Subsonic and intersonic mode II crack propagation with a rate-dependent cohesive zone

A recent experimental study has demonstrated the attainability of intersonic shear crack growth along weak planes in otherwise homogeneous, isotropic, linear elastic solids subjected to remote loading conditions (Rosakis et al., Science 284 (5418) (1999) 1337). The relevant experimental observations are summarized briefly here and the conditions governing the attainment of intersonic crack speeds are examined. Motivated by experimental observations, subsonic and intersonic mode II crack propagation with a rate-dependent cohesive zone is subsequently analyzed. A cohesive law is assumed, wherein the cohesive shear traction is either a constant or varies linearly with the local sliding rate. Complete decohesion is assumed to occur when the crack tip sliding displacement reaches a material-specific critical value. Closed form expressions are obtained for the near-tip fields. With a cohesive zone of finite size, it is found that the dynamic energy release rate is finite through out the intersonic regime. Crack tip stability issues are addressed and favorable speed regimes are identified. The influence of shear strength of the crack plane and of a rate parameter on crack propagation behavior is also investigated. The isochromatic fringe patterns predicted by the analytical solution are compared with the experimental observations of Rosakis et al. (1999) and comments are made on the validity of the proposed model.     

混凝土黏聚开裂模型若干进展

黏聚模型是用来描述混凝土断裂行为的基本模型, 首先介绍了混凝土的黏聚开裂模型的基本概念,总结了确定黏聚区的本构方程的各种方法,即直接单轴拉伸测试、J积分方法、R曲线法、柔度法和逆推法.然后介绍了黏聚模型在I型和复合型裂纹问题、疲劳断裂问题中的应用以及黏聚模型与混凝土尺寸效应的关系.最后对黏聚开裂模型与桥联模型、带状裂缝模型进行了比较和总结, 指出了该模型存在的问题, 并对其以后的发展方向提出了建议.      

Sudden deceleration or acceleration of an intersonic shear crack

Sudden jumps in the crack tip velocity were revealed by numerical simulation (in both continuum/cohesive element and molecular dynamics approaches) and experiments for rapid shear cracking. The cracking velocity may accelerate from a sub-Rayleigh speed to the intersonic range, or from an intersonic speed to a higher one, when the reflected impact wave reloads the crack tip. On the other hand, the cracking velocity may decelerate from an intersonic speed to a lower one or recede to the sub-Rayleigh range when the fracture driving force declines. The velocity change encountered during intersonic cracking plays a different role from that in the acceleration or deceleration of a subsonic crack. A crack propagating at an intersonic speed would leave a shear wave trailing behind. When the crack decelerates or accelerates, the effect of the trailing wave will lead to a transition period from one steady-state solution of crack tip singularity to another. This investigation aims at quantifying these processes. The full field solution of an intersonic mode II crack whose speed changed suddenly from one velocity (intersonic or subsonic) to another (intersonic or subsonic) is given in closed form. The solution is facilitated via superposing a series of propagating crack problems that are loaded by dislocations to seal the unwanted crack-face sliding or by concentrated forces moving at various speeds to negate the crack-face traction. In contrast to the subsonic solution, the results in the intersonic case indicate that the elastic fields around the crack tip depend on the deceleration or acceleration history that is traced back over a long time. Singularity matching dictates the jump that may actually take place.     

Tensile and mixed-mode strength of a thin film-substrate interface under laser induced pulse loading

Laser induced stress waves are used to characterize intrinsic interfacial strength of thin films under both tensile and mixed-mode conditions. A short-duration compressive pulse induced by pulsed-laser ablation of a sacrificial layer on one side of a substrate is allowed to impinge upon a thin test film on the opposite surface. Laser-interferometric measurements of test film displacement enable calculation of the stresses generated at the interface. The tensile stress at the onset of failure is taken to be the intrinsic tensile strength of the interface. Fused-silica substrates, with their negative nonlinear elasticity, cause the compressive stress wave generated by the pulse laser to evolve a decompression shock, critical for generation of the fast fall times needed for significant loading of surface film interfaces. By allowing the stress pulse to mode convert as it reflects from an oblique surface, a high amplitude shear wave with rapid fall time is generated and used to realize mixed-mode loading of thin film interfaces. We report intrinsic strengths of an aluminum/fused silica interface under both tensile and mixed-mode conditions. The failure mechanism under mixed-mode loading differs significantly from that observed under pure tensile loading, resulting in a higher interfacial strength for the mixed-mode case. Inferred strengths are found to be independent, as they should be, of experimental parameters.     

Dynamic behaviors of mode III interfacial crack under a constant loading rate

In an earlier study on intersonic crack propagation, Gao et al. (J. Mech. Phys. Solids 49: 2113–2132, 2001) described molecular dynamics simulations and continuum analysis of the dynamic behaviors of a mode II dominated crack moving along a weak plane under a constant loading rate. The crack was observed to initiate its motion at a critical time after the onset of loading, at which it is rapidly accelerated to the Rayleigh wave speed and propagates at this speed for a finite time interval until an intersonic daughter crack is nucleated at a peak stress at a finite distance ahead of the original crack tip. The present article aims to analyze this behavior for a mode III crack moving along a bi-material interface subject to a constant loading rate. We begin with a crack in an initially stress-free bi-material subject to a steadily increasing stress. The crack initiates its motion at a critical time governed by the Griffith criterion. After crack initiation, two scenarios of crack propagation are investigated: the first one is that the crack moves at a constant subsonic velocity; the second one is that the crack moves at the lower shear wave speed of the two materials. In the first scenario, the shear stress ahead of the crack tip is singular with exponent ?1/2, as expected; in the second scenario, the stress singularity vanishes but a peak stress is found to emerge at a distance ahead of the moving crack tip. In the latter case, a daughter crack supersonic with respect to the softer medium can be expected to emerge ahead of the initial crack once the peak stress reaches the cohesive strength of the interface.     

Three-dimensional mixed-mode stress intensity factors for cracks in functionally graded materials using enriched finite elements

Three-dimensional enriched finite elements are used to compute mixed-mode stress intensity factors (SIFs) for three-dimensional cracks in elastic functionally graded materials (FGMs) that are subject to general mixed-mode loading and constraint conditions. The method, which advantageously does not require special mesh configuration/modifications and post-processing of finite element results, is an enhancement of previous developments applied so far on isotropic homogeneous and isotropic interface cracks. The spatial variation of FGM material properties is taken into account at the level of element integration points. To validate the developed method, two- and three-dimensional mixed-mode fracture problems are selected from the literature for comparison. Two-dimensional cases include: inclined central crack in a large FGM medium under uniform tensile strain and stress loadings, a slanted crack in a finite-size FGM plate under exponentially varying tensile stress loading and an edge crack in a finite-size plate under shear traction load. The three-dimensional example models a deflected surface crack in a finite-size FGM plate under uniform tensile stress loading. Comparisons between current results and those from analytical and other numerical methods yield good agreement. Thus, it is concluded that the developed three-dimensional enriched finite elements are capable of accurately computing mixed-mode fracture parameters for cracks in FGMs.     

Loss of adhesion at the tip of an interface crack

A model is constructed to analyze adhesive bond failure at the tip of an interface crack. The model is based on the assumption that there are zones of bounded cohesive tensile and shear stresses near a crack tip. Within the context of certain broad a-priori assumptions on the distributions of certain stress and displacement components in the cohesive zones, the requirement thatall stresses in the two materials remain bounded provides a method to compute the specific details for these zones. It is assumed that bond failure occurs when the extension of the bond fiber at the crack tip exceeds a critical value. For an interface crack in a uniform tension field computations for two alternate formulations suggest that this failure criterion is independent of the precise distribution of the cohesive stresses, but rather depends only upon their averaged values. Combined loading with a dominant tensile component has also been analyzed. If the critical extension of bond fibers and the maximum value of the cohesive tensile stress are known, the model provides the maximum allowable interface stresses for given crack dimension and material parameters.     

Mixed-mode failure of thin films using laser-generated shear waves

A new test method is developed for studying mixed-mode interfacial failure of thin films using laser generated stress waves. Guided by recent parametric studies of laser-induced tensile spallation, we successfully extend this technique to achieve mixed-mode loading conditions. By allowing an initial longitudinal wave to mode convert at an oblique surface, a high amplitude shear wave is generated in a fused silica substrate and propagated toward the thin-film surface. A shear wave is obtained with amplitude large enough to fail an Al film/fused silica interface and the corresponding shear stress calculated from high-speed interferometric displacement measurements. Examination of the interfaces failed under mixed-mode conditions reveals significant wrinkling and tearing of the film, in great contrast to blister patterns observed in similar Al films failed under tensile loading.     

Numerical analysis of the dual actuator load test applied to fracture characterization of bonded joints

The dual actuator load test was numerically analyzed in order to assess its adequacy for fracture characterization of bonded joints under different mixed-mode loading conditions. This test enables asymmetric loading of double cantilever beam specimens, thus providing a large range of mixed-mode combinations. A new data reduction scheme based on specimen compliance, beam theory and crack equivalent concept was proposed to overcome several difficulties inherent to the test. The method assumes that the dual actuator test can be viewed as a combination of the double cantilever beam and end loaded split tests, which are used for pure modes I and II fracture characterization, respectively. A numerical analysis including a cohesive mixed-mode damage model was performed considering different mixed-mode loading conditions to evaluate the test performance. Some conclusions were drawn about the advantages and drawbacks of the test.     

Transition of mode II cracks from sub-Rayleigh to intersonic speeds in the presence of favorable heterogeneity

Understanding sub-Rayleigh-to-intersonic transition of mode II cracks is a fundamental problem in fracture mechanics with important practical implications for earthquake dynamics and seismic radiation. In the Burridge-Andrews mechanism, an intersonic daughter crack nucleates, for sufficiently high prestress, at the shear stress peak traveling with the shear wave speed in front of the main crack. We find that sub-Rayleigh-to-intersonic transition and sustained intersonic propagation occurs in a number of other models that subject developing cracks to intersonic loading fields. We consider a spontaneously expanding sub-Rayleigh crack (or main crack) which advances, along a planar interface with linear slip-weakening friction, towards a place of favorable heterogeneity, such as a preexisting subcritical crack or a small patch of higher prestress (similar behavior is expected for a small patch of lower static strength). For a range of model parameters, a secondary dynamic crack nucleates at the heterogeneity and acquires intersonic speeds due to the intersonic stress field propagating in front of the main crack. Transition to intersonic speeds occurs directly at the tip of the secondary crack, with the tip accelerating rapidly to values numerically equal to the Rayleigh wave speed and then abruptly jumping to an intersonic speed. Models with favorable heterogeneity achieve intersonic transition and propagation for much lower prestress levels than the ones implied by the Burridge-Andrews mechanism and have transition distances that depend on the position of heterogeneity. We investigate the dependence of intersonic transition and subsequent crack propagation on model parameters and discuss implications for earthquake dynamics.     

Integrated numerical–experimental analysis of interfacial fatigue fracture in SnAgCu solder joints

In ball grid array (BGA) packages, solder balls are exposed to cyclic thermo-mechanical strains arising from the thermal mismatch between package components. Thermo-mechanical fatigue crack propagation in solder balls is almost always observed at the chip side of the bump/pad junction. The objective of the experimental part of this study is to characterize the bump/pad interface under fatigue loading. Fatigue specimens are prepared by reflowing Sn3.8Ag0.5Cu lead-free solder alloy on Ni/Au substrates. Obtained results show that fatigue damage evolution strongly depends on the microstructure. Applied strain and solder volume both have an influence on the fatigue damage mechanism. In the numerical part of the study, fatigue experiments are modeled using the finite element technique. A cohesive zone approach is used to predict the fatigue damage evolution in soldered connections. Crack propagation is simulated by an irreversible linear traction–separation cohesive zone law accompanied by a non-linear damage parameter. Cohesive zone elements are placed where failure is experimentally observed. Damage evolution parameters for normal and tangential interaction are scrutinized through dedicated fatigue tests in pure tensile and shear directions. The proposed cohesive zone model is quantitatively capable of describing fatigue failure in soldered joints, which can be further extended to a numerical life-time prediction tool in microelectronic packages.     

Analytical representation of the non-square-root singular stress field at a finite angle sharp notch

The stress field near the tip of a finite angle sharp notch is singular. However, unlike a crack, the order of the singularity at the notch tip is less than one-half. Under tensile loading, such a singularity is characterized by a generalized stress intensity factor which is analogous to the mode I stress intensity factor used in fracture mechanics, but which has order less than one-half. By using a cohesive zone model for a notional crack emanating from the notch tip, we relate the critical value of the generalized stress intensity factor to the fracture toughness. The results show that this relation depends not only on the notch angle, but also on the maximum stress of the cohesive zone model. As expected the dependence on that maximum stress vanishes as the notch angle approaches zero. The results of this analysis compare very well with a numerical (finite element) analysis in the literature. For mixed-mode loading the limits of applicability of using a mode I failure criterion are explored.     

The simulation of dynamic crack propagation using the cohesive segments method

The cohesive segments method is a finite element framework that allows for the simulation of the nucleation, growth and coalescence of multiple cracks in solids. In this framework, cracks are introduced as jumps in the displacement field by employing the partition of unity property of finite element shape functions. The magnitude of these jumps are governed by cohesive constitutive relations. In this paper, the cohesive segments method is extended for the simulation of fast crack propagation in brittle solids. The performance of the method is demonstrated in several examples involving crack growth in linear elastic solids under plane stress conditions: tensile loading of a block; shear loading of a block and crack growth along and near a bi-material interface.     

Mixed-mode crack initiation at a v-notch in presence of an adhesive joint

The failure criterion for v-notched specimens developed for mixed-mode loadings by Yosibash et al. [Yosibash, Z., Priel, E., Leguillon, D., 2006. A failure criterion for brittle elastic materials under mixed mode loading. Int. J. Fract. 141(1), 289–310.] is generalised in order to consider the influence of the shear stresses and the mode-dependence of the toughness. This is demonstrated to be important in some cases with adhesive joints under complex loadings for instance. The original criterion assumes that an abrupt onset of a crack with a finite extension occurs when two conditions are fulfilled simultaneously: first the normal traction all along the presupposed crack path reaches a critical value, and second the onset is energetically allowed. The influence of the shear stresses is now considered through a mixed law involving critical shear and tensile stresses as well as the mode-dependent toughness introducing a new equivalent SIF. This extended criterion is applied to the simulation of an Arcan test on v-notched compact tension shear (CTS) specimens made of two parts bonded together along the geometric plane of symmetry of the specimen. The competition between two modes of failure is studied: crack initiation along the weak joint which may constitute a privileged fracture surface and initiation in the homogeneous material at an optimum angle that minimizes the critical load to failure.     

Subcritical crack growth in weak interface under mixed mode in two dimension

Constitutive equations, which govern subcritical crack growth within a weak brittle interface, are derived assuming mixed-mode loading, i.e., both tensile and shear stresses acting at the crack tip. The subcritical crack growth is assumed to be caused by the classical activation mechanism of fluctuation fracture kinetics. To derive the constitutive equations, two approaches are developed. The first approach is process-zone-detail-independent (PZI), which ignores any details of the process zone, i.e., a near-crack-tip zone of significant damage resulting in fracture, and takes into account only the process zone length. The second approach is process-zone-detail-dependent (PZD), which takes into account some details of the process-zone structure. After some general considerations including 3D case, the detailed consideration is given for 2D case, particularly, for plane strain. Illustrative calculated examples of the obtained theoretical results are presented.     

Dynamic crack growth along a polymer composite-Homalite interface

Dynamic crack growth along the interface of a fiber-reinforced polymer composite-Homalite bimaterial subjected to impact shear loading is investigated experimentally and numerically. In the experiments, the polymer composite-Homalite specimens are impacted with a projectile causing shear dominated interfacial cracks to initiate and subsequently grow along the interface at speeds faster than the shear wave speed of Homalite. Crack growth is observed using dynamic photoelasticity in conjunction with high-speed photography. The calculations are carried out for a plane stress model of the experimental configuration and are based on a cohesive surface formulation that allows crack growth, when it occurs, to emerge as a natural outcome of the deformation history. The effect of impact velocity and loading rate is explored numerically. The experiments and calculations are consistent in identifying discrete crack speed regimes within which crack growth at sustained crack speeds is possible. We present the first conclusive experimental evidence of interfacial crack speeds faster than any characteristic elastic wave speed of the more compliant material. The occurrence of this crack speed was predicted numerically and the calculations were used to design the experiments. In addition, the first experimental observation of a mother-daughter crack mechanism allowing a subsonic crack to evolve into an intersonic crack is documented. The calculations exhibit all the crack growth regimes seen in the experiments and, in addition, predict a regime with a pulse-like traction distribution along the bond line.     

Effect of adhesive thickness,adhesive type and scarf angle on the mechanical properties of scarf adhesive joints

The effects of adhesive thickness, adhesive type and scarf angle, which are determined as the main control parameters by the dimensional analysis, on the mechanical properties of a scarf adhesive joint (SJ) subjected to uniaxial tensile loading are examined using a mixed-mode cohesive zone model (CZM) with a bilinear shape to govern the interface separation. Particularly, the adhesive-dependence of the vital cohesive parameters of CZM, which mainly include initial stiffness, total fracture energy and separation strength, is introduced emphatically. The numerical results demonstrate that the ultimate tensile loading increases as the adhesive thickness decreases. Cross the ultimate tension, the joint loses the load-bearing capacity when adopting the brittle adhesive but sustains partial load-bearing capacity while selecting the ductile adhesive. In addition, for the joint with the ductile adhesive, the maximum applied displacement until the complete failure of it is directly proportional to the adhesive thickness, which is different from the case using the brittle adhesive. Taking the combination of the ultimate loading and applied displacement into account, failure energy is employed to evaluate the joint performances. The results show that the failure energy of the joint with the brittle adhesive increases as the adhesive thickness decreases. Conversely, the situation of the joint using the ductile adhesive is vice versa. Moreover, the effect of the adhesive thickness becomes more noticeable with decreasing the scarf angle owing to the variation of the proportion of each component of the mixed-mode. Furthermore, all the characteristic parameters (the ultimate tensile loading, the maximum applied displacement and the failure energy) that adopted to describe the performances of SJ increase as the scarf angle decreases. Finally, the numerical method employed in this study is validated by comparing with existing experimental results.