Application of the biaxial Iosipescu method to mixed-mode fracture of unidirectional composites

In this paper the biaxial Iosipescu test method has been used, employing specimens with a central precrack placed along the notch-root axis, to study the intralaminar failure properties of a unidirectional carbon/epoxy composite under mixed-mode (dominated by shear) loadings. A linear finite element analysis has been performed to determine the energy release rates and stress intensity factors for the central crack under various biaxial loading conditions. In addition, a series of simple and biaxial fracture experiments have been performed on the composite material. Numerical results indicate that the method is capable of generating a wide range of mixed-mode loading conditions at the crack tip for various loading angles and crack lengths. Using the numerical results, in conjunction with experimental data, the biaxial intralaminar failure process in the cracked Iosipescu specimens has been explained.     

Mixed-mode fatigue crack growth behaviour in aluminium alloy

Fatigue crack propagation tests in compact mixed-mode specimens were carried out for several stress intensity ratios of mode I and mode II, K_I/K_II, in AlMgSi1-T6 aluminium alloy with 3 mm thickness. The tests were performed in a standard servo-hydraulic machine. A linkage system was developed in order to permit the variation of the K_I/K_II ratio by changing the loading angle. Crack closure loads were obtained through the compliance technique. A finite element analysis was also done in order to obtain the K_I and K_II values for the different loading angles. Crack closure increases under mixed-mode loading conditions in comparison to mode-I loading due the friction between the crack tip surfaces. Moreover, the crack closure level increases with the K_I/K_II ratio decrease. Correlations of the equivalent values of the effective stress intensity factor with the crack growth rates are also performed. Finally, an elastic–plastic finite element analysis was performed to obtain the plastic zones sizes and shapes and model the effect of mixed-mode loading on crack closure.     

Three-dimensional fracture analysis of thin-film debonding

A simplified mixed-mode fracture analysis combining nonlinear thin-plate stress solutions with crack-tip elasticity results has been developed to account for local variations of G _I, G _II and G _III in thin-film debond problems associated with large film deformations. Membrane and bending stresses from the plate analysis are matched with the crack-tip singularity solution over a small boundary region at the crack tip where the effect of geometric nonlinearity is small. Local variations in each of the individual components of the energy release rate are directly related to the jump in these stresses across the crack border.Specific results are presented for 1-D and elliptical planeform cracks. Deformations were induced either by a pressure acting normal to the film surface or biaxial compression or tension stresses applied to the substrate in which the loading axes and debond axes coincide. The latter type of loading involves buckling of the delaminated film. The model predictions compare well with more rigorous solutions provided the film thickness is small compared to the debond dimensions. In all cases analyzed, G _III was negligible. The ratio G _I/G _II typically decreases with increasing load or film deformation, the rate was moderate for pressure loading while generally sharp for compression loading. Film-substrate overlap may occur for certain debond geometry and loading conditions. Prevention of this by the substrate may critically increase the energy available for crack propagation.     

Fatigue crack propagation in short-carbon-fiber reinforced plastics evaluated based on anisotropic fracture mechanics

The influence of fiber orientation on crack propagation was studied with single edge-notched specimens cut from injection-molded plates of fiber-reinforced polyphenylene sulfide (PPS). Fracture mechanics parameters were calculated by FEM based on anisotropic elasticity. For mode I crack propagation in specimens parallel (MD) and perpendicular (TD) to molding direction, difference in crack propagation rate, dc/dN, among specimens becomes small when correlated to a crack-tip-opening radius parameter, H_IΔG_I, where H_I is a compliance parameter. Including crack propagation under mixed loading, all the data tend to merge a single relation when correlated to total energy-release-rate range divided by Young’s modulus, ΔG_total/E_θ.     

The kinked interface crack

The integral equation of the kinked interface crack is solved numerically. The values of K _I, K _II and G for an interface crack with an infinitesimal kink are used to predict the kinking angle for two different material combinations under uniaxial tension.     

Fracture criterion for kinking cracks in a tri-material adhesively bonded joint under mixed mode loading

The fracture behavior of a composite/adhesive/steel bonded joint was investigated by using double cantilever beam specimens. A starter crack is embedded at the steel/adhesive interface by inserting Teflon tape. The composite adherend is a random carbon fiber reinforced vinyl ester resin composite while the other adherend is cold rolled steel. The adhesive is a one-part epoxy that is heat cured. The Fernlund-Spelt mixed mode loading fixture was employed to generate five different mode mixities. Due to the dissimilar adherends, crack turning into the adhesive (or crack kinking) associated with joint failure, was observed. The bulk fracture toughness of the adhesive was measured separately by using standard compact tension specimens. The strain energy release rates for kinking cracks at the critical loads were calculated by a commercial finite element analysis software ABAQUS in conjunction with the virtual crack closure technique. Two fracture criteria related to strain energy release rates were examined. These are (1) maximum energy release rate criterion (G_max) and, (2) mode I facture criterion (G_II = 0). They are shown to be equivalent in this study. That is, crack kinking takes place at the angle close to maximum G or G_I (also minimum G_II, with a value that is approximately zero). The average value of G_IC obtained from bulk adhesive tests using compact tension specimens is shown to be an accurate indicator of the mode I fracture toughness of the kinking cracks within the adhesive layer. It is concluded that the crack in tri-material adhesively bonded joint tends to initiate into the adhesive along a path that promotes failure in pure mode I, locally.     

The behaviour of cracks in elastic-plastic materials under plane normal and shear loadings

Cracks in structures are often subjected to complex loading conditions. The direction of the crack extension depends on the normal and the shear components of the load. This paper is based on the kinking behaviour of cracks taking elastic-plastic behaviour of materials into account. The J-integral and the mixed-mode components J _I and J _II were determined after having performed several finite element analyses for different loading conditions. The path independence of J, J _I and J II is investigated for both, the line integral proposed by Rice and the volume integral proposed by deLorenzi. For correctly determined crack deflection angles the J _II-component vanishes when a FE-model with a kinked crack is considered. Hence, cracks propagate perpendicularly to the local mode I load.     

Subcritical interlaminar crack growth in fibre composites exhibiting a risingR-curve

Subcritical crack growth behaviour has been evaluated in composite laminates based on uniaxial carbon fibres in poly(ether-ether ketone) matrices. Double cantilever beam (DCB) specimens have been employed to give mode I loading and it is first shown that the materials exhibit a risingR-curve, i.e. the value of the interlaminar fracture energy,G _IC, increases as the crack propagates through the specimens. Secondly, when a DCB specimen is held at a constant displacement, subcritical crack growth is found to occur. The velocity of the subcritical crack growth,v, has been measured using a load-relaxation technique. Hence, values of the crack velocity,v, have been obtained as a function of the strain-energy release rate,G _I applied during subcritical crack growth. Owing to the presence of theR-curve, these data have been measured at various stages during the development of theR-curve. The relationships betweenv andG _I are modelled using power-law expressions. Finally, it is considered that theR-curve behaviour is most likely caused by the fibre bridging which develops behind the crack tip as the delamination propagates through the specimen. Fibre bridging allows stress to be transferred across the crack faces, behind the advancing crack tip, and so results in a shielding of the stress field at the crack tip from the applied stress. Therefore, the expression ascertained for the relationship between the velocity,v, of subcritical crack growth and the corresponding value ofG _I has been further refined and modelled to account for the presence of fibre bridging.     

Short fatigue crack growth behavior under mixed-mode loading

Mixed-mode loading represents the true loading condition in many practical situations. In addition, most of the fatigue life of many components is often spent in the short crack growth stage. The study of short crack growth behavior under mixed-mode loading has, therefore, much practical significance. This work investigated short crack growth behavior under mixed-mode loading using a common medium carbon steel. The effects of load mixity, crack closure, and load ratio on short crack growth behavior were evaluated by conducting experiments using four-point bending specimens with several initial K _II /K _I mixed-mode ratios and two load ratios. Cracks were observed to grow along the paths with very small K _II /K _I ratios (i.e. mode I). The maximum tangential stress criterion was used to predict the crack growth paths and the predictions were found to be close to the experimental observations. Several parameters including equivalent stress intensity factor range and effective stress intensity factor range were used to correlate short crack growth rates under mixed-mode loading. Threshold values for short cracks were found to be lower than those for long cracks for all the mixed-mode loading conditions. Crack closure was observed for the entire crack length regime with all load mixity conditions at R ≈ 0.05 and for short crack regime under high load mixity condition at R = 0.5. Several models were used to describe mean stress effects and to correlate crack growth rate data.     

The Transfer of Matrix Toughness to Composite Mode I Interlaminar Fracture Toughness in Glass-Fibre/vinyl Ester Composites

The transfer of matrix toughness to composite mode I interlaminar fracture toughness (G _Ic ) has been investigated in unidirectional glass-fibre reinforced composites with brittle and rubber-toughened vinyl ester matrices. Single-edge-notch bend (SENB) and double cantilever beam (DCB) specimens were used for matrix and composite G _Ic characteristion, respectively. The initial crack opening displacement rate was used as the parameter for comparison of G _Ic results. Matrix G _Ic was completely transferred to composite G _Ic for crack initiation (G _Ic-init) in the brittle-matrix composites, but in the toughened composites transfer was only partial due to the presence of fibres. The conclusion is that the maximum contribution to energy absorption by the matrix is more accurately reflected by G _Ic-init, and should be used for further assessment of the enhancing effect of fibre bridging during steady-state crack propagation, instead of matrix G _Ic . A plot of composite G _Ic for steady-state crack propagation, G _Ic-prop versus G _Ic-init indicates that the enhancing effect of fibre bridging is greater in the toughened composites. This enhancement is related to a larger deformation zone size in the toughened matrices.     

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.     

Accuracy assessment of a three-dimensional,crack tip element based approach for predicting delamination growth in stiffened-skin geometries

Experimental observations of delamination growth in two stiffened-skin geometries are compared to predictions made using a three-dimensional crack tip element based approach. Each geometry consists of a six-ply graphite/epoxy skin co-cured to a six-ply, hat-shaped stiffener containing a preimplanted teflon delamination between the skin and stiffener at the stiffener termination point. One stiffened-skin geometry was loaded in three-point bending and the other had in-plane tension loads applied to the skin. To predict delamination growth, a three-dimensional crack tip element analysis was first performed on each geometry in order to determine the total energy release rate, G, as well as its mode I, II and III components, G_I, G_II and G_III, respectively. These results were used to define a mode mix at each point along the delamination front, G_s/G, where G_s=G_II + G_III. To obtain the delamination toughness, G_c, it was assumed that G_c exhibits the same dependence on G_s/G as on G_II/G, where the results for G_c versus G_II/G were taken from an earlier experimental study. Next, a comparison of the energy release rate to the toughness at each position along the delamination front was performed, and these results were scaled appropriately in order to predict the sequence of loads and corresponding locations at which the delamination will advance. The predictions were then compared to experimental results that included c-scan images of the test specimens taken at each increment of observed growth, and very good quantitative and qualitative correlations were obtained for both geometries. These results indicate the practicality of, and considerable computational savings that may be achieved by, employing crack tip element analyses for delamination growth predictions in realistic structural geometries.     

Analytical and Numerical Solution of the Stress Field and the Dynamic Stress Intensity Factors in a Cracked Plate under Sinusoidal Loading

A cracked plate subjected to a sinusoidal loading perpendicular to its plane is considered, and the analytical solution of the dynamic vibration behavior of a plate, which allowed the determination of the stress field near the crack tip, is developed. A mixed mode of loading near the crack tip has been established and described with dynamic stress intensity factors K _I (z,t) and K _II (z,t) associated with modes I and II crack openings, respectively. To validate the analytical results, a finite element analysis (FEA) of a 1 × 1 m square plate with a thickness of 1 cm, having a middle crack of 10 cm in length, is made. The results have shown significant agreement between analytical and FEA findings.     

Weight functions for bimaterial interface cracks

An efficient approach using the analytically decoupled near-tip displacement solution for bimaterial interface cracks presented in this paper involves: (1) the calculation of the decoupled strain energy release rates G _I and G _II associated respectively with the decoupled stress intensity factors K _I and K _II and (2) the extension of Rice's displacement derivative representation of Bueckner's weight function vectors beyond the homogeneous media. It is shown that the stress intensity factors for a bimaterial interface crack predicted by the present approach agree very well with those solutions available in the literature. The computational efficiency is enhanced through the use of singular elements in the crack-tip neighborhood.As reported in the homogeneous case, the calculated weight function for a bimaterial interface crack is load-independent but depends strongly on geometry and constraint conditions. Due to the coupling nature of the stress intensity factors of a bimaterial interface crack, the invariant characteristics of the dimensionless weight function vectors are different from those of a crack in homogeneous material. In addition, the elastic constants of two constituents can significantly alter the weight function behavior for a cracked bimaterial medium.Due to the load-independent characteristic of the weight functions, the stress intensity factors for a bimaterial interface crack can be obtained accurately and inexpensively by performing the sum of worklike products between the applied loads and the weight functions for the cracked bimaterial body under any loading conditions once the weight functions are explicitly predetermined. The same calculation can also be applied for the identical cracked bimaterial medium with different constraint conditions by including the self-equilibrium forces that contain both the external loads and the reaction forces induced at the constraint locations. Moreover, the physical interpretation of the weight functions can provide a guidance for damage tolerant design application.     

Fatigue cracking simulation based on crack closure effects in Al-based sheet materials subjected to biaxial cyclic loads

Fatigue crack growth experiments have been carried out on cruciform specimens in the range of thickness 1.2–10 mm of Al-based alloys, loaded under constant (regular) and variable (irregular) amplitudes of uniaxial and biaxial loads, including sequences of various overloads. Different cases for crack closure effects are considered because of shear lips development, crack-growth direction re-orientation after multiparameters change of cyclic loads, by examining plastic blunting effect at a crack tip during an overload and interaction effects analyzing the crack retardation length and associated parameters together with their relationships. Crack closure effect because of rotation instability of material mesovolumes under biaxial compression–tension has suggested to analyse semi-elliptical cracks. Under biaxial cyclic loads in the range of load ratio-1.4 < λ < +1.5, and R-ratios from 0.05 to 0.8, for frequency variations ?, fatigue striation formation takes place beyond a crack-growth rate close to 4 × 10⁻⁸ m/cycle. The striation spacing and the crack-growth rate increase as the ?-angle of the out-of-phase biaxial loads increases (in the range of ? from 0° to 180°). Cycle loading parameters must be taken into account in order to describe the crack growth period when using a unified method that involves an equivalent stress intensity factor K_e=K_IF(λ,R,?,?). The values of F(λ,R,?,?) are determined. The calculated crack growth period (predicted using F(λ,R,?,?)) in regular and irregular cases of cyclic loads, including material cracking after overloads, is correlated with the experimental data, and the error is of the order of 15%.     

Critical Examination of the Iosipescu Shear Test as Applied to 0degrees Unidirectional Composite Materials

Several issues regarding the application of the shear and biaxial Iosipescu tests for the shear strength characterization of unidirectional composite materials are addressed in this article. First, the nonlinear effects of specimen sliding and geometric nonlinearity on the mechanical response of 0degrees standard unidirectional graphite/polyimide Iosipescu specimens with different loading conditions and loading block geometries have been investigated. Second, an attempt has been made to improve the Iosipescu shear test to eliminate normal compressive stresses in the specimen gauge section and at the same time prevent axial splitting. Finally, several Iosipescu shear and biaxial experiments have been performed to select proper specimen geometry and loading conditions for the shear strength measurements of unidirectional composites. The nonlinear effects are examined with respect to various coefficients of friction, displacements, loading angles, and fixtures (biaxial with short and modified biaxial with long loading blocks) using nonlinear finite-element techniques. It is shown that the effect of nonlinearity is small on the stresses at the center of the standard Iosipescu specimen, but significant for the stresses near the notch root up to 2 mm applied displacements. In some cases, significant differences in the stresses calculated for different coefficients of friction have been observed. All of these results are somewhat consistent for both fixtures, but with the stress components sigma x, sigma y, and sigma xy significantly lower in the standard Iosipescu specimens tested in the fixture with the long blocks. Numerical load/displacement diagrams show that specimen sliding and geometric nonlinearity have a negligible effect on reaction forces in the biaxial fixture, and a significant effect on the reaction forces in the modified biaxial fixture. Since the various combinations of the loading conditions evaluated in this study do not eliminate transverse compressive stresses in the gauge section of the standard Iosipescu specimens, a major improvement to the Iosipescu shear test has been proposed. Using an optimized specimen geometry subjected to biaxial shear/tension loading conditions, a state of almost uniform pure shear stress can be generated in 0degrees unidirectional composite Iosipescu specimens without the possibility of axial splitting along the fibers at the roots of the notches. However, it is shown in the experimental part of this study that for the optimized Iosipescu specimen, crushing at the inner loading blocks can significantly affect the shear intralaminar failure process. Only by reducing the cross-sectional area of the optimized Iosipescu specimen can the effect of crushing on the failure process be reduced without, however, highquality shear stress fields present in the gauge section at failure.     

An end-notched beam test for specific fracture energy in mixed mode in timber

A simple method for achieving stable crack propagation in a beam notched at a support is presented. The method allows the measurement of fracture energy in mixed modeG _f,mix. Results from a small number of laminated veneer lumber specimens suggest that there is a relationship betweenG _f,mix and density and that the ratioG _f,mix/(G _f,I+G _f,II) is about 0.35. Calculations of fracture energy in mode IG _f,I and mode, IIG _f,II did not coincide with values from mode-specific tests, indicating that an adjustment is necessary to take account of the interaction between the modes.     

Mixed mode fatigue and fracture in planar geometries: Observations on Keq and crack path modelling

The mixed mode I‐II fatigue and fracture is briefly reviewed, addressing experimental and numerical modelling aspects, and focusing on planar specimens. One major challenge concerns the determination of equivalent stress intensity factor (K_eq) in mixed mode situations. Several approaches were compared through the determination of K_eq/K_I over a wide range of values of K_I/K_II or K_II/K_I. Whereas all different approaches converge to the same value as K_I/K_II increases, the same does not happen for large K_II/K_I, where differences between values of K_eq persist. In the regions of 0 < K_I/K_II < 2 and 0 < K_II/K_I < 2, no stable trend of results can be defined. Experimental fatigue crack growth results are presented for Al alloy AA6082‐T6. Compact tension specimens, modified with holes, and four‐point bending specimens under asymmetrical loading promoting mixed mode situations, were subjected to fatigue crack growth tests, where crack path and crack growth rate were measured. The presentation of the fatigue crack growth data was made using a Paris law based upon K_eq. Differences in the Paris law constants were found for the different K_eq criteria. Recent developments in numerical techniques, as the implementation of the extended finite element method (XFEM) in finite element software packages allows to determine accurately crack paths in mixed mode fracture. This article highlights concepts for mixed‐mode fatigue and fracture and supporting data, identifying challenges still to be overcome.     

Interlaminar fracture analysis of composite DCB specimens

Mode I interlaminar fracture was studies using double cantilever beam (DCB) specimens of unidirectional carbon fiber/epoxy composites. An improved analytical model was introduced to study the crack opening displacement (COD), compliance (C) and fracture toughness (G _I) as a function of the material stiffness () ahead of the crack front. The COD expression was derived and compared with the COD profiles near the crack tip measured by moiré interferometry. Results showed that the COD expression can predict all the important qualitative features of the measured COD profiles; also, the quantitative agreement at the loading point was very good. It was found that plays an important role in evaluating the values of COD, C and GI.     

Calculation of transient strain energy release rates under impact loading based on the virtual crack closure technique

This paper describes an interface element to calculate the strain energy release rates based on the virtual crack closure technique (VCCT) in conjunction with finite element analysis (FEA). A very stiff spring is placed between the node pair at the crack tip to calculate the nodal forces. Dummy nodes are introduced to extract information for displacement openings behind the crack tip and the virtual crack jump ahead of the crack tip. This interface element leads to a direct calculation of the strain energy release rate (both components G_I and G_II) within a finite element analysis without extra post-processing. Several examples of stationary cracks under impact loading were examined. Dynamic stress intensity factors were converted from the calculated transient strain energy release rate for comparison with the available solutions by the others from numerical and experimental methods. The accuracy of the element is validated by the excellent agreement with these solutions. No convergence difficulty has been encountered for all the cases studied. Neither special singular elements nor the collapsed element technique is used at the crack tip. Therefore, the fracture interface element for VCCT is shown to be simple, efficient and robust in analyzing crack response to the dynamic loading. This element has been implemented into commercial FEA software ABAQUS^® with the user defined element (UEL) and should be very useful in performing fracture analysis at a structural level by engineers using ABAQUS^®.