A Direct Energy Balance Approach for Determining Energy Release Rates in Three and Four Point Bend End Notched Flexure Tests

A direct energy balance approach has been developed and used to determine energy release rates in three and four point bend end notched flexure tests. This study was performed in the context of the larger goal of understanding the wide variation in mode II toughnesses that have been obtained by the two tests when used on the same material. The primary motivation for developing the direct energy balance approach was to fully account for the effects of friction, large deformations, and other geometric nonlinearities that occur during these tests. The direct energy balance approach simulates crack advance as it occurs in physical testing. Most significantly, this approach accounts for frictional dissipation that occurs during crack advance, which is an effect that has been neglected in previous analyses of these tests. The direct energy balance approach is used to show that, for most cases of practical interest, the virtual crack closure technique is quite accurate, and predictions by this latter approach are only in error when moderately large geometric nonlinearities occur prior to crack advance. Based on these results, a “cut-off value,” expressed in terms of the maximum slope in the specimen as predicted by classical beam theory, is suggested for the upper limit of applicability of the virtual crack closure technique.     

Examination of the 4-ENF test for measuring the mode III R-curve of wood

The four-point bend end-notched flexure (4-ENF) test, which was originally developed for measuring the mode II R-curve, is thought to be applicable for measuring the mode III R-curve. In this study, a 4-ENF fracture test of spruce was conducted for obtaining the mode III R-curve, and the test method was numerically and experimentally analyzed. In the numerical analysis, three-dimensional finite element calculations were conducted to determine the distribution of the strain energy release rate along the delamination front by the virtual crack closure technique (VCCT). In the experimental analysis, the mode III R-curve was examined by the modified beam theory and compliance calibration methods of data reduction, which have been conventionally used for analyzing the mode I or mode II R-curve. In addition to these conventional data reduction methods, the strain at each loading point was measured, as was the loading-line displacement and critical load for crack propagation, and the R-curve was obtained by the combination of loading-line compliance, load-longitudinal strain compliance, and critical load for crack propagation, which is named the “compliance combination method”. The finite element analyses suggested that the pure mode III fracture state existed in the mid-section of the specimen in spite of the existence of a small mode II component at the free edges of the delamination front, and the mode III strain energy release rate component calculated by the VCCT coincided well with those obtained by the data reduction methods examined here. The actual R-curve obtained by the compliance combination method coincided well with those by the modified beam theory and compliance calibration methods when the strain was appropriately measured. From these results, therefore, the 4-ENF fracture test is a promising means for obtaining the mode III R-curve of wood.     

Strain-gauge compliance measurements near the crack tip for crack closure evaluation: Applicability and accuracy

The “plasticity-induced crack closure” phenomenon is the leading mechanism which controls the main effects on fatigue crack growth (e.g. stress ratio and load interaction effects) in metallic materials. Experimental tests, consisting mainly in global and/or local compliance measurements of the considered specimen, are usually carried out to quantify the physical phenomenon, but some aspects concerning the elaboration of acquired local compliance signals are not yet clear. From the analytical point of view, the so-called “Strip-Yield” model has proven to be the most versatile and powerful tool for estimating crack closure levels, but its application to structural steels is not yet straightforward due to the delicate calibration process.The present work tries to add some new ideas on the elaboration of local compliance experimental data, obtained from a M(T) specimen, simulating the measurements by means of an optimised Strip-Yield model implementation enriched by a novel module based on the Westergaard’s elastic complex potentials. The application of the method to the calibration of the Strip-Yield model has been already successfully faced elsewhere, so here the analytical results gave the possibility, together with dedicated FEM analyses, to investigate some of the different parameters and to state some conclusions about the reliability and applicability of local compliance measurements.     

A new method for determining mode II R-curve by the End-Notched Flexure test

A new method for obtaining the mode II R-curve in a End-Notched Flexure test is proposed in the present work. New compliance and energy release rate equations have been derived incorporating shear, local deformation and bending rotation effects.Mode II R-curve, which represents energy release rate as a function of crack extension, is obtained without optical determination of crack tip position. Crack length and energy release rate are determined at each point of the test based on experimental compliance until unstable advance occurs. In order to confirm the theoretical models, unidirectional carbon/epoxy specimens have been tested. Experimental data are evaluated by means of two reduction schemes: an existing data method named Corrected Beam Theory with effective crack length and the new method named Beam Theory including Bending Rotation effects. Shear and local deformation effects are included in both reduction schemes.Results concerning the determination of crack length without crack advance and during stable crack propagation are presented. The agreement between experimental values and theoretical results obtained by the new approach is excellent. Based on the accurate crack length determination at each point of the test, energy release rate is determined point to point and therefore R-curve is obtained.     

Computation analysis of interlaminar fracture of laminated composites

The interlaminar fracture behavior of laminated composites has been investigated. Contact and friction along the crack surfaces is taken into account in the finite element modeling of the delamination crack growth. Mode I, mode II and mixed mode loading conditions at the crack tip have been analyzed. For the cracks with contact and friction along the crack surfaces the virtual crack closure integral method is used in order to calculate separated energy release rates. Computational modeling and analysis of cross-ply laminates in three-point bending has been performed. Contact elements were used in order to prevent the material interpenetration along the crack surfaces. Comparison of the results obtained with and without using contact elements has been carried out and significant differences between the correlated values of the energy release rates have been found. The influence of the coefficient of friction on the energy release rates was found to be significant for short delamination crack lengths but insignificant for long cracks. Numerical analyses of experimental data obtained for unidirectionally reinforced glass fiber composites by double cantilever beam tests and by notched flexure tests have been carried out. For the double cantilever beam test geometric linear and nonlinear finite element analyses have been performed and critical energy release rates were calculated. For the end notched flexure test the contact problem has been solved taking into account that adjacent to the support contact and friction will occur. For the double cantilever beam test the critical energy release rates obtained by linear and nonlinear finite element solution has been compared with those from four different analytical data reduction methods (the area method, the Berry method, the modified beam analysis, the compliance method). For the end notched flexure test the critical energy release rates, calculated by the finite element analysis and taking into account contact and friction along the crack surfaces, have been compared with those obtained by conventional beam analysis. This revised version was published online in July 2006 with corrections to the Cover Date.     

Application of virtual crack closure integral method for interface cracks in low-k integrated circuit devices under thermal load

The virtual crack closure integral (VCCI) method is used to evaluate the stress intensity factor (SIF) and energy release rate (ERR) of an interface crack under thermal load. The VCCIs used in this work include the traditionally known “Mode I” and “Mode II” VCCIs and an additional coupling VCCI. The singularity element is used in the finite element method (FEM) implementation. The SIF and ERR calculated by the FEM are compared to the exact solution in the case of a joint dissimilar semi-infinite plates with double edge crack under thermal loading. The FEM result agrees well with the exact solution for relatively coarse meshes. The contribution of the mesh density and material mismatch to the FEM error is also explored. The VCCI method is used with the multi-scale FEM in a delamination risk assessment of a low-k integrated circuits device in flip-chip plastic ball grid array packages. The ERR is calculated for different package configurations and the prediction of the delamination risk is confirmed by reliability tests.     

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.     

Characterization of a cracked specimen with full-field measurements: direct determination of the crack tip and energy release rate calculation

The crack characterization in a pre-cracked aluminum specimen is investigated in this study using the grid method. The images of this grid are analyzed to provide the crack tip location as well as the displacement and strain fields on the surface of the specimen during a tensile test. Experimental data are used to calculate the energy release rate with the compliance method A fracture analysis is also performed using the invariant M $\uptheta $ integral in which both real and virtual displacement fields are introduced. This integral is implemented in the finite element software Cast3M. Both approaches give similar results in 2D case.     

A new fracture mechanics theory for orthotropic materials like wood

A new fracture mechanics theory is derived based on a new orthotropic-isotropic transformation of the Airy stress function, making the derivation of the Wu-“mixed mode I-II” fracture criterion possible, based on the failure criterion of a flat elliptic crack. As a result of this derivation, the right fracture energy and theoretical relation between mode I and II stress intensities and energy release rates are obtained.     

Fracture toughness testing with sections of cylinders

Finite element procedures were used to generate stress intensity, crack mouth compliance, and load point compliance for a segment from a cylindrical shell. Stress intensity and crack mouth compliance were found to be independent of span length for a large range of crack lengths when suitably normalized. Load point compliance was not truly independent of span length. The experimental procedures developed made it possible to perform tests with ease and reliability. Fracture test results showed good correlation with analytical predictions for the two alloys considered.     

Linear and nonlinear finite element analyses of unidirectional, symmetric single leg four-point bending tests

Geometrically linear and nonlinear finite element analyses are used to determine the energy release rate and mode ratio in simulated tests of unidirectional, symmetric, single leg bending specimens under four-point bending. It is shown that the finite diameter loading rollers that are typically used in practical test set-ups cause this test to be inherently nonlinear. The differences between the linear and nonlinear results are presented parametrically as a function of material properties, specimen thickness, roller diameter, crack length, and inner and outer span length. The perceived advantages and disadvantages of this test are compared to those of the more commonly used three-point single leg bending test. It is concluded that the four-point test provides an attractive alternative, as it can use the same type of test specimens and will produce toughnesses with essentially the same accuracy. Moreover, it allows non-precracked and precracked toughnesses, as well as R-curve data, to be obtained from each specimen tested.     

Influence of moisture on functional properties of climbing ropes

Recently developed experimental-numerical-analytical (ENA) methodology presented in Ref. [13] by Emri et al. based on a simple non-standard falling weight experiment, was used for mechanical characterization of “dry” and “wet” climbing ropes. Analysis of the maximum impact force; the visco-plastic component of rope deformation; the amount of stored, retrieved and dissipated energy; the stiffness of the rope; and the maximum value of the first derivative of the de-acceleration (jolt) showed that moisture significantly affects the functionality and durability of ropes. “Wet” ropes create larger maximum force, dissipate less energy, and generate larger retrieved energy that propels climbers in the opposite vertical direction. Properties of “wet” ropes are also more sensitive to number of repeated drops. Major changes of all physical quantities are, as a rule, observed during the first three to four drops. It has been shown that for the safety of climbers the most indicative properties are dissipated energy and jolt (first derivative of climber de-acceleration). The ratio of dissipated and retrieved energy, ψ=Wdys/Wret, could be used as a criterion for evaluation of the quality of climbing ropes.     

Simulation of cup-cone fracture using the cohesive model

The cohesive model is used for the prediction of the crack path during stable crack extension in ductile materials. The problem of crack-path deviation is investigated by means of simulation of crack propagation in a round tensile bar. The respective phenomenon is known as cup-cone fracture. It is shown that the model is able to predict the failure mechanism, which consists of normal fracture in the center and combined normal/shear fracture in the so-called “shear lips” at the specimen’s rim. The damage evolution and crack path predicted by the model are presented. The effect of the normal and shear failure parameters on the crack-deflection point and several aspects of the finite element mesh are discussed.     

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.     

A compliance-based approach to measure fracture resistance curve for surface cracked steel plates

This paper proposes a compliance-based approach to determine the fracture resistance $J$ – $R$ curve for surface cracks in high-strength steel (S690) plate specimens in a four-point-bend set up. This study extends the $\eta $ -approach used in the fracture test for the typical specimens with a through-thickness crack, to the surface cracks in plate specimens in calculating the energy release rate from the area below the measured moment versus the crack-plane rotation. The energy release rate, computed from the detailed finite element models using the domain integral approach, confirms a constant $\eta $ value for surface cracked steel plates. Coupled with the post-test sectioning, the unloading compliance method quantifies the extended crack-front profiles ahead of the fatigue pre-cracked surface notch, using the crack-size versus the compliance relationship derived from linear-elastic finite element analyses. The fracture resistance curves thus obtained remain similar at different locations along the crack front and comparable with the fracture resistance measured using a standard side-grooved compact tension specimen at a finite crack extension.     

低延性陶瓷材料高温KR-曲线行为

在高温下采用加载－卸载技术与柔度标定相结合的方法测量了SiC陶瓷材料的KR－曲线行为。结果表明：在高温下陶瓷材料KR随裂纹长度的增加而提高。随温度提高，陶瓷材料韧化指数m减小，裂纹阻力降低，其KR曲线越平坦。陶瓷材料在高温下增大的KR曲线行为，与裂纹的转向及裂纹尾区裂纹表面之间的相互作用密切相关，加载－卸载技术与柔度标定相结合的试验方法，对测量陶瓷材料高温KR－曲线是一种有效简便的方法。     

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.     

Effect of friction between fiber and matrix on the fracture toughness of the composite interface

A method of evaluating the interfacial fracture toughness using a single-fibre composite test is proposed. In contrast with the existing techniques, the method takes into account the phenomenon of friction between the fibre and matrix in the debonding zone. A general mathematical solution of the problem and modelling of the friction phenomenon are presented. Finite-element analysis using a contact statement is utilized for numerical evaluation of the stress–strain state. The influence of the coefficient of friction and interfacial debonding length is analysed in detail. It is shown that the friction reduces the calculated value of the elastic strain energy release rate for a given debonding length, relative to that obtained when friction is neglected. The magnitude of the difference depends on the coefficient of friction, the elastic properties of the fibre and matrix, and the characteristics of the debonding mechanism. Experimental data on debonding in a series of glass-epoxy single-fibre composites are analysed using the proposed numerical technique to obtain the effects of fibre surface treatments and fibre strain-to-break on the interfacial fracture toughness. © Kluwer Academic Publishers     

Derivation of effective stress intensity factors from measured crack face displacements

When a crack is subjected to cyclic shear-mode loading, crack faces interference wedge the crack open and reduce the effective ΔK_II. The methods proposed in the literature to prevent it or to derive the effective ΔK_I and ΔK_II are discussed. It is shown that when crack tip plasticity becomes important it tends to make displacements larger than those predicted by LEFM and to “hide” friction effects. Finite element simulations combining friction and plasticity can separate these two effects, but the analysis of force-sliding displacement loops derived from displacement field measurements based on image correlation is a more straightforward and efficient method.     

Influence of the crack plane asymmetry over GII results in carbon epoxy ENF specimens

In this paper unidirectional carbon epoxy End Notch Flexure samples (ENF) with different crack plane positions have been analyzed in order to study the influence of the degree of asymmetry over the load state at the crack tip. This study has been performed from different points of view: experimental, numerical approaches (Finite Element Models) and fractographic observations.From these studies it has been concluded that the deviation of the crack plane from the specimen midplane has no influence or little influence over the corresponding critical value of the energy release rate in mode II (G_II).In other words, the asymmetry of the crack plane does not induce mode I load state at the crack tip while the energy release rate related to the mode II load remains almost constant as the crack plane moves from the midplane.Fractographic observations have shown “hackle markings” structures typical from mode II fracture.Finally, a good agreement was found between experimental and numerical procedures as errors were always lower than 7%.