R-curve Effect Research Articles

A simulation strategy for fatigue modeling of delamination in composite structures under multiple loading conditions considering loading history and [formula omitted]-curve effects

This work evaluates the ability of cohesive zone modeling-based approaches to predict delamination in composite materials that develop large process zones under complex loading conditions. The R-curve effects subjected to static and fatigue loading under multiple loading modes, considering the loading history, are analyzed. To this end, the delamination predictions of a state-of-the-art CZM-based simulation strategy are evaluated by blind simulation of a validation benchmark test. The validation test promotes a non-self-similar delamination scenario, including a process zone that evolves under different loading mode conditions with a non-straight leading delamination front. Good delamination prediction accuracy is achieved. In addition, insights into the relationship between the features of the simulation strategy and the physics of the delamination process are discussed. With regard to the limitations of the simulation strategy, particular attention should be paid to modeling the contribution of an evolving process zone based on the loading mode history.

An extended multi-linear cohesive zone model for mixed-mode I/II delamination growth simulation in composite laminates with R-curve effects

An extended multi-linear cohesive zone model for mixed-mode I/II delamination growth simulation in composite laminates with R-curve effects

The anomalous crack growth behaviour of an elastic-brittle octet-truss architected solid

Strongly rising R-curves are observed for octet-truss architected material specimens comprising ∼1 million unit cells and made from an elastic-brittle parent material. The measurements are supported by finite element (FE) calculations where the octet-truss is modelled discretely and show that the pseudo-ductility is not related to the usual toughening mechanisms such as crack bridging or crack-tip plasticity. Rather, the toughening is attributed to the three-dimensional (3D) structure of the octet-truss specimen: remarkably, no R-curve effect is observed in a quasi two-dimensional (2D) octet-truss specimen. This anomalous behaviour is clarified via FE calculations of the indentation stiffness of the octet-truss which reveal a strong indentation size effect in 3D with the stiffness decreasing with decreasing indenter size relative to the cell size of the octet-truss. In the 3D octet-truss, the size independent continuum stiffness is only attained for indenter sizes greater than about 20 unit cells while the size effect is almost absent in the quasi-2D octet-truss. The strong elastic softening in the 3D octet-truss in the presence of strain gradients gives rise to a large elastic crack-tip process zone. This in turn implies that a specimen geometry independent fracture toughness (i.e., a material property) can only be measured in specimens with large numbers of unit cells. We report a fracture mechanism map that reveals that measurements of specimen geometry independent fracture toughness can only be made in specimens with more than ∼10 million unit cells. This map is also used to rationalise the measured rising R-curves in the 3D specimens with ∼ 1 million unit cells. We end with a discussion on the implication of these findings for the laboratory measurement of the fracture toughness of both architected materials as well as biological cellular materials such as bone.

Propagation rate transients in J-controlled fatigue characterization of adhesives

Fatigue analyses of adhesively bonded composite laminates were conducted to investigate discrepancies in experimentally measured rates of crack propagation in double cantilever beam specimens subjected to increasing or decreasing load histories. A cohesive fatigue damage model that accounts for the quasi-static traction-separation response and bridging was used in a finite element model of the test to identify transient effects caused by the evolution of the process zone in the adhesive, as well as the long-range, low-intensity bridging induced by the knit supporting carrier inside the adhesive. In addition to the quasi-static cohesive properties, the fatigue model requires two parameters that were obtained by fitting the rates of propagation at two experimental data points. The results indicate that the analyses accurately reproduce transient and steady-state rates of propagation observed in the experimental results, as well as the threshold of propagation. Transient rates of propagation consistently exceed those in steady state, which underscores the need to understand transients and account for them in fatigue characterization. The results also highlight that a Paris curve for steady-state propagation can be obtained most quickly using a decreasing load function, but that this curve is an unconservative bound to the rates of propagation. A lateral shift in the Paris curve is proposed for a conservative characterization that removes the effect of bridging and other R-curve effects.

Numerical Investigation of the R-Curve Effect in Delamination of Composite Materials Using Cohesive Elements

This paper presents a numerical investigation of the R-curve effect in delamination propagation in composite materials. The R-curve effect refers to the phenomenon whereby resistance to crack propagation increases with the advancement of the delamination, due to toughening mechanisms, such as fiber bridging. Numerical models often neglect this effect assuming a constant value of the fracture toughness. A numerical approach based on cohesive elements and on the superposition of two bilinear traction-separation laws is adopted here to accurately predict the R-curve effect in skin-doubler composite specimens subjected to three-point bending tests. The carbon-epoxy material presents two different sensitivities to the fiber bridging phenomenon resulting in two different R-curves. Comparisons with literature experimental data, in terms of load and delaminated area vs. applied displacement, and ultrasonic C-scan images show the effectiveness of the adopted approach in simulating the R-curve effect. The predicted numerical stiffness aligns with the experimental scatter, although the maximum load is slightly underestimated by approximately 15% compared with the average experimental results. The numerical model accurately predict the R-curve effect observed in the experimental data, demonstrating a 31% increase in the maximum load for the material configuration exhibiting greater sensitivity to fiber bridging.

Experimental characterization and cyclic cohesive zone modeling of mode I delamination growth in glass/epoxy composite laminates under high cycle fatigue

Large-scale fiber bridging makes a significant impact on fatigue delamination growth (FDG) in fiber-reinforced polymer composites. As bilinear cyclic cohesive zone modeling (CCZM) is not a promising modeling tool in this case, another solution is needed. The present study aims to evaluate a new easily-calibrating trilinear CCZM framework for modeling the FDG in glass/epoxy double cantilever beam (DCB) laminates with large-scale fiber bridging. First, mode I delamination growth, characterized by a significant R-curve effect, is experimentally determined under both quasi-static and high cycle fatigue regimes. Next, trilinear forms of the Turon et al. and the Kawashita-Hallett damage models are developed to predict such fatigue delamination behavior. Results show that the accuracy and efficiency of the proposed trilinear CCZM framework are enhanced by increasing the bridging length of the composite structure. Additionally, the Kawashita-Hallett trilinear model yields the most consistent and precise predictions with the least computational time.

Numerical modelling of the mode I fracture behavior in composite laminates with significant R-curve effect

Significant R-curve effect is very common in the mode I fracture of composite laminates. Accurate simulation of fracture with R-curve effect is important for damage tolerance design of composite structures. However, traditional numerical methods usually do not consider the R-curve effect. In this work, four methods, including three phenomenological methods and cohesive zone model with multi-linear constitutive relation are presented. The three phenomenological methods are based on bilinear cohesive zone model, virtual crack closure technique and extended finite element method respectively with the consideration of R-curve. It is interesting to know which method is more suitable for modelling delamination growth with significant R-curve effect. To achieve this goal, double cantilever beam specimen is modeled and delamination growth is simulated using above four methods. Comparisons between the numerical results and the experimental results of laminates with different material types and stack sequences are made for these methods. Finally, the accuracy, computational efficiency and parameter acquisition of these methods are discussed with the goal of finding a suitable method for delamination modelling in laminated composites. It is shown that the cohesive zone model with multi-linear constitutive relation shows good accuracy in all studied cases.

Combined experimental–numerical mode I fracture characterization of the pultruded composite bars

In this paper, pultruded GFRP bars are investigated to determine their fracture properties. The double cantilever beam test (DCB) is used to assess fracture behavior under mode I loading conditions. However, due to the presence of the R-curve effect (variable fracture energy dependent on the length of the crack), it is necessary to introduce a nonstandard approach to determine fracture properties. The mixed experimental–numerical approach is proposed to deal with this issue. Numerical simulations were carried out in Simulia Abaqus, and with Python scripting it was possible to generate models and obtain R-curve for the material. The numerical model built based on the experimental results has very good agreement with it (force–displacement and delamination length–time characteristics) which allows the use of the mentioned model in the analysis of more complex structures. Acoustic emission analysis was introduced as an auxiliary technique. The delamination obtained from both the numerical model and the experiment complies with the registered acoustic emission events. The proposed method can be used in preparing a material model for other composite materials, which display the presence of the R-curve effect.

Characterization and analysis of the mode I interlaminar fatigue behaviour of thermoplastic composites considering [formula omitted]-curve effects

Through the application of innovative production processes, thermoplastic composites might help the aviation industry become more sustainable. However, there is currently not much experimental understanding on the fatigue behaviour, and validated analysis methodologies on thermoplastic composites are rather limited. In this work, the fatigue onset and propagation interlaminar properties of AS4D/PEKK-FC thermoplastic composite were characterized under mode I loading. Testing costs were reduced by using a multi-fatigue testing rig that allows loading 6 specimens at the same time. The offset in the fatigue crack growth rate curves tested at different severities was explained by the fatigue R-curve effects. The offset was modelled by means of superposed fatigue cohesive laws. Different approaches on how to consider fatigue damage were compared with experimental data, giving an insight on how fatigue damage evolves, and developing a novel modelling strategy based on a robust method for fatigue model parameter identification for fatigue delamination when several failure mechanisms interact.

A step toward bio-inspired dental composites

This feasibility study aimed to develop a new composite material of aligned glass flakes in a polymer resin matrix inspired by the biological composite nacre. The experimental composite was processed by an adapted method of pressing a glass flake and resin monomer system. By pressing and allowing the excess monomer to flow out, the long axis of the flakes was aligned. The resultant anisotropic composite with silanized and non-silanized glass flakes were subjected to fracture toughness tests. We observed increasing fracture toughness with increasing crack extension (Δa) known as resistance curve (R-curve) behavior. Silanized specimens had higher stress intensity KR-Δa over non-silanized specimens, whereas non-silanized specimens had a much lower Young’s modulus, and higher nonlinear plastic-elastic JR-Δa R-curve. In comparison with conventional composites, flake-reinforced composites can sustain continued crack growth for more significant extensions. The primary toughening mechanism seen in flake-reinforced composites was crack deviation and crack branching. We produced an anisotropic model of glass flake-reinforced composite showing elevated toughening potential and a prominent R-curve effect. The feasibility of flake reinforcement of dental composites has been shown using a relatively efficient method. The use of a biomimetic, nacre-inspired reinforcement concept might guide further research toward improvement of dental restorative materials.

Increasing crack growth resistance for through-thickness matrix cracking and its role in suppressing ply cracking in thin-ply laminates

A new analysis is proposed, and verified, to quantitively explain the recent experimental observations of a transition from unstable to slow, stable through-thickness crack growth in cross-ply laminates, as the ply thickness is decreased to 40 µm. The approach is prompted by the observed stable rise of crack opening under increasing applied strain in micromechanical model simulations. Estimates of the energy release rates associated with these crack opening profiles clearly indicate an increasing value of crack growth resistance with crack extension, i.e. an R-curve effect. To verify that the thickness dependence of the matrix crack growth behaviour can be attributed to an R-curve effect, the R-curves were independently determined by micromechanical modelling and employed to predict the progression of through-thickness cracking in constrained plies of various thicknesses. These predictions are shown to agree closely with the reported experimental observations. The benefit of this new approach is that only one simulation is required to generate the R-curve, instead of requiring separate simulations for each ply thickness, thus considerably reducing the computational burden. This is particularly valuable for parametric studies to investigate the dependence on various material properties. As an illustrative example, the effects of changing the fracture energy of the epoxy matrix and the volume fraction of fibres present in the composite were investigated, and simple linear relationships were identified for both the steady-state value of the crack-growth resistance and the crack extension required to reach the steady state. Such relationships can further reduce the computational cost of micromechanical modelling, especially when the relevant material properties may not be available from direct measurements but can be reasonably estimated, as for cryogenic applications.

Characterization and analysis of the interlaminar behavior of thermoplastic composites considering fiber bridging and R-curve effects

Thermoplastic composites can enable the development of new manufacturing techniques to make the aviation industry more sustainable while at the same time greatly benefit cost-efficient and high-volume production. One of the thermoplastic composite materials that can enable this transition is AS4D/PEKK-FC. In this work, the interlaminar properties of AS4D/PEKK-FC thermoplastic composite are characterized and analyzed by means of Mode I, II and Mixed Mode I/II at 50:50 tests, while considering fiber bridging and R-curve effects. In order to achieve stable crack propagation the test configurations are adjusted to account for the large fracture process zone ahead of the crack tip and an appropriate data reduction method is selected. The experimental data is reduced using an inverse methodology to extract cohesive laws based on only the load–displacement curves. Additionally, the use of this methodology provides new insights into the validity of two different mode II tests and the influence of fiber bridging on the mixed-mode interlaminar behavior. The interlaminar damage mechanisms are investigated by means of scanning electron microscopy. The resulting cohesive laws are implemented in commercial finite element software in tabular form, without the need for user-subroutines. All experimental test configurations are analyzed using a single material card and it is shown that fiber bridging and R-curve effects are well captured.

Determination of fibre tension fracture toughness of composite laminates at high loading rate

The dynamic fracture toughness of composite laminates in fibre tension failure mode is characterized here with compact tension (CT) sample, at loading rate of 25 m/s using bi-directional electromagnetic Hopkinson bars. Digital image correlation (DIC) with high-speed imaging is employed for obtaining the full-field strain fields. A dynamic J-integral method is proposed to analyse the dynamic fracture toughness, the kinetic energy and inertia effects involved in dynamic loading and crack propagation process has been analysed and calibrated. The fracture toughness of fibre tension failure at high loading rate is found to be about 25% lower than that in quasi-static load regime, and the R-curve effect under high loading rate is not significant than that in quasi-static load regime. It is suggested from X-ray CT and SEM images that the fibre pull-out length and size of failure process zone is reduced with increase in loading rate.

An R-curve effect-included delamination growth criterion for mixed-mode I/II delamination predictions of composite laminates

Most existing mixed-mode I/II delamination growth criteria are purely empirical or semi-empirical fitting functions of the experimental fracture toughness, which limit their wide range applications. In this work, a novel delamination growth criterion is proposed for mixed-mode I/II delamination predictions considering the significant R-curve effects in composite laminates. Its adequacy and accuracy are validated by the available experimental data for two different composite materials from literature. Both the trend and the magnitude of the predicted failure loci show fairly good agreements with the experimental ones. Only two experimentally obtained parameters, i.e. the pure mode I and mode II fracture toughness curves are required in the proposed criterion, which makes the proposed criterion easy to use. The proposed criterion can be applied for the evaluation of the mixed-mode I/II delamination growth behavior and thus the prediction of the structural strength, which may have important theoretical significance and engineering guidance value for the damage-tolerant design of composite materials and structures.

Holistic determination of physical fracture toughness values and numerical parameters for delamination analysis considering multidirectional-interfaces

Delaminations are a concern for the structural integrity of fibre composite structures. The decisive material parameter for delaminations is fracture toughness. Structures usually have a multidirectional ply stacking. However, interface orientation dependant fracture toughness values and R-curve effects are neglected in numerical delamination analysis. This work investigates interface orientation specific fracture toughness values and considers R-curve effects in mode I to improve simulation accuracy. Therefore, mode I, mode II and mixed mode fracture toughness values of different interface ply orientations are determined experimentally and verified using numerical analysis of the characterisation specimens with cohesive zone modelling. In this way, an engineering methodology is provided for the experimental characterisation and comprehensive numerical modelling of delaminations in mesoscale progressive damage analysis of multidirectional composite structures. In addition, a full parameter set for the simulation of four different interfaces of M21-T700GC prepreg material is given. It can be shown that the use of standard 0°//0°-values leads to very conservative results. The use of interface specific values increases the accuracy. Ultrasonic scans of the DCB specimens are used to compare the crack front shapes for validation. Not only the load displacement curves of the characterisation specimens are well captured, but also the crack front shapes. This demonstrates that by smearing the microscale effects, the material behaviour can be captured phenomenologically correct by mesoscale modelling suitable for industrial use.

Resistance-curve envelopes for dental lithium disilicate glass-ceramics

Recent investigations on the mechanics of dental lithium disilicates have gathered mounting evidences of a probable R-curve behavior. Here we use an indirect approach to obtain the contours of the R-curve using the relationship between strength and initial crack size obtained from quasi-static fracture toughness tests. The tangent condition formed by the multiple KI,appl (ai,j) curves give a smoothened polygonal envelope for the R-curves. The R-curves obtained this way were extended toward an initial value at the onset of stable crack extension, obtained hereusing a microcantilever specimen featuring a sharp notch produced by focused-ion-beam milling. The presence of an R-curve effect was confirmed for the three evaluated lithium disilicates, being more pronounced for the injection-molded material having an oriented elongated Li2Si2O5 microstructural architecture, with crystal fraction playing an important role for low-aspect ratio randomly-oriented microstructures.

Initial R-curves for trans-laminar fracture of quasi-isotropic carbon/epoxy laminates from specimens with increasing size

In this paper, a set of initial R-curves were measured for three different in-plane sizes using scaled Over-height Compact Tension (OCT) specimens and two different stacking sequences. The data reduction scheme was based on a Virtual Crack Closure Technique which is informed by the measured failure load and effective crack length. The R-curve effects were studied by using X-ray Computed Tomography to measure the crack length throughout fracture propagation. It was found that the initial R-curves measured from all three specimen sizes follow the same linearly increasing trend, with similar results for both stacking sequences. However, no plateau was seen on any of the measured R-curves, implying that even the largest specimens are not large enough to generate a full R-curve.

Three-parameter (3P) weibull distribution for characterization of strength of ceramics showing R-Curve behavior

Strength distribution of advanced ceramics is commonly characterized by two-Parameter (2P) Weibull distribution. However, deviation of strength distribution from 2P-Weibull distribution may occur in ceramics due to various mechanisms. R-curve behavior is one of these mechanisms where increase of fracture resistance with the extension of crack occurs. In such cases, 2P-Weibull distribution may not be the best fitting distribution function based on the goodness-of-fitness tests. This article examines the effectiveness of three-parameter (3P) Weibull distribution function for fitting the strength variation due to R-curve effect by using experimental and virtual strength data. The effect of Weibull parameters, degree of increase in crack resistance and number of samples on effectiveness of fitting via 3P-Weibull distribution is investigated. It is reported that 3P-Weibull distribution function fits the strength distribution better than 2P-Weibull distribution function for materials showing R-curve behavior when the crack resistance curve is steep and Weibull modulus is high. Furthermore, it is shown that at least 100 samples should be used for a reliable estimate when the material exhibits R-curve behavior.

Modeling the R-curve effects in laminate composites using the interface thick level set method

This paper presents a new model for simulating delamination growth in composites in presence of the resistance curve (R-curve) effects. This model is extended from the interface thick level set (ITLS) method which is developed recently. In this approach the common definition of damage in the constitutive relationship of the interface elements is replaced with a new damage definition based on the thick level set (TLS) method. This new damage definition provides a direct link between fracture and damage mechanics that allows use of interface elements without a predefined cohesive law. In this paper the ITLS method is adopted to incorporate the R-curve effects caused by extended bridging, in modeling delamination. The fracture energy in the developed model is defined as a function of the crack length which allows simple use of experimentally measured R-curve as a direct input of this model. This is in contrast with the popular cohesive models that need to develop some mathematical relationships to extract the cohesive law parameters from experimental R-curve. The proposed 3D model is validated against experimental data obtained from literature to support the accuracy of obtained results.

A simple procedure for the determination of the cohesive law in 4-ENF test with consideration of the friction and R-curve effect

The measurement of initiation and propagation fracture toughness is important for characterizing the delamination behavior of composite laminates, especially in the case with distinct R-curve behavior which can be caused by fiber bridging, shear deformation of matrix, etc. Four-Point End Notched Flexure (4-ENF) is one of the widely adopted tests to measure the mode II toughness values, while the friction between the crack surfaces may have a significant impact on the results. In this paper, a simple procedure to determine the cohesive law in 4-ENF test with consideration of the friction and R-curve effect is proposed. Firstly, determining the equivalent crack advance according to the compliance variation of the specimen is performed, without the need for monitoring the crack length during the test. Next, the Energy Release Rate in mode II (GII) and crack sliding displacement (δt) are derived based on the point-friction assumption and beam theory. Finally, the cohesive law is obtained via the J-integral approach. To obtain suitable values of the penalty stiffness and interface strength, the influence of these parameters on the prediction of 4-ENF test is studied. Comparisons of the numerical results and the experimental data show a good correlation. The effect of variation of friction coefficients on derived cohesive stress and initiate energy release rate is discussed as well.