Intralaminar Damage Model for Laminates Subjected to Membrane and Flexural Deformations

Prediction of transverse damage initiation and evolution for not necessarily symmetric laminates under membrane and/or bending loads is the subject of this work. The laminate stiffness reduction is computed via crack opening displacement (COD) methods and the generalization to multiple cracking laminas is made via continuum damage mechanics (CDM) concepts. Using available COD solutions combined with homogenization techniques leads to an analytical constitutive model capable of predicting the initiation and evolution of crack density versus applied strain, as well as laminate modulus degradation, not only for symmetric laminates subjected to membrane deformation but also for general laminates subjected to flexural deformations as well. To adjust the model parameters, experimental data is required in the form of crack density, or modulus reduction, versus strain for two laminates of the same material system. Then, the model is capable of predicting crack density and modulus degradation for other laminate stacking sequences. The model takes into account crack closure, which is important under flexure, as well as the case of the center lamina straddling the neutral axis. The effect of thermal stresses is incorporated in the formulation.     

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。     

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.     

Modeling of facesheet crack growth in titanium-graphite hybrid laminates. Part II: Experimental results

Titanium-graphite hybrid composite laminates exhibit a coupled damage growth mode of facesheet cracking and delamination. Part I of this work modeled the growth of the coupled damage mode. Fatigue experiments were conducted on single edge notch tension specimens to measure the crack growth rate. This paper compares the model predictions with experimental data. The three-dimensional finite element model successfully captured the damage growth behavior for two of the lay-ups ([Ti/0/90/0₂]_s and [Ti/90/0/90₂]_s) in the experimental program. However, in a third lay-up, [Ti/0/90/±30]_s, the underlying damage modes were found to be sufficiently different than the other two lay-ups and the model did not capture the steady-state growth behavior. The effects of temperature and specimen size were also investigated for TiGr laminates. Except for the effects of the load ratio, elevated temperatures did not affect the crack growth rate significantly. For wider specimens, the steady-state fatigue crack growth behavior was similar to the narrow specimens, indicating that the steady-state facesheet crack growth behavior is independent of specimen size.     

纤维增强复合材料层合板缺口尺寸及形状效应数值模拟

通过考虑基体裂纹、纤维断裂、层内劈裂和层间脱层等破坏形式,建立三维有限元模型研究含中心圆孔和中心裂缝的准各向同性复合材料层合板([45/0/-45/90]_(2S))在拉伸载荷下的缺口尺寸效应及缺口形状效应。模拟结果显示:随着缺口尺寸的增大,层合板的破坏强度逐渐降低,然而,在本文研究范围内含中心裂缝的层合板破坏强度始终高于对应的含中心圆孔的层合板破坏强度。进一步分析有限元模拟结果表明,含中心裂缝的层合板亚临界损伤发生得更早,并且亚临界损伤范围更大,亚临界损伤会大大缓解缺口尖端的应力集中,从而使含中心裂缝层合板表现出更高的破坏强度。     

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。     

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.     

Some aspects of crack growth and failure in fibre reinforced composites

A theoretical analysis, previously developed to deal with the machanics of matrix cracking in unidirectional composites and with transverse ply cracking in cross ply laminates, has been developed further to deal with the tensile failure of unidirectional fibrous composites in with the fibres have a known distribution of strengths. It is proposed that, under the application of a tensile load, stable transverse cracks are formed which originate from regions of initial damage and which become unstable at some critical strain value. The model takes account of various parameters including the interfacial fibre/matrix debonding energy, the residual frictional shear strength of the debonded interface and the elastic properties of fibres and matrix. Comparisons are made between the predictions of the model and the observed failing strains of the 0° plies in carbon fibre polymer matrix laminates. The relevance of the model to the study of delayed fracture in fibrous composites is discussed. The modification of this model, previously developed to describe crack growth in the transverse plies of 0°/90° laminates, is used to predict the initial cracking strains for a wide range of CFRP laminate geometries and initial crack sizes. Some aspects of the mechanics of crack extension across interply interfaces are discussed.     

Damage mechanics characterization of transverse cracking behavior in quasi-isotropic CFRP laminates with interlaminar-toughened layers

Microscopic damage behavior in quasi-isotropic CFRP laminates with interlaminar-toughened layers under tensile fatigue loading is investigated. Damage observation is conducted using an optical microscope and soft X-ray radiography. The material used is CFRP with interlaminar-toughened layers, T800H/3900-2. The laminate configurations are quasi-isotropic [45/0/−45/90]_s, [0/45/−45/90]_s and [45/−45/0/90]_s to discuss the effect of stacking sequence on microscopic fatigue damages. A damage mechanics analysis is used to obtain the energy release rate for transverse cracking which is correlated to the transverse crack density growth rate. The modified Paris-law analysis proves to be valid for characterization of transverse crack multiplication when the effect of other damage is small.     

Simulation of multiple delaminations in impacted cross-ply laminates using a finite element model based on cohesive interface elements

The paper investigates the capability of a finite element model based on cohesive interface elements to simulate complex three-dimensional damage patterns in composite laminates subjected to low-velocity impact. The impact response and the damage process of cross-ply laminated plates with grouped ([0₃/90₃]s and [90₃/0₃]s) and interspersed ([0/90]₃s) ply stacking was simulated using a FE model developed by the authors in a previous study and the numerical results were compared to experimental observations. The model provided a correct simulation of size, shape and location of the principal fracture modes occurring in impacted [0₃/90₃]s and [90₃/0₃]s laminates. In [0/90]₃s laminates, characterized by a complex spatial damage distribution, the model was able to predict the approximately circular shape of the overall projected damage area and to capture the typical shape features of individual delaminations; significant discrepancies between experiments and predictions were however observed in terms of delamination sizes at single interfaces. Further investigations are needed to clarify the main reasons of these discrepancies.     

Effect of Off – Axis Matrix Cracking on Stiffness of Symmetric Angle-Ply Composite Laminates

Matrix cracking models developed for cross-ply composite laminates cannot easily be applied to more complicated geometries. In this paper a detailed analysis of the effect of matrix cracking on the longitudinal Young’s Modulus of a [0/45]s plate under uniaxial tension is attempted. The theoretical approach, based on a semi-analytical generalized plane strain model, is compared to experimental data obtained by microscopic strain measurements on a fiber sensor using the technique of laser Raman Spectroscopy. The experimental results are in a good agreement with theoretical stiffness degradation predictions obtained using the semi-analytical model.     

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.     

碳-铝正交层板疲劳行为研究

本研究用真空热压法制备了两种铺层的C/AI正交层板(0/90/0)_s及(90/0/90)_s.在MTS NEW810上进行了一系列疲劳损伤及破坏试验.疲劳损伤的行为研究工作包括:以刚度下降为损伤参数对C/AI正交层板进行了降级应力分析,并由此来预计在△S_h以下,正交层板不会发生疲劳损伤累积.研究了C/Al正交层板在同一应力水平而不同应力范围作用下的疲劳响应,发现试样在疲劳损伤时其刚度下降值相近似,即疲劳破坏的门槛值依赖于所施加的应力水平.依据MMC对各种循环载荷的不同响应,基体的疲劳损伤状态在S-N平面上可分为三种不同的区域:无损伤区,损伤累积区和断裂区.利用扫描电镜及金相显微镜分别对其疲劳断口形貌、基体裂纹进行观察,对该正交层板的疲劳破坏行为进行分析及讨论.结果表明:C/Al正交层板的疲劳断口呈脆断型,其中主承力层(0°铺层)断口平齐,偏轴层(90°铺层)断口平齐最差,层间损伤形式有局部分层、界面连续开裂及复合丝之间基体开裂等三种形式;其疲劳破坏主导因素是层间局部严重损伤及主承力层中复合丝大量断裂由于其疲劳裂纹沿垂直于载荷方向迅速扩展,寻找适中的界面结合强度对改善C/Al层板的疲劳性能有很大影响.     

Damage Assessment of CFRP [90/±45/0] Composite Laminates over Fatigue Cycles

The present paper develops a stiffness-based model to characterize the progressive fatigue damage in quasi-isotropic carbon fiber reinforced polymer (CFRP) [90/±45/0] composite laminates with various stacking sequences. The damage model is constructed based on (i) cracking mechanism and damage progress in matrix (Region I), matrix-fiber interface (Region II) and fiber (Region III) and (ii) corresponding stiffness reduction of unidirectional plies of 90°, 0° and angle-ply laminates of ±45° as the number of cycles progresses. The proposed model accumulates damages of constituent plies constructing [90/±45/0] laminates by means of weighting factor η ₉₀, η ₀ and η ₄₅. These weighting factors were defined based on the damage progress over fatigue cycles within the plies 90°, 0° and ±45° of the composite laminates. Damage model has been verified using CFRP [90/±45/0] laminates samples made of graphite/epoxy 3501-6/AS4. Experimental fatigue damage data of [90/±45/0] composite laminates have fell between the predicted damage curves of 0°, 90° plies and ±45°, 0/±45° laminates over life cycles at various stress levels. Predicted damage results for CFRP [90/±45/0] laminates showed good agreement with experimental data. Effect of stacking sequence on the model of stiffness reduction has been assessed and it showed that proposed fatigue damage model successfully recognizes the changes in mechanism of fatigue damage development in quasi-isotropic composite laminates.     

纤维金属层板低速冲击试验和数值仿真

为开展纤维金属层板（FML）低速冲击有限元数值仿真研究,改进了传统的连续损伤力学（CDM）模型,然后对FML落锤低速冲击试验进行数值仿真,并与实验结果进行对比验证。分别采用5.11 J 和10.33 J冲击能量对FML进行落锤低速冲击试验,得到冲击载荷、位移和能量时程曲线,分析FML的动态响应和失效模式。建立了考虑塑性应变、压缩刚度衰减特征和纤维拉伸断裂损伤的新CDM模型,描述S2-玻璃纤维/环氧树脂（S2-galss/epoxy）复合材料的损伤本构,并编写VUMAT子程序,通过ABAQUS/Explicit求解器对FML落锤冲击试验进行数值仿真。研究结果表明：低能量冲击条件下,FML背面主要为鼓包和裂纹等失效模式,位移峰值随冲击能量的提高而增加,冲击载荷峰值在穿透前也随冲击能量的提高而增加;采用改进的CDM模型描述FML中S2-galss/epoxy复合材料铺层后,有限元数值计算可以较好地预测FML低速冲击载荷下的动态响应;有限元数值仿真结果表明,FML中第2层复合材料铺层发生的纤维断裂损伤比第1层的更严重。     

Residual Stiffness of Cracked Cross-Ply Composite Laminates Under Multi-Axial In-plane Loading

In this paper, the Equivalent Constraint Model (ECM) together with a 2-D shear lag stress analysis approach is applied to predict residual stiffness properties of polymer and ceramic matrix [0/90 _n /0] cross-ply laminates subjected to in-plane biaxial loading and damaged by transverse and longitudinal matrix cracks. It is found that the longitudinal Young’s modulus, shear modulus and major Poisson’s ratio undergo large degradation as the matrix crack density increases, with Poisson’s ratio appearing to be the most affected by transverse cracking. In cross-ply laminates with thick 90° layer strip-shaped delaminations begin to initiate and grow from the tips of matrix cracks at the 0°/90° interface. These delaminations contribute to further stiffness degradation of such laminates, and hence have to be taken into account in failure analysis models. The thickness of the 90° layer plays an important role; the thicker the 90° layer, the bigger stiffness reduction suggesting a size (volume) effect at ply level. In SiC/CAS cross-ply laminates reduction in the longitudinal modulus occurs mainly due to transverse cracks, while the shear modulus appears to be the most affected by the presence of longitudinal cracks. The shear modulus reduction ratio predicted previously by a semi-empirical formula is, in the most of cases, within 10% of the current ECM/2-D shear lag approach value. In some cases, though, the error of the semi-empirical finite element expression can be as big as 20% since it fails to capture damage mode interaction.     

Multi-directional stiffness degradation induced by matrix cracking in composite laminates

A model was developed for predicting the stiffness degradation of fiber reinforced plastics (FRP), with ply configuration [0_m/±θ_n]_S, induced by matrix cracking under in-plane tension. The model assumes that the cracks in off-axis plies are uniformly distributed and a damage variable D is defined. Based on the theory of fracture mechanics, the elastic moduli of cracked matrix are obtained and indicated by the damage variable D, then the reduction of elastic moduli of laminates caused by the matrix cracks was studied. Comparison with experimental values for the glass/epoxy [90₃/0]_S, [0/90]_S and [0/±45]_S laminates shows good agreement with the theoretical prediction given by the presented model.     

Modelling Strategies for Simulating Delamination and Matrix Cracking in Composite Laminates

The composite materials are nowadays widely used in aeronautical domain. These materials are subjected to different types of loading that can damage a part of the structure. This diminishes the resistance of the structure to failure. In this paper, matrix cracking and delamination propagation in composite laminates are simulated as a part of damage. Two different computational strategies are developed: (i) a cohesive model (CM) based on the classical continuum mechanics and (ii) a continuous damage material model (CDM) coupling failure modes and damage. Another mixed methodology (MM) is proposed using the continuous damage model for delamination initiation and the cohesive model for 3D crack propagation and mesh openings. A good agreement was obtained when compared simple characterization tests and corresponding simulations.     

基于断裂韧性预报复合材料层合板的开孔拉伸强度

开孔层合板的强度预报往往取决于孔边的临界长度,它不仅与材料性能,而且与铺层、孔径都有关。本文基于线弹性断裂力学,提出了一种预报对称铺层层合板开孔拉伸强度的新方法,只需提供正交层合板的断裂韧性和无缺口层合板的拉伸强度,显著降低对实验数据的依赖性。首先,将临界长度表作为层合板断裂韧性和无缺口拉伸强度的函数,再通过正交层合板[90/0]_8s的紧凑拉伸试验和虚拟裂纹闭合技术,确定出0°层断裂韧性,进而计算得到任意对称铺层层合板的断裂韧性。本文测试了T300/7901层合板[0/±45/90]_2s和[0/±30/±60/90]_s的开孔拉伸强度,孔径分别为3 mm、6 mm和9 mm。理论预报结果与试验值吻合较好,最大误差为15.2%,满足工程应用需求。      

Interaction between matrix cracking and edge delamination in composite laminates

Matrix cracking and edge delamination are two main damage modes in continuous-fibre composite laminates. They are often investigated separately, and so the interaction between two damage modes has not yet been revealed. In this paper, a simple parallel-spring model is introduced to model the longitudinal stiffness reduction due to matrix cracking and edge delamination together. The energy release rate of edge delamination eliminating the matrix crack effect and the energy release rate of matrix cracking in the presence of edge delamination are then obtained. Experimental materials include carbon- and glass-fibre-reinforced bismaleimide composite laminates under static tension. The growth of matrix cracks and edge delaminations was recorded by means of NDT techniques. Results show that matrix cracks may initiate before or after edge lamination. This depends on the laminate layup, and especially on the thickness of the 90° plies. Edge delamination may also induce matrix cracking. Matrix cracking has a significant effect on the stiffness reduction in GRP laminates. The present model can predict the stiffness reduction in a laminate containing both matrix cracks and edge delaminations. The mixed-mode delamination fracture toughness obtained from the present model shows up to 50% differences compared with O'Brien's model for GRP laminates. However, matrix cracking has a small effect on the mixed-mode interlaminar fracture toughness of the CFRP laminates.