hard", "quasi-isotropic", "soft") and scaled specimens with delamination-dominated failure mechanisms, where the interaction between intra-ply matrix cracks and delamination is accurately captured.
This paper describes two new additions to the CompDam fiber kinking model: 1) a failure index for fiber kinking to facilitate interaction between kinking and matrix cracking and 2) a strain-based criterion for fiber rupture to account for fiber breakage before or after fiber kinking. The model formulation is described with the aid of the graphical Considère construction, which is used to determine the conditions for fiber kinking. Verification studies are conducted with single element models to show the behavior of the model in light of the new developments and to evaluate the effect of numerical parameters of the model. The model is used to analyze open hole compression (OHC) specimens with soft, quasi-isotropic, and hard layups. Analysis results are shown for a conventional continuum damage mechanics model using a trilinear constitutive law for fiber damage and for the proposed model to highlight the differences between the two. Comparison with test data highlights the capabilities and remaining gaps in the quantitative prediction of the damage and failure response. While both models can predict many features of the OHC response, accurate prediction of kinking onset in the OHC specimens remains elusive. The results suggest that it is necessary to predict the combined effects of compression and shear loading as well as the structural effects (e.g., ply thickness, strain gradient effect) accurately to match the experimental observations of fiber kinking onset in the three OHC laminates.
Failure loads and modes of standard test coupons of general unidirectional laminates are predicted by a reliable virtual testing toolset based on the finite element method. This computational mechanics framework combines a cohesive-frictional constitutive formulation coupled with the kinematics of penalty-based contact surfaces to simulate interlaminar damage, and a sophisticated three-dimensional continuum damage model to predict intraply damage. This virtual testing laboratory was previously validated for conventional laminates and can be applied to composite materials screening, design, and certification. In the present work, the computational framework is enhanced with features to increase realism of the virtual tests and to allow generalization to a larger range of configurations. A relevant new feature is stochastic simulation that allows replicating realistic test campaigns. The demonstration of the enhanced stochastic virtual testing laboratory is performed on dispersed-ply laminates that, by not being constrained to conventional 0°, 90°, and ±45° ply arrangements, allow widening the design space of composites.
Uncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling-based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time-consuming if the simulation-based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and corresponding high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different setting of MLMC and MFMC, aims to address the issue of UQ and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small data sets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. An algorithm summary and the corresponding code implementation are provided for each of the modern MC methods. The extension and application of these methods are discussed in detail. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Monte Carlo Methods Statistical and Graphical Methods of Data Analysis > Sampling.
Part I [1] of this two-part paper presented the formulation of a novel progressive failure model for pultruded fibre reinforced polymer (FRP) composites, allowing for the 3D simulation of quasi-orthotropic FRP plates as a homogenized material, as well as the model calibration based on a set of standardized material characterization tests. Part II presents the application of that (calibrated) model to the simulation of two case studies: (i) transverse compact tensile (CT) tests; and (ii) web-crippling tests for two load configurations, external two-flanges (ETF) and internal two-flanges (ITF). The CT test, which is often used to determine the (tensile) fracture energy of FRP materials, is especially interesting as it allows assessing the quality of the simulations for a combination of in-plane transverse tensile and shear stresses in a geometry with a sharp singularity. The web-crippling test, on the other hand, is often used to determine the strength of FRP shapes under concentrated transverse loads, a real structural problem involving combined in-plane compressive and shear stresses. In this paper these two relatively complex case studies are used to assess the quality of the simulation in the presence of combined in-plane stresses. The numerical results showed an excellent agreement with their CT test counterparts; the simulation of these experiments were also used to demonstrate the need for using a mesh regularization scheme when modelling problems with singularities. The models were also well able to simulate both web-crippling load configurations, only slightly underestimating the maximum load – this was likely due to the slight underestimation of shear strength for combined in-plane shear and moderate transverse compressive stresses, as discussed in Part I [1], and/or non-quasi-orthotropic behaviour of the web-flange junction. Overall, the numerical results showed a good agreement with the experimental data, even for relatively coarse meshes, attesting the feasibility and precision of the proposed damage progression model.
In this paper, the low velocity impact (LVI) on a quasi-isotropic laminate [45/0/-45/90]3s (QIL) is studied to predict the deformation response and damage state of the laminate. This stacking of a QIL is a benchmark case that results in a “rotating-fan” pattern of delamination damage due to the impact. Drop-tower tests were performed with an impact energy of 25J and an impactor mass of 7.5 Kg. 3D digital image correlation (3D DIC) was carried out to measure the in-situ deformation of the laminate. Non-destructive inspection (NDI) including ultrasound C-scanning and X-ray micro computed tomography (micro-CT) were done to characterize the overall damage footprint and the internal detailed damage morphology. The computational model is an extension and refinement of the model developed in [76], [74]. Enhanced Schapery Theory (EST) is used as the constitutive model and implemented with a user material subroutine in the commercial code Abaqus. The EST uses Schapery theory for pre-peak damage and the crack band model for post peak failure. The major contributions reported in this paper are as follows; in the experimental study, the damage mechanisms have been illustrated with high-resolution micro-CT scanning, while in the numerical study, the “rotating-fan” pattern, damage-free cone and damage modes interaction have been accurately and efficiently captured with a uniform, non-fiber-aligned mesh.
In situ experimentation using synchrotron radiation computed tomography (SRCT) is instrumental to observe subsurface mechanical behavior of composite materials in real time. However, the investigation areas of SRCT are often limited by its experimental configurations. In the present study, an ex situ observation is carried out to complement the in situ test results reported previously in the literature. Previous SRCT-based in situ experiment on the single-edge-notched symmetric cross-ply laminate has visualized various failure modes at the notch, which are sequentially occurring inside the specimen. The tested specimen is further examined here using a laboratory-scale X-ray CT device with a dye penetrant. Delamination failures interacting with longitudinal and transverse cracks, which have not been detected by the in situ test, are found with other failure modes distributed over the entire specimen area. The ex situ and in situ experimental results are complementary and both of them are required to comprehensively understand the various and complex failure mechanisms of the composite.
Interactive failure mechanisms of a single-edge notched (SEN) symmetric cross-ply laminate subjected to tensile loading are studied in situ using synchrotron X-ray tomography and a DIC technique at the micro- and macro-scales. The SEN configuration guarantees the repeatability and consistency of the progressive and interactive failure behavior at the two different scales. The evolution of various fracture modes in different layers is observed through synchrotron radiation CT images. Due to the geometrically asymmetric configuration, all the initial cracks sequentially occurring near the notch tip can be captured in a narrow field of view. Stress relaxation due to the subsequent fracture initiations is indirectly captured by measuring the crack lengths and opening displacements from the 3D CT images at different loading steps. The stress relaxation is directly measured by DIC analysis using the macro-scale test data. DIC results show strains relaxed in the region between existing and emerging transverse cracks.
A materials failure postulate is established giving upper and lower bounds on the numbers of failure parameters involved with any associated failure criterion. The failure postulate is based upon the symmetry properties for the material, any macroscopically homogeneous material form. It is of significant and decisive help in deriving failure criteria. A wide variety of examples of its usage are given but most of them are aimed at the standard forms for fiber composite materials. Specific failure criteria are the end result for everything from unidirectional, highly anisotropic composites at one extreme to the quasi-isotropic laminate form at the other extreme. Almost all of the work is for fiber composites with any degree of anisotropy. Section 7 and part of Section 4 have highly anisotropic conditions taken as appropriate to carbon fiber/polymeric matrix composites. The results are summarized in Section 8 and they are responsive to the field's many years of searching for the failure criteria of composite materials.
Failure prediction for carbon fiber reinforced polymer (CFRP) composites has been a longstanding challenge. In this study, we address this challenge by first applying a well-established computational micromechanics model based on representative volume element to predict the failure envelopes of unidirectional (UD) CFRP composites.
Then, these failure envelopes are compared with the classical failure criteria. We have evaluated the performances of these failure criteria and identified the aspects for further improvement in their accuracies for
the UD CFRP composites studied herein. Based on the failure mechanisms from computational analyses and the comparisons between predicted failure envelopes and classical failure criteria, a new set of homogenized failure criteria is proposed. The newly proposed failure criteria show significant improvement according to our computational and experimental results. Furthermore, we have compared the proposed failure criteria with existing experimental data and computational results available in the literature for different types of composites. Good agreements are generally observed.
In the current study a high fidelity analysis approach is used to predict the failure process of notched composite structures. Discrete cracking is explicitly modelled by incorporating cohesive interface elements along potential failure paths. These elements form an interconnected network to account for the interaction between interlaminar and intralaminar failure modes. Finite element models of these configurations were created in the commercial analysis software ABAQUS and a user defined material subroutine (UMAT) was used to describe the behaviour of the cohesive elements. The material subroutine ensured that the model remained stable despite significant damage, which is a significant challenge for implicit damage simulations. Two analysis approaches were adopted using either the as-measured or modified (in-situ) ply strengths. Both approaches were capable of closely predicting the mean ultimate strength for a range of hole diameters. However, using the measured ply properties resulted in extensive matrix cracking in the surface ply which caused a deviation from the experimentally measured surface strain. The results demonstrate that high fidelity physically based modelling approaches have the ability to complement or replace certain experimental programs focussed on the design and certification of composite structures.
Intra-inter crack band model (I2CBM) is proposed for studying the progressive damage and failure of laminated composite bolted joints. The model combines Schapery theory for matrix microcrack modeling with crack band theory for lamina macroscopic failure modeling in a standard 3D finite element framework and is implemented as material laws at element integration points. Three different failure planes defined by material orthotropy are considered for the modeling of macroscopic failure using crack band theory. This procedure allows the model to be used either as an intraply element or as an interply element of finite thickness by an appropriate choice of the crack planes of interest. Localized bearing failure, observed in bolted joints, is modeled using a residual strength approach in the post-peak response of individual ply elements. Simulation results for single lap shear and double lap shear bolted joint problems are compared against experiments for model validation.
Countersunk holes are used in bolted joints for creating non-protruding smooth surfaces. As a first step towards a detailed progressive damage and failure analysis of bolted joint configurations, a new model referred to as the Intra-inter crack band model (I2CBM) is used to study open hole tensile/compressive and filled hole tensile/compressive progressive damage and failure. The I2CBM is a unified approach for modeling intralaminar and interlaminar progressive damage and failure in polymer matrix composites. 3D finite elements in the I2CBM formulation can be adapted for modeling individual lamina elements or to model interface delamination elements. A path for communication between the intralaminar and interlaminar failure mechanisms is enabled in the model to overcome limitations associated with homogenized element modeling and to capture complex interactions in a physically correct manner. A non-local crack spacing method is implemented for tracking matrix cracks. Comparisons between test results and I2CBM predictions for the open hole tensile/compressive and filled hole tensile/compressive cases with a countersunk hole configuration are discussed. The effect of bolt pretension on the filled hole failure analysis case is also presented and the failure mechanisms are discussed.
Graphite-epoxy cross-ply laminates generally show multiple fracture of the transverse ply at higher applied stress. This phenomenon is described by means of a Monte Carlo simulation method based on the assumption that the strength of the transverse ply obeys a two-parameter Weibull distribution function. The main results show that the smaller the scatter of strength of the 90°-ply (i.e. the larger the shape parameter at a constant mean strength of the Weibull distribution), the higher becomes the threshold for the multiple fracture to occur, and the more rapidly the length of 90°-ply segments decreases with increasing applied stress once multiple fracture takes place. The methods to determine the shape and scale parameters of the Weibull distribution for the strength of the 90°-ply proposed by Manders et al. and Peters are proved to be useful even for a small number of test specimens. When the interfacial bond strength between 0°- and 90°-plies is low, saturation of 90°-ply cracking occurs at higher applied stress. The stress-carrying capacity and stiffness of the composites as a whole are reduced by multiple fracture of the 90°-ply. This reduction is more pronounced at increasing applied stress or at a larger number of transverse cracks, especially when the interfacial bond strength is low.
Reducing the timeline for development and certification for composite structures has been a long standing objective of the aerospace industry. This timeline can be further exacerbated when attempting to integrate new fiber-reinforced composite materials due to the large number of testing required at every level of design. Computational progressive damage and failure analysis (PDFA) attempts to mitigate this effect; however, new PDFA methods have been slow to be adopted in industry since material model evaluation techniques have not been fully defined. This study presents an efficient evaluation framework which uses a piecewise verification and validation (V&V) approach for PDFA methods. Specifically, the framework is applied to evaluate PDFA research codes within the context of intralaminar damage. Methods are incrementally taken through various V&V exercises specifically tailored to study PDFA intralaminar damage modeling capability. Finally, methods are evaluated against a defined set of success criteria to highlight successes and limitations.
The topic of Uncertainty Quantification (UQ) has witnessed massive developments in response to the promise of achieving risk mitigation through scientific prediction. It has led to the integration of ideas from mathematics, statistics and engineering being used to lend credence to predictive assessments of risk but also to design actions (by engineers, scientists and investors) that are consistent with risk aversion. The objective of this Handbook is to facilitate the dissemination of the forefront of UQ ideas to their audiences. We recognize that these audiences are varied, with interests ranging from theory to application, and from research to development and even execution.
A homogenization approach to simulate the curing process in textile composites is presented. Internal stresses build-up in a thermosetting polymer matrix during cure because of thermal expansion/contraction mismatch, chemical shrinkage and lack of uniformity in the thermal field. The matrix cures within the fiber tows as well as in the pockets outside the tows. Although the explicit modeling of each fiber and matrix is preferred, it is computationally prohibitive for tows containing large numbers of fibers. Instead, a homogenization approach that is presented here models the effective curing of the tow by extending the cure hardening instantaneously linear elastic (CHILE) model, originally developed for matrix curing. The possibility of damage and/or failure in the matrix during cure is modeled using the Bažant-Oh crack band model. Using cure parameters for the IM7-8552 material system, the effectiveness of the homogenization procedure is assessed by comparing the curing of a homogenized tow with the curing of discrete fiber-matrix tow, followed by the virtual curing of a 8-harness satin (8HS) weave. Subsequently, the 8HS weave mechanical response is computed. Lastly, the 8HS mechanical response when cure induced effects (manufacturing process) are not taken into consideration are assessed.
This study predicted transverse cracking progression in laminates including 90° plies. The refined stress field (RSF) model, which takes into account thermal residual strain for plies including transverse cracks is formulated, and the energy release rate associated with transverse cracking is calculated using this model. For comparison, the energy release rate based on the continuum damage mechanics (CDM) model is formulated. Next, transverse cracking progression in CFRP cross-ply laminates including 90° plies is predicted based on both stress and energy criteria using the Monte Carlo method. The results indicated that the RSF model and the CDM model proposed in this study can predict the experiment results for the relationship between transverse crack density and ply strain in 90° ply. The models presented in this paper can be applied to an arbitrary laminate including 90° plies.
This study seeks to establish a high-fidelity mesoscale simulation methodology that can predict the progressive damage and resultant failure of carbon fiber reinforced plastic laminates (CFRPs). In the proposed scheme, the plastic behavior (i.e., pre-peak nonlinear hardening in the local stress-strain response) is characterized through the pressure-dependent elasto-plastic constitutive law. The evolution of matrix cracking and delamination, which result in post-peak softening in the local stress-strain response, is modeled through cohesive zone models (CZM). The CZM for delamination is introduced through an interface element, but the CZM for matrix cracking is introduced through an extended finite element method (XFEM). Additionally, longitudinal failure, which is dominated by fiber breakage and typically depends on the specimen size, is modeled by the Weibull criterion. The validity of the proposed methodology was tested against an off-axis compression (OAC) test of unidirectional (UD) laminates and an open-hole tensile (OHT) test of quasi-isotropic (QI) laminates. Finally, sensitivity studies were performed to investigate the effect of plasticity and thermal residual stress against the prediction accuracy in the OHT simulation.
