Predicting the failure of ultrasonic spot welds by pull-out from sheet metal

A methodology for determining the cohesive fracture parameters associated with pull-out of spot welds is presented. Since failure of a spot weld by pull-out occurs by mixed-mode fracture of the base metal, the cohesive parameters for ductile fracture of an aluminum alloy were determined and then used to predict the failure of two very different spot-welded geometries. The fracture parameters (characteristic strength and toughness) associated with the shear and normal modes of ductile fracture in thin aluminum alloy coupons were determined by comparing experimental observations to numerical simulations in which a cohesive-fracture zone was embedded within a continuum representation of the sheet metal. These parameters were then used to predict the load–displacement curves for ultrasonically spot-welded joints in T-peel and lap-shear configurations. The predictions were in excellent agreement with the experimental data. The results of the present work indicate that cohesive-zone models may be very useful for design purposes, since both the strength and the energy absorbed by plastic deformation during weld pull-out can be predicted quite accurately.     

Ductile–brittle transitions in the fracture of plastically deforming,adhesively bonded structures. Part II: Numerical studies

To enable the effective and reliable use of structural adhesive bonding in automotive applications, the cohesive properties of a joint need to be determined over a wide range of loading rates. In this paper, a strategy for determining these properties has been described and used to analyze a set of experimental results presented in a companion paper. In the particular system studied, a crack growing in a toughened quasi-static mode could make a catastrophic transition to a brittle mode of fracture. The cohesive parameters for both the toughened and brittle modes of crack growth were determined by comparing numerical predictions from cohesive-zone simulations to the results of experimental tests performed using double-cantilever beam specimens and tensile tests. The cohesive parameters were found to be essentially rate-independent for the toughened mode, but the toughness dropped by a factor of four upon a transition to the brittle mode. The results of wedge tests were used as an independent verification of the cohesive parameters, and to verify that the quasi-static properties remained rate-independent to very high crack velocities corresponding to conditions of low-velocity impact. The effects of friction, and the use of the wedge test to determine cohesive parameters, were also explored.     

Extraction of cohesive-zone laws from elastic far-fields of a cohesive crack tip: a field projection method

A solution method of an inverse problem is developed to extract cohesive-zone laws from elastic far-fields surrounding a crack-tip cohesive zone. The solution method is named the “field projection method (FPM).” In the process of developing the method a general form of cohesive-crack-tip fields is obtained and used for eigenfunction expansions of the plane elastic field in a complex variable representation. The closing tractions and the separation-gradients at the cohesive zone are expressed in terms of orthogonal polynomial series expansions of the general-form complex functions. The series expansion forms a set of cohesive-crack-tip eigenfunctions, which is complete and orthogonal in the sense of the interaction J-integral in the far field as well as at the cohesive-zone faces. The coefficients of the eigenfunctions in the J-orthogonal representation are extracted directly, using interaction J-integrals in the far field between the physical field of interest and auxiliary probing fields. The path-independence of the interaction J-integral enables us to identify the cohesive-zone variables, i.e. tractions and separations, and thus the cohesive-zone constitutive laws uniquely from the far-field data. A set of numerical algorithms is developed for the inversion method and the results from numerical experiments suggest that the proposed algorithms are well suited for extracting cohesive-zone laws from the far-field data. The set includes methods to find the position and size of a cohesive zone. Further included are discussions on error analysis and stability of the inversion scheme.     

Bonded lap joints of composite laminates with tapered edges

This study presents a semi-analytical solution method to analyze the geometrically nonlinear response of bonded composite lap joints with tapered and/or non tapered adherend edges under uniaxial tension. The solution method provides the transverse shear and normal stresses in the adhesives and in-plane stress resultants and bending moments in the adherends. The method utilizes the principle of virtual work in conjunction with von Karman’s nonlinear plate theory to model the adherends and the shear lag model to represent the kinematics of the thin adhesive layers between the adherends. Furthermore, the method accounts for the bilinear elastic material behavior of the adhesive while maintaining a linear stress–strain relationship in the adherends. In order to account for the stiffness changes due to thickness variation of the adherends along the tapered edges, the in-plane and bending stiffness matrices of the adherents are varied as a function of thickness along the tapered region. The combination of these complexities results in a system of nonlinear governing equilibrium equations. This approach represents a computationally efficient alternative to finite element method. The numerical results present the effects of taper angle, adherend overlap length, and the bilinear adhesive material on the stress fields in the adherends, as well as the adhesives of a single- and double-lap joint.     

Cohesive zone modeling of interfacial stresses in plated beams

The elastic analysis of interfacial stresses in plated beams has been the subject of several investigations. These studies provided both first-order and higher-order solutions for the distributions of interfacial shear and normal stresses close to the plate end in the elastic range. The notable attention devoted to this topic was driven by the need to develop predictive models for plate end debonding mechanisms, as the early models of this type adopted debonding criteria based on interfacial stresses. Currently, approaches based on fracture mechanics are becoming increasingly established. Cohesive zone modeling bridges the gap between the stress- and energy-based approaches. While several cohesive zone analyses of bonded joints subjected to mode-II loading are available, limited studies have been conducted on cohesive zone modeling of interfacial stresses in plated beams. Moreover, the few available studies present complex formulations for which no closed-form solutions can be found. This paper presents an analytical cohesive zone model for the determination of interfacial stresses in plated beams. A first-order analysis is conducted, leading to closed-form solutions for the interfacial shear stresses. The mode-II cohesive law is taken as bilinear, as this simple shape is able to capture the essential properties of the interface. A closed-form expression for the debonding load is proposed, and the comparison between cohesive zone modeling and linear-elastic fracture mechanics predictions is discussed. Analytical predictions are also compared with results of a numerical finite element model where the interface is described with zero-thickness contact elements, using the node-to-segment strategy and incorporating decohesion and contact within a unified framework.     

Analytical solutions for adhesive composite joints considering large deflection and transverse shear deformation in adherends

This paper presents a novel formulation and analytical solutions for adhesively bonded composite single lap joints by taking into account the transverse shear deformation and large deflection in adherends. On the basis of geometrically nonlinear analysis for infinitesimal elements of adherends and adhesive, the equilibrium equations of adherends are formulated. By using the Timoshenko beam theory, the governing differential equations are expressed in terms of the adherend displacements and then analytically solved for the force boundary conditions prescribed at both overlap ends. The obtained solutions are applied to single lap joints, whose adherends can be isotropic adherends or composite laminates with symmetrical lay-ups. A new formula for adhesive peel stress is obtained, and it can accurately predict peel stress in the bondline. The closed-form analytical solutions are then simplified for the purpose of practical applications, and a new simple expression for the edge moment factor is developed. The numerical results predicted by the present full and simplified solutions are compared with those calculated by geometrically nonlinear finite element analysis using MSC/NASTRAN. The agreement noted validates the present novel formulation and solutions for adhesively bonded composite joints. The simplified shear and peel stresses at the overlap ends are used to derive energy release rates. The present predictions for the failure load of single lap joints are compared with those available in the literature.     

Stress-function variational method for interfacial stress analysis of adhesively bonded joints

High interfacial stresses at the free edges of adherends are responsible for the debonding failure of adhesively bonded joints (ABJs). In this paper, a general stress-function variational method is formulated to determinate the interfacial shear and normal (peeling) stresses in ABJs in high accuracy. By extending authors’ prior work in stress analysis of bonded joints (Wu and Jenson, 2011), all the planar stress components in the adherends and adhesive layer of an ABJ are expressed in terms of four unknown interfacial stress functions, which are introduced at the upper and lower surfaces of the adhesive layer. A set of governing ordinary differential equations (ODEs) of the four interfacial stress functions is obtained via minimizing the complimentary strain energy of the ABJ, which is further solved by using eigenfunctions. The obtained semi-analytic stress field can satisfy all the traction boundary conditions (BCs) of the ABJ, especially the stress continuity across the bonding lines and the shear-free condition at the ends of adherends and adhesive layer. As an example, the stress field in an adhesively single-sided strap joint is determined by the present method, whose numerical accuracy and reliability are validated by finite element method (FEM) and compared to existing models in the literature. Parameter studies are performed to examine the dependencies of the interfacial stresses of the exemplified ABJ upon the geometries, moduli and temperature change of the adherends and adhesive layer, respectively. The present method is applicable for scaling analysis of joint strength, optimal design of ABJs, etc.     

An investigation into the stresses in double-lap adhesive joints with laminated composite adherends

The mechanics of double-lap joints with unidirectional ([0₁₆]) and quasi-isotropic ([0/90/?45/45]_2S) composite adherends under tensile loading are investigated experimentally using moiré interferometry, numerically with a finite element method and analytically through a one-dimensional closed-form solution. Full-field moiré interferometry was employed to determine in-plane deformations of the edge surface of the joint overlaps. A linear-elastic two-dimensional finite element model was developed for comparison with the experimental results and to provide deformation and stress distributions for the joints. Shear-lag solutions, with and without the inclusion of shear deformations of the adherend, were applied to the prediction of the adhesive shear stress distributions. These stress distributions and mechanics of the joints are discussed in detail using the results obtained from experimental, numerical and theoretical analyses.     

Effects of adhesive thickness on global and local Mode-I interfacial fracture of bonded joints

The interfacial fracture of adhesively bonded structures is a critical issue for the extensive applications to a variety of modern industries. In the recent two decades, cohesive zone models (CZMs) have been receiving intensive attentions for fracture problems of adhesively bonded joints. Numerous global tests have been conducted to measure the interfacial toughness of adhesive joints. Limited local tests have also been conducted to determine the interface traction-separation laws in adhesive joints. However, very few studies focused on the local test of effects of adhesive thickness on the interfacial traction-separation laws. Interfacial toughness and interfacial strength, as two critical parameters in an interfacial traction-separation law, have important effect on the fracture behaviors of bonded joints. In this work, the global and local tests are employed to investigate the effect of adhesive thickness on interfacial energy release rate, interfacial strength, and shapes of the interfacial traction-separation laws. Basically, the measured laws in this work reflect the equivalent and lumped interfacial fracture behaviors which include the cohesive fracture, damage and plasticity. The experimentally determined interfacial traction-separation laws may provide valuable baseline data for the parameter calibrations in numerical models. The current experimental results may also facilitate the understanding of adhesive thickness-dependent interface fracture of bonded joints.     

EFFECT OF Z-PINS＇ DIAMETER, SPACING AND OVERLAP LENGTH ON CONNECTING PERFORMANCE OF CMC SINGLE LAP JOINT

The effect of through-thickness reinforcement by composite pins （Z-pins） on the static tensile strength and failure mechanisms of the joints made from ceramic matrix composite （CMC） is investigated. Overlap length of the single lap joint is 15 mm, 20 mm, 23 mm, 37 mm, and 60 mm, respectively. The experimental results indicate that the final failure modes of the joints can be divided into two groups, （a） the bond-line stops debonding until crack encounters Z-pins; and then the adherends break at the location of Z-pins, when overlap length is more than 20 mm; （b） the bond-line detaches entirely and Z-pins are drawn from adherends, when overlap length is equal to 15 mm. A simple efficient computational approach is presented for analyzing the benefit of through-thickness pins for restricting failure in the single lap joints. Here, the mechanics problem is simplified by representing the effect of the pins by tractions acting on the fracture surfaces of the cracked bond-line. The tractions are prescribed as functions of the crack displacement, which are available in simple forms that summarize the complex deformations to a reasonable accuracy. The resulting model can be used to track the evolution of complete failure mechanisms, for example, bond-line initial delamination and ultimate failure associated with Z-pin pullout, ultimate failure of the adherends. The paper simulates connecting performance of the single lap joints with different Z-pins＇ diameter, spacing and overlap length; the numerical results agree with the experimental results; the numerical results indicate enlarging diameter and decreasing spacing of Z-pins are in favor of improving the connecting performance of the joints. By numerical analysis method, the critical overlap length that lies between two final failure modes is between 18 mm and 19 mm, when Z-pins＇ diameter and spacing are 0.4 mm, 5 mm, respectively.     

Direct Measurement of the Cohesive Law of Adhesives Using a Rigid Double Cantilever Beam Technique

Engineering adhesive joints are being increasingly used in industry because of the advantages they offer over other joining methods such as fastening or welding. The development and the use of adhesives in a design environment require accurate mechanical tests in order to measure their strength and toughness. Standard techniques such as the shear lap test are commonly used to measure shear strength, but the results they produce generally depend on geometry and on initial defects within the bond line. Fracture tests such as the double cantilever beam (DCB) tests overcome these limitations, but rely on elasticity models and assumptions to determine toughness. In this study, we present a novel technique to directly determine the mode I fracture toughness of engineering adhesive joints as well as their full cohesive law, without any initial assumption on its shape. Our new method is remarkably simple in terms of experimental setup, execution and analysis. It is similar to the standard double cantilever beam (DCB) test with the difference that the material and dimensions of the beams are chosen so that they are assumed to be rigid compared to the bond line. In this rigid DCB (RDCB) technique the crack opening is known everywhere along the interface, which we use to compute the cohesive law of the adhesive directly from the load-displacement data obtained from experiment and the geometry of the RDCB specimen. The RDCB method is validated and applied to three typical commercial adhesives (polyurethane, epoxy, and silicone), to determine their cohesive law and fracture toughness.     

Cohesive-zone laws for void growth — I. Experimental field projection of crack-tip crazing in glassy polymers

A hybrid framework for inverse analysis of crack-tip cohesive-zone model is developed in this two-part paper to measure cohesive-zone laws of void growth in polymers by combining analytical, experimental, and numerical approaches. This paper focuses on experimental measurements of the cohesive-zone laws for two nonlinear fracture processes in glassy polymers, namely multiple crazing in crack-growth toughening of rubber-toughened high-impact polystyrene (HIPS) and crazing of steady-state crack growth in polymethylmethacrylate (PMMA) under a methanol environment. To this end, electronic speckle pattern interferometry (ESPI) is first applied to measure the crack-tip displacement fields surrounding the fracture process zones in these polymers. These fields are subsequently equilibrium smoothed and used in the extraction of the cohesive-zone laws via an analytical solution method of the inverse problem, the planar field projection method (P-FPM) [Hong, S., Kim, K.-S., 2003. Extraction of cohesive-zone laws from elastic far-fields of a cohesive crack tip: a field projection method. Journal of the Mechanics and Physics of Solids 51, 1267-1286]. Results show that the proposed framework of the P-FPM could provide a systematic way of finding the shape of the cohesive-zone laws governed by the different micro-mechanisms in the fracture processes. In HIPS, inter-particle multiple crazing develops and the craze zone broadens ahead of a crack-tip under mechanical loading. The corresponding cohesive-zone relationship of the multiple-craze zone is found to be highly convex, which indicates effectiveness of rubber particle toughening. It is also observed that the effective peak traction, 7 MPa, in the crack-tip cohesive zone of HIPS (30% rubber content) is lower than the uniaxial yield stress of 9 MPa, presumably due to stress multi-axiality effects. In contrast, in PMMA, methanol localizes the crack-tip craze, weakening the craze traction for craze-void initiation to about 9 MPa and the fibril pull-out stress to less than 6 MPa. This reduction in cohesive traction, coupled with a strongly concave traction-separation cohesive-zone relationship, signifies environmental embrittlement of PMMA. These experimentally determined cohesive-zone laws are compared with detailed numerical analyses of effective microscale-void growth ahead of a crack tip in Part II.     

EFFECT OF Z-PINS'' DIAMETER, SPACING AND OVERLAP LENGTH ON CONNECTING PERFORMANCE OF CMC SINGLE LAP JOINT

The effect of through-thickness reinforcement by composite pins (Z-pins) on the static tensile strength and failure mechanisms of the joints made from ceramic matrix composite (CMC) is investigated.Overlap length of the single lap joint is 15 mm,20 mm,23 mm,37 mm,and 60 mm,respectively.The experimcntal results indicate that the final failure modes of the joints can be divided into twó groups,(a) the bond-line stops debonding until crack encounters Z-pins;and then the adherends break at the location of Z-pins,when overlap length is more than 20 mm;(b) the bond-line detaches entirely and Z-pins are drawn from adherends,when overlap length is equal to 15 mm.A simple efficient computational approach is presented for analyzing the benefit of through-thickness pins for restricting failure in the single lap joints.Here,the mechanics problem is simplified by representing the effect of the pins by tractions acting on the fracture surfaces of the cracked bond-line.The tractions are prescribed as functions of the crack displacement,which are available in simple forms that summarize the complex deformations to a reasonable accuracy.The resulting model can be used to track the evolution of complete failure mechanisms,for example,bond-line initial delamination and ultimate failure associated with Z-pin pullout,ultimate failure of the adherends.The paper simulates connecting performance of the single lap joints with different Z-pins' diameter,spacing and overlap length;the numerical results agree with the experimental results;the numerical results indicate enlarging diameter and decreasing spacing of Z-pins are in favor of improving the connecting performance of the joints.By numerical analysis method,the critical overlap length that lies between two final failure modes is between 18 mm and 19 mm,when Z-pins' diameter and spacing are 0.4 ram,5 ram,respectively.     

Nonlinear interface shear fracture of end notched flexure specimens

The nonlinear analytical solutions of an end notched flexure adhesive joint or fracture test specimen with identical or dissimilar adherends are investigated. In the current study, a cohesive zone model (with arbitrary nonlinear cohesive laws) based analytical solution is obtained for the interface shear fracture of an end notched flexure (ENF) specimen with sufficiently long bond length. It is found that the scatter and inconsistency in calculating Mode II toughness may be significantly reduced by this model. The present work indicates that the Mode II toughness G_IIc under pure shear cracking condition is indeed very weakly dependent on the initial crack length. And this conclusion is well supported by the experimental results found in the literature. The parametric studies show that the interface shear strength is the most dominant parameter on the critical load. It is also interesting to note that with very short initial crack length and identical interface shear strength, higher Mode II toughness indeed cannot increase the critical load. Unlike the high insensitivity of critical load to the detailed shape of the cohesive law for Mode I peel fracture, the shape of the cohesive law becomes relatively important for the critical load of joints under pure Mode II fracture conditions, especially for joints with short initial crack length. The current study may help researchers deepen the understanding of interface shear fracture and clarify some previous concepts on this fracture mode.     

Fully-coupled nonlinear analysis of single lap adhesive joints

This paper presents novel closed-form and accurate solutions for the edge moment factor and adhesive stresses for single lap adhesive bonded joints. In the present analysis of single lap joints, both large deflections of adherends and adhesive shear and peel strains are taken into account in the formulation of two sets of nonlinear governing equations for both longitudinal and transverse deflections of adherends. Closed-form solutions for the edge moment factor and the adhesive stresses are obtained by solving the two sets of fully-coupled nonlinear governing equations. Simplified and accurate formula for the edge moment factor is also derived via an approximation process. A comprehensive numerical validation was conducted by comparing the present solutions and those developed by Goland and Reissner, Hart-Smith and Oplinger with the results of nonlinear finite element analyses. Numerical results demonstrate that the present solutions for the edge moment factor (including the simplified formula) and the adhesive stresses appear to be the best as they agree extremely well with the finite element analysis results for all ranges of material and geometrical parameters.     

Cohesive–frictional interface constitutive model

In the framework of numerical analysis of joined bodies, the present paper is devoted to the constitutive modeling, via an interface kinematic formulation, of mechanical behaviour of internal adhesive layers. The proposed interface constitutive model couples a cohesive behaviour, based on the damage mechanics theory, with a frictional one, defined in a non-associative plasticity framework. Namely, the interface formulation follows the transition of the adhesive material from the sound elastic condition to the fully cracked one. This formulation is able to model, by means of a specific interpretation of the damage variable and in a relevant mathematical setting, the interface intermediate mechanical properties, during the microcracks spreading process up to the discontinuity surface formation (macrocrack). The constitutive modeling is performed in fully compliance with the thermodynamic principles, in order to ensure the thermodynamic consistency requirement. In the present work, various monotonic and cyclic loading conditions are examined in order to show the main features of the constitutive formulation as well as several significant differences with respect to other existing models. Computational efficiency of the interface constitutive model is tested in a numerical application by FEM resolution strategy approach.     

软材料粘接结构界面破坏研究综述

软材料已经在软机器人、生物医学及柔性电子等各个领域得到广泛的应用. 实际应用中, 软材料多需要粘附于不同类型的基底上, 与之共同组成工程构件进而实现特定的功能, 粘接界面性能对构件的结构完整性与功能可靠性起着关键性作用. 本文对目前软材料粘接结构界面破坏行为方面的研究进行了系统总结. 首先通过与传统粘接结构的对比, 指出了“软界面”与“软基体”两种软材料粘接结构界面破坏行为的独特性及其物理本质. 接着分别总结了“软界面”与“软基体”两种粘接结构界面破坏行为的实验表征方面的研究成果, 对界面及基体黏弹性耗散对界面破坏机理的影响分别进行了分析. 然后从理论角度, 介绍了针对两种软材料粘接结构界面破坏行为的理论分析方法, 并对已建立的相关理论模型进行了总结. 之后以内聚力模型方法为基础, 介绍了软材料粘接结构界面破坏行为数值模拟方面的相关研究进展. 最后基于已有的研究成果, 提出了目前研究所面临的挑战, 并对可能的软材料粘接结构界面破坏的未来研究方向进行了讨论和展望.      

Delamination of an elastic film from an elastic–plastic substrate during adhesive contact loading and unloading

Adhesive contact between a rigid sphere and an elastic film on an elastic–perfectly plastic substrate was examined in the context of finite element simulation results. Surface adhesion was modeled by nonlinear springs obeying a force-displacement relationship governed by the Lennard–Jones potential. A bilinear cohesive zone law with prescribed cohesive strength and work of adhesion was used to simulate crack initiation and growth at the film/substrate interface. It is shown that the unloading response consists of five sequential stages: elastic recovery, interface damage (crack) initiation, damage evolution (delamination), film elastic bending, and abrupt surface separation (jump-out), with plastic deformation in the substrate occurring only during damage initiation. Substrate plasticity produces partial closure of the cohesive zone upon full unloading (jump-out), residual tensile stresses at the front of the crack tip, and irreversible downward bending of the elastic film. Finite element simulations illustrate the effects of minimum surface separation (i.e., maximum compressive surface force), work of adhesion and cohesive strength of the film/substrate interface, substrate yield strength, and initial crack size on the evolution of the surface force, residual deflection of the elastic film, film-substrate separation (debonding), crack-tip opening displacement, and contact instabilities (jump-in and jump-out) during a full load–unload cycle. The results of this study provide insight into the interdependence of contact instabilities and interfacial damage (cracking) encountered in layered media during adhesive contact loading and unloading.     

Characterization of a cohesive-zone model describing damage and de-cohesion at bonded interfaces. Sensitivity analysis and mode-I parameter identification

The identification of mode-I parameters of a cohesive-zone model for the analysis of adhesive joints is presented. It is based on an experimental–numerical methodology whereby the optimal parameters are obtained as the solution of a nonlinear programming problem. The data set for inverse analysis is provided either by local kinematic data, by global static data, or a combination of the two. Parameter sensitivities are computed via direct differentiation and identification exercises are discussed that show the effectiveness of the procedure and its stability with respect to noise and time–space sampling.     

A simple technique for determining yield strength of thin films

A technique to measure the yield strength of thin films has been developed which combines experimental observations of deflection and plastic deformation with finite element predictions of stress. This technique relies on integrated circuit technology to build bridge and cross beam test structures with a range of dimensions. Each structure is deflected in increments of 1 μm until the structure no longer elastically recovers upon release. In tandem with experimentally verified numerical predictions of force and stress, the yield strength of the thin film can be bounded between the highest elastic stress result and the lowest plastic stress result. For our test material of copper, this method provides a yield strength between 2.80 and 3.09 GPa, a value significantly larger than that for bulk copper, but consistent with thin film theory.