Size effect in quasibrittle failure: Analytical model and numerical simulations using cohesive zone model

The recent rewriting of the Ba?ant’s size effect law (Morel, 2008) which has suggested the existence of an additional asymptotic regime for intermediate structure sizes is now compared to numerical simulations of fracture of geometrically similar notched structures of different sizes extending over 2.4 decades. The quasibrittle fracture behavior is simulated through cohesive zone model (bilinear softening) using a constant set of cohesive parameters whatever the specimen size D is. The R-curves resulting from the load–displacement responses are estimated and appear as size-independent. On this basis, the different asymptotic regimes expected for the size effect on fracture properties at peak load such as the relative crack length, the resistance to crack growth and the nominal strength are shown in fair agreement with the size effect observed on the results obtained from numerical simulations.     

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.     

Theoretical development and experimental validation of a thermally dissipative cohesive zone model for dynamic fracture of amorphous polymers

A thermally dissipative cohesive zone model is developed for predicting the temperature increase at the tip of a crack propagating dynamically in a nominally brittle material exhibiting a cohesive-type failure such as crazing. The model assumes that fracture energy supplied to the crack tip region that is in excess of that needed for the creation of new free surfaces during crack advance is converted to heat within the cohesive zone. Bulk dissipation mechanisms, such as plasticity, are not accounted for. Several cohesive traction laws are examined, and the model is then used to make predictions of crack tip heating at various crack propagation speeds in the nominally brittle amorphous polymer PMMA, observed to fail by a crazing-type mechanism. The heating predictions are compared to experimental data where the temperature field surrounding a high speed crack in PMMA was measured. Measurements are made in real time using a multi-point high speed HgCdTe infrared radiation detector array. At the same time as temperature, simultaneous measurement of fracture energy is made by a strain gauge technique, and crack tip speed is monitored through a resistance ladder method. Material strength can be estimated through uniaxial tension tests, thus minimizing the need for parameter fitting in the stress-opening traction law. Excellent agreement between experiments and theory is found for two of the cohesive traction law temperature predictions, but only for the case where a single craze is active during the dynamic fracture of PMMA, i.e. crack tip speed up to approximately 0.2c_R. For higher speed fracture where subsurface damage becomes prominent, the line dissipation model of a cohesive zone is inadequate, and a distributed damage model is needed.     

In-plane fracture of laminated fiber reinforced composites with varying fracture resistance: Experimental observations and numerical crack propagation simulations

A series of experimental results on the in-plane fracture of a fiber reinforced laminated composite panel is analyzed using the variational multiscale cohesive method (VMCM). The VMCM results demonstrate the influence of specimen geometry and load distribution on the propagation of large scale bridging cracks in the fiber reinforced panel. Experimentally observed variation in fracture resistance is substantiated numerically by comparing the experimental and VMCM load–displacement responses of geometrically scaled single edge-notch three point bend (SETB) specimens. The results elucidate the size dependence of the traction-separation relationship for this class of materials even in moderately large specimens, contrary to the conventional understanding of it being a material property. The existence of a “free bridging zone” (different from the conventional “full bridging zone”) is recognized, and its influence on the evolving fracture resistance is discussed. The numerical simulations and ensuing bridging zone evolution analysis demonstrates the versatility of VMCM in objectively simulating progressive crack propagation, compared against conventional numerical schemes like traditional cohesive zone modeling, which require a priori knowledge of the crack path.     

Cohesive zone model of intermediate crack-induced debonding of FRP-plated concrete beam

External bonding of FRP plates or sheets has emerged as a popular method for strengthening reinforced concrete structures. Debonding along the FPR–concrete interface can lead to premature failure of the structures. In this study, debonding induced by a flexural crack in a FRP-plated concrete beam is analyzed through a nonlinear fracture mechanics method. The concrete beam and FRP plate are modeled as linearly elastic simple beams connected together through a thin layer of FRP–concrete interface. A bi-linear cohesive (bond-slip) law, which has been verified by experiments, is used to model the FRP–concrete interface as a cohesive zone. Thus a cohesive zone model for intermediate crack-induced debonding is established with a unique feature of unifying the debonding initiation and growth into one model. Closed-form solutions of interfacial stress, FRP stress and ultimate load of the plated beam are obtained and then verified with the numerical solutions based on finite element analysis. Parametric studies are carried out to demonstrate the significant effect of FRP thickness on the interface debonding. The bond-slip shape is examined specifically. In spite of its profound effect on softening zone size, the bond-slip shape has been found to have little effect on the ultimate load of the plated beam. By making use of such a unique feature, a simplified explicit expression is obtained to determine the ultimate load of the plated concrete beam with a flexural crack conveniently. The cohesive zone model in this study also provides an efficient and effective way to analyze more general FRP–concrete interface debonding.     

Direct Evaluation of Cohesive Law in Mode I of Pinus pinaster by Digital Image Correlation

The direct identification of the cohesive law in pure mode I of Pinus pinaster is addressed in this work. The approach couples the double cantilever beam (DCB) test with digital image correlation (DIC). Wooden beam specimens loaded in the radial-longitudinal (RL) fracture propagation system are used. The strain energy release rate in mode I (G _I) is uniquely determined from the load–displacement curve by means of the compliance-based beam method (CBBM). This method relies on the concept of equivalent elastic crack length (a _eq) and therefore does not require the monitoring of crack propagation during test. DIC measurements are processed with two different purposes. Firstly, the physical evidence of a _eq is discussed with regard to actual estimation of the crack length based on post-processing full-field displacement measurements. Secondly, the crack tip opening displacement in mode I (w _I) is determined from the displacements near the initial crack tip. The cohesive law in mode I (σ _I???w _I) is then identified by numerical differentiation of the G _I???w _I relationship. The methodology and accuracy on this reconstruction are addressed. Moreover, the proposed procedure is validated by finite element analyses including cohesive zone modelling. It is concluded that the proposed data reduction scheme is adequate for assessing the cohesive law in pure mode I of P. pinaster.     

基于弹性-塑性内聚力模型的纤维拔出界面宏观行为分析

用解析法分析了单纤维从聚合物基体中的拔出过程,采用弹性—塑性内聚力模型模拟裂纹的扩展和界面失效,确定了临界纤维埋入长度,该值区分两种不同长度的纤维拔出过程. 在纤维拔出过程,界面经历不同的阶段. 纤维埋长小于临界长度时,界面的脱粘载荷与纤维的埋长成正比;超过临界长度后,界面的脱粘载荷近似为常数. 分析了界面参数对脱粘载荷的影响:增加界面的剪切强度和界面的断裂韧性,或减小界面裂纹萌生位移,均能提高界面的脱粘载荷;界面脱粘后无界面摩擦应力时,拔出载荷—位移曲线的峰值载荷等于界面的脱粘载荷;界面摩擦应力存在时,使峰值载荷大于脱粘载荷,需要较长的纤维埋入长度和较大的界面摩擦应力.      

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.     

Pure mode II fracture characterization of composite bonded joints

A new data reduction scheme is proposed for measuring the critical fracture energy of adhesive joints under pure mode II loading using the End Notched Flexure test. The method is based on the crack equivalent concept and does not require crack length monitoring during propagation, which is very difficult to perform accurately in these tests. The proposed methodology also accounts for the energy dissipated at the Fracture Process Zone which is not negligible when ductile adhesives are used. Experimental tests and numerical analyses using a trapezoidal cohesive mixed-mode damage model demonstrated the good performance of the new method, namely when compared to classical data reduction schemes. An inverse method was used to determine the cohesive properties, fitting the numerical and experimental load–displacement curves. Excellent agreement between the numerical and experimental R-curves was achieved demonstrating the effectiveness of the proposed method.     

压电复合材料粘接界面断裂有限元模拟

根据数字化FRMM(Fix-Ratio Mix-Mode)断裂试验,得到了压电复合材料试件的断裂韧性和位移及应变场。本文在试验的基础上,通过非线性有限元软件ABAQUS及用户子程序UMAT进行了模拟分析,采用基于损伤力学的粘聚区模型(CZM)对压电复合材料界面的起裂和脱胶扩展进行了分析,并与VCCT方法进行了比较。计算得到的荷载位移曲线更接近于试验结果,但在裂纹扩展路径上的吻合需要对粘聚区法则进一步修正。通过进一步对CZM参数进行分析,表明界面粘结强度和界面刚度对计算结果的影响很大。研究结果表明,粘聚区模型可以很好地表征压电复合材料弱粘接界面脱胶断裂问题。     

Molecular-dynamics simulation-based cohesive zone representation of intergranular fracture processes in aluminum

A traction-displacement relationship that may be embedded into a cohesive zone model for microscale problems of intergranular fracture is extracted from atomistic molecular-dynamics (MD) simulations. An MD model for crack propagation under steady-state conditions is developed to analyze intergranular fracture along a flat Σ99 [1 1 0] symmetric tilt grain boundary in aluminum. Under hydrostatic tensile load, the simulation reveals asymmetric crack propagation in the two opposite directions along the grain boundary. In one direction, the crack propagates in a brittle manner by cleavage with very little or no dislocation emission, and in the other direction, the propagation is ductile through the mechanism of deformation twinning. This behavior is consistent with the Rice criterion for cleavage vs. dislocation blunting transition at the crack tip. The preference for twinning to dislocation slip is in agreement with the predictions of the Tadmor and Hai criterion. A comparison with finite element calculations shows that while the stress field around the brittle crack tip follows the expected elastic solution for the given boundary conditions of the model, the stress field around the twinning crack tip has a strong plastic contribution. Through the definition of a Cohesive-Zone-Volume-Element—an atomistic analog to a continuum cohesive zone model element—the results from the MD simulation are recast to obtain an average continuum traction-displacement relationship to represent cohesive zone interaction along a characteristic length of the grain boundary interface for the cases of ductile and brittle decohesion.     

内聚力模型裂纹问题分析的解析奇异单元

内聚力模型已经被广泛应用于需要考虑断裂过程区的裂纹问题当中,然而常用的数值方法应用于分析内聚力模型裂纹问题时还存在着一些不足,比如不能准确的给出断裂过程区的长度、需要网格加密等。为了克服这些缺点,论文构造了一个新型的解析奇异单元,并将之应用于基于内聚力模型的裂纹分析当中。首先将虚拟裂纹表面处的内聚力用拉格拉日插值的方法近似表示为多项式的形式,而多项式表示的内聚力所对应的特解可以被解析地给出。然后利用一个简单的迭代分析,基于内聚力模型的裂纹问题就可以被模拟出来了。最后,给出二个数值算例来证明本文方法的有效性。     

Cohesive zone laws for fatigue crack growth: Numerical field projection of the micromechanical damage process in an elasto-plastic medium

Cohesive zone failure models are widely used to simulate fatigue crack propagation under cyclic loading, but the model parameters are phenomenological and are not closely tied to the underlying micromechanics of the problem. In this paper, we will inversely extract the cohesive zone laws for fatigue crack growth in an elasto-plastic ductile solid using a field projection method (FPM), which projects the equivalent tractions and separations at the cohesive crack-tip from field information outside the process zone. In our small-scale yielding model, a single row of discrete voids is deployed directly ahead of a crack in an elasto-plastic medium subjected to cyclic mode I K-field loading. Damage accumulation under cyclic loading is captured by the growth of voids within the micro-voiding zone ahead of the crack, while the evolution of the cohesive zone law representing the micro-voiding zone is inversely extracted via the FPM. We show that the field-projected cohesive zone law captures the essential micromechanisms of fatigue crack growth in the ductile medium: from loading and unloading hysteresis caused by void growth and plastic hardening, to the softening damage locus associated with crack propagation via a void by void growth mechanism. The results demonstrate the effectiveness of the FPM in obtaining a micromechanics-based cohesive zone law in-place of phenomenological models, which opens the way for a unified treatment of fatigue crack problems.     

Direct Extraction of Cohesive Fracture Properties from Digital Image Correlation: A Hybrid Inverse Technique

The accuracy of an adopted cohesive zone model (CZM) can affect the simulated fracture response significantly. The CZM has been usually obtained using global experimental response, e.g., load versus either crack opening displacement or load-line displacement. Apparently, deduction of a local material property from a global response does not provide full confidence of the adopted model. The difficulties are: (1) fundamentally, stress cannot be measured directly and the cohesive stress distribution is non-uniform; (2) accurate measurement of the full crack profile (crack opening displacement at every point) is experimentally difficult to obtain. An attractive feature of digital image correlation (DIC) is that it allows relatively accurate measurement of the whole displacement field on a flat surface. It has been utilized to measure the mode I traction-separation relation. A hybrid inverse method based on combined use of DIC and finite element method is used in this study to compute the cohesive properties of a ductile adhesive, Devcon Plastic Welder II, and a quasi-brittle plastic, G-10/FR4 Garolite. Fracture tests were conducted on single edge-notched beam specimens (SENB) under four-point bending. A full-field DIC algorithm was employed to compute the smooth and continuous displacement field, which is then used as input to a finite element model for inverse analysis through an optimization procedure. The unknown CZM is constructed using a flexible B-spline without any “a priori” assumption on the shape. The inversely computed CZMs for both materials yield consistent results. Finally, the computed CZMs are verified through fracture simulation, which shows good experimental agreement.     

Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials

A numerical method is developed to simulate complex two-dimensional crack propagation in quasi-brittle materials considering random heterogeneous fracture properties. Potential cracks are represented by pre-inserted cohesive elements with tension and shear softening constitutive laws modelled by spatially-varying Weibull random fields. Monte Carlo simulations of a concrete specimen under uni-axial tension were carried out with extensive investigation of the effects of important numerical algorithms and material properties on numerical efficiency and stability, crack propagation processes and load-carrying capacities. It was found that the homogeneous model led to incorrect crack patterns and load–displacement curves with strong mesh-dependence, whereas the heterogeneous model predicted realistic, complicated fracture processes and load-carrying capacity of little mesh-dependence. Increasing the variance of the tensile strength random fields with increased heterogeneity led to reduction in the mean peak load and increase in the standard deviation. The developed method provides a simple but effective tool for assessment of structural reliability and calculation of characteristic material strength for structural design.     

Fatigue and fracture of bovine dentin

In this paper, the fatigue and fracture properties of bovine dentin are evaluated usingin vitro experimental analyses. Double cantilever beam (DCB) specimens were prepared from bovine maxillary molars and subjected to zeroto-tension cyclic loads. The fatigue crack growth rate was evaluated as a function of the dentin tubule orientation using the Paris law. Wedge-loaded DCB specimens were also prepared and subjected to monotonic opening loads. Moiré interferometry was used to acquire the in-plane displacement field during stable crack growth, and the instantaneous wedge load and crack length were acquired to evaluate the crack growth resistance and crack tip opening displacement (CTOD) with crack extension. The rate of fatigue crack growth was generally larger for crack propagation occurring perpendicular to the dentin tubules. The Moiré fringe fields documented during monotonic crack growth exhibited non-linear deformation occurring within a confined region adjacent to the crack tip. Both the wedge load and CTOD response provided evidence that a fracture process zone contributes to energy dissipation during crack extension and that dentin exhibits a risingR-curve behavior. Results from this preliminary investigation are being used as a guide for an evaluation of the fatigue and fracture properties of human dentin.     

单轴荷载下饱水岩石静态和动态抗压强度的细观力学分析

由于单轴荷载下饱水岩石的动态力学特性与静态力学特性存在很大差异,从宏观上进行力学分析 存在局限性。根据岩石受压全应力应变曲线的细观机制,分析了静态及动态单轴荷载条件下孔隙水影响饱水 岩石裂纹扩展的情况。在静态单轴压缩条件下,初始裂隙受压使自由水产生孔隙水压力,自由水对翼裂纹有 向外挤压的应力,促进裂纹扩展。在动态单轴压缩条件下,自由水会产生粘结力,抑制裂纹扩展。根据翼裂纹 受压扩展原理,推导出饱水单轴条件下动态抗压强度、静态抗压强度的计算公式,在相同断裂韧度下,饱水岩 石静态抗压强度风干岩石静态抗压强度饱水岩石动态抗压强度。对自然风干和饱水砂岩进行单轴静态、 动态压缩实验,结果与理论模型的结果相符。     

Three-dimensional cohesive fracture modeling of non-planar crack growth using adaptive FE technique

In this paper, the three-dimensional adaptive finite element modeling is presented for cohesive fracture analysis of non-planer crack growth. The technique is performed based on the Zienkiewicz–Zhu error estimator by employing the modified superconvergent patch recovery procedure for the stress recovery. The Espinosa–Zavattieri bilinear constitutive equation is used to describe the cohesive tractions and displacement jumps. The 3D cohesive fracture element is employed to simulate the crack growth in a non-planar curved pattern. The crack growth criterion is proposed in terms of the principal stress and its direction. Finally, several numerical examples are analyzed to demonstrate the validity and capability of proposed computational algorithm. The predicted crack growth simulation and corresponding load-displacement curves are compared with the experimental and other numerical results reported in literature.