A simplified beam analysis of the end notched flexure mode II delamination specimen

A modified classical beam theory solution is developed for the end notched flexure (ENF) specimen, one of the most widely used mode II delamination tests for fibre reinforced composite materials. The effect of crack tip deformation is analyzed by assuming that a region of certain length close to the crack tip rests on an elastic shear spring foundation. The mathematical procedure of the present analysis is simple and clear, and the resulting solutions for the compliance and the strain energy release rate of the ENF specimen are highly accurate. Excellent agreement is obtained over a wide range of material and geometrical properties of ENF specimens when the current results are compared with finite element results and other analytical analyses that include the effect of crack tip deformation in their solutions. The success of the present analysis indicates that the effect of crack tip deformation is the most important factor that must be considered when calculating the relationship between G_II and the load in the ENF specimen.     

弹性地基上的4ENF试件柔度分析

基于Timoshenko梁理论,考虑了剪切变形和裂纹尖端变形的影响,建立了双参数弹性地基上的II型加载末端切口四点弯曲试件(4-point bending end-notched flexure specimen,简称4ENF)的柔度和能量释放率模型(BEF)。4ENF柔度与裂纹长度成正比,柔度变化率、能量释放率与裂纹长度无关,因而4ENFII型断裂实验无需测量裂纹的扩展长度,根据临界荷载便可求得临界能量释放率,从而大大简化了实验手段。对FRP-木4ENF试件II型加载情况下的BEF模型、Timoshenko梁理论模型和有限元结果比较证明:BEF模型的4ENF柔度在裂纹扩展的一定范围内与有限元吻合很好;而Timoshenko梁理论模型的柔度小于有限元结果,精度较差。该模型可用于复合材料界面断裂分析、确定断裂参数以及作为断裂试验数据分析的依据。     

Beam analysis of angle-ply laminate end-notched flexure specimens

Analysis and experiments on quasi-unidirectional and angle-ply laminate end-notched flexure specimens are presented. The analysis is based on laminated beam theory incorporating first-order shear deformation theory. Compliance and strain-energy release rate determined for relatively thin unidirectional and angle-ply laminate ENF specimens were in good agreement with a previous classical plate theory formulation. For thicker laminates, however, effects of shear deformation on the compliance of the ENF specimen become significant. An experimental study on glass/polyester quasi-unidirectional and angle-ply laminate ENF specimens was conducted. Specifically, [0]₆, [±30]₅ and [±45]₅ laminates with mid-plane delaminations were considered. Experimental compliance data agreed well with analytical predictions. The fracture toughness increased with increased angle θ at the ±θ interface. This is attributed to the fracture work associated with the debonding of transversely oriented fiber bundles in the quasi-unidirectional plies. The angle-ply laminates displayed more yarn debonding than the quasi-unidirectional laminate. For all laminates it was observed that the crack propagated in a non-uniform manner which is correlated with elastic coupling effects with cracked regions of the laminate beams.     

Bending effect of through-thickness reinforcement rods on mode II delamination toughness of ENF specimen: Elastic and rigid–perfectly plastic analyses

In this paper, a new simple metallic z-rod model is proposed to study the bending effect of the metallic z-rods on mode II delamination toughness of laminated composites. A new transverse shear force–deformation relationship for a metallic z-rod is obtained by using the classical beam theory and modeling its surrounding matrix as linearly elastic, rigid–perfectly plastic or linearly elastic–perfectly plastic springs. The bridging traction provided by a metallic z-rod to the mode II delamination toughness is assumed to be only the shear force carried by a z-rod created by the relative slippage between two substrate beams in an end-notched flexure (ENF) specimen, whereas the longitudinal sliding friction is assumed to make negligible contribution to the bridging traction. Mode II strain energy release rate (SERR) is employed to evaluate the influence of the metallic z-rods on the interlaminar fracture toughness of end-notched flexure (ENF) specimens. A parametric study of ENF specimens reinforced with the z-rods is conducted to demonstrate the effect of the new bridging mechanism by the metallic z-rods on the mode II delamination toughness.     

Finite element analysis of the end notched flexure specimen for measuring mode II fracture toughness

This paper presents a finite element analysis of the End Notched Flexure (ENF) test specimen for Mode II interlaminar fracture testing of composite materials. Virtual crack closure and compliance techniques employed to calculate strain energy release rates from linear elastic two-dimensional analysis indicate that the ENF specimen is a pure Mode II fracture test within the constraints of small deflection theory. Furthermore, the ENF fracture specimen is shown to be relatively insensitive to process induced cracks offset from the laminate midplane. Frictional effects are investigated by including the contact problem in the finite element model. A parametric study investigating the influence of delamination length, span, thickness and material properties is presented to assess the accuracy of beam theory expressions for compliance and strain energy release rate, G_II. Finite element results indicate that data reduction schemes based upon beam theory underestimate G_II by approximately 20–40% for typical unidirectional graphite fiber composite test specimen geometries. Consequently, an improved data reduction scheme which retains the simplicity of beam theory but also includes the accuracy of the finite element method is proposed.     

Mode II fracture characterization of a hybrid cork/carbon-epoxy laminate

Fracture characterization under mode II loading of a hybrid laminate composed by a unidirectional carbon fiber-epoxy composite and cork was performed using the End Notched Flexure test. A data reduction scheme based on equivalent crack length concept, specimen compliance and Timoshenko beam theory was applied to evaluate fracture toughness under mode II loading of a composed beam (cork and carbon-epoxy composite). The adopted procedure depends exclusively on the data issuing from load–displacement (P–δ) curve and does not require crack length monitoring during the test which is a difficult task to be accomplished with the necessary accuracy in the ENF test. A numerical analysis using cohesive zone modeling and an inverse procedure was performed to assess the mode II cohesive law that simulates the material fracture under shear loading. It was concluded that hybridization is advantageous relative to monolithic carbon-epoxy laminate in which concerns the observed failure mode, which altered from typically brittle to very ductile thus contributing to avoid sudden shear failures in real applications.     

Mode II fracture energy characterization of brittle adhesives using compliance calibration method

The end‐notched flexure (ENF) test is widely used for measuring the Mode II critical strain energy release rate of adhesively bonded joints (ABJs). Unstable crack growth in ENF joints with brittle adhesives is a common phenomenon. Classic data reduction methods like the direct beam theory (DBT) and the compliance‐based beam method (CBBM) usually result in unacceptable scatter when crack grows unstable. In this study, the application of a compliance calibration method (CCM) for ENF adhesive joints with a brittle adhesive is experimentally investigated. For this purpose, ENF specimens were manufactured and tested. Different data reduction methods were considered for treating the results. Afterwards, the obtained fracture energies were used as an input parameter in a finite element (FE) analysis with a cohesive zone model to evaluate the validity of the experimental data. It is shown that the fracture loads obtained by the CCM have the best agreement with the experimental ones comparing with the other data reduction approaches. To study the effect of geometry on the CCM results, ENF specimens with different adhesive thicknesses, substrate thicknesses and span lengths were also considered in this study, and some general conclusions are made about the geometrical parameters effect on the Mode II fracture energy.     

Numerical analysis of the ENF and ELS tests applied to mode II fracture characterization of cortical bone tissue

The objective of this work is to verify numerically the adequacy of the ENF and the ELS tests to determine the fracture toughness under mode II loading of cortical bovine bone tissue. A data‐reduction scheme based on the specimen compliance and the equivalent crack concept is proposed to overcome the difficulties inherent to crack monitoring during its growth. A cohesive damage model was used to simulate damage initiation and growth, thus assessing the efficacy of the proposed data‐reduction scheme. The influences of the initial crack length, local strength and toughness on the measured fracture energy were analysed, taking into account the specimen length restriction. Some limitations related to spurious influence on the fracture process zone of the central loading in the ENF test, and clamping conditions in the ELS test were identified. However, it was verified that a judicious selection of the geometry allows, in both cases, a rigorous estimation of bone toughness in mode II.     

Tapered beam on elastic foundation model for compliance rate change of TDCB specimen

A modified beam theory is developed to predict compliance rate change of tapered double cantilever beam (TDCB) specimens for mode-I fracture of hybrid interface bonds, such as polymer composites bonded to wood. The analytical model treats the uncracked region of the specimen as a tapered beam on generalized elastic foundation (TBEF), and the effect of crack tip deformation is incorporated in the formulation. A closed-form solution is obtained to compute the compliance and compliance vs. crack length rate change. The present TBEF model is verified with finite element analyses and experimental calibration data of compliance for wood-wood and wood-composite bonded interfaces. The compliance rate change can be used with experimental critical fracture loads to determine the respective critical strain energy release rates or fracture toughness of interface bonds. The present analytical model, which accounts for the crack tip deformation, can be efficiently and accurately used for compliance and compliance rate-change predictions of TDCB specimens and reduce the experimental calibration effort that is often necessary in fracture studies. Moreover, the constant compliance rate change obtained for linear-slope TDCB specimens can be applied with confidence in mode-I fracture tests of hybrid material interface bonds.     

Analysis of beam-type fracture specimens with crack-tip deformation

A novel bi-layer beam model is developed to account for local effects at the crack tip of a bimaterial interface by modeling a bi-layer composite beam as two separate shear deformable beams. The effect of interface stresses on the deformations of sub-layers, which is referred to as the elastic foundation effect in the literature, is considered in this model by introducing two interface compliance coefficients; thus a flexible joint condition at the crack tip is considered in contrast to the rigid joint condition used in the conventional bi-layer model. An elastic crack tip deformable model is presented, and the closed-form solutions of local deformation at the crack tip are then obtained. By applying this novel crack tip deformation model, the new terms due to the local deformations at the crack tip, which are missing in the conventional composite beam solutions of compliance and energy release rate (ERR) of beam-type fracture specimens, are recovered. Several commonly used beam-type fracture specimens are examined under the new light of the present model, and the improved solutions for ERR and mode mixity are thus obtained. A remarkable agreement achieved between the present and available solutions illustrates the validity of the present study. The significance of local deformation at the crack tip is demonstrated, and the improved solutions developed in this study provide highly accurate predictions of fracture properties which can actually substitute the full continuum elasticity analysis such as the finite element analysis. The new and improved formulas derived for several specimens provide better prediction of ERR and mode mixity of beam-type fracture experiments.*Author for correspondence (E-mail address: qiao@uakron.edu)     

Analysis of stitched laminated ENF specimens for interlaminar mode II fracture toughness

Two simple micromechanics based models are proposed to predict the effect of through-thickness reinforcement (stitching) on the improvement of delamination crack growth resistance in end-notched flexure (ENF) specimens. In the first model, it is assumed that stitches stretch elastically and then rupture when the load carried approaches the failure load. In the second model, it is assumed that stitches are discontinuous and that the stitch thread-matrix interface is completely frictional. Approximate closed form solutions for energy release rates are obtained, and the effects of stitch density, matrix-stitch thread interfacial shear stress, stitch thread diameter, volume fraction of stitches, critical energy release rate and Young's modulus are then examined. A simple design study for sizing the ENF specimen to minimise geometric nonlinear response is presented. The influences of interlaminar shear deformation and friction between the crack surfaces on the strain energy release rate are examined.     

Analysis of unidirectional (0°) fiber-reinforced laminated composite double cantilever beam specimen using higher order beam theories

Mathematical models, for the stress analysis of unidirectional (0°) fiber-reinforced laminated composite double cantilever beam (DCB) specimen using classical beam theory, first and higher order shear deformation beam theories, have been developed to determine the mode I strain energy release rate (SERR) for unidirectional composites. In the present study, appropriate matching conditions at the crack tip of the DCB specimen have been derived by using variational principles. SERR has been calculated using compliance method. In general, the performance of shear deformation beam models of DCB specimen with variationally derived matching conditions at the crack tip is good in determining the SERR for medium to long crack lengths. Performance of higher order shear deformation beam model (having quadratically varying transverse displacement over the thickness) of DCB specimen, with non-variationally derived matching conditions at the crack tip, is good in determining the SERR for all the crack lengths in comparison with the available theoretical and finite element solutions in the literature. Higher order shear deformation beam theories having varying transverse displacement over the thickness are more appropriate to analyze DCB specimen as they predict the appropriate nature of the interlaminar normal stress at the crack tip and its distribution ahead of the crack tip.     

Fracture characterization of Carbon fiber-reinforced polymer-concrete bonded interfaces under four-point bending

A combined analytical and experimental approach is presented to characterize both mode-II and mixed mode fracture of Carbon fiber-reinforced polymer-concrete bonded interfaces under four-point bending load, and closed-form solutions of compliance and energy release rate of the mode-II (four-point symmetric end-notched flexure) and mixed (four-point asymmetric end-notched flexure) mode fracture specimens are provided. The transverse shear deformation in each sub-layer of bi-material bonded beams is included by modeling each sub-layer as an individual first order shear deformable beam, and the effect of interface crack tip deformation on the compliance and energy release rate are taken into account by applying the interface deformable bi-layer beam theory (i.e., the flexible joint model). The improved accuracy of the present analytical solutions for both the compliance and energy release rate is illustrated by comparing with the solutions predicted by the conventional rigid joint model and finite element analysis. The fracture of Carbon fiber-reinforced polymer-concrete bonded interface is experimentally evaluated using both the four-point symmetric and asymmetric end-notched flexure specimens, and the corresponding values of critical energy release rates are obtained. Comparisons of the compliance rate-changes and resulting critical energy release rates based on the rigid joint model, the present theoretical model, and numerical finite element analysis demonstrate that the crack tip deformation plays an important role in accurately characterizing the mixed mode fracture toughness of hybrid material bonded interfaces under four-point bending load. The improved solution of energy release rates for the four-point symmetric and asymmetric end-notched flexure specimens by the flexible joint model can be used to effectively characterize hybrid material interface, and the fracture toughness values obtained for the Carbon fiber-reinforced polymer-concrete interface under mode-II and mixed mode loading can be employed to predict the interface fracture load of concrete structures strengthened with composites.     

Experimental Determination of Fracture Toughness of CFRP in Mode II by Raman Spectroscopy

Fracture toughness of Mode II of carbon fiber reinforced plastic (CFRP) was investigated using end notched flexure (ENF) specimens and a Raman coating method. Distribution of shear strain near the crack tip of CFRP was measured by Raman spectroscopy. A thin film of PbO on the measured surface of the ENF specimen was deposited by physical vapor deposition (PVD) as pretreatment to measure the strain by Raman spectroscopy. Fracture toughness of CFRP in Mode II was determined using the Raman results. The results coincided closely with those measured by the compliance method and FEM analyses (finite element method).     

Slice synthesis of a three dimensional “work of fracture” specimen

In the evaluation of brittle materials such as ceramics, the “work of fracture” specimen is enjoying considerable interest. That specimen, basically a beam bending specimen with rectangular cross section has inclined notches machined such that the remaining ligament is an isoceles triangle with the apex on the tension surface. The specimen is of particular interest because it tends to provide for slow stable crack propagation in such materials. The usual use of the test involves the recording of the load-deformation curve and the identification of the total area under the curve with the “work of fracture”. This then in turn is related to the fracture toughness.The present paper presents the results of an attempt to analyze this truly three dimensional problem in an approximate “two dimensional” fashion, treating the specimen as a series of slices and neglecting in effect the inter slice shear effects. Both beam bending and beam shear effects on compliance are considered. Plasticity effects are considered negligible for the class of brittle materials for which the test specimen is most frequently utilized. Comparison of experimental and analytical results is discussed. The observed stability characteristics of this “work of fracture” specimen is discussed in light of these results.     

Analytical modelling of dynamic fracture and crack arrest in DCB specimens under fixed displacement conditions

An analytical solution via the beam theory considering shear deformation effects is developed to solve the static and dynamic fracture problem in a bounded double cantilever beam (DCB) specimen. Fixed displacement condition is prescribed at the pin location under which crack arrest occurs. In the static case, at first, the compliance function of a DCB specimen is obtained and shows good agreement with the experimental results cited in the literature. Afterward, the stress intensity factor is determined at the crack tip via the energy release rate formula. In the dynamic case, the obtained governing equations for the model are solved supposing quasi‐static treatment for unstable crack propagation. Finally, a closed form expression for the crack propagation velocity versus beam parameters and crack growth resistance of the material is found. It is shown that the reacceleration of crack growth appears as the crack tip approaches the finite boundary. Also, the predicted maximum crack propagation velocity is significantly lower than that obtained via the Euler–Bernoulli theory found in the literature.     

Analysis of mode II interlaminar fracture and damage behavior in end notched flexure testing of GFRP woven laminates at cryogenic temperatures

Summary Interlaminar fracture and damage behavior of glass fiber reinforced polymer (GFRP) woven laminates at cryogenic temperatures is investigated for end notched flexure (ENF) pure Mode II configuration. The corrected beam theory (CBT) and finite element analysis (FEA) are used to calculate the Mode II interlaminar fracture toughness of ENF specimen at room temperature (RT), liquid nitrogen temperature (77 K) and liquid helium temperature (4 K). A FEA coupled with damage is also employed to study the damage distributions within the specimen and to examine the effect of damage on the Mode II energy release rate. The numerical results show that damage occurs at the matrix and causes a decrease in the energy release rate. The technique presented can be efficiently used for the characterization of cryogenic Mode II interlaminar fracture and damage behavior of woven laminate ENF specimens.     

Simultaneous binding and ex situ toughening concept for textile reinforced pCBT composites: Influence of preforming binders on interlaminar fracture properties

A new concept consisting of binding and ex situ toughening is proposed for manufacturing and toughening of textile reinforced pCBT composites. The present study assesses the influence of various preforming binders on interlaminar fracture properties. Interlaminar fracture toughness of textile reinforced pCBT composites was investigated under mode I and mode II deformation. A standard double cantilever beam (DCB) test and an end notched flexure (ENF) test based on a three-point bending test were applied to evaluate the interlaminar fracture toughness in mode I and mode II, respectively. The effect of binder type, filling content and preparation concept on fracture properties under the mentioned two deformation modes were discussed on the basis of morphology analysis of fracture sections with scanning electric microscopy. Flexural properties of the textile reinforced pCBT laminates prepared using the selected preforming binder were characterized for further verification of the performance of the proposed concept.     

Effect of compressive transverse normal stress on mode II fracture toughness in polymeric composites

Effect of transverse normal stress on mode II fracture toughness of unidirectional fiber reinforced composites was studied experimentally in conjunction with finite element analyses. Mode II fracture tests were conducted on the S2/8552 glass/epoxy composite using off-axis specimens with a through thickness crack. The finite element method was employed to perform stress analyses from which mode II fracture toughness was extracted. In the analysis, crack surface contact friction effect was considered. It was found that the transverse normal compressive stress has significant effect on mode II fracture toughness of the composite. Moreover, the fracture toughness measured using the off-axis specimen was found to be quite different from that evaluated using the conventional end notched flexural (ENF) specimen in three-point bending. It was found that mode II fracture toughness cannot be characterized by the crack tip singular shear stress alone; nonsingular stresses ahead of the crack tip appear to have substantial influence on the apparent mode II fracture toughness of the composite.     

MWCNTs对碳纤维非褶皱无纺布/环氧层合板力学性能的影响

将羧基化多壁碳纳米管(MWCNTs)添加到TDE85环氧树脂中,然后与碳纤维非褶皱无纺布(C-NCF)复合,制备成[0°/90°/+45°/-45°]_S层合板。采用三点弯曲、短梁剪切和单边切口弯曲测试方法以及动态力学性能分析方法,研究了不同含量的MWCNTs对层合板弯曲性能、层间剪切强度(ILSS)、Ⅱ型层间断裂韧性(G_ⅡC,以及玻璃态转变温度(T_g)的影响。并采用SEM对Ⅱ型试样的断面进行分析。结果表明,MWCNTs的加入显著提高了NCF层合板的力学性能。与空白试样相比,当MWCNTs在树脂中的质量分数为2.0%时,弯曲强度和模量分别提高了约26%和6%;当MWCNTs的质量分数为0.5%时,ILSS、G_ⅡC、T_g分别提高约14%、27%和14%。