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.     

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。     

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)     

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.     

Shear and crack tip deformation correction for the double cantilever beam and three-point end-notched flexure specimens for mode I and mode II fracture toughness measurement of wood

Using specimens of western hemlock with various depths, double cantilever beam and three-point bend end-notched flexure tests were conducted to obtain the mode I and mode II fracture toughness. To correct the deflection caused by shearing and crack tip deformation, four conventional data reduction methods were examined as well as the compliance combination method, which was proposed by the author. In addition to the actual fracture tests, finite element analyses were conducted and the validity of the data reduction methods were also examined. The compliance combination method was more suitable for the data reduction than the others examined here because the fracture toughness was determined simply and appropriately while correcting the deflection caused by shearing as well as the crack tip deformation.     

Interface crack between two interface deformable piezoelectric layers

Based on an interface deformable piezoelectric bi-layer beam model, a bonded piezoelectric bi-material beam with an interface crack perpendicular to the poling axis is analyzed within the framework of the theory of linear piezoelectricity. The layer-wise approximations of both the elastic displacements and electric potential are employed, and each sub-layer is modeled as a single linearly elastic Timoshenko beam perfectly bonded together through a deformable interface. Using the impermeable crack assumption, the closed form solutions for the energy release rate (ERR) and crack energy density (CED) are derived for the layered piezoelectric beam subjected to combined uniformly distributed electromechanical loading. Based on superposition principle, both the ERR and CED and their components are all reduced to the functions of the crack tip loading parameters. Loading dependence of the total CED with respect to the applied electric field is manifested with the analytical results, showing that there is a transformation from an even dependence to an odd dependence for the normalized CED when the applied mechanical loading increases. Compared with the commonly used equivalent single layer model, the proposed analysis augments the crack driving force by alleviating the stress concentration along the interface and thus increases the loading parameters at the crack tip. The proposed model provides improved solutions for fracture analysis of piezoelectric layered structures and sheds light on the loading dependence of the fracture parameters (i.e., the ERR and CED) with respect to the applied electromechanical loadings.     

A theoretical note on mode-I crack branching and kinking

An energy-based fracture mode has been derived for the mode-I crack branching and kinking. The classic J_i-integral has been further explored by a new partial integral path and the analytical solution of the energy release rate for crack branching and kinking from a mode-I crack tip has been established. The crack branching/kinking angle has also been analytically derived. It shows that the Griffith’s theorem and conservation law can be applied to both mode-I crack extension and mode-I crack branching and kinking. The branching mechanism for quasi-static mode-I crack has been theoretically investigated. The branching toughness and the K-based criterion for crack branching have been defined. The crack branching phenomena predicted by the present model are in well agreement with the experimental observations reported in the literatures.     

Influence of dynamic fracture toughness on dynamic crack propagation

A dynamic finite element code was used in its “propagation mode” to assess the differences in dynamic crack propagation in a wedge-loaded (WL) single-edged notch (SEN) specimen, a tapered double cantilever beam (TDCB) specimen and a rectangular double cantilever beam (RDCB) specimen. The dynamic fracture toughness, K_ID, vs the crack velocity, a, relations determined experimentally for WL-SEN, WL-TDCB and WL-RDCB specimens machined from Araldite B were used as dynamic fracture criteria and the resultant k_id variations with crack propagations in the three specimens were compared with the corresponding experimental results. While the specific K_ID vs /.a relations established for each specimen obviously yielded calculated k_ID which were in best agreement with the experimental K_ID for the respective specimen, the K_ID vs /.a relation for the large WL-SEN specimen provided the best overall fit between the calculated and measured K_ID variations with crack propagation in all three specimens.     

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.     

Dynamic fracture mechanics analysis of failure mode transitions along weakened interfaces in elastic solids

Dynamic fracture mechanics theory was employed to analyze the crack deflection behavior of dynamic mode-I cracks propagating towards inclined weak planes/interfaces in otherwise homogenous elastic solids. When the incident mode-I crack reached the weak interface, it kinked out of its original plane and continued to propagate along the weak interface. The dynamic stress intensity factors and the non-singular T-stresses of the incident cracks were fitted, and then dynamic fracture mechanics concepts were used to obtain the stress intensity factors of the kinked cracks as functions of kinking angles and crack tip speeds. The T-stress of the incident crack has a small positive value but the crack path was quite stable. In order to validate fracture mechanics predictions, the theoretical photoelasticity fringe patterns of the kinked cracks were compared with the recorded experimental fringes. Moreover, the mode mixity of the kinked crack was found to depend on the kinking angle and the crack tip speed. A weak interface will lead to a high mode-II component and a fast crack tip speed of the kinked mixed-mode crack.     

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.     

Delamination of thin bonded cement-based overlays: analytical analysis

This paper deals with an analytical approach for the prediction of debonding initiation between cement-based overlay and old concrete substrate under monotonous mechanical loading. Based on the linear elastic fracture mechanics, an available analytical model has been used. The calculations take into account the interlocking between two crack surfaces in the overlay. To validate the model, three point static bending tests on composite specimens were carried out. Assuming that the debonding initiation just occurs after the crack cutting the overlay layer reaches the overlay–substrate interface, the stress intensity factor of the debonding tip can be calculated, allowing prediction of stress fields near the interface debonding tip. Then with a criterion of debonding initiation and propagation depending on the interface tensile strength, the load associated could be determined and might be interesting for the design of thin bonded cement-based overlays. The adequateness of this analytical approach was verified by both experimental data and finite element calculations.     

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.     

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.     

Effect of adhesive thickness on fatigue and fracture of toughened epoxy joints – Part II: Analysis and finite element modeling

The effect of bondline thickness on the fatigue and fracture of aluminum adhesive joints bonded using a rubber-toughened epoxy adhesive was studied using finite element analysis. The fatigue data of Part I examined the dependence of the fatigue threshold and cyclic crack growth rate on the adhesive thickness under both mode-I and mixed-mode loading. The fracture data of Part I illustrated the relation between the adhesive thickness and the quasi-static crack initiation and steady-state critical strain energy release rates. These experimental trends are explained in terms of the effects of the adhesive thickness and the applied strain energy release rate on the stress distribution in the bondline, the stress triaxiality at the crack tip, and the plastic zone size in the adhesive layer.     

Novel beam analysis of end notched flexure specimen for mode-II fracture

A novel beam model of end notched flexure (ENF) specimen for mode-II fracture testing is presented. By applying the principle of superposition, the ENF specimen is modeled as two sub-problems: (1) an un-cracked beam under three-point bending; and (2) a skew symmetric cracked beam under shear traction on the crack surface. Due to skew-symmetry of sub-problem two, only the upper half of the beam is analysed, and based on compatibility of deformation, a shear compliance coefficient is introduced to establish beam deformation equation. Explicit and simple closed form solutions of compliance and strain energy release rate are obtained, and they compare well with existing finite element analyses. Compared to other available analytical methods of the ENF specimen, the present beam model is relatively simple and easy to use; further, it can be applied to other beam fracture specimen analysis (e.g., mixed mode fracture and bi-material interface specimen).     

The calculation of adhesive fracture energies in mode I: revisiting the tapered double cantilever beam (TDCB) test

Analytical corrections have been derived for a beam theory analysis for the adhesively bonded tapered double cantilever beam test specimen to account for the effects of beam root rotation and for the real, as opposed to idealised, profile of the beam as required experimentally. A number of adhesive-substrate combinations were tested according to a new test protocol and the new analysis method for data reduction is compared critically with the existing simple beam theory and experimental compliance approaches. Correcting the beam theory for root rotation effects is shown to be more important than correcting only for the effects of shear deformation of the substrates. Results from a finite element analysis, using a cohesive zone model, also showed close agreement with the proposed new corrected beam theory analysis method.     

The fracture behaviour of structural adhesives under high rates of testing

Structural adhesive joints were subjected to high loading rates in mode I and their resulting fracture behaviour was studied in detail. Joints were formed between unidirectional carbon-fibre epoxy composites and between aluminium alloy substrates bonded with a tough, single-part automotive adhesive (XD4600) from Dow Automotive. Double cantilever beam (DCB) and tapered double cantilever beam (TDCB) tests were performed, from quasi-static loading rates up to 15 m/s, and a test rig was developed incorporating high-speed video acquisition for the high-speed tests. A detailed data reduction strategy was developed to account for (i) the types of different fracture behaviour regimes encountered, (ii) the dynamic effects in the test data, and (iii) the contribution of kinetic energy in the specimen arms to the energy balance. Using the above data reduction strategy, increasing the test rate over six decades (from 10⁻⁵ to 10¹ m/s) was found to lead to a reduction in the value of the adhesive fracture energy, G_Ic, by about 40% of its quasi-static value, i.e. from 3.5 to about 2.2 kJ/m². Further, at quasi-static loading rates, the measured adhesive fracture energies were independent of substrate material and test geometry (i.e. DCB or TDCB). However, at faster loading rates, the TDCB tests induced higher crack velocities for a given loading rate compared with the DCB test geometry, and neither the test rate nor the crack velocity were found to be the parameter controlling the variation in G_Ic with increased test rate. Thus, an isothermal–adiabatic model was developed and it was demonstrated that such a model could unify the DCB and TDCB test results. Indeed, when the G_Ic values were plotted as a function of 1/√time, where the time was defined to be from the onset of loading the material to that required for the initiation of crack growth, the results collapsed onto a single master curve, in agreement with the isothermal–adiabatic model.     

An analytical investigation of debonding problems in beams strengthened using composite plates

This work deals with an enhanced analytical model for the analysis of typical edge debonding problems in concrete or steel beams strengthened/repaired with externally bonded composite laminated plates induced by beam/adhesive interface fracture phenomena. The strengthened system is viewed as composed by three physical different layers: the strengthened beam, the adhesive layer and the bonded plate. On the other hand, the structural model consists of two shear deformable mathematical layers, the upper one representing the beam and the lower one incorporating the adhesive layer and the bonded plate. Bonding conditions between layers are simulated by using the Lagrangian multipliers method and governing equations are obtained by a variational approach. In the context of a fracture mechanics approach, analytical solutions for both total and mode components of energy release rate are obtained by using stress resultant and strain discontinuities across at the crack tip. Closed form solutions are obtained for specific loading conditions and geometric configurations. Comparisons with predictions from very careful FE investigations point out the effectiveness of the proposed results which may form the basis for a design process taking into account properly of debonding failure modes triggered by interface fracture at the edge of the repairing composite plate. Finally, the significance of the paper relies in the analytical approach to the problem, which avoids the complexities commonly shared by FE-based methodologies, related to stress singularities and differences in length scales and in mechanical properties of the single components of the system.     

FRACTURE ENERGY OF WEAKLY REINFORCED CONCRETE BEAMS

Abstract— An experimental investigation was conducted on a series of simply supported concrete beams with an aim to determine the fracture energy of the composite beam and apply the concept of fracture mechanics to predict flexural strength. In order that a single crack from the tip of the premolded central notch would propagate, the beams were lightly reinforced with a large cover so that the moment capacity of the beam as unreinforced section would be greater than that of the reinforced section. Based on test data, compliance calibration and energy release rate ( G ₁) curves are presented for progressive cracking and a correlation between the critical energy release rate and ultimate moment capacity is suggested.