A cohesive edge crack

The nucleation and growth of a cohesive edge crack is studied. This topic is germane to the initiation and early stage of crack growth during unnotched beam testing, the growth of short edge cracks in finite test pieces, and the formation of tension cracks of geological origin. This paper focuses on an edge crack in a semi-infinite plane, under a uniform far-field tensile stress acting parallel to the plane boundary. Expressions for the Mode I stress-intensity-factor and crack-opening-displacement for an edge crack subjected to arbitrary crack face loading are determined via the weight function method. All of the constants needed to define the weight function and associated integrals are themselves explicit functions of just two constants: f_r and ψ. Two types of softening behavior in the cohesive zone are examined: rectangular softening, and linear softening. In each case the process zone size, energy-release-rate, crack-opening displacement and load-ratio are examined. The different test behavior exhibited under load-control versus fixed-grip displacement control is explored. The test control conditions alter the fracture behavior significantly. For a linear softening cohesive edge crack, it is found out that under fixed-grip control (load-control), the process zone size decreases (increases) steadily with increasing traction-free crack length, approaching the semi-infinite crack asymptote from above (below). The differences between load-control versus fixed-grip control decrease rapidly with increasing traction-free crack length.     

坝基界面在非线性水压力驱动下的非线性断裂过程模拟

基于多边形比例边界有限元法和粘聚裂缝模型提出了混凝土坝坝基界面在随缝宽非线性变化的水压力驱动下的非线性断裂数值模型。混凝土和基岩采用多边形比例边界单元模拟,界面裂缝的断裂过程区采用粘性界面单元模拟。因为界面裂缝总是处于复合断裂模态,故同时引入了法向和切向的界面单元,且考虑了裂纹面作用有法向和切向任意荷载时的应力强度因子求解。以裂尖为原点,裂尖附近的位移场和应力场在径向上解析求解,在环向具有有限元精度。因此无需在裂尖附近加密网格或采用富集技术即可求得高精度的解。对于界面断裂,可模拟出与两种材料差异性相关的非1/2奇异性。断裂过程区的水压力随缝面宽度变化,采用指数函数的形式进行表征,通过参数调整可实现不同分布的水压力的模拟。水压力与粘聚力考虑为与裂缝宽度相关的组合函数,便于非线性迭代的实现。结合多边形网格生成和重剖分技术,可方便地模拟界面裂缝在水力驱动下的扩展过程。算例研究表明了该文模型的有效性,从中也可看出考虑缝内水压及其具体分布形式对研究坝的稳定性具有重要影响。     

The cohesive band model: a cohesive surface formulation with stress triaxiality

In the cohesive surface model cohesive tractions are transmitted across a two-dimensional surface, which is embedded in a three-dimensional continuum. The relevant kinematic quantities are the local crack opening displacement and the crack sliding displacement, but there is no kinematic quantity that represents the stretching of the fracture plane. As a consequence, in-plane stresses are absent, and fracture phenomena as splitting cracks in concrete and masonry, or crazing in polymers, which are governed by stress triaxiality, cannot be represented properly. In this paper we extend the cohesive surface model to include in-plane kinematic quantities. Since the full strain tensor is now available, a three-dimensional stress state can be computed in a straightforward manner. The cohesive band model is regarded as a subgrid scale fracture model, which has a small, yet finite thickness at the subgrid scale, but can be considered as having a zero thickness in the discretisation method that is used at the macroscopic scale. The standard cohesive surface formulation is obtained when the cohesive band width goes to zero. In principle, any discretisation method that can capture a discontinuity can be used, but partition-of-unity based finite element methods and isogeometric finite element analysis seem to have an advantage since they can naturally incorporate the continuum mechanics. When using interface finite elements, traction oscillations that can occur prior to the opening of a cohesive crack, persist for the cohesive band model. Example calculations show that Poisson contraction influences the results, since there is a coupling between the crack opening and the in-plane normal strain in the cohesive band. This coupling holds promise for capturing a variety of fracture phenomena, such as delamination buckling and splitting cracks, that are difficult, if not impossible, to describe within a conventional cohesive surface model.     

FRACTURE MODELLING OF BRITTLE-MATRIX COMPOSITES WITH SPATIALLY DEPENDENT CRACK BRIDGING

Abstract In brittle-matrix composites cracking of the matrix is often accompanied by bridging of the crack surfaces. The bridging will reduce the net stress intensity factor at the crack tip and consequently increase the toughness of the composite material. The bridging mechanism is due to for example unbroken whiskers, fibres, ductile particles or interlocking grains. Analysis of the bridging mechanism in cracked structures is conveniently carried out using the concept of cohesive zone modelling. In this case the action of the bridging elements is replaced by a distribution of forces, so called cohesive forces trying to close the crack. The commonly used approach in such modelling has been to replace the action from individual bridging elements by a continuous spatially independent distribution of closing tractions whose magnitude is a function of the crack opening displacement only. In this paper the influence of the spatial distribution of bridging elements is considered for plane crack problems. The cross section of the bridging elements is assumed to be circular and the distance between the different bridging elements is determined by the volume fraction, the radius and the geometrical distribution of the bridging elements. Damage resistance curves have been calculated for typical whiskers-reinforced ceramic composites, and the results from the present spatially dependent models are compared with results from calculations with spatially independent models. The influence of the radius of the bridging element, the volume fraction of whiskers and the material properties are illustrated and the use of spatially independent models is discussed.     

Modelling of crack propagation of gravity dams by scaled boundary polygons and cohesive crack model

Crack propagation in concrete gravity dams is investigated using scaled boundary polygons coupled with interface elements. The concrete bulk is assumed to be linear elastic and is modelled by the scaled boundary polygons. The interface elements model the fracture process zone between the crack faces. The cohesive tractions are modelled as side-face tractions in the scaled boundary polygons. The solution of the stress field around the crack tip is expressed semi-analytically as a power series. It reproduces the singular and higher-order terms in an asymptotic solution, such as the William’s eigenfunction expansion when the cohesive tractions vanish. Accurate results can be obtained without asymptotic enrichment or local mesh refinement. The stress intensity factors are obtained directly from their definition and provide a convenient and accurate means to assess the zero-K condition, which determines the stability of a cohesive crack. The direction of crack propagation is determined from the maximum circumferential stress criterion. To accommodate crack propagation, a local remeshing algorithm that is applicable to any polygon mesh is augmented by inserting cohesive interface elements between the crack surfaces as the cracks propagate. Three numerical benchmarks involving crack propagation in concrete gravity dams are modelled. The results are compared to the experimental and other numerical simulations reported in the literature.     

Application of cohesive zone model in crack propagation analysis in multiphase composite materials

A nonlinear cohesive stress distribution function is employed by relating the cohesive stress to the cohesive zone size (CZS) and the distance from the crack tip to investigate the elastic-plastic fracture behaviors. A crack-inclusion interaction problem is taken as an example to explore the fracture process in the cohesive zone area. The CZS and crack surface opening displacement are evaluated numerically. It is found that for different cohesive parameter combinations, the normalized CZS and crack surface opening displacements change drastically. By reducing the current model to the famous Dugdale model, the results obtained match well with the existing ones.     

Fast fracture and crack arrest according to the Dugdale model

Dynamic effects near a propagating crack tip in a ductile material have been investigated on the basis of a model with a strip-zone of yielding. In the analysis of fast fracture the unknown variables are the speeds of the leading and trailing edges of the yield zone, where the latter defines the position of the actual crack tip. Propagation of the crack tip is governed by two conditions: the usual one that the cleavage stress is bounded at the leading edge of the yield zone, and a second condition which involves the yield stress and the stretch of the fiber at the trailing edge of the yield zone. By combining well-known results for transient dynamic stress-intensity factors and crack-opening displacements corresponding to external loads, with steady-state dynamic results for the fields corresponding to the cohesive tractions in the yield zone, the dynamic problem of fast fracture has been analyzed for both the Mode III and the Mode I case. The results can be used to investigate crack arrest when a propagating crack tip enters a region of higher ductility.     

Incremental‐secant modulus iteration scheme and stress recovery for simulating cracking process in quasi‐brittle materials using XFEM

In this paper, an incremental‐secant modulus iteration scheme using the extended/generalized finite element method (XFEM) is proposed for the simulation of cracking process in quasi‐brittle materials described by cohesive crack models whose softening law is composed of linear segments. The leading term of the displacement asymptotic field at the tip of a cohesive crack (which ensures a displacement discontinuity normal to the cohesive crack face) is used as the enrichment function in the XFEM. The opening component of the same field is also used as the initial guess opening profile of a newly extended cohesive segment in the simulation of cohesive crack propagation. A statically admissible stress recovery (SAR) technique is extended to cohesive cracks with special treatment of non‐homogeneous boundary tractions. The application of locally normalized co‐ordinates to eliminate possible ill‐conditioning of SAR, and the influence of different weight functions on SAR are also studied. Several mode I cracking problems in quasi‐brittle materials with linear and bilinear softening laws are analysed to demonstrate the usefulness of the proposed scheme, as well as the characteristics of global responses and local fields obtained numerically by the XFEM. Copyright © 2006 John Wiley & Sons, Ltd.     

The fracture toughness of medium density fiberboard (MDF) including the effects of fiber bridging and crack–plane interference

The fracture toughness of medium density fiberboard (MDF) as a function of crack length (R curve) was measured. Fracture toughness was determined from force–displacement and crack length data using a new energy analysis procedure that avoids the scatter of prior discrete analysis methods. Because crack lengths were difficult to observe, they were measured using digital image correlation (DIC). The R curves for two different densities of MDF, two thicknesses, and for both in-plane and through-the-thickness cracks all increased linearly with crack length. The increase was interpreted as the development of a fiber-bridging process zone. Numerical modeling methods were used to determine the cohesive stress of the fiber-bridging zone.     

Two equal symmetric circular arc cracks in an elastic medium – the Dugdale approach

The Dugdale model for two equal, symmetrically situated coplanar circular arc cracks contained in an infinite elastic perfectly-plastic plate is proposed. Biaxial loads are applied at the infinite boundary of the plate. Consequently, the rims of the cracks open in Mode I and develop a plastic zone ahead of each of the cracks. These plastic zones are then closed by the distribution of uniform normal closing stresses over the rims of the plastic zones. Based on the complex-variable technique and the superposition principle, the solution for the above problem is obtained. Closed-form analytic expressions are obtained for the determination of the sizes of the plastic zones and the crack-opening displacement (COD) at the tip of the crack. Numerical studies are carried out to calculate the load ratio (load applied at infinity/yield point stress applied at the rims of the plastic zones) required for the closure of the plastic zones, for various radii of arc cracks and for various angles subtended by them at the centre. The crack-opening displacement is also investigated with respect to these parameters.     

On the Cohesive Zone Model for a Finite Crack

The paper concentrates on the development of the crack tip model with the cohesive zone in an infinite plate with a finite crack of mode I. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration according to Westergaard's theory with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. The calculation results of the crack tip opening displacement are compared with the Dugdale solution for the plane stress. This revised version was published online in July 2006 with corrections to the Cover Date.     

Two cracks with coalesced interior plastic zones — The generalised Dugdale model approach

The problem investigated is of an elastic-perfectly plastic infinite plate containing two equal collinear and symmetrically situated straight cracks. The plate is subjected to loads at infinity inducing mode I type deformations at the rims of the cracks. Consequently, plastic zones are formed ahead of the tips of the cracks. The loads at infinity are increased to a limit such that the plastic zones formed at the neighbouring interior tips of the cracks get coalesced. The plastic zones developed at the tips of the cracks are closed by applying normal cohesive quadratically varying stress distribution over their rims. The opening of the cracks is consequently arrested. Complex variable technique is used in conjugation with Dugdale’s hypothesis to obtain analytical solutions. Closed form analytical expressions are derived for calculating plastic zone size and crack opening displacement. An illustrative numerical example is discussed to study the qualitative behaviour of the loads required to arrest the cracks from opening with respect to parameters viz. crack length, plastic zone length and inter-crack distance. Crack opening displacement at the tip of the crack is also studied against these parameters.     

Finite element modeling of plasticity-induced crack closure with emphasis on geometry and mesh refinement effects

Two-dimensional, elastic-perfectly plastic finite element analyses of middle-crack tension (MT) and compact tension (CT) geometries were conducted to study fatigue crack closure and to calculate the crack-opening values under plane-strain and plane-stress conditions. The behaviors of the CT and MT geometries were compared. The loading was selected to give the same maximum stress intensity factor in both geometries, and thus approximately similar initial forward plastic zone sizes. Mesh refinement studies were performed on both geometries with various element types. For the CT geometry, negligible crack-opening loads under plane-strain conditions were observed. In contrast, for the MT specimen, the plane-strain crack-opening stresses were found to be significantly larger. This difference was shown to be a consequence of in-plane constraint. Under plane-stress conditions, it was found that the in-plane constraint has negligible effect, such that the opening values are approximately the same for both the CT and MT specimens.     

Instability during cohesive zone growth

Tensile microcracking of quasi-brittle materials is studied by means of micromechanics, based on (i) an elasto-damaging cohesive zone model accounting for cohesive softening and (ii) a dilute distribution of non-interacting microcracks of uniform orientation and size. Considering virgin microcracks (initially without cohesive zones), macroscopic tensile load increase results in growth of cohesive zones ahead of stationary (non-propagating) cracks and, subsequently, in crack propagation which, notably, will be encountered before the cohesive zones are fully developed i.e. onset of instable cohesive zone growth will be encountered at a load level (i) at which tractions are still transmitted across the inner edges of the cohesive zones and (ii) at which the separation at the inner edges of the cohesive zones is smaller than its critical value. Focusing on onset of instable cohesive zone growth, the chosen approach allows for accessing quantities characterizing the stability limit (e.g., the intensity of the macroscopic loading and the opening at the inner edges of the cohesive zones), without raising the need for non-linear Finite Element analyses. It is shown that the tensile macrostrength of materials containing virgin microcracks is larger than the one related to cracks with already initially fully developed cohesive zones, and related strength differences are quantified for a wide class of cohesive softening behavior. The proposed model is validated by comparing model predictions with an exact solution (available for the special case of constant cohesive tractions) and with results from reliable Finite Element analyses. The paper will be of interest for engineers involved in testing and/or in modeling of quasi-brittle media including cementitious materials and rock.     

混凝土裂缝端部粘聚力的计算

混凝土裂缝端部断裂过程区的粘聚力分布是导致混凝土断裂呈现非线性特性的重要原因。基于混凝土的断裂特性和虚拟裂缝端部存在粘聚力的分析模型，并通过分布函数的特性分析，提出了粘聚力分布函数的两种简化表达式：一为单参数待定式，另一为双参数待定式。由变形体叠加原理，推导出计算单参数待定函数公式和计算双参数待定函数代数方程组。进而通过裂缝张开位移实测数据即可求得粘聚力分布，并且给出了适当的算例分析和讨论。     

A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks

A cracked elastostatic structure is artificially divided into subdomains of simpler topology such that the well-developed classic dual integral equations can be applied appropriately to each domain. Applying the continuity and equilibrium conditions along artificial boundaries and properties of the integral kernels a single-domain dual-boundary-integral equation formulation is derived for a cracked elastic structure. A cohesive zone model is used to model the crack tip processes and is coupled with the single-domain dual-boundary-integral equation formulation; the resulting nonlinear equations are solved using the iterative method of successive-over-relaxation. The constitutive law used for a crack includes three parts: a law relating cohesive force to crack displacement difference when a crack is opening, a characterization of tangential interaction between crack surfaces when the crack surfaces are in contact, and a maximum principal stress criterion of crack advance. Incorporation of local unloading effect of the cohesive zone material has enabled a simulation of fracture with initial damage, partial development of the failure process zone at structural instability and multiple crack interaction. Some of the features of the method are demonstrated by considering three examples. The first problem is a single-edge-cracked specimen that exhibits a snap-back instability. The second example is the development of wing cracks from an angled crack under compression. The last example demonstrates the capability to consider mixed-mode crack growth and interaction of cracks. Thus, the problem of crack growth has been reduced to the determination of the cohesive model for the fracture process. This revised version was published online in July 2006 with corrections to the Cover Date.     

A distributed dislocation stress analysis for crazes and plastic zones at crack tips

A distributed dislocation method is developed to obtain analytically the applied stress as well as the surface stress profile along narrow plastic zones at the tip of a crack in a homogeneous tensile stress field. Replacing the plastic zone by a continuous array of mathematical dislocations, the stress field solution of this mixed boundary value problem (the displacement profile of the plastic zone is fixed while the tensile stresses are zero across the crack) can be solved. A computer program based on this stress field solution has been constructed and tested using the analytical results of the Dugdale model. The method is then applied to determining the surface stress profiles of crazes and plane-stress plastic deformation zones grown from electron microprobe cracks in polystyrene and polycarbonate respectively. The necessary craze and zone surface displacement profiles are determined by quantitative analysis of transmission electron micrographs. The surface stress profiles, which show small stress concentrations at the craze or zone tip falling to an approximately constant value which is maintained to the crack tip, are compared with those previously computed using an approximate Fourier transform method involving estimation of the displacement profile in the crack. The agreement between the approximate method and the exact distributed dislocation method is satisfactory.     

Inverse extraction of cohesive zone laws by field projection method using numerical auxiliary fields

The cohesive zone law relates the cohesive tractions with the cohesive separations within the fracture process zone of a material and is used to quantify the strength and toughness of the material. Determining the material's cohesive zone law, however, is a nontrivial inverse problem of finding unknown tractions and separations from measurement data. Previously, a field projection method was established to extract the cohesive zone laws from far‐field data using interaction J‐integrals between the physical field of interest and auxiliary analytical probing fields. Here, we extend the universality of the field projection method and its ease of numerical implementation by using numerical auxiliary fields. These numerical fields are generated by systematically imposing uniform surface tractions element‐by‐element along the crack faces in finite element models. Then, interaction J‐ and M‐integrals between these auxiliary probing fields and the measurement field are used to reconstruct the traction and separation relationship along the crack faces. The effectiveness of this method in extracting the cohesive zone law from measured displacements in the far‐field region is demonstrated through numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Superposition method for calculating singular stress fields at kinks, branches and tips in multiple crack arrays

A method is developed for calculating stresses and displacements around arrays of kinked and branched cracks having straight segments in a linearly elastic solid loaded in plane stress or plain strain. The key idea is to decompose the cracks into straight material cuts we call `cracklets', and to model the overall opening displacements of the cracks using a weighted superposition of special basis functions, describing cracklet opening displacement profiles. These basis functions are specifically tailored to induce the proper singular stresses and local deformation in wedges at crack kinks and branches, an aspect that has been neglected in the literature. The basis functions are expressed in terms of dislocation density distributions that are treatable analytically in the Cauchy singular integrals, yielding classical functions for their induced stress fields; that is, no numerical integration is involved. After superposition, nonphysical singularities cancel out leaving net tractions along the crack faces that are very smooth, yet retaining the appropriate singular stresses in the material at crack tips, kinks and branches. The weighting coefficients are calculated from a least squares fit of the net tractions to those prescribed from the applied loading, allowing accuracy assessment in terms of the root-mean-square error. Convergence is very rapid in the number of basis terms used. The method yields the full stress and displacement fields expressed as weighted sums of the basis fields. Stress intensity factors for the crack tips and generalized stress intensity factors for the wedges at kinks and branches are easily retrieved from the weighting coefficients. As examples we treat cracks with one and two kinks and a star-shaped crack with equal arms. The method can be extended to problems of finite domain such as polygon-shaped plates with prescribed tractions around the boundary.     

The determination of crack bridging forces

A method is presented for determining the bridging tractions acting on the fracture surfaces of cracks from measurements of the crack opening profile. The tractions may be expressed either as a function (x) of position in the crack or a function p(u) of the crack opening displacement. The feasibility of deducing (x) or p(u) from noisy displacement data is demonstrated by numerical simulations. It is found that the most complete information is contained in profiles of cracks growing from notches. Improved estimates of p(u) can also be found by analzying data from several cracks at different stress levels simultaneously.