Cohesive zone modelling of interface fracture near flaws in adhesive joints

A cohesive zone model is suggested for modelling of interface fracture near flaws in adhesive joints. A shear-loaded adhesive joint bonded with a planar circular bond region is modelled using both the cohesive zone model and a fracture mechanical model. Results from the models show good agreement of crack propagation on the location and shape of the crack front and on the initial joint strength. Subsequently, the cohesive zone model is used to model interface fracture through a planar adhesive layer containing a periodic array of elliptical flaws. The effects of flaw shape are investigated, as well as the significance of fracture process parameters. The results from simulations of fracture in a bond containing circular flaws show that localization of crack propagation in the vicinity of a flaw has significant effect on the joint strength and crack front shape. The localization effects are highly dependent on the fracture process zone width relative to the flaw dimensions. It is also seen that with increasing fracture process zone width, the strength variation with the flaw shape decreases, however, the strength is effected over a wider range of propagation.     

Cohesive Fracture Model Based on Necking

A linear hardening model together with a linear elastic background material is first used to discuss some aspects of the mathematical and physical limitations and constraints on cohesive laws. Using an integral equation approach together with the cohesive crack assumption, it is found that in order to remove the stress singularity at the tip of the cohesive zone, the cohesive law must have a nonzero traction at the initial zero opening displacement. A cohesive zone model for ductile metals is then derived based on necking in thin cracked sheets. With this model, the cohesive behavior including peak cohesive traction, cohesive energy density and shape of the cohesive traction–separation curve is discussed. The peak cohesive traction is found to vary from 1.15 times the yield stress for perfectly plastic materials to about 2.5 times the yield stress for modest hardening materials (power hardening exponent of 0.2). The cohesive energy density depends on the critical relative plate thickness reduction at the root of the neck at crack initiation, which needs to be determined by experiments. Finally, an elastic background medium with a center crack is employed to re-examine the shape effect of cohesive traction–separation curve, and the relation between the linear elastic fracture mechanics (LEFM) and cohesive zone models by considering the cohesive zone development and crack growth in the infinite elastic medium. It is shown that the shape of the cohesive curve does affect the cohesive zone size and the apparent energy release rate of LEFM for the crack growth in the elastic background material. The apparent energy release rate of LEFM approaches the cohesive energy density when the crack extends significantly longer than the characteristic length of the cohesive zone.     

Prediction of interfacial fracture between concrete and fiber reinforced polymer (FRP) by using cohesive zone modeling

Interfacial debonding between concrete and fiber reinforced polymer (FRP) is investigated through integrating experiments and computations. An experimental program is designed to evaluate interfacial fracture parameters of mode-I through cutting and bonding specimens with an FRP sheet. The evaluated fracture parameters, i.e. the fracture energy and the bonding strength, are confirmed by predicting FRP debonding failure with the cohesive zone modeling approach. In the cohesive zone model, a traction-separation relation for FRP debonding is proposed with a shape index while providing various initial descending slopes. Computational results of the cohesive zone model agree well with three-point bending test results for both FRP debonding and plain concrete fracture. Furthermore, both experimental and computational results demonstrate that the fracture energy and the cohesive strength are essential fracture parameters for the prediction of FRP debonding behavior.     

On the interfacial constitutive laws of mixed mode fracture with various adhesive thicknesses

Interfacial toughness and interfacial strength, as two critical parameters in an interfacial traction–separation law, have important effect on the fracture behaviors of adhesively bonded joints. In this work, the global and local fracture 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 under the mixed I/II mode loading condition. Basically, the measured interface laws based on the single leg bending (SLB) specimens reflect the equivalent and lumped interfacial fracture behaviors which include the cohesive fracture, damage and plasticity. Several new and interesting experimental results were obtained. 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 SLB joints.     

Derivation of Path Independent Coupled Mix Mode Cohesive Laws from Fracture Resistance Curves

A generalised approach is presented to derive coupled mixed mode cohesive laws described with physical parameters such as peak traction, critical opening, fracture energy and cohesive shape. The approach is based on deriving mix mode fracture resistance curves from an effective mix mode cohesive law at different mode mixities. From the fracture resistance curves, the normal and shear stresses of the cohesive laws can be obtained by differentiation. Since, the mixed mode cohesive laws are obtained from a fracture resistance curve (potential function), path independence is automatically satisfied. The effective mix mode cohesive law can have different shape and cohesive law parameters at different mode mixities so that the approach can be applied to various material failure models.     

Modelling the quasi-static behaviour of bituminous material using a cohesive zone model

This paper investigates the applicability of a cohesive zone model for simulating the performance of bituminous material subjected to quasi-static loading. The Dugdale traction law was implemented within a finite volume code in order to simulate the binder course mortar material response when subjected to indirect tensile loading. A uniaxial tensile test and a three-point bend test were employed to determine initial stress-strain curves at different test rates and the cohesive zone parameters (specifically, fracture energy and cohesive strength). Numerical results agree well with the experimental data up to the peak load and onset of fracture, demonstrating the value of the cohesive zone modelling technique in successfully predicting fracture initiation and maximum material strength.     

Influence of root curvature on the fracture energy of adhesive layers

Previously performed experiments to study the mode I behavior of an adhesive layer revealed an apparent increase in the fracture toughness when the adherends deformed plastically. Attempts to simulate the experiments are made; both with elastically and plastically deforming adherends. Thus, effects of the size of the process zone and the deformation of the adherends are revealed. The adhesive layer is modeled using finite elements with different approaches; cohesive elements and representative volume elements. The adherends are modeled with solid elements. With a long process zone, all models give good results as compared to the experiments. However, only the model with representative volume elements gives good agreement for large root curvatures and correspondingly short process zones. The results are interpreted by analyzing the deformation and mechanisms of crack propagation in the representative volume elements. It is shown that with large root curvature of the adherends, the in-plane stretching of the adhesive layer gives a substantial contribution to the fracture energy. A simple formula is derived and shown to give an accurate prediction of the effects of the root curvature. This result indicates the limits of conventional cohesive zone modeling of an adhesive layer of finite thickness.     

Effects of bondline thickness on Mode-I nonlinear interfacial fracture of laminated composites: An experimental study

Delamination or interfacial fracture is one of the most important issues for laminated composite structures. For the recent two decades, cohesive zone models (CZMs) have been receiving intensive attentions for modeling interfacial fracture of composite structures. Numerous global fracture tests have been conducted to measure the interfacial toughness of laminated composite plates. Some local tests have also been conducted to determine the interfacial traction–separation laws in adhesively bonded joints. However, most of these local studies focused on the interfacial fracture between metal adherends. Limited studies have addressed the local test of interfacial fracture between laminated composite plates. While it was well known that the bondline thickness has an important effect on interfacial fracture, very few studies investigated its effects on the local interfacial traction–separation laws of laminated composite plates in the literatures. In this work, both global and local tests are employed to investigate the effect of bondline thickness on the interfacial energy release rate, interfacial strength, and shape of the local interfacial traction–separation laws. Basically, the measured laws in the present work reflect the equivalent and lumped interfacial fracture behaviors which include the cohesive fracture, damage and plasticity. The experimentally determined traction–separation laws provide valuable baseline data for parameter calibrations in numerical models. The experimental results may also facilitate the understanding of bondline thickness dependent delamination of composite structures.     

Silica–silane coupling agent interphase properties using molecular dynamics simulations

In this paper, strength of the interphase between silica and glycidoxypropyltrimethoxy silane (GPS) coupling agent has been studied using molecular dynamics (MD) simulations. Silica–GPS interphase model is created by coupling the hydroxylated silica surface with monolayer-hydroxylated GPS molecules. The interphase model is subjected to mode-I (normal), mode-II (shear) and mixed-mode (normal–shear) mechanical loading to determine the interphase cohesive traction–separation (T–S) response (i.e., cohesive traction law). In MD simulations, atomic interactions are modeled with the reactive force field ReaxFF. Effects of interphase thickness and GPS bond density on the T–S response are studied. Simulation results indicate that interphase strength decreases with increase in the interphase thickness before attaining a plateau level at higher thickness. For a particular thickness, strength improves significantly with increase in the GPS bond density with the silica surface. Damage mode is adhesive at the silica interface at lower thickness and transitions to mixed mode and cohesive failure within the silane interphase at higher thickness. Mixed-mode T–S responses are bounded by the mode-I and mode-II responses. Characteristic parameters of the continuum-level potential-based cohesive zone model (PPR–CZM) are determined by fitting the MD-based mode-I and mode-II T–S responses with PPR–CZM functional. Development of the PPR–CZM parameters enables bridging length scales from the MD to the continuum scale for fracture modeling of the fiber–matrix interphase in composites subjected to mixed-mode loading. Results on mode-I and mode-II unloading are also presented.     

An experimental analysis of the fracture behavior of composite bonded joints in terms of cohesive laws

Modeling adhesive joints by means of cohesive models relies on the definition of cohesive laws. Although cohesive laws are known to be dependent on the loading mode, there is a lack of experimental evidences to describe this dependence. At the same time, the adherend and adhesive thicknesses are known to affect the fracture toughness of the bond, but their effect on the cohesive law has not been clarified. In this work, an experimental characterization of an epoxy adhesive is presented. The effect that the mode mixity has on the bond toughness and its cohesive law is compared against the effect of the adhesive and adherend thicknesses. The impact of these two latest parameters is shown to be minor if compared to the influence of the mode mixity, which mainly defines the cohesive law shape. Finally, the implications of these experimental findings on the numerical simulation of adhesive joints are discussed.     

Prediction of the Work of Separation and Implications to Modeling

Following Barenblatt's idea about the modeling of cracks by the use of a cohesive zone has attracted considerable attention. Recently, this model has also been applied to the prediction of ductile crack growth. For this case the present investigation aims to compare the predictions of a cohesive zone model with the predictions of the more physically based modified Gurson relation. The results demonstrate that in case of ductile fracture the parameters cohesive strength and energy may only be regarded as material properties within a small range of stress triaxialities. This finally leads to the conclusion that special care has to be taken if a cohesive zone model is used for the analysis of ductile fracture. The use of the modified Gurson relation predicts that the cohesive energy and strength do not remain constant throughout a crack growth analysis and their change is not known a priori. Improved cohesive zone models that take a coupling to the surrounding material into account may overcome this problem. This revised version was published online in July 2006 with corrections to the Cover Date.     

Influence of temperature and strain rate on cohesive properties of a structural epoxy adhesive

Effects of temperature and strain rate on the cohesive relation for an engineering epoxy adhesive are studied experimentally. Two parameters of the cohesive laws are given special attention: the fracture energy and the peak stress. Temperature experiments are performed in peel mode using the double cantilever beam specimen. The temperature varies from −40 to + 80°C. The temperature experiments show monotonically decreasing peak stress with increasing temperature from about 50 MPa at −40°C to about 10 MPa at + 80°C. The fracture energy is shown to be relatively insensitive to the variation in temperature. Strain rate experiments are performed in peel mode using the double cantilever beam specimen and in shear mode, using the end notch flexure specimen. The strain rates vary; for peel loading from about 10⁻⁴ to 10 s⁻¹ and for shear loading from 10⁻³ to 1 s⁻¹. In the peel mode, the fracture energy increases slightly with increasing strain rate; in shear mode, the fracture energy decreases. The peak stresses in the peel and shear mode both increase with increasing strain rate. In peel mode, only minor effects of plasticity are expected while in shear mode, the adhesive experiences large dissipation through plasticity. Rate dependent plasticity, may explain the differences in influence of strain rate on fracture energy between the peel mode and the shear mode.     

An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi‐brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well‐known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.     

Experimental and numerical investigations of mixed mode crack growth resistance of a ductile adhesive joint

Polymeric adhesive joints are extensively employed in various industrial and technological applications. It has been observed that in ductile adhesive joints, interface fracture is a common mode of failure which may involve stable crack propagation followed by catastrophic growth. The objectives of this paper are to investigate the effects of bondline thickness and mode mixity on the steady state energy release rate J_ss of such a joint. To this end, a combined experimental and numerical investigation of interfacial crack growth is carried out using a modified compact tension shear specimen involving two aluminium plates bonded by a thin ductile adhesive layer. A cohesive zone model along with a simple traction versus separation law is employed in the finite element simulations of crack growth. It is observed that J_ss increases strongly as mode II loading is approached. Also, it enhances with bondline thickness in the above limit. These trends are rationalized by examining the plastic zones obtained from the numerical simulations. The numerically generated J_ss values are found to agree well with the corresponding experimental results.     

成层混凝土的剪切强度和Ⅱ型断裂韧度

该文对成层混凝土的层间结合性能进行研究,通过直接剪切试验和双边缺口单边加载断裂试验测得混凝土层间的抗剪强度参数和Ⅱ型断裂韧度,研究了混凝土浇筑的层间间隔时间对成层混凝土层间结合性能的影响。试验结果表明:Ⅱ型断裂韧度K_Ⅱc与粘聚力c具有较好的线性相关性,而其与摩擦系数f不具有明显相关性。同时根据Ⅱ型断裂的定义假定断裂时满足摩尔库伦准则,推导出K_Ⅱc与c具有线性相关性。因此Ⅱ型断裂韧度可根据粘聚力直接估计出,避免了进行复杂而难以控制试验要素的Ⅱ型断裂试验。此方法为成层混凝土日后的断裂力学研究,提供了可参考的参数。     

Higher order Cauchy–Born rule based multiscale cohesive zone model and prediction of fracture toughness of silicon thin films

In this work, we extend the multiscale cohesive zone model (MCZM) (Zeng and Li in Comput Methods Appl Mech Eng 199:547–556, 2010), in which interatomic potential is embedded into constitutive relation to express cohesive law in fracture process zone, to include the hierarchical Cauchy–Born rule in the process zone and to simulate three dimensional fracture in silicon thin films. The model has been applied to simulate fracture stress and fracture toughness of single-crystal silicon thin film by using the Tersoff potential. In this study, a new approach has been developed to capture inhomogeneous deformation inside the cohesive zone. For this purpose, we introduce higher order Cauchy–Born rules to construct constitutive relations for corresponding higher order process zone elements, and we introduce a sigmoidal function supported bubble mode in finite element shape function of those higher order cohesive zone elements to capture the nonlinear inhomogeneous deformation inside the cohesive zone elements. Benchmark tests with simple 3D models have confirmed that the present method can predict the fracture toughness of silicon thin films. Interestingly, this is accomplished without increasing of computational cost, because the present model does not require quadratic elements to represent heterogeneous deformation, which is the inherent weakness of the previous MCZM model. Quantitative comparisons with experimental results are performed by computing crack propagation in non-notched and initially notched silicon thin films, and it is found that our model can reproduce essential material properties, such as Young’s modulus, fracture stress, and fracture toughness of single-crystal silicon thin films.     

The prediction of dynamic fracture evolution in PMMA using a cohesive zone model

A cohesive zone model was used in conjunction with the finite volume method to model the dynamic fracture of single edge notched tensile specimens of PMMA under essentially static loading conditions. In this study, the influence of the shape of the cohesive law was investigated, whilst keeping the cohesive strength and separation energy constant. Cohesive cells were adaptively inserted between adjacent continuum cells when the normal traction across that face exceeded the cohesive strength of the material. The cohesive constitutive law was therefore initially rigid, and the effective elasticity of the material was unaltered prior to insertion of the cohesive cells. Notch depths ranging from 2.0 to 0.1 mm were considered. The numerical predictions were compared with experimental observations for each notch depth and excellent qualitative and quantitative agreement was achieved in most cases. Following an initial period of rapid crack tip acceleration up to terminal velocities well below the Rayleigh wave speed, subsequent propagation took place at a constant rate under conditions of increasing energy flux to an expanding process region. In addition, attempted and successful branching was predicted for the shorter notches. It was found that the shape of the cohesive law had a significant influence on the dynamic fracture behaviour. In particular, the value of the initial slope of the softening function was found to be an important parameter. As the slope became steeper, the predicted terminal crack speed increased and the extent of the damage decreased.     

A cohesive XFEM model for simulating fatigue crack growth under mixed-mode loading and overloading

Structures are subjected to cyclic loads that can vary in direction and magnitude, causing constant amplitude mode I simulations to be too simplistic. This study presents a new approach for fatigue crack propagation in ductile materials that can capture mixed-mode loading and overloading. The extended finite element method is used to deal with arbitrary crack paths. Furthermore, adaptive meshing is applied to minimize computation time. A fracture process zone ahead of the physical crack tip is represented by means of cohesive tractions from which the energy release rate, and thus the stress intensity factor can be extracted for an elastic-plastic material. The approach is therefore compatible with the Paris equation, which is an empirical relation to compute the fatigue crack growth rate. Two different models to compute the cohesive tractions are compared. First, a cohesive zone model with a static cohesive law is used. The second model is based on the interfacial thick level set method in which tractions follow from a given damage profile. Both models show good agreement with a mode I analytical relation and a mixed-mode experiment. Furthermore, it is shown that the presented models can capture crack growth retardation as a result of an overload.     

A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk material

A bilinear cohesive zone model (CZM) is employed in conjunction with a viscoelastic bulk (background) material to investigate fracture behavior of asphalt concrete. An attractive feature of the bilinear CZM is a potential reduction of artificial compliance inherent in the intrinsic CZM. In this study, finite material strength and cohesive fracture energy, which are cohesive parameters, are obtained from laboratory experiments. Finite element implementation of the CZM is accomplished by means of a user-subroutine which is employed in a commercial finite element code (e.g., UEL in ABAQUS). The cohesive parameters are calibrated by simulation of mode I disk-shaped compact tension results. The ability to simulate mixed-mode fracture is demonstrated. The single-edge notched beam test is simulated where cohesive elements are inserted over an area to allow cracks to propagate in any general direction. The predicted mixed-mode crack trajectory is found to be in close agreement with experimental results. Furthermore, various aspects of CZMs and fracture behavior in asphalt concrete are discussed including: compliance, convergence, and energy balance.     

FRP-混凝土三点受弯梁损伤粘结模型有限元分析

该文采用双线形损伤粘结模型研究带切口FRP-混凝土三点受弯梁(3PBB)I型加载下的界面断裂性能。通过有限元参数分析,详细讨论了界面粘结强度、界面粘结能、混凝土抗拉强度、混凝土断裂能对3PBB受力性能的影响。数值模拟表明,FRP-混凝土界面有两种破坏形式,包括FRP-混凝土界面的损伤脱粘和界面混凝土的损伤脱粘破坏,与实验所观察到的现象一致。两种破坏形式尽管在宏观上均表现为界面脱粘,但破坏机制却不同。FRP-混凝土界面的损伤粘结模型与混凝土的拉伸塑性损伤模型相结合,不但再现了3PBB的宏观力学性能,数值分析得到的荷载-位移曲线接近实验结果,而且还能详细展示FRP-混凝土界面的损伤、断裂破坏过程以及损伤在FRP-混凝土界面和界面混凝土之间的转移,能够预测构件的承载力,有助于界面优化设计,这是单纯以能量判据预测裂纹发展的经典断裂力学方法所无法做到的。