Analytical solutions for beam specimens used in fracture-mechanics testing of composites and adhesively-bonded joints typically use a beam on an elastic foundation model which assumes that a non-infinite, linear-elastic stiffness exists for the beam on the elastic foundation in the region ahead of the crack tip. Such an approach therefore assumes an elastic-stiffness model but without the need to assume a critical, limiting value of the stress,σmax, for the crack tip region. Hence, they yield asingle fracture parameter, namely the fracture energy,Gc. However, the corresponding value ofσmax that results can, of course, be calculated from knowledge of the value ofGc. On the other hand, fracture models and criteria have been developed which are based on the approach thattwo parameters exist to describe the fracture process: namelyGcandσmax. Hereσmax is assumed to be a critical,limiting maximum value of the stress in the damage zone ahead of the crack and is often assumed to have some physical significance. A general representation of the two-parameter failure criteria approach is that of the cohesive zone model (CZM). In the present paper, the two-parameter CZM approach has been coupled mainly with finite-element analysis (FEA) methods. The main aims of the present work are to explore whether the value ofσmax has a unique value for a given problem and whether any physical significance can be ascribed to this parameter. In some instances, both FEA and analytical methods are used to provide a useful crosscheck of the two different approaches and the two different analysis methods.

The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints

Pure mode II fracture characterization of composite bonded joints

The addition of liquid-filled urea-formaldehyde (UF) microcapsules to an epoxy matrix leads to significant reduction in fatigue crack growth rate and corresponding increase in fatigue life. Mode-I fatigue crack propagation is measured using a tapered double-cantilever beam (TDCB) specimen for a range of microcapsule concentrations and sizes: 0, 5, 10, and 20% by weight and 50, 180, and 460 μm diameter. Cyclic crack growth in both the neat epoxy and epoxy filled with microcapsules obeys the Paris power law. Above a transition value of the applied stress intensity factor ΔKT, which corresponds to loading conditions where the size of the plastic zone approaches the size of the embedded microcapsules, the Paris law exponent decreases with increasing content of microcapsules, ranging from 9.7 for neat epoxy to approximately 4.5 for concentrations above 10 wt% microcapsules. Improved resistance to fatigue crack propagation, indicated by both the decreased crack growth rates and increased cyclic stress intensity for the onset of unstable fatigue-crack growth, is attributed to toughening mechanisms induced by the embedded microcapsules as well as crack shielding due to the release of fluid as the capsules are ruptured. In addition to increasing the inherent fatigue life of epoxy, embedded microcapsules filled with an appropriate healing agent provide a potential mechanism for self-healing of fatigue damage.

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Fatigue crack propagation in microcapsule-toughened epoxy

The analysis of adhesively bonded joints started in 1938 with the closed-form model of Volkersen. The equilibrium equation of a single lap joint led to a simple governing differential equation with a simple algebraic equation. However, if there is yielding of the adhesive and/or the adherends and substantial peeling is present, a more complex model is necessary. The more complete is an analysis, the more complicated it becomes and the more difficult it is to obtain a simple and effective solution. The finite element (FE) method, the boundary element (BE) method and the finite difference (FD) method are the three major numerical methods for solving differential equations in science and engineering. These methods have also been applied to adhesive joints, especially the FE method. This book deals with the most recent numerical modelling of adhesive joints. Advances in damage mechanics and extended finite element method are described in the context of the FE method with examples of application. The classical continuum mechanics and fracture mechanics approach are also introduced. The BE method and the FD method are also discussed with indication of the cases they are most adapted to. There is not at the moment a numerical technique that can solve any problem and the analyst needs to be aware of the limitations involved in each case.

Advances in Numerical Modelling of Adhesive Joints

The determination of the mode II adhesive fracture resistance, GIIC, of structural adhesive joints: an effective crack length approach

Measuring the mode I adhesive fracture energy, GIC, of structural adhesive joints: the results of an international round-robin

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

In a previous letter [1] it was reported that the value of the adhesive fracture energy, Gc, was dependent upon the type of substrate material which was employed for the test specimens. Even though the locus of joint failure was observed to be cohesive, approximately in the center of the adhesive layer, and the measurements were made using specimens which obeyed the requirements of linear-elastic fracture-mechanics (LEFM) tests. The results from this earlier study are given in Table I. (A single-part, heat-curing rubber-toughened epoxypaste adhesive, designed for general-purpose use, was employed to bond the following substrate materials: mild steel, aluminum alloy and CFRP (a unidirectional carbon-fiber reinforced-plastic epoxy composite). Both double cantilever beam (DCB) and tapered double cantilever beam (TDCB) specimens were employed [1, 2]. The thickness of the adhesive layer was 0.4 mm.) The results given in Table I were explained in terms of the shape and size of the plastic zone, ahead of the crack tip within the adhesive layer, changing the value of Gc. The shape and size of the plastic zone was, in turn, considered to be affected by the value of the transverse modulus, E22, of the substrates used to form the joint. However, in a recent study involving the same grade of general-purpose epoxy-paste adhesive and range of substrates [3, 4], a far more pronounced effect of the substrate material on the value of Gc was noted in the case of the CFRP joints. All tests were again valid LEFM tests, and the crack always grew cohesively, approximately through the center of the adhesive layer. (Hence, as in the previous work, any differences in the values of Gc from using the different substrates cannot be explained in terms of any changes in the interfacial adhesion as the type of substrate was varied.) The results from this second study revealed no significant differences in the values of Gc for the mild steel and aluminum alloy joints, but the value of Gc for the CFRP joints was even lower than that recorded in the first study. For the CFRP joints the value of Gc was now only 202± 23 Jm−2. (In this second study, the adhesive joints were all manufactured in the same laboratory by the same personnel, but testing was carried out by twelve different laboratories, according to a protocol [2, 4], as part of a “round-robin” series of tests on structural adhesives tested under Mode I (tensile) loading.) It was considered that the very low value of Gc recorded for the CFRP joints from this second study was far too low to be explained by the mechanics arguments advanced previously [1]. Now, in the previous study, the adhesive had always been cured under the same temperature versus time cycle of 150 ◦C for 70 min in a fan-assisted oven, whereupon the heaters were turned off and the joints allowed to cool slowly over a period of about 5 h in the oven. This relatively long curing time was employed to try to ensure that the same state of cure was experienced by the adhesive layer in all the different types of joints. Thermocouples were buried into the adhesive layer to monitor accurately the temperature of the adhesive, during the curing cycle, in combination with the various types of substrate materials. Indeed, very similar cure cycles were recorded for the different types of joints. However, it was decided to investigate the extent of cure of the epoxy adhesive in more detail by measuring the glass transition temperature, Tg, of the cured adhesive layer. This was undertaken for adhesive samples removed from the different types of joints by using differential scanning calorimetry (DSC) and employing a heating rate of 20 ◦C min−1. Samples of adhesive were therefore carefully removed from the fracture surfaces of the failed joints and these samples were then analyzed. The values of Tg were defined on the normalized heat flow endotherms as the transition half-width obtained from a second heating cycle. For the “round robin” exercise, the values of the glass transition temperatures, Tg, for the samples of adhesive taken (a) from the mild steel joints was 103.0± 0.9 ◦C, (b) from the aluminum alloy joints was 100.6± 0.6 ◦C and (c) from the CFRP joints was 87.6± 1.3 ◦C. Thus, clearly the most striking feature is the very low value for the Tg for the adhesive layer in the CFRP joints, and these joints also possessed the correspondingly very low value of Gc of 202± 23 Jm−2.

The effect of the substrate material on the value of the adhesive fracture energy, Gc: Further considerations

This paper reports the results of research studies on measuring the mode II fracture toughness, GIIC, of structural adhesive joints via a Linear Elastic Fracture Mechanics (LEFM) test method. Adhesive joints were manufactured using carbon-fibre reinforced plastic substrates and these were bonded with one of two commercial structural epoxy adhesives. Mode II loading was achieved using the end-loaded split test geometry and the specimen dimensions were controlled to ensure the conditions of LEFM were not violated. Data analysis was achieved by the use of corrected beam theory and experimental compliances approaches. Values of GIIC were plotted against the measured crack length to yield the apparent mode II resistance curves (i.e. the mode II R-curves). These curves revealed a characteristic shape that required detailed interpretation. Uncertainties in the measured crack length values were identified as the most likely cause of the disagreement between the values of GIIC deduced via corrected beam theory and experimental compliance approaches. It is suggested that these uncertainties are due to the problem of misinterpreting microcracking as the main crack growing.

M. Paraschi

Papers

The determination of the mode II adhesive fracture resistance, GIIC, of structural adhesive joints: an effective crack length approach

Measuring the mode I adhesive fracture energy, GIC, of structural adhesive joints: the results of an international round-robin

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

The effect of the substrate material on the value of the adhesive fracture energy, Gc: Further considerations

On the Mode II Loading of Adhesive Joints