Delamination dynamics in through-thickness reinforced laminates with application to DCB specimen

Bridged crack models using beam theory formulations have proved to be effective in modeling quasi-static delamination crack growth under large scale bridging conditions in through-thickness reinforced structures. In this paper, beam theory is used to study dynamic mode I crack propagation in through-thickness reinforced laminar structures. In particular, steady state dynamic crack growth for a Double Cantilever Beam (DCB) loaded with a flying wedge is examined. The steady state crack propagation characteristics are mapped out in terms of controllable loading and material parameters including the crack velocity and the properties of the through-thickness reinforcement. For small crack velocities, the through-thickness reinforcement considerably enhances the delamination resistance of the structure. At higher velocities, the kinetic energy term dominates the overall energetics and the relative effect of the reinforcement on the delamination resistance is insignificant. The model suggests a simple fracture test for estimating the properties of the through-thickness reinforcement under dynamic loading conditions.     

Constitutive Model for a Fiber Tow Bridging a Delamination Crack

A simple model is introduced for a fibrous tow, stitch, or rod that bridges a delamination crack in a laminate. The model is introduced for mode II delamination cracks but is intended for general mixed-mode cracks. Modeling is guided by prior observations and measurements on laminates reinforced through the thickness by stitches or short rods. Salient phenomena include shear deformation of the bridging tow, its debonding from and sliding relative to the surrounding laminate, and its sideways displacement through the laminate. The tow is represented as a beam that can shear and extend axially. Its axial displacement relative to the laminate is resisted over its debonded periphery by friction. The forces associated with its sideways displacement are estimated by regarding it as a punch being driven through a plastic medium (the laminate). Thus the mechanics of the whole problem are reduced to a set of one-dimensional equations. The distinction between continuous stitches and discontinuous rod reinforcement consists of a boundary condition. With realistic values assigned to undetermined parameters, experimental data for stitches are reproduced over the whole range of displacements up to ultimate failure of the stitch. The model generates a bridging traction law that can be used for optimal design of through-thickness reinforcement for damage tolerance in a wide variety of structures.     

Mixed-mode quasi-static failure criteria for adhesively-bonded pultruded GFRP joints

The strain energy release rates of adhesively-bonded pultruded GFRP joints were determined experimentally. The crack propagated in the adherend along paths outside the symmetry plane accompanied by fiber bridging. A new method, designated the “extended global method”, was introduced to facilitate mode partitioning in the mixed-mode experiments. Non-linear finite element models were developed in order to quantify the effect of the observed fiber bridging on crack propagation. An exponential traction-separation cohesive law was used to model the fiber bridging zone and calculate the energy release rate due to the fiber bridging, while the virtual crack closure technique was used for calculation of the fracture components at the crack tip. Experimental, analytical and numerical analyses were used to establish quasi-static mixed-mode failure criteria for crack initiation and propagation. The derived mixed-mode failure criteria can be used for simulating progressive crack propagation in other joint configurations comprising the same adhesive and adherends.     

ON THE EQUIVALENCE OF STRESS INTENSITY AND ENERGY APPROACHES IN BRIDGING ANALYSIS

Abstract— Through-thickness reinforcement is effective in suppressing delamination in composite laminates. It provides bridging in the crack wake during delamination crack growth. The closure traction behind the crack tip due to this bridging increases the delamination toughness significantly. The effect of bridging may be analysed using the stress intensity approach or the Griffith energetic approach. In this paper, delamination crack growth resistance K _R,(Δa) of a double-cantilever-beam specimen, with through-thickness reinforcement, under Mode I delamination is determined using these two approaches. Additionally, the J-integral analysis is also employed. A comparison between the theoretical results predicted by these three methods is provided using the example of stitched CFRP.     

Failure criteria for mixed mode delamination in glass fibre epoxy composites

Critical strain energy release rate of glass/epoxy laminates using the virtual crack closure technique for mode I, mode II, mixed-mode I + II and mode III were determined. Mode I, mode II, mode III and mixed-mode I + II fracture toughness were obtained using the double cantilever beam test, the end notch flexure test, the edge crack torsion test and the mixed-mode bending test respectively. Results were analysed through the most widely used criteria to predict delamination propagation under mixed-mode loading: the Power Law and the Benzeggagh and Kenane criteria. Mixed-mode fracture toughness results seem to represent the data with reasonable accuracy.     

基于扩展有限元法的钢筋混凝土梁复合断裂过程模拟研究

针对钢筋混凝土梁裂纹扩展问题,基于扩展有限元法,建立了预置裂纹的简支混凝土梁三维模型,用粘聚裂纹模型描述裂纹面间的力学行为,采用线性的软化曲线表示裂纹尖端断裂过程区的应变软化行为,分别对素混凝土梁和钢筋混凝土梁的复合断裂过程进行模拟,分析了纵向钢筋对裂纹扩展路径、荷载-挠度和荷载 -CMOD (裂缝开口处张开位移)曲线的影响,并与文献中的试验结果进行对比,计算结果与试验结果吻合良好,展示了扩展有限元法在结构断裂破坏分析方面的独特优势。     

Cohesive models for damage evolution in laminated composites

A trend in the last decade towards models in which nonlinear crack tip processes are represented explicitly, rather than being assigned to a point process at the crack tip (as in linear elastic fracture mechanics), is reviewed by a survey of the literature. A good compromise between computational efficiency and physical reality seems to be the cohesive zone formulation, which collapses the effect of the nonlinear crack process zone onto a surface of displacement discontinuity (generalized crack). Damage mechanisms that can be represented by cohesive models include delamination of plies, large splitting (shear) cracks within plies, multiple matrix cracking within plies, fiber rupture or microbuckling (kink band formation), friction acting between delaminated plies, process zones at crack tips representing crazing or other nonlinearity, and large scale bridging by through-thickness reinforcement or oblique crack-bridging fibers. The power of the technique is illustrated here for delamination and splitting cracks in laminates. A cohesive element is presented for simulating three-dimensional, mode-dependent process zones. An essential feature of the formulation is that the delamination crack shape can follow its natural evolution, according to the evolving mode conditions calculated within the simulation. But in numerical work, care must be taken that element sizes are defined consistently with the characteristic lengths of cohesive zones that are implied by the chosen cohesive laws. Qualitatively successful applications are reported to some practical problems in composite engineering, which cannot be adequately analyzed by conventional tools such as linear elastic fracture mechanics and the virtual crack closure technique. The simulations successfully reproduce experimentally measured crack shapes that have been reported in the literature over a decade ago, but have not been reproduced by prior models.     

Characterizing Mode II Delamination Cracks in Stitched Composites

This paper deals with characterizing the bridging mechanisms developed across delamination cracks by through-thickness reinforcement, using stitched carbon/epoxy laminates under mode II loading as a prime example. End Notched Flexure (ENF) tests are performed which show that stitching can provide stable crack growth. The bridging law, which characterizes the bridging action of the stitches, is deduced from both crack profile measurements and load vs. deflection curves. Consistent results are obtained from the two methods. The inferred laws imply that delamination cracks will commonly grow in conditions that are neither accurately nor properly described by linear elastic fracture mechanics. Large scale bridging calculations are required, in which the essential material property is the bridging traction law. The level of detail in which the law must be determined can be inferred from the sensitivity of predicted crack growth to variations in the law. It is recommended that the required parametric traction law be deduced in engineering practice from load vs. deflection data from the standard ENF (or similar) test, with due regard to selecting the notch size and other specimen dimensions to ensure that crack growth is stable in the test. This revised version was published online in August 2006 with corrections to the Cover Date.     

Z-pin增强复合材料Ⅰ型断裂韧性数值分析

采用细观力学方法以及虚拟裂纹闭合法(VCCT)对含有Z-pin增强复合材料双悬臂梁(DCB)结构Ⅰ型断裂韧性进行了研究。利用有限元法建立了结构模型,采用实体单元模拟复合材料层压板结构和非线性弹簧元模拟Z-pin。通过计算应变能释放率对含有不同体积分数Z-pin的复合材料层压板Ⅰ型断裂韧性与不含Z-pin的复合材料层压板Ⅰ型断裂韧性进行了对比分析。研究表明,含有Z-pin增强复合材料双悬臂梁(DCB)结构Ⅰ型断裂韧性在裂纹扩展过程中受到Z-pin桥联作用的影响而显著增强,且其增强效果与Z-pin的体积分数、处在桥联区的Z-pin数目均相关,这表明Z-pin增强方法能够有效提高复合材料层压板的分层扩展阻力。     

Composite toughness enhancement with interlaminar reinforcement

An experimental investigation of a newly proposed through-thickness reinforcement approach aimed to increase interlaminar toughness of laminated composites is presented. The approach alters conventional methods of creating three-dimensional fiber-reinforced polymer composites in that the reinforcing element is embedded into the host laminate after it has been cured. The resulting composite is shown to possess the benefits of a uniform surface quality and consolidation of the original unreinforced laminate. This technique was found to be highly effective in suppressing the damage propagation in delamination double-cantilever beam (DCB) test samples under mode I loading conditions. Pullout testing of a single reinforcing element was carried out to understand the bridging mechanics responsible for the improved interlaminar strength of reinforced laminate and stabilization and/or arrest of delamination crack propagation. The mode I interlaminar fracture of reinforced DCB samples was modeled using two-dimensional cohesive finite-element scheme to support interpretation of the experiments.     

Mixed-mode crack propagation through reinforced concrete

This paper presents the results of a very recent experimental research program aimed at investigating mixed-mode fracture of reinforced concrete. The tests were designed so that only one single mixed-mode crack generates and propagates through the specimen, as opposed to the usual dense crack pattern found in most of the tests in scientific literature. The specimens were three-point-bend beams of three different sizes. They were notched asymmetrically and reinforced with various ratios of longitudinal and inclined reinforcement. These experiments may help to understand the mechanisms of crack initiation and propagation through reinforcing bars under mixed-mode load conditions.     

Numerical study of mixed-mode fracture in concrete

In the present paper, a finite element code based on the microplane model for concrete is used for the analysis of typical mixed-mode geometries: a Single-Edge-Notched beam, a Double-Edge-Notched specimen and a Dowel-Disk specimen. A local smeared fracture finite element analysis is carried out. As a regularization procedure, the crack band method is used. The principal objective of the study was to investigate whether the smeared fracture finite element code is able to predict mixed-mode fracture of concrete with no optimisation of the material model parameters. Comparison between experimental and numerical results shows that the used code predicts structural response and crack patterns realistically for all cases investigated. Moreover, it is shown that for most of the studied geometries a mixed-mode fracture mechanism dominates at crack initiation, however, with increase of the crack length mode-I fracture becomes dominant and the specimens finally failed in mode-I fracture.     

A revised virtual crack closure technique for physically consistent fracture mode partitioning

The virtual crack closure technique (VCCT) is a well-established method for computing the energy release rate when analysing fracture problems via the finite element method. For mixed-mode fracture problems, the VCCT is also commonly used to partition the fracture modes, i.e. to determine the energy release rate contributions related to the three classical fracture modes. A perhaps little known fact, however, is that in some circumstances the standard VCCT predicts physically inconsistent, negative values for the modal contributions to the energy release rate. Focusing on I/II mixed-mode problems, this paper presents a revised VCCT which furnishes a physically consistent partitioning of fracture modes by associating the mode I and II contributions to the amounts of work done in a suitably defined two-step process of closure of the virtually extended crack. Deeper investigation pinpoints the origins of the physically inconsistent predictions of the standard VCCT in the lack of energetic orthogonality between the crack-tip force components used to compute the modal contributions. Further insight into the problem is offered by a geometric construction, which introduces the ‘ellipse of crack-tip flexibility’. In closing, the phenomena of contact, interpenetration, and friction between the crack surfaces are briefly touched upon.     

Stress intensity factors as the fracture parameters for delamination crack growth in composite laminates

A “mutual integral” approach is used to calculate the mixed-mode stress intensity factors for a free-edge delamination crack in a laminate under tensile loading conditions. This “mutual integral” approach, for generalized plane strain conditions, is based on the application of the path-independent J integral to a linear combination of three solutions: one, the problem of the laminate to be solved using the quasi 3-D finite element method, the second, an “auxiliary” solution with a known asymptotic singular solution, and the third, the particular solution due to the out-of-plane loading. A comparison with the exact solutions is made to determine the accuracy and efficiency of this numerical method. With this “mutual integral” approach, it was found that the calculated mixed-mode stress intensity factors of the free-edge delamination crack remain relatively constant as the crack propagates into the laminate. It was also found that the fracture criterion based on the mixed-mode stress intensity factors is more consistent with the experimental observations than the criterion based on the total energy release rate, and hence demonstrates the importance of the ability to calculate each individual component of the stress intensity factors. Furthermore, it was found that the fracture toughness measurements from double cantilever beam specimens can be used directly to predict the onset of delamination crack growth between two dissimilar laminae. Using these fracture toughness measurements from the double cantilever beam specimens, some examples are given to show that the fracture criterion based on the mixed-mode stress intensity factors can accurately predict the failure load for various laminates under tensile loading conditions.     

A review of some aspects of fatigue crack growth under variable amplitute loading

The literature on some aspects of the influence of variable amplitude loading on fatigue crack growth has been reviewed. In particular the importance of residual stresses, fatigue crack closure, microstruture, geometry and environment on the fatigue crack growth of long, through-thickness cracks following overloads, underloads and overload-underload combinations in Mode 1 opening have been identified. Other behaviour, including the influence of temperature, frequency and the effects of mixed-mode loading, is beyond the scope of this review. Areas of work requiring further investigation have been proposed.     

A fracture mechanics problem of a functionally graded layered structure with an arbitrarily oriented crack crossing the interface

In this paper, the fracture mechanics problem for an arbitrarily oriented crack crossing the interface in a functionally graded layered structure is investigated. The elastic modulus is assumed to be continuous at the interface, but its derivative may be discontinuous. Applying the superposition principle and Fourier integral transform, the stress fields and displacement fields are derived. A group of auxiliary functions defined in both layers are introduced and then the mixed-mode crack problem is turned into solving a group of singular integral equations. The mixed-mode stress intensity factors (SIFs) are obtained by solving the singular integral equations. The influences of the material nonhomogeneity parameter, normalized crack length and crack angle on the SIFs are investigated. It is found that the mixed-mode SIFs can be affected greatly by the crack angle. Moreover, the mixed-mode SIFs usually attain their extremum when the crack tips get to the interface during one crack moves from one layer into another layer. The present work may form the basic work for establishing a multi-layered fracture mechanics model of FGMs with an arbitrarily oriented crack and general mechanical properties.     

Experimental determination of the mode I delamination fracture and fatigue properties of thin 3D woven composites

The mode I delamination fracture toughness and fatigue strength of thin-section three-dimensional (3D) woven composite materials is experimentally determined. The non-crimp 3D orthogonally woven carbon–epoxy composites were thin (2 mm) and consequently their through-thickness z-binder yarns were inclined at a very steep angle (about 70°) from the orthogonal direction. The steep z-binder angle has a marked effect on the delamination toughening and fatigue strengthening mechanisms. Experimental testing revealed that the fracture toughness and fatigue resistance increased progressively with the volume content of z-binders. However, the steep angle caused the z-binder yarns bridging the delamination crack to deform and fail in shear and through-thickness tension, rather than in-plane tension which usually occurs in thick 3D woven composites. Mode I pull-off tests on a single woven z-binder yarn embedded within the composite revealed that the crack bridging traction load, strain energy absorption and failure mechanism were strongly affected by the steep angle.     

Finite element calculation of stress intensity factors for interfacial crack using virtual crack closure integral

This paper presents a successful implementation of the virtual crack closure integral method to calculate the stress intensity factors of an interfacial crack. The present method would compute the mixed-mode stress intensity factors from the mixed-mode energy release rates of the interfacial crack, which are easily obtained from the crack opening displacements and the nodal forces at and ahead of the crack tip, in a finite element model. The simple formulae which relate the stress intensity factors to the energy release rates are given in three separate categories: an isotropic bimaterial continuum, an orthotropic bimaterial continuum, and an anisotropic bimaterial continuum. In the example of a central crack in a bimaterial block under the plane strain condition, comparisons are made with the exact solution to determine the accuracy and efficiency of the numerical method. It was found that the virtual crack closure integral method does lead to very accurate results with a relatively coarse finite element mesh. It has also been shown that for an anisotropic interfacial crack under the generalized plane strain condition, the computed stress intensity factors using the virtual crack closure method compared favorably with the results using the J integral method applied to two interacting crack tip solutions. In order for the stress intensity factors to be used as physical variables, the characteristic length for the stress intensity factors must be properly defined. A study was carried out to determine the effects of the characteristic length on the fracture criterion based the mixed-mode stress intensity factors. It was found that the fracture criterion based on the quadratic mixture of the normalized stress intensity factors is less sensitive to the changes in characteristic length than the fracture criterion based on the total energy release rate along with the phase angle.This work has been supported by ONR, with Dr. Y. Rajapakse as the program official.     

Mode I interlaminar fracture toughness properties of advanced textile fibreglass composites

The Mode I interlaminar fracture toughness properties of vinyl ester-based composites reinforced with fibreglass manufactured by the advanced textile technologies of braiding, knitting, stitching and through-the-thickness weaving are assessed in comparison to a variety of traditional composites made from fibreglass such as unidirectional or woven rovings. The interlaminar fracture toughness (G_Ic) of braided and knitted composites are higher than traditional composites by factors of more than two and four, respectively. Toughening in these textile composites was caused by extensive crack branching as the interlaminar crack was forced to follow a tortuous path through the complex fibre architectures. The G_Ic values of the composites reinforced in the through-thickness direction by weaving or stitching were higher than traditional composites by factors of nearly two and three, respectively, with the main toughening mechanism being crack bridging by the through-thickness binder yarns/stitches. A review of Mode I interlaminar fracture data collected from papers shows that advanced textile techniques are capable of manufacturing composites with substantially improved delamination resistance.     

Mixed-mode I/II wood fracture characterization using the mixed-mode bending test

In this work fracture characterization of wood under mixed-mode I/II loading is addressed. The mixed-mode bending test is used owing to its aptitude for easier alteration of mode ratio. Experimental tests were performed covering a wide range of mode ratios in order to obtain a mixed-mode fracture criterion for the maritime pine (Pinus pinaster Ait.) in the RL crack propagation system. A data reduction scheme based on beam theory and crack equivalent concept was used to overcome some difficulties inherent to the test. The method does not require crack length monitoring during propagation and provide an entire resistance curve allowing easier identification of the fracture energy. A numerical analysis using cohesive elements was also performed to validate the method. The linear energetic fracture criterion was proved to be the most adequate to describe the failure envelop of this wood species.