A method for calculating stress intensities in bimaterial fracture

A numerical method is presented for obtaining the values of K^* ₁,K ^* _II and K^* _III in the elasticity solution at the tip of an interface crack in general states of stress. The basis of the method is an evaluation of theJ-integral by the virtual crack extension method. Individual stress intensities can then be obtained from further calculations ofJ perturbed by small increments of the stress intensity factors. The calculations are carried out by the finite element method but minimal extra computations are required compared to those for the boundary value problem. Very accurate results are presented for a crack in the bimaterial interface and compared with other methods of evaluating the stress intensity factors. In particular, a comparison is made with stress intensity factors obtained by computingJ by the virtual crack extension method but separating the modes by using the ratio of displacements on the crack surface. Both techniques work well with fine finite element meshes but the results suggest that the method that relies entirely on J-integral evaluations can be used to give reliable results for coarse meshes.

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Résumé On présente une méthode numérique en vue d'obtenir les valeurs de K^* ₁, K^* _II et K^* _III relatives à la solution élastique d'application à l'extrémité d'une fissure d'interface sujette à un état de contraintes général. La méthode repose sur l'évaluation de l'intégraleJ par la technique d'extension virtuelle de la fissure. On peut ensuite obtenir les intensités de contraintes individuelles à partir de calculs deJ subséquents, correspondant à des perturbations introduites par de petits accroissements des facteurs d'intensité de contraintes.Les calculs sont accomplis par la méthode des éléments finis, mais, par rapport aux calculs à mettre en oeuvre dans le problème des valeurs aux limites, il ne faut procéder qu'à quelques calculs supplémentaires.On présente des résultats très précis pour le cas d'une fissure dans un interface entre deux matériaux, et on les compare avec ceux provenant d'autres méthodes d'évaluation des facteurs d'intensité de contraintes.En particulier, on fait une comparaison pour des facteurs d'intensité de contraintes obtenus en calculant J par la méthode d'extension virtuelle d'une fissure, mais en séparant les modes selon le rapport des déplacements de la surface de la fissure.Les deux techniques fonctionnent de manière satisfaisante avec des maillages fins d'éléments finis; cependant, les résultats suggèrent que la méthode qui repose entièrement sur les évaluations de l'intégraleJ peut être utilisée afin d'obtenir des résultats fiables dans les réseaux à mailles grossières.

     

Progressive delamination growth analysis using discontinuous layered element

In this paper, a fracture mechanic approach is used to analyze delamination propagation between layers of composite laminates. A finite element method based on layer-wise theory is extended for the analysis of delamination growth. In this approach, delamination is modeled by jump discontinuity conditions at the interfaces. The layer-wise finite element is developed to calculate the strain energy release rates based on the virtual crack closure technique (VCCT). A procedure is proposed to handle the progressive delamination of laminates. Finally, analyses of the edge delamination propagation for several composite laminates are performed and the corresponding failure stresses are calculated. The predicted results are compared with the available experimental and numerical results. It is shown that the predicted failure stresses using this method are comparable with those obtained using interface elements.     

A generalized micromechanical approach for the analysis of transverse crack and induced delamination in composite laminates

The previously developed micromechanical approaches for the analysis of transverse cracking and induced delamination are limited for laminates with specific lay-ups such as cross-ply and specific loading conditions. In this paper a new micromechanical approach is developed to overcome such shortcomings. For this purpose, a unit cell in the ply level of composite laminate including transverse cracking and delamination is considered. Then, the governing equations for the stress and displacement fields of the unit cell are derived. The obtained approximate stress field is used to calculate the energy release rate for the propagation of transverse cracking and induced delamination. To show the capability of the new method, it is employed for the analyses of general laminates with [0/90]_s, [45/−45]_s, [30/−30]_s and [90/45/0/−45]_s lay-ups under combined loadings to calculate the energy release rate due to the transverse cracking and induced delamination. It is shown that the obtained energy release rates for transverse cracking and delamination initiation are in good agreement with the available results in the literature and finite element method. Furthermore, the occurrence priority of further transverse cracks and/or delamination at each damage state of the laminates will be discussed.     

Variational approach development in analysis of matrix cracking and induced delamination of cross-ply composite laminates subjected to in-plane shear loading

Matrix microcracking and induced delamination propagating from the edge of microcracks in cross-ply composite laminates with [0_n/90_m]_s and [90_m/0_n]_s layups under in-plane static shear loading are investigated. An admissible stress field, which satisfies all of equilibrium equations, boundary conditions, and continuity of interfaces, is approximated. Then using the principle of the minimum complementary energy, the stress state is obtained from calculations of variation. The calculated stress state gives the stiffness reduction and the total strain energy of the laminated composite structure. Finally, the strain energy release rate of a general cross-ply laminate due to initiation and propagation of matrix cracking and induced delamination can be deduced. Results of the developed approach are in good agreement with experimental observations and finite element analyses, which confirms its accuracy.     

Fracture mechanics analysis of anisotropic plates by the boundary element method

This paper presents an analysis of mixed mode fracture mechanics problems arising in anisotropic composite laminates. The boundary element method (BEM) and the J _k integral are presented as accurate techniques to compute the stress intensity factors K _I and K _II of two dimensional anisotropic bodies. Using function of a complex variable a decoupling procedure is derived to obtain the stress intensity factors. The procedure is based on the computation of the J ₁-integral and of the ratio of relative displacements at the crack faces, near the crack tip. Applications are presented for unidirectional and symmetric laminates of glass, boron and graphite-epoxy materials. Numerical examples of problems of pure mode I and mixed mode deformations are given, in order to demonstrate the accuracy of the method.     

Interaction of laminate damage and adhesive disbonding in composite scarf joints subjected to combined in-plane loading and impact

Impact tests were carried out on composite laminates and composite scarf repairs, while both were subjected to in-plane loading with tensile pre-strain levels up to 5000 microstrain. The results show that pre-straining of the composite laminates has no noticeable influence on the size of the delamination area for the given impact energy of 8 J, which represents a typical barely-visible impact on thin-skin composite structures. For composite scarf joints, however, resulting damage has been found to be a combination of adhesive disbonding and matrix cracking (delamination and intraply cracking) in the composite laminate. The size of this mixed type of damage increases significantly with increasing pre-strain levels. A finite element model was developed to investigate the interaction between adhesive disbonding and composite delamination. The computational results reveal that both delamination and adhesive disbonding are dominated by the mode II fracture. Since the critical mode II fracture energy release rate for composite laminates (G_IIC = 1.08 kJ/m²) is much less than that pertinent to the adhesive (G_IIC = 3.73 kJ/m²), delamination tends to occur first in the composite laminates, which then shield the growth of disbonding in the adhesive.     

Fracture mechanics of delamination in ARALL laminates

Fracture mechanics of delamination in ARALL laminates is examined by using a finite element method employing special singular elements. Since these special elements contain the exact stress singularity, the delamination stress intensity factors and energy release rates can be evaluated conveniently. Solution convergence is studied to demonstrate the efficiency of this method. To ensure the validity of the result, the numerical prediction is compared with experimental results. Very good agreement is obtained.     

Finite element evaluation of mixed mode stress intensity factors in functionally graded materials

This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co‐ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed‐mode two‐dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path‐independent J^*_k‐integral, modified crack‐closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

含分层复合材料层板的压缩性能

使用商用有限元软件建立了含分层复合材料层板的有限元模型,采用Hashin失效准则对层板内单元进行损伤判断,并编写程序对失效单元进行刚度折减,使用cohesive单元模拟层间区域,并对缺陷区域进行弱化处理,利用应力失效判据和能量释放准则判断层板内起始分层与分层的扩展。对完好以及含分层缺陷复合材料单向层板试验件进行压缩实验研究,实验结果给出了分层位置和尺寸及对材料压缩性能的影响。研究表明,有限元模拟结果与实验结果具有良好的一致性。     

Stress distribution in damaged composite laminates under transverse impact

Studies on damage in composite laminates subjected to central and normal impact are conducted by a 3-D finite element analysis. The stress analysis is carried out by developing a constitutive equation of composite laminates coupled with the damage. Effects of the damage on the stress distribution in the laminates are investigated in details. The obtained contact force history correlates well with the results reported in literatures. Stress distributions across the thickness of the elastic non-damaged laminate show a probable distribution of delamination. The simulated result for delamination is coincided with the observation of experiments. Stress distributions for the damaged laminates show that the damage releases strain energy and lessens stress concentration.     

FINITE ELEMENT ANALYSIS OF DELAMINATION CRACKS IN BENDING OF CROSS-PLY LAMINATES

SUMMARY

A study of delamination crack growth due to bending in cross-ply laminates is presented. For the understanding of interlaminar fracture behaviour of laminated composites the modelling of delamination crack growth induced by bending and shear cracks in three point bending specimens is carried out. A plane strain two-dimensional (2-D) finite element analysis is used to determine the strain energy release rates during delamination of the laminated beam. Contact elements were used to prevent the material interpenetration on the crack surfaces. The solution of the contact problem taking into account friction along crack surfaces is obtained. Energy release rates G_I and G_II for Mode I and Mode II fracture are calculated by virtual crack closure integral (VCCI) methods. Comparison of total energy release rates, obtained by local energy methods, with an analytical solution based on the beam theory and a global energy method have been carried out. Good agreement of the results obtained by various methods have been observed. Comparison of the results obtained by the solution of the contact problem and without contact elements have been performed. Significant differences between the values of energy release rates obtained with and without using contact elements have been observed. The influence of the coefficient of friction on the energy release rates is insignificant.     

Stress intensity factor analyses of interface cracks between dissimilar anisotropic materials using the finite element method

Delamination along an interface between dissimilar materials is the primary cause of failure in microstructures like electronic packages, micro-electro-mechanical systems (MEMS), and so on. Fracture mechanics is a powerful tool for the evaluation of delamination. However, many materials used in microstructures such as composite materials and single crystals are anisotropic materials. Stress intensity factors of an interface crack between dissimilar anisotropic materials, which were proposed by Hwu, are useful for evaluating the reliability of microstructures. However, numerical methods that can analyze the stress intensity factors of an interface crack between anisotropic materials have not been developed. We propose herein a new numerical method for the analysis of an interface crack between dissimilar anisotropic materials. The stress intensity factors of an interface crack are based on the generalized plane strain condition. The energy release rate is obtained by the virtual crack extension method in conjunction with the finite element method for the generalized plane strain condition. The energy release rate is separated into individual modes of the stress intensity factors K_I, K_II, and K_III, using the principal of superposition. The target problem to be solved is superposed on the asymptotic solution of displacement in the vicinity of an interface crack tip, which is described using the Stroh formalism. Analyses of the stress intensity factors of center interface cracks between semi-infinite dissimilar anisotropic media subjected to concentrated self-balanced loads on the center of crack surfaces and to uniform loads are demonstrated. The present method accurately provides mode-separated stress intensity factors using relatively coarse meshes for the finite element method.     

Delamination growth mechanisms in woven glass fiber reinforced polymer composites under Mode II fatigue loading at cryogenic temperatures

The purpose of this research is to characterize the cryogenic delamination growth behavior in woven glass fiber reinforced polymer (GFRP) composite laminates subjected to Mode II fatigue loading. Mode II fatigue delamination tests were performed at room temperature, liquid nitrogen temperature (77 K) and liquid helium temperature (4 K) using the four-point bend end-notched flexure (4ENF) test method, and the delamination growth rate data for the woven GFRP laminates were obtained. The energy release rate range was determined by the finite element method. Microscopic examinations of the specimen sections and fracture surfaces were also carried out. The present results are discussed to obtain an understanding of the fatigue delamination growth mechanisms in the woven GFRP laminates under Mode II loading at cryogenic temperatures.     

Finite element analysis of postbuckling and delamination of composite laminates using virtual crack closure technique

The two-dimensional and three-dimensional parametric finite element analysis (FEA) of composite flat laminates with two through-the-width delamination types: 0₄/(±θ)₆//0₄ and 0₄//(±θ)₆//0₄ (θ = 0°, 45°, and “//” denotes the delaminated interface) under compressive load are performed to explore the effects of multiple delaminations on the postbuckling properties. The virtual crack closure technique which is employed to calculate the energy release rate (ERR) for crack propagation is used to deal with the delamination growth. Three typical failure criteria: B-K law, Reeder law and Power law are comparatively studied for predicting the crack propagation. Effects of different mesh sizes and pre-existing crack length on the delamination growth and postbuckling properties of composite laminates are discussed. Interaction between the delamination growth mechanisms for multiple cracks for 0₄//(±θ)₆//0₄ composite laminates is also investigated. Numerical results using FEA are also compared with those by existing models and experiments.     

纤维增强复合材料层合板分层破坏的研究

本文对纤维增强复合材料层合板的分层破坏进行了大量的试验,同时用三维有限元进行应力分析。试验和分析结果表明此类层合板的分层总是发生在θ/90界面上,该界面上不仅层间剪应力大而且层间正应力也大。通过对不同θ/90界面的临界能量释放率的测定表明,对层合板不同的θ/90分层界面的G_ⅠC和G_ⅡC是随θ角的变化而变化。文中对一个Ⅰ型,Ⅱ型耦合型能量释放率分层判据的应用作了改进,试验结果表明此改进是有效的。     

A three-dimensional multilayer composite finite element for stress analysis of composite laminates

A three-dimensional multilayer composite finite element method has been developed based on a composite variational functional which takes three in-plane strains ε_x, ε_x, ε_xy and three transverse stresses σ_z, σ_yz, σ_xz as the basic variables. The continuity of the transverse stresses σ_z, σ_yz, σ_xz across the laminate thickness is assured a priori by introducing a partial stress field parameter α which is associated with the lower and upper surfaces of a lamina in a laminate. A method has been developed to form the partial stress field based on the assumed displacement field. With this method, a three dimensional (3-D) multilayer composite finite element is formulated for stress analysis of composite laminates. A numerical example is given, which shows some advantages of this composite element.     

Interlaminar stress recovery for three-dimensional finite elements

An accurate evaluation of interlaminar stresses in multilayer composite laminates is crucial for the correct prediction of the onset of delamination. In general, three-dimensional finite element models are required for acceptable accuracy, especially near free edges and stress concentrations. Interlaminar stresses are continuous both across and along layer interfaces. Nonetheless, the continuity of interlaminar stresses is difficult to enforce in C⁰$C^0$ interpolated elements. Nodal values of the stresses are usually retrieved using extrapolation techniques from super-convergent points, if known, or Gauss points inside the element. Stress fields within an element can be deduced using either constitutive relations or variationally consistent procedures. In either case, spurious oscillations in stress fields may be encountered leading to a reduced accuracy of the recovered stresses at nodes. In this paper, an efficient interlaminar stress recovery procedure for three-dimensional finite element formulations is presented. The proposed procedure does not rely on extrapolation techniques from super-convergent or integration points. Interlaminar stress values are retrieved directly at nodes and stress continuity at the inter-element boundary is automatically satisfied. Several benchmark problems were analysed. Comparisons with finite element software and available solutions in the literature confirmed the accuracy of the procedure. Accurate interlaminar stresses were obtained using coarser meshes compared to customary recovery procedures.     

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.     

Effect of initial crack length on the measured bridging law of unidirectional E-glass/epoxy double cantilever beam specimens

In this paper, the effect of initial delamination length is experimentally investigated on obtaining the mode I bridging law of unidirectional E-glass/epoxy double cantilever beam (DCB) specimens manufactured by hand layup method. To this end, an experimental test set-up is established for accurate measurement of crack tip opening displacement (CTOD) using digital image processing method. DCB tests are performed for three different delamination lengths and the corresponding bridging laws are calculated using J-integral approach. Results showed that the maximum bridging stress, the shape of bridging law and energy dissipation in bridging zone are slightly affected by changing initial crack length. In other words, the measured bridging law acts independent of initial delamination length. Therefore, the obtained bridging law can be used with the cohesive elements available in the commercial finite element software to simulate the delamination propagation behavior in unidirectional DCB specimens.     

Fracture mechanics analysis of a crack in a residual stress field

The standard definition of the J integral leads to a path dependent value in the presence of a residual stress field, and this gives rise to numerical difficulties in numerical modelling of fracture problems when residual stresses are significant. In this work, a path independent J definition for a crack in a residual stress field is obtained. A number of crack geometries containing residual stresses have been analysed using the finite element method and the results demonstrate that the modified J shows good path-independence which is maintained under a combination of residual stress and mechanical loading. It is also shown that the modified J is equivalent to the stress intensity factor, K, under small scale yielding conditions and provides the intensity of the near crack tip stresses under elastic-plastic conditions. The paper also discusses two issues linked to the numerical modelling of residual stress crack problems-the introduction of a residual stress field into a finite element model and the introduction of a crack into a residual stress field.