A 3D ductile constitutive mixed-mode model of cohesive elements for the finite element analysis of adhesive joints

In this paper, a new traction–separation law is developed that represents the constitutive relation of ductile adhesive materials in Modes I, II, and III. The proposed traction–separation laws model the elastic, plastic, and failure material response of a ductile adhesive layer. Initially, the independent-mode proposed laws (loading and fracture in Modes I, II, and III) are mathematically described and then introduced in a developed formulation that simulates the interdependency of the mixed-mode coupled laws. Under mixed-mode conditions, damage initiation is predicted with the quadratic stress criterion and damage propagation with the linear energetic fracture criterion. For verification and validation purposes of the proposed laws and mixed-mode model, steel adherends have been adhesively bonded with a structural ductile adhesive material in order to fabricate a series of single and double strap adhesive joint configurations. The specimens have been tested under uni-axial quasi-static load and the respective force and displacement loading history have been recorded. Corresponding numerical and experimental results have been compared for each joint case, respectively. Additionally, the developed stress fields (peel, in-plane, and out-of-plane shear) are presented as they evolve during the loading of both joint cases.     

Influence of Layer Thickness on Cohesive Properties of an Epoxy-Based Adhesive—An Experimental Study

Cohesive laws are determined for different layer thicknesses of an engineering adhesive. The shape of the cohesive law depends on the adhesive layer thickness. Of the two parameters of the cohesive law—the fracture energy and the strength—the fracture energy is more sensitive to thickness variation than the strength. The fracture energy in peel mode (Mode I) increases monotonically as the thickness is increased from 0.1 to about 1.0 mm. At an adhesive thickness of 1.5 mm, the fracture energy is slightly lower than for a 1.0 mm adhesive thickness, indicating a maximum between 1.0 and 1.5 mm. In shear mode (Mode II), the thickness dependence is not as strong, but an increasing trend in fracture energy with increasing adhesive thickness is evident. A slight decrease in strength with increasing adhesive thickness is found in both loading modes.     

Characterization of the mode I fracture energy of adhesive joints

This investigation was aimed at improving the calculation of the Mode I fracture energy, G_IC, of adhesive joints by incorporating the elasticity of the adhesive layer. It was also aimed at proposing ways to improve the calculation of G_IC over the existing standard for the measurement of that material property.

Our experiments were performed with a variety of aluminum specimens bonded with a large number of different adhesives in various thicknesses. All specimens were tested under mode I loading. In calculating G_IC, it is common to neglect the properties of the adhesive layer, especially when the compliance of the system is considered. The validity of this attitude was tested in the present study, and it was found to be in accord with the experimental results but only for joints bonded with a relatively thin layer of adhesive. A method for improving the calculation of the fracture energy of standard specimens, using a theoretical model for the compliance is proposed. This method comprises all the adhesive parameters and is appropriate for linear elastic joints.     

Effect of Flaws in the Adhering Interface on the Strength of Adhesive-bonded Butt Joints

This paper presents the strength of metal-to-metal bonded joints with a flaw in the interface between the adhesive layer and the adhering surface of adherend. The test specimens of butt joints are prepared by bonding two thin-wall metal tubes. The materials are carbon steel, aluminum alloy, brass and copper. The adhesive is epoxy resin. The tensile and shear strength of the joints are experimentally determined by subjecting the specimens to axial load and torsion for various flaw sizes and thickness of adhesive layers. Linear elastic fracture mechanics is applied to the experimental results. The stress intensity factors for a layered composite with a flaw in the interface are numerically calculated in terms of flaw size and loading by using Erdogan's formulas. The fracture stresses of joints with a flaw are predicted at the critical values of the stress intensity factors. The strength of joints without a flaw is also correlated with the stress intensity factors by use of a concept of “effective flaw size”.     

Effect of Bondline Thickness on Mixed-Mode Fracture of Adhesively Bonded Joints

The effect of bondline thickness on the critical strain energy release rate (CSERR) was investigated using aluminum adherends and an epoxy adhesive. Complete mixed Mode I/II fracture envelopes for adhesive thicknesses ranging from 0.203 to 1.52 mm were developed using double cantilever beam (DCB), mixed-mode bending (MMB), and end notch flexure (ENF) tests methods. Bondline thickness strongly affects the CSERR for different amounts of Mode II component. Pure Mode II had the largest CSERR and showed monotonic increase with bondline thickness, whereas pure Mode I had the lowest CSERR and exhibited non monotonic relationship with bondline thickness. The shape and size of the plastic zone that develops prior to crack propagation was predicted by finite element analysis. Variation of CSERR as a function of adhesive layer thickness was shown to be related to the size of the plastic zone for Mode I and MMB 50% fracture. No correlation was observed for the MMB 75% and Mode II, however.     

Failure load prediction of structural adhesive joints Part 1: Analytical method

This paper describes an analytical method for calculating the strain energy release rate of cracked adhesive joints. The calculations proceed from a knowledge of the reactions in the adherends at the end of the joint overlap. For joints with equal adherends, a simple method exists for determining the Mode I and Mode II components of the energy release rate. The equations make it relatively easy to apply fracture mechanics failure criteria to arbitrarily loaded adhesive joints. In a subsequent paper, it is shown that by treating uncracked joints as having a crack, with the crack tip coinciding with the location of the spew fillet, the load required to propagate a crack in a cracked joint serves as a reliable conservative estimate of the load required to propagate a crack in an uncracked joint. The present method is suitable, therefore, for failure load predictions of structural adhesive joints in design applications.     

Temperature and Loading Rate Effects on the Fracture Behaviour of Adhesively Bonded GFRP Nylon-6,6

The combined effect of varying test temperature and loading rate on the Mode II fracture toughness of plasma-treated GFRP Nylon-6,6 composites bonded using a silica-reinforced epoxy adhesive has been studied. End notch flexure tests have shown that the adhesive system used in this study offers a wide range of fracture energies that are extremely sensitive to changes in temperature and loading rate. Increasing the test temperature resulted in a substantial reduction in the Mode II fracture toughness of the adhesive, with the value of G_IIc at 60°C being approximately one-half of the room temperature value. In contrast, increasing the crosshead displacement rate at a given temperature has been shown to increase the value of G_IIc by up to 250%. Compression tests performed on bulk adhesive specimens revealed similar trends in the value of [sgrave]_y with temperature and loading rate. In addition, it was found that the plasma treatment employed in this study resulted in stable crack propagation through the adhesive layer under all testing conditions.

A more detailed understanding of the effect of varying temperature and loading rate on the failure mechanisms occurring at the crack tip was achieved using the double end notch flexure (DENF) geometry, which was considered in tandem with the fracture surface morphologies. Here, changes in the degree of matrix shear yielding and particle-matrix debonding were used to explain the trends in [sgrave]_y and G_IIc.     

Mode II Fracture Toughness of a Brittle and a Ductile Adhesive as a Function of the Adhesive Thickness

The main goal of this study was to evaluate the effect of the thickness and type of adhesive on the Mode II toughness of an adhesive joint. Two different adhesives were used, Araldite ® AV138/HV998 which is brittle and Araldite 2015 which is ductile. The end notched flexure (ENF) test was used to determine the Mode II fracture toughness because it is commonly known to be the easiest and widely used to characterize Mode II fracture. The ENF test consists of a three-point bending test on a notched specimen which induces a shear crack propagation through the bondline. The main conclusion is that the energy release rate for AV138 does not vary with the adhesive thickness whereas for Araldite 2015, the fracture toughness in Mode II increases with the adhesive thickness. This can be explained by the adhesive plasticity at the end of the crack tip.     

Mode-II fracture properties of CFRP/steel composite structure reinforced by short aramid fibers

Abstract

To obtain a good bonding strength of steel/CFRP adhesive joint, the steel surface was machined by grooving process. Short aramid fibers were mixed into the adhesive layer to achieve the further adhesion strength. In the pressing process of steel/CFRP specimen preparation, short aramid fibers with the diameter of several micrometers could be embedded in the grooved gap and the rough surface of CFRP. The higher strength aramid fibers had been not only improved interfacial strength of steel/epoxy and CFRP/epoxy, but also reinforced the adhesive layer due to the bridging activities of aramid fibers. In this study, Mode II fracture strength of grooved-steel/CFRP adhesive joints was investigated by end-notch bending test. The ultimate load and fracture energy of specimens have been improved by 15.7 and 6.8%, in comparison to specimens with smooth steel surface, respectively. The reinforcing mechanisms of CFRP/steel bonding joint as a result of short aramid fibers were discussed according to the failure modes of specimens, and scanning electron microscopy observation and experimental results were carried out.     

Measurement of Mode I and Mode II Fracture Properties of Wood-Bonded Joints

In this work, the fracture characterisation of wood-bonded joints under pure mode I and mode II loading was performed. The tested material was maritime pine (Pinus pinaster Ait.) bonded with an epoxy adhesive. Two fracture mechanical tests were chosen: the double cantilever beam (DCB) for opening mode I loading, and the end-notched flexure (ENF) for sliding mode II loading. The compliance-based beam method (CBBM) was used for both mode I and mode II fracture, since the Resistance-curves can be obtained directly from the global mechanical response of the specimens (load–displacement curve), without crack monitoring during propagation. This data reduction scheme was validated by direct comparison with the modified experimental compliance method (MECM).     

Characterising bonded joints with a thick and flexible adhesive layer–Part 1: Fracture testing and behaviour

Adhesively bonded structural joints have increasingly found applications in automotive primary structures, joining dissimilar lighter-weight materials. Low-modulus rubbery adhesives are attracting rising interest as an alternative to conventional rigid structural adhesives due to benefits such as the excellent impact resistance they provide. This paper is the first of two parts that investigate, both experimentally and numerically, the mechanical behaviour of a rubbery adhesive and the bonded joints to be used in a lightweight automobile structure. This part 1 paper characterises the fracture behaviour of the flexible adhesive layer with thick bondlines and presents a way to reliably determine the fracture mechanics parameters under a range of loading modes. Assessment of the various fracture tests indicated that DCB and SLB should provide mode I and mixed mode fracture energies but that the conventional ENF for mode II would not be practical for such compliant adhesive layers. Instead a cracked thick adherend shear specimen was developed and used. Reliable fracture energies were obtained from these specimens and a mixed mode fracture criterion developed for application in the part 2 paper.     

Mixed-mode fracture behaviour of refractories with asymmetric wedge splitting test. Part II: Experimental case study

The asymmetric wedge splitting test for performing mixed-mode loading and its numerical evaluation has been presented in a companion paper (Part I). In this work (Part II), the influences of various levels of mode II loading on damage behaviour of refractories with different brittleness were experimentally investigated by comparing mode I and mixed-mode fractures under symmetric and asymmetric wedge splitting loading with seven different wedge angles. The digital image correlation technique was also used for strain maps visualization as well as the deformation parameters acquisition.With the increase of asymmetric wedge angle, the fracture behaviour becomes unstable what is associated with steeper load-displacement curves, more instantaneous energy release and restrained fracture process zone development. The in-plane shear loading contributes to the accelerated extension of the crack tip and its deviation from central plane. Meanwhile, the co-existing local shear stresses caused by the refractory's heterogeneity lead to crack path deflection as well.     

Mixed-Mode Fracture Toughness of Ceramic Materials

An experimental technique whereby pure mode I, mode II, and combined mode I-mode II fracture toughness values of ceramic materials can be determined using four-point bend specimens containing sharp, through-thickness precracks is discussed. In this method, notched and fatigue-precracked specimens of brittle solids are subjected to combined mode I-mode II and pure mode II fracture under asymmetric four-point bend loading and to pure mode I under symmetric bend loading. A detailed finite element analysis of the test specimen is performed to obtain stress intensity factor calibrations for a wide range of loading states. The effectiveness of this method to provide reproducible combined mode I-mode II fracture toughness values is demonstrated with experimental results obtained for a polycrystalline Al₂O₃. Multiaxial fracture mechanics of the Al₂O₃ ceramic in combined modes I, II, and III are also described in conjunction with the recent experimental study of Suresh and Tschegg (1987). While the mode II fracture toughness of the alumina ceramic is comparable to the mode I fracture toughness K _Ic, the mode III fracture initiation toughness is 2.3 times higher than K _Ic. The predictions of fracture toughness and crack path based on various mixed-mode fracture theories are critically examined in the context of experimental observations, and possible effects of fracture abrasion on the apparent mixed-mode fracture resistance are highlighted. The significance and implications of the experimental methods used in this study are evaluated in the light of available techniques for multiaxial fracture testing of brittle solids.     

Effect of Joint Thickness and Residual Stresses on the Properties of Ceramic Adhesive Joints: II, Experimental Results

The residual stresses in joints were varied by varying the thickness and thermal expansion mismatch of joints prepared using alumina adherends and silicate glass adhesives. An apparent fracture toughness was measured by the single-edge notchedbeam method; strengths were measured in flexure, and the fracture surfaces were studied. Stress distributions, determined using the finite element method in Part I, together with the results of literature analyses for stresses in joints subjected to externally applied loads, were used to aid in interpreting the experimental observations. The measured fracture toughness and strength of ceramic adhesive joints increase with decreasing adhesive thickness and decrease with increasing thermal expansion mismatch (residual stress), both positive and negative.     

Failure behavior of adhesively bonded joints with a rubber modified epoxy adhesive under tensile-shear combined cyclic loading

Rubber-modified epoxy adhesives are used widely as structural adhesive owing to their properties of high fracture toughness. In many cases, these adhesively bonded joints are exposed to cyclic loading. Generally, the rubber modification decreases the static and fatigue strength of bulk adhesive without flaw. Hence, it is necessary to investigate the effect of rubber-modification on the fatigue strength of adhesively bonded joints, where industrial adhesively bonded joints usually have combined stress condition of normal and shear stresses in the adhesive layer. Therefore, it is necessary to investigate the effect of rubber-modification on the fatigue strength under combined cyclic stress conditions. Adhesively bonded butt and scarf joints provide considerably uniform normal and shear stresses in the adhesive layer except in the vicinity of the free end, where normal to shear stress ratio of these joints can cover the stress combination ratio in the adhesive layers of most adhesively bonded joints in industrial applications.

In this study, to investigate the effect of rubber modification on fatigue strength with various combined stress conditions in the adhesive layers, fatigue tests were conducted for adhesively bonded butt and scarf joints bonded with rubber modified and unmodified epoxy adhesives, wherein damage evolution in the adhesive layer was evaluated by monitoring strain the adhesive layer and the stress triaxiality parameter was used for evaluating combined stress conditions in the adhesive layer. The main experimental results are as follows: S–N characteristics of these joints showed that the maximum principal stress at the endurance limit indicated nearly constant values independent of combined stress conditions, furthermore the maximum principal stress at the endurance limit for the unmodified adhesive were nearly equal to that for the rubber modified adhesive. From the damage evolution behavior, it was observed that the initiation of the damage evolution shifted to early stage of the fatigue life with decreasing stress triaxiality in the adhesive layer, and the rubber modification accelerated the damage evolution under low stress triaxiality conditions in the adhesive layer.     

Analysis and design of adhesively bonded tee joints with a single support plus angled reinforcement

In this study, stress and stiffness analyses of adhesively bonded tee joints with a single support plus angled reinforcement were carried out using the finite element method. It was assumed that the adhesive had linear elastic properties. In actual bonded joints, some amount of adhesive, called the spew fillet, accumulated at the free ends of the adhesive layer; therefore, the presence of the adhesive fillet at the adhesive free ends was taken into account. The tee joints were analysed for two boundary conditions: a rigid base and a flexible base. In addition, each boundary condition was analysed for four loading conditions: tensile, compressive, and two side loadings. The stress analysis showed that both side loading conditions resulted in higher stress levels in the joint region in which the vertical plate and supports are bonded to each other, as well as in the adhesive layer in this region for both rigid and flexible base boundary conditions. In adhesively bonded joints, the joint failure is expected to initiate in the adhesive regions subjected to high stress concentrations; therefore, the peak adhesive stresses were evaluated in these critical regions. In the case of the rigid base, the peak adhesive stresses occurred at the corner of the vertical plate, which was bent at right angles, for the tensile and compressive loading conditions, and in the adhesive fillet at the upper free end of the vertical adhesive layer-vertical support interface for both the left and the right side loading conditions. However, in case of the flexible base, the peak adhesive stresses occurred in the adhesive fillet at the right free end of the horizontal adhesive layer-horizontal support interface for the tensile, compressive, and the right side loading conditions, and in the vertical adhesive fillet at the upper free end of the vertical adhesive layer-vertical support interface for the left side loading condition. Furthermore, the adhesive stresses showed a nonlinear variation in the direction of the adhesive thickness for all boundary and loading conditions. The left side loading condition, among the present loading conditions, which results in the highest adhesive stresses is the most critical loading condition for both boundary conditions. The effects of horizontal and vertical support lengths on the peak adhesive stresses and on the joint stiffness were also investigated and the appropriate support dimensions relative to the plate thickness were determined based on the stress and stiffness analyses.     

Analysis and design of adhesively bonded corner joints

In this study, the stress and stiffness analyses of corner joints with a single corner support, consisting of two plates, one of which plain and the other bent at right angles, have been carried out using the finite element method. It was assume that the plates and adhesive had linear elastic properties. Corner joints without a fillet at the free ends of the adhesive layer were considered. The joint support was analysed under three loading conditions, two linear and one bending moment. In the stress analysis, it was found that for loading in the y-direction and by bending moment, the maximum stresses occurred around the lower end of the vertical adhesive layer/ vertical plate interface; for loading in the x-direction, the maximum stresses occurred around the right free end of the horizontal adhesive layer/vertical plate interface. The effects of upper support length, support taper length and adhesive thickness on the maximum stresses have been investigated. Since the peel stresses are critical for this type of joint, a second corner joint with double corner support (i.e., one in which the horizontal plate is reinforced by a support that is an extension of the vertical plate) was investigated which showed considerable decreases in the stresses, particularly peel stresses. A third type of corner joint with single corner support plus an angled reinforcement member was investigated as an alternative to the previous two configurations. It was found that increasing the length and particularly the thickness of the angled reinforcement reduced the high peel stresses around the lower free end of the adhesive/vertical plate interface, but resulted in higher compressive stresses. In the stiffness analysis, the effects of the geometry of the joints, relative stiffness of adhesive/adherends and adhesive thickness were investigated under three loading conditions. For three types of corner joint, results were compared and recommended designs were determined based on the overall static stiffness of the joints and on the stress analysis.     

Fracture characterization of bonded joints using the dual actuator load apparatus

Mixed-mode I?+?II fracture characterization tests of steel-bonded joints were carried out with the dual actuator load apparatus using a previously developed data reduction scheme in order to obtain the fracture envelope. This test involves independent loading of the specimen arms of a clamped double cantilever beam, which allows for easy variation of the I?+?II mode mixity in fracture characterization through altering the applied displacement rates. Difficulties inherent to crack monitoring during its propagation and imperfections of initial crack manufacture are well managed with the proposed method. Three different cases corresponding to different mode mixities were tested. The experimental results revealed that a linear energetic criterion performs well in describing the fracture envelope of these bonded joints.     

Stress redistributions in adhesively bonded double-lap joints, with elastic–perfectly plastic adhesive behavior, subjected to axial lap-shear cyclic loading

A shear-lag model is developed in order to evaluate stress redistributions in double-lap joints under axial (tensile) lap-shear cyclic loading. The adherend materials exhibit linear elastic behavior, whereas the material of the adhesive layer satisfies the elastic–perfectly plastic shear stress–strain constitutive relation. The reference state (from which the stresses are redistributed) is based on the standard elastic–perfectly plastic shear-lag analysis for double-lap joints. The main conclusion of the current analysis is that, during unloading, shear stresses of opposite sign may develop in the plastic zones of the adhesive layer, at the ends of the overlap, without reversing the direction of the applied load. A simple model for evaluating the variation of the maximum peel stress in the adhesive layer, based on the variation of the peak shear stress, demonstrates that the sign of peel stresses may alternate, as well. Under cyclic (fatigue) loading, the range of the peak stresses in the adhesive layer is the basic parameter for the evaluation of the variation of the energy release rate and the associated crack growth rate in the overlap. In this framework, the current simplified analysis may provide a reference model for comparisons with experimental data or with results which are based on more complex numerical models. The current model can be readily extended to cover the cases of development of plastic zones in the adhesive layer with shear stresses and plastic strains of opposite sign (during unloading or during load direction change).     

A new curvature-driven delamination test for measuring the mode II toughness of composites

The measurement of the delamination toughness of composites requires fracture tests with well-characterized geometries. Because the delamination toughness is frequently used as a material selection parameter, it is necessary to differentiate between experimental artifacts, data scatter, and changes in the mechanism of delamination. This new Curvature-Driven Delamination (CDD) test for Mode II provides a direct, steady-state measurement of the pure Mode II delamination toughness without the compliance calculations inherent in other delamination test protocols. Like the Wedge-Driven Delamination (WDD) test for Mode I, the CDD Mode II test measures the toughness at controlled crack growth rates. The CDD Mode II test correlates well with the results of the established End-Notch Flexure (ENF) Mode II test but is not subject to the geometric instabilities of that test. Changes in the delamination mechanism are easily observed by the direct, continuous measurement of the toughness in the CDD Mode II test.