A probabilistic approach for transverse crack evolution in a composite laminate under variable amplitude cyclic loading

This paper presents a theoretical approach for predicting transverse cracking behavior in a cross-ply laminate with a thick transverse ply under variable amplitude loads for which the cracks grow instantaneously, or very quickly, across the specimen width. The transverse crack density was derived on the basis of the slow crack growth (SCG) concept using the Paris law in conjunction with the Weibull distribution for a brittle material subjected to multi-stage cyclic loading. A fracture criterion obtained was related with the empirical rules by Miner and Broutman & Sahu. Next, the probabilistic SCG model was applied to transverse cracking in a cross-ply laminate under multi-stage cyclic loading. The two-stage fatigue tests with various loading sequences and amplitudes were conducted for carbon fibre reinforced plastic (CFRP) cross-ply laminates in addition to single-stage fatigue tests for various maximum stresses. The experiment results were compared with the predictions to verify the validity of the model.     

Estimation of dynamic fatigue strengths in brittle materials under a wide range of stress rates

This paper aims to statistically estimate the dynamic fatigue strength in brittle materials under a wide range of stress rates. First, two probabilistic models were derived on the basis of the slow crack growth (SCG) concept in conjunction with two-parameter Weibull distribution. The first model, Model I, is a conventional probabilistic delayed-fracture model based on a concept wherein the length of the critical crack growth due to SCG is enough larger than the initial crack length. For the second model, Model II, a new probabilistic model is derived on the basis of a concept wherein the critical cracks have widely ranging lengths. Next, a four-point bending test using a wide range of stress rates was performed for soda glass and alumina ceramics. We constructed fracture probability–strength–time diagrams (F–S–T diagrams) with the experimental results of both materials using both models. The F–S–T diagrams described using Model II were in good agreement with plots of the fracture strength and the fracture time of both materials more so than Model I.     

Prediction for progression of transverse cracking in CFRP cross-ply laminates using Monte Carlo method

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.     

Simulation of fatigue performance of cross-ply composite laminates

The fatigue life of cross-ply composite laminates was evaluated using a statistical model. A modified shear-lag analysis was applied to describe the cycle-number-dependent stiffness reduction and consequent stress redistribution processes in the laminates resulted from both progressive transverse matrix cracking in transverse plies and local delamination at tips of transverse cracks. From the strength degradation behaviour and the static strength distribution of 0° plies as well as the fatigue behaviour of 90° plies, the fatigue life of cross-ply laminates with various types of lay-up can be simulated from the model. Predictions of fatigue performance are compared with experimental data for [0/90₂] _s , [02/90₂] _s and [0₂/90₄] _s graphite/epoxy cross-ply laminates: good agreements are obtained.     

The behaviour of Kevlar fibre-epoxy laminates under static and fatigue loading—Part 2: Modelling

A model has been developed for the modulus reduction of cross-ply Kevlar laminates under static loading as a function of applied strain. The effects of strain-rate and temperature have also been considered. The ‘stiffening’ of Kevlar fibres and Kevlar fibre-epoxy (KFRP) laminates under creep or fatigue conditions has been modelled using a kinetic approach. This has enabled stiffening effects to be subtracted out of the residual modulus-with-cycles behaviour of cross-ply KFRP laminates under fatigue loading, leaving a modulus-reduction-with-cycles curve which reflects the damage due to matrix cracking. The analyses compare well with experimental data reported in Part 1.     

Microscopic fatigue damage progress in CFRP cross-ply laminates

Damage progress in toughened-type carbon fibre-reinforced plastic (CFRP) cross-ply laminates under tensile fatigue loading was measured using the replica technique. The laminate configuration was [0/90_m/0], where m = 4, 8 and 12. The damage parameters, transverse crack density and delamination ratio, were determined. A power-law model was proposed, relating the cyclic strain range and the number of cycles at transverse crack initiation. Based on experimental data, a simple shear-lag analysis combined with the modified Paris law was conducted to model the transverse crack multiplication. An extension of the shearlag analysis for laminates containing delaminations initiating from the tips of the transverse cracks was used to conduct a modified Paris law analysis for delamination growth.     

Specificity of matrix cracking development in CFRP laminates under mechanical or thermal loadings

Fatigue tests at room temperature and thermal cycling experiments have been performed on carbon/epoxy laminates of cross-ply and complex stacking sequences. An ‘equivalent’ fatigue loading level has been evaluated in order to impose on the 90° plies the same amplitude of transverse stress σ₂₂ than in thermal cycling. A comparison of the matrix crack development throughout both types of tests has been undertaken: they have been found analogous, but with very far much faster kinetics under thermal cycling. Moreover, the fatigue test frequency has a significant influence on crack onset and development. However, it seems that the parameters ‘time’ and ‘transverse ply stress amplitude’ are not sufficient to completely explain the very fast cracking kinetics observed under thermal cycling.     

A computational continuum damage mechanics model for predicting transverse cracking and splitting evolution in open hole cross‐ply composite laminates

The present study focuses on a computational constitutive model which predicts the matrix cracking evolution and fibre breakage in cross‐ply composite laminates with open hole under in‐plane loading. To consider the effects of matrix cracking on the nonlinear response of laminates, a simplified crack density based model is applied which evaluates the representative damage parameters of matrix cracking. Furthermore, a developed subroutine based on continuum damage mechanics concepts is applied in ANSYS code which is capable to consider the transverse cracking/splitting evolution and predict the final failure load of mentioned laminate under monotonic loading in a progressive damage analyses. It is shown that the obtained stress–strain behaviours and the damage evaluation of considered laminates are in good agreement with the available experimental results.     

Fatigue damage mechanisms and residual properties of graphite/epoxy laminates

Damage mechanisms and accumulation, and associated stiffness and residual strength reductions were studied in cross-ply graphite/epoxy laminates under cyclic tensile loading. Stress-life data were fitted by a two-parameter wearout model and by a second-degree polynomial on a log-log scale. The fatigue sensitivity is highest for the unidirectional laminates and it decreases for the crossply laminates with increasing number of contiguous 90° plies. Five different damage mechanisms were observed: transverse matrix cracking, dispersed longitudinal cracking, localized longitudinal cracking, delaminations along transverse cracks, and local delaminations at the intersection of longitudinal and transverse cracks. Failure patterns vary with cyclic stress level and number of cycles to failure. Under monotonie loading, failure is brittle-like and concentrated. At high stress amplitudes and short fatigue lives failure results from few localized flaws, whereas at lower stress amplitudes and longer fatigue lives failure results from more dispersed flaws. The residual modulus shows a sharp reduction initially, followed by a more gradual decrease up to failure. The residual strength showed a sharp reduction initially, followed by a plateau or even some increase in the middle part of the fatigue life, and a rapid decrease in the last part of the fatigue life. A tentative cumulative damage model is proposed based on residual strength and the concept of equal damage curves.     

Evaluation of long-term durability in high temperature resistant CFRP laminates under thermal fatigue loading

Thermal fatigue tests were conducted on high temperature resistant carbon fiber reinforced plastics cross-ply laminates to evaluate microscopic damage progress which affects macroscopic mechanical behavior of the laminates. Materials system used were thermoplastic polyetheretherketone based, AS4/PEEK and thermoset bismaleimide based, G40-800/5260. Several types of laminate configuration were used to clarify the effect of ply thickness on microscopic damage progress. Microscopic damages were observed using optical microscopy and soft X-ray radiography. Energy release rate associated with transverse cracking was calculated using variational analysis. The modified Paris law was used to predict transverse cracking. From comparison to mechanical fatigue test results, it is clarified that transverse crack accumulation rate was larger under thermal fatigue loading at same energy release rate range due to the dependence of the fracture toughness on temperature.     

A novel statistical model for predicting matrix cracking in high temperature polymer composite laminates

This paper studied the progressive matrix cracking in high temperature polymer composite laminates that could be used for next generation high speed transport airframe structures and aircraft engine components exposed to elevated temperatures. Damage mechanisms of matrix cracking were identified by X-ray radiography at room temperature and in-test photography technique at high temperature. It was found that the non-deterministic scenario is always involved in the procedure of transverse matrix cracking. Monte Carlo simulations using experimentally obtained materials properties were applied to simulate the multiple transverse cracking and compared with the experiment data. Finally, a novel statistical model combining Weibull theory with shear lag model was proposed to predict the matrix cracking based upon the previously obtained probability density function of crack spacing. It is shown that the predictions of this statistical model agree well with the experimental results and can be used to have an in-depth understanding of the random matrix cracking problem in composite laminates.     

On multiple transverse cracking in glass fibre epoxy cross-ply laminates

An investigation has been made of multiple transverse cracking in glass fibre epoxy cross-ply laminates. Four laminates of differing transverse ply thicknesses were investigated. Transverse crack spacing was found to decrease with increasing applied stress and decreasing transverse ply thickness. Very close agreement has been found between the experimental results and a multiple cracking theory based on shear lag analysis in which the plies remain essentially elastically bonded. In these composites a small modulus change is observed at a strain lower than that at which cracking initiated. This phenomenon is associated with a visual, under some circumstances reversible, whitening effect.     

界面脱粘对正交铺设SiC/CAS复合材料基体开裂的影响

采用细观力学方法研究了正交铺设SiC/CAS复合材料在单轴拉伸载荷作用下界面脱粘对基体开裂的影响。采用断裂力学界面脱粘准则确定了0°铺层纤维/基体界面脱粘长度, 结合能量平衡法得到了主裂纹且纤维/基体界面发生脱粘(即模式3)和次裂纹且纤维/基体界面发生脱粘(即模式5)的临界开裂应力, 讨论了纤维/基体界面剪应力、 界面脱粘能对基体开裂应力的影响。结果表明, 模式3和模式5的基体开裂应力随纤维/基体界面剪应力、 界面脱粘能的增加而增加。将这一结果与Chiang考虑界面脱粘对单向纤维增强陶瓷基复合材料初始基体开裂影响的试验研究结果进行对比表明, 该变化趋势与单向SiC增强玻璃陶瓷基复合材料的试验研究结果一致。     

The fatigue damage mechanics of notched carbon fibre/PEEK laminates

A model is presented for the strength, post-fatigue residual strength and damage propagation in notched, cross-ply carbon fibre/polyetheretherketone (PEEK) laminates. Fracture mechanics principles are used to predict quasi-static damage growth, and the application of a Paris law permits extension to fatigue damage. Strength is predicted by applying a failure criterion based on the tensile stress distribution in the 0° plies, as modified by damage (either quasi-static or fatigue). The volume dependence of strength is included by using a simple Weibull distribution. The parameters of the model are determined from independent experiments. Good agreement with experimental results is obtained. Comparisons are made with previous results from carbon fibre/epoxy laminates. The behaviour of the carbon fibre/PEEK is similar, although the extent of delamination and matrix cracking is reduced owing to the higher inherent toughness of the matrix.     

Prediction of initiation of transverse cracks in cross-ply CFRP laminates under fatigue loading by fatigue properties of unidirectional CFRP in 90° direction

A fatigue life to the initiation of transverse cracks in cross-ply carbon fiber-reinforced plastic (CFRP) laminates has been predicted using properties of the fatigue strength of unidirectional CFRP in the 90° direction. In the experiments, unidirectional [90]₁₂ laminates were used to obtain a plot of maximum stress versus number of cycles to breaking, and two types of cross-ply laminates of [0/90₄]_S and [0/90₆]_S were used to evaluate the initiation and multiplication of transverse cracks under fatigue loading. Transverse cracks were studied by optical microscopy and soft X-ray photography. Analytical and experimental results showed good agreement, and the fatigue life for transverse crack initiation in cross-ply laminates was predicted successfully from the fatigue strength properties of the unidirectional CFRP in the 90° direction. The prediction results showed a conservative fatigue life than the experimental results.     

Monte-Carlo simulation of multiple fracture in the transverse ply of cross-ply graphite-epoxy laminates

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 Manderset 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.     

The behaviour of cross-ply hybrid matrix composite laminates: Part 1: Experimental results

Two types of cross-ply laminate have been made from prepreg: (a) hybrid matrix laminates consisting of longitudinal plies of glass fibres in epoxy resin and transverse plies of glass fibres in epoxy resin/urethane elastomer blend; and (b) uniform matrix laminates with the same resin in both the longitudinal and transverse plies. The presence of the urethane in the transverse plies increases the applied strains necessary for the initiation and development of transverse cracking during the extension of both hybrid matrix and uniform matrix laminates. The effect is greater with increasing amounts of urethane. The cracking data, stress/strain behaviour, acoustic emission response and ply thickness effects on crack development are discussed in the light of existing theories concerning transverse cracking.     

Predicting evolution of ply cracks in composite laminates subjected to biaxial loading

An energy-based model is developed to predict the evolution of sub-critical matrix crack density in symmetric multidirectional composite laminates for the case of multiaxial loading. A finite element-based numerical scheme is also developed to evaluate the critical strain energy release rate, G_Ic, associated with matrix micro-cracking, a parameter that previously required fitting with experimental data. Furthermore, the prediction scheme is improved to account for the statistical variation of G_Ic within the material volume by using a two-parameter Weibull distribution. The variation of G_Ic with increasing crack density is also accounted for based on reported experimental evidence. The simulated results for carbon/epoxy and glass/epoxy cross-ply laminates demonstrate the ability of the improved model to predict the evolution of multidirectional ply cracking. By integrating this damage evolution model with the synergistic damage mechanics approach for stiffness degradation, the stress-strain response of the studied laminates is predicted. Finally, biaxial stress envelopes for ply crack initiation and pre-determined stiffness degradation levels are predicted to serve as representative examples of stiffness-based design and failure criterion.     

Statistical model of the transverse ply cracking in cross-ply laminates by strength and fracture toughness based failure criteria

Cross-ply laminate subjected to tensile loading provides a relatively well understood and widely used model system for studying progressive cracking of the transverse ply. This test allows to identify material strength and/or toughness characteristics as well as to establish relation between damage level and the composite stiffness reduction. The transverse ply cracking is an inherently stochastic process due to the random variability of local material properties of the plies. The variability affects both crack initiation (governed by the local strength) and propagation (governed by the local fracture toughness). The primary aim of the present study is elucidation of the relative importance of these phenomena in the fragmentation process at different transverse and longitudinal ply thickness ratios. The effect of the random crack distribution on the mechanical properties reduction of the laminate is also considered. Transverse ply cracking in glass fiber/epoxy cross-ply laminates of the lay-ups [0₂/90₂]_s, [0/90₂]_s, and [0/90₄]_s is studied. Several specimens of each lay-up were subjected to uniaxial quasistatic tension to obtain crack density as a function of applied strain. Crack spacing distributions at the edge of the specimen also were determined at a predefined applied strain. Statistical model of the cracking process is derived, calibrated using crack density vs. strain data, and verified against the measured crack spacing distributions.     

Analysis of stiffness loss in cross-ply composite laminates

The behaviour of laminated composite plates beyond first-ply failure has been the subject of much research work. It is well known that generally, the load-bearing capability of laminated composite plates can remain significant despite the presence of some damage in the plies. Traditionally, the ply-discount method has been used among analysts and designers, although the approach is generally regarded as too conservative. It is therefore desirable to develop models for the prediction of the mechanical properties of damaged composite laminates at various applied loads, and to be able to correlate the changes in properties with the amount of damage and cracking within each constituent ply. Generally, if the models are to be useful as predictive tools, they must be capable of not only sufficiently describing the damage state but also the nature of the damage evolution with loading. This ‘evolution law’ is often obtained through fracture analysis, although it should be noted that the diffused nature of cracks and the multiplicity of failure modes in composites in general greatly complicates the analysis. The problem of transverse matrix cracking in cross-ply laminates under uniaxial tension is considerably simpler because it is essentially dominated by mode I fracture. Thus it is necessarily the first step for any model aiming to predict stiffness losses in composite laminates. In this paper, a constitutive model of the damage state for composite laminates, first proposed by Allen et al., is used with a damage evolution criterion based on strain energy to predict the stiffness loss due to matrix cracking in cross-ply laminated composite plates. Although the constitutive model does not require the determination of many constants, the state of damage is described by a vector of internal state variables (ISV), which contains information on the crack geometry and fracture modes. A series of parametric finite element analyses was performed to determine the effects of relative ply thicknesses, crack density and crack opening profile on the vector of ISVs. A computer algorithm was written for the analysis of cross-ply laminates based on the damage evolution criterion proposed in this work. The results of the analysis compare favourably with experimental measurements of progressive stiffness loss in damaged cross-ply graphite-epoxy laminates obtained from other researchers.