A discrete constitutive model for transverse and shear damage of symmetric laminates with arbitrary stacking sequence

A damage constitutive model in conjunction with a 2-D finite element discretization is presented for predicting onset and evolution of matrix cracking and subsequent stiffness reduction of symmetric composite laminates with arbitrary stacking sequence subjected to membrane loads. The formulation uses laminae crack densities as the only state variables, with crack growth driven by both mechanical stress and residual stress due to thermal expansion. The formulation is based on fracture mechanics in terms of basic materials properties, lamina moduli, and critical strain energy release rates G_IC and G_IIC, only. No additional adjustable parameters are needed to predict the damage evolution. Spurious strain localization and mesh size dependence are intrinsically absent in this formulation. Thus, there is no need to define a characteristic length. Comparison of model results to experimental data is presented for various laminate stacking sequences. Prediction of crack initiation, evolution, and stiffness degradation compare very well to experimental data.     

In‐plane progressive matrix cracking analysis of symmetric cross‐ply laminates with holes

Most of the previously performed damage analyses in composite laminates have been restricted to model the plain laminates without geometry discontinuities. In this study, a micromechanical damage model is combined with the finite element formulation and is implemented in the integration points to perform progressive damage analyses of composite laminates. A micromechanical damage model based on the stress transfer method is used to find the degradation of mechanical properties of composite laminates. Crack density is also used as an only state variable representing the damage in each Gauss point of every layer of the laminate. The strain energy and critical energy release rate criterion is also used to predict the damage initiation and evolution in each layer. A finite element discretization is used in conjunction with the user element definition capability of ABAQUS commercial software. To verify the developed procedure, a single element is analysed, and the obtained results are compared with available results in the literature. Progressive damage analyses are also performed for several symmetric cross‐ply laminates with and without geometry discontinuity subjected to matrix cracking damage mechanism under in‐plane loading conditions. The obtained mechanical response and variations of matrix crack density versus the applied load are also discussed.     

Modelling off-axis ply matrix cracking in continuous fibre-reinforced polymer matrix composite laminates

The fracture process of composite laminates subjected to static or fatigue tensile loading involves sequential accumulation of intra- and interlaminar damage, in the form of transverse cracking, splitting and delamination, prior to catastrophic failure. Matrix cracking parallel to the fibres in the off-axis plies is the first damage mode observed. Since a damaged lamina within the laminate retains certain amount of its load-carrying capacity, it is important to predict accurately the stiffness properties of the laminate as a function of damage as well as progression of damage with the strain state. In this paper, theoretical modelling of matrix cracking in the off-axis plies of unbalanced symmetric composite laminates subjected to in-plane tensile loading is presented and discussed. A 2-D shear-lag analysis is used to determine ply stresses in a representative segment and the equivalent laminate concept is applied to derive expressions for Mode I, Mode II and the total strain energy release rate associated with off-axis ply cracking. Dependence of the degraded stiffness properties and strain energy release rates on the crack density and ply orientation angle is examined for glass/epoxy laminates. Suitability of a mixed mode fracture criterion to predict the cracking onset strain is also discussed.     

Time-dependent damage in carbon fibre-reinforced polymer composites

Time-dependent damage (matrix cracks) evolution in AS4/3501-6 cross-ply laminates was studied using constant strain rate and constant stress tests. First ply failure stress and strain as well as the matrix crack density at a given stress level were found to be strongly dependent on strain rate. Matrix crack density increased with creep time at a constant stress level. The compliance and creep rate of the laminate increased in the presence of these cracks. These results emphasize the importance of the knowledge of time-dependent damage evolution in a lamina/laminate of a polymer composite for reliable prediction of creep and creep rupture.     

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.     

Stiffness and fracture analysis of laminated composites with off-axis ply matrix cracking

Matrix cracks parallel to the fibres in the off-axis plies is the first intralaminar damage mode observed in laminated composites subjected to static or fatigue in-plane tensile loading. They reduce laminate stiffness and strength and trigger development of other damage modes, such as delaminations. This paper is concerned with theoretical modelling of unbalanced symmetric laminates with off-axis ply cracks. Closed-form analytical expressions are derived for Mode I, Mode II and the total strain energy release rates associated with off-axis ply cracking in [0/θ]_s laminates. Stiffness reduction due to matrix cracking is also predicted analytically using the Equivalent Constraint Model (ECM) of the damaged laminate. Dependence of the degraded stiffness properties and strain energy release rates on the crack density and ply orientation angle is examined for glass/epoxy and carbon/epoxy laminates. Suitability of a mixed mode fracture criterion to predict the cracking onset strain is also discussed.     

A mechanistic model for transverse damage initiation,evolution, and stiffness reduction in laminated composites

A constitutive model to predict stiffness reduction due to transverse matrix cracking is derived for laminae with arbitrary orientation, subject to in-plane stress, embedded in laminates with symmetric but otherwise arbitrary laminate stacking sequence. The moduli of the damaged laminate are a function of the crack densities in the damaging laminae, which are analyzed one by one. The evolution of crack density in each lamina is derived in terms of the calculated strain energy release rate and predicted as function of the applied load using a fracture mechanics approach. Unlike plasticity-inspired formulations, the proposed model does not postulate damage evolution functions and thus there is no need for additional experimental data to adjust material parameters. All that it is needed are the elastic moduli and critical energy release rates for the laminae. The reduction of lamina stiffness is an integral part of the model, allowing for stress redistribution among laminae. Comparisons with experimental data and some results from the literature are presented.     

缝合复合材料层板低速冲击损伤数值模拟

建立了缝合复合材料层板在低速冲击载荷下的渐进损伤分析模型。模型中采用空间杆单元模拟缝线的作用;采用三维实体单元模拟缝合层板,通过基于应变描述的Hashin准则,结合相应的材料性能退化方案模拟层板的损伤和演化;采用界面单元模拟层间界面,结合传统的应力失效判据和断裂力学中的应变能释放率准则判断分层的起始和扩展规律。通过对碳800环氧树脂复合材料（T800/5228）层板的数值仿真结果和试验结果相比较,验证了模型的正确性,同时讨论了不同冲击能量下缝合层板的损伤规律。研究结果表明：缝线能够有效地抑制层板的分层损伤扩展;相同冲击能量下缝合与未缝合层板的基体损伤和纤维损伤在厚度分布上相似,缝合层板的损伤都要小于未缝合层板。     

基于应变能耗散的复合材料层合板面内缺口强度分析CDM模型

基于伴随能量释放的渐进损伤演化思想,建立了复合材料层合板面内失效分析的连续介质损伤力学（CDM）分析模型,该模型包含损伤表征、损伤起始判定和损伤演化法则3个方面。基于CDM模型,通过引入损伤状态变量表征损伤,建立了平面应力状态下的材料损伤本构模型。采用损伤参量 f_E改写Hashin准则,以判定损伤的起始。损伤演化由特征长度内的应变能释放密度控制,建立了损伤状态变量关于等效应变的渐进损伤演化法则。模型中还同时考虑了面内剪切非线性和网格敏感性,并进行了对比分析。对含缺口的[90/0/±45]_3s和[（±θ）₄]_s 2类典型复合材料层合板的面内拉伸失效进行了分析,结果表明,本文中的模型能有效预测复合材料层合板的面内拉伸强度。     

Damage evolution and characterisation of crack types in CF/EP laminates loaded at low temperatures

Tensile testing of CF/EP AS4/8552 cross-ply laminates at room (RT) and cryogenic (around −150 °C) temperatures has been performed to study the effect of temperature on damage (intralaminar cracking) evolution. Microscopy studies of the specimen edges showed a significant difference in damage pattern for the two different temperatures. At the low temperature (LT), more complex crack types were obtained that could not be found in specimens tested at the RT. The effect of these crack types on the laminate tensile modulus was studied by FEM. In analytical stiffness modelling complex shape crack was replaced by an “effective” normal (straight) crack with an “effective” crack opening displacement (COD) that leads to the same reduction in laminate stiffness. A crack efficiency factor was introduced to characterize the significance of complex crack shapes for stiffness reduction. The reduction of tensile modulus for a laminate damaged at low temperature was measured and compared with model predictions.     

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.     

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.     

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.     

Engineering expressions for thermo-elastic constants of laminates with high density of transverse cracks

Thermo-elastic constants of symmetric and balanced laminates with intralaminar cracks in 90-layers depend on the opening displacement (COD) of the crack. The COD dependence on the interaction between cracks in the same layer is studied using FEM. The COD dependence on crack density is described by interaction function in form of tanh(). This interaction function multiplied with COD of non-interactive crack is the input parameter in analytical model for thermo-elastic properties of damaged symmetric and balanced laminates. Predictions performed for cross-ply laminates with cracks in inside and in surface layers and for quasi-isotropic laminates with different position of the 90-layer are in a very good agreement with direct FEM calculations.     

A variational method for analysis of laminates with parallel arrays of intralaminar cracks

This paper presents a robust variational method for prediction of the effective thermoelastic properties of laminates with parallel, but not necessarily coplanar intralaminar cracks. Arbitrary laminate layup, crack patterns, and any in-plane forces and bending moments, and temperature change can be treated simultaneously. The method is based on the principle of minimum complementary energy. The admissible in-plane stress components are assumed to be linear through the ply thickness. Simple matrix expressions are derived for the effective compliance matrix, thermal expansions and curvatures, and specific heat of a cracked laminate. Predictions for the Young's modulus, Poisson's ratio, shear modulus, flexural rigidity, and thermal expansion as functions of crack density for various laminates of symmetric and nonsymmetric layups demonstrated excellent agreement with experimental results of seven independently published studies. Accuracy and robustness of the method are complemented by the fact that no experimentally fitted parameters are required. It is shown that the same formalism can be applied to analysis of closed intralaminar cracks, as well as nonuniform and nonsymmetrical distributions of cracks.     

Numerical analysis of influence factors on low-velocity impact damage of stitched composite laminates

Abstract

A 3D dynamic finite-element model is proposed in this paper to simulate the damage development process of stitched laminates subjected to low-velocity impact. The strain-based Hashin criteria and a sudden degradation scheme are employed to determine the intra-laminar damage initiation and evolution; a mix-mode bilinear constitutive model is adopted to evaluate the inter-laminar delamination damage. The predicted numerical results agree well with the available experimental data, thus validating the proposed damage analysis model. Moreover, the influence factors, including the thickness of laminates, stitching density, diameter of stitching thread and strength of stitching thread, are analyzed and discussed in detail.     

Numerical Modeling of Combined Matrix Cracking and Delamination in Composite Laminates Using Cohesive Elements

Sub-laminate damage in the form of matrix cracking and delamination was simulated by using interface cohesive elements in the finite element (FE) software ABAQUS. Interface cohesive elements were inserted parallel to the fiber orientation in the transverse ply with equal spacing (matrix cracking) and between the interfaces (delamination). Matrix cracking initiation in the cohesive elements was based on stress traction separation laws and propagated under mixed-mode loading. We expanded the work of Shi et al. (Appl. Compos. Mater. 21, 57–70 2014) to include delamination and simulated additional [45/?45/0/90]_s and [0₂/90_n]_s {n?=?1,2,3} CFRP laminates and a [0/90₃]_s GFRP laminate. Delamination damage was quantified numerically in terms of damage dissipative energy. We observed that transverse matrix cracks can propagate to the ply interface and initiate delamination. We also observed for [0/90_n/0] laminates that as the number of 90° ply increases past n?=?2, the crack density decreases. The predicted crack density evolution compared well with experimental results and the equivalent constraint model (ECM) theory. Empirical relationships were established between crack density and applied stress by linear curve fitting. The reduction of laminate elastic modulus due to cracking was also computed numerically and it is in accordance with reported experimental measurements.     

Off-axis fatigue cracking behaviour in notched fibre metal laminates

The off‐axis fatigue cracking behaviour of notched fibre metal laminates under constant amplitude loading conditions was investigated experimentally and numerically. It was found that the off‐axis fatigue crack initiation life decreased as the off‐axis angles increased. This indicated that the off‐axis laminates raised the applied stress level in the aluminium (Al) layer and subsequently resulted in earlier cracking in the Al layer. The off‐axis fatigue crack initiation lives of notched fibre metal laminates were predicted using lamination theory and an energy‐based critical plane fatigue damage analysis from the literature. After a crack initiated in the Al layer, it was observed that the crack path angles of the off‐axis specimens were neither perpendicular to the fibre nor to the loading direction. A finite‐element model was established for predicting the crack path angles.     

Analysis of a buffer strip laminate with fiber and matrix damage and interlaminar debonding

A shear lag solution for a hybrid buffer strip laminate containing initial damage in the form of a rectilinear notch, matrix splitting and interlaminar debonding is presented. The model is a unidirectional monolayer with two symmetric constraint layers that represent angle plies. The intent of the analysis is to estimate the remote strain required to propagate the initial damage and/or to fail the laminate catastrophically. The analytical solution has a set of integral equations in which material and geometric parameters appear explicitly. Some typical results are presented for a graphite/epoxy panel having either high strength and low modulus or low strength and low modulus buffer strips. Matrix damage, angle plies, and interlaminar debonding are shown to affect the damage tolerance capability of buffer strip laminates.     

复合材料层合板基体裂纹的协同损伤演化模型

首先,为研究复合材料层合板在准静态载荷下的基体裂纹演化特征,提出了一个基于能量的协同损伤演化模型。然后,通过模型对损伤进行了多尺度分析:从微观角度,采用三维有限元方法求得裂纹表面位移;从宏观角度,结合裂纹表面位移,推导了萌生基体裂纹的能量释放率。最后,根据裂纹萌生准则对基体裂纹的演化过程进行预测。模型考虑了演化过程中损伤的相互影响、残余应力、基体材料非线性、材料初始损伤分布及损伤演化的不均匀性。根据演化分析流程计算了[±θ/904]s铺层玻璃纤维复合材料的基体裂纹演化过程。结果表明:这一模型能够准确地预测准静态载荷下复合材料层合板基体裂纹的损伤演化规律。