Effect of shear and rotary inertia on dynamic fracture of a beam or plate in tensile loading

The dynamic fracture response of a long beam of brittle material subjected to tensile loading is studied. If the magnitude of the applied tensile loading is increased to a critical value, a crack will propagate from one of the longitudinal surfaces of the beam. As an extension of previous work, the effect of shear and of rotary inertia on the tensile loading and the induced bending moment at the fracturing section is included in the analysis. Thus an improved formulation is presented by means of which the crack length, crack tip velocity, bending moment and axial force at the fracture section are determined as functions of time after crack initiation. It is found that the rotary effect diminishes the bending moment effect and retards total fracture time whereas the shear has an opposite effect. Thus by combining the two effects (to simulate to first order the Timoshenko beam) overall fracture is retarded. The results also apply for plane strain fracture of a plate in tensile loading provided the value of the elastic modulus is appropriately modified.     

Effect of bending moment on the dynamic fracture of a beam or plate under tensile loading

The dynamic fracture response of a long beam of brittle elastic material under tensile loading is studied. If the magnitude of the applied loading is increased to a critical value, a crack is assumed to propagate across the beam cross section. As an extension of previous work, an induced bending moment generated during fracture is incorporated into the analysis and this improved formulation is presented. The crack length, crack tip speed, axial force and bending moment on the fracturing section are determined as functions of time after crack initiation. It is found that the bending moment has a significant effect on the fracture process in that it tends to retard fracture and causes a drastic change in the slope of the loading curve for large crack depths. Finally, by appropriate change of the elastic modulus, the results may be applied to plane strain fracture of a plate in pure tensile loading.     

Dynamic fracture of a beam or plate under tensile loading

The dynamic fracture response of a long beam of brittle elastic material under tensile loading is studied. If the magnitude of the applied loading is increased to a critical value, a crack is assumed to propagate across the beam cross section. In a parallel analysis to [t] the crack length and applied loading at the fracture face are determined as functions of time measured from fracture initiation. The results of the analysis are shown in graphs of crack length, crack tip speed and fracturing section tensile loading vs time. As found in [1], the crack tip accelerates very quickly to a speed near the characteristic terminal speed for the material, travels at this speed through most of the beam thickness, and then decelerates rapidly in the final stage of the process. Finally, by appropriate change of the elastic modulus, the results may be applied to plane strain fracture of a plate under pure tensile loading.     

A continuous-discontinuous model for crack branching

A new continuous-discontinuous model for fracture that accounts for crack branching in a natural manner is presented. It combines a gradient-enhanced damage model based on nonlocal displacements to describe diffuse cracks and the extended finite element method (X-FEM) for sharp cracks. Its most distinct feature is a global crack tracking strategy based on the geometrical notion of medial axis: the sharp crack propagates following the direction dictated by the medial axis of a damage isoline. This means that, if the damage field branches, the medial axis automatically detects this bifurcation, and a branching sharp crack is thus easily obtained. In contrast to other existing models, no special crack-tip criteria are required to trigger branching. Complex crack patterns may also be described with this approach, since the X-FEM enrichment of the displacement field can be recursively applied by adding one extra term at each branching event. The proposed approach is also equipped with a crack-fluid pressure, a relevant feature in applications such as hydraulic fracturing or leakage-related events. The capabilities of the model to handle propagation and branching of cracks are illustrated by means of different two-dimensional numerical examples.     

用3D离散元实现Ⅰ/Ⅱ型拉剪混合断裂的模拟

将变形体离散元分别与弥散式旋转裂缝模型和分离式裂缝模型耦合起来,采用上述两种非线性断裂力学模型分析了混凝土、岩石等准脆性材料的Ⅰ/Ⅱ型拉剪混合模式的开裂行为,以实现从连续介质到非连续介质转化的数值模拟。基于变形体离散元方法的界面接触本构关系提出了混凝土Ⅰ/Ⅱ混合型开裂的拉剪分区开裂准则;基于缝面法向开度增大对刚度强度的逐渐折减实现了界面裂缝扩展的模拟。作为数值验证,通过单边切口非对称三点弯梁试验的数值计算与试验结果对比,表明两种断裂力学模型均能够合理预测裂缝的起裂和扩展,在混合形式的荷载条件下,裂缝通常以Ⅰ/Ⅱ型拉剪混合模式起裂,而后以Ⅰ型为控制方式稳定扩展。文末指出,基于离散单元法的分离裂缝模型能够实现系统在外载作用下破坏全过程仿真。     

The interaction between the wellbore and pressure-induced fractures

A method based on modeling a crack by a continuous distribution of dislocations has been used to study the interaction between the wellbore and pressure-induced fractures. This method allows the stress intensity factor and opening of the fractures to be calculated as a function of fracture length, in-situ state of stress, fracture pressure and rock properties. It also allows one to determine the wellbore pressure during quasi-static propagation. Two configurations have been considered in this report; a fracture in which acts a pressure equal to the pressure in the hole (hydraulic fracturing using a perfectly penetrating fluid) and a stress free fracture (i.e. sleeve fracturing or hydraulic fracturing using a non penetrating fluid). These two cases give respectively the lower bound and upper bound for the pressure during a micro-hydraulic fracture experiment.The pressure behaviour has been found to be strongly dependent on fracture toughness and ₂ for short cracks, but also on the mode of loading (i.e. pressurized crack vs. stress free crack). The influence of fracture toughness and ₂ on the opening has been found to increase with fracture length when the fluid pressurizes the crack and to be relatively small during sleeve fracturing. These various behaviours could be used to determine the value of ₂, ₃, G/(1-) and fracture toughness if the crack opening at the wellbore is measured during a pressure test: the model is efficient enough to be used in an inversion procedure.The model has also shown that the use of Kirsch's solution to predict breakdown pressure is only valid if the fluid does not penetrate the fracture.     

定向射孔多缝压裂工艺案例分析

一种新型的压裂工艺在中国长庆油田试验成功,该工艺在层内压裂形成了2条独立裂缝,大幅增加了裂缝与油藏的接触面积,提高了产量。长庆油田开发的超低渗透油藏岩芯分析渗透率一般在0.5 mD以下,且受到储层条件、注采井网、压裂工艺等多重限制,常规压裂工艺改造难以实现该类油藏的有效开发。历经多年研究并结合自身实际,长庆油田于2006年提出了体积压裂的理念,通过在油层内造多缝扩大油藏的泄流体积以提高单井产量。室内物模试验研究射孔对压裂影响时观察到,当射孔方向与最大水平主应力方位存在夹角的情况下裂缝发生转向,即裂缝均先沿射孔孔眼方向起裂,后转向最大主应力方向。在同一油层内上下各射一排孔,两排孔方位与最大主应力方向呈一定夹角,一排偏右,一排偏左。分别对两排孔眼进行压裂,这样由于初始裂缝转向导致两条裂缝在井眼处虚拟相交,但在地层内不会重合,形成近井类似于"X"形4条裂缝,远井形成2条近于平行的裂缝。提出了一种产生多裂缝的新型压裂工艺———定向射孔多裂缝压裂。开展了大量物模试验,证实了思路的可行性,并且长庆油田在A井开展了现场压裂试验。井下微地震裂缝测试表明:形成了"X"形多裂缝;产量数据表明:定向射孔多缝压裂井相比邻井产量提高38.1%。     

On a relationship between stretched zone parameters and fracture toughness of ductile structural steels

Quantitative stereofractographic analysis of the stretched zone at the crack tip is performed on specimens of two ductile steels used for determining fracture toughness. The effect of temperature, loading rate, stress-state mode and specimens size on the stretched zone formation is studied. A correlation is obtained of the stretched-zone height and width with fracture toughness of materials and the crack tip opening displacement value which is determined by means of various mathematical models.     

Strain-rate effects on concrete behavior

In this paper, a previously developed meso-scale model for concrete, called the Confinement Shear Lattice (CSL) model, is extended in order to include the effect of loading rate on concrete strength and fracturing behavior. The rate dependence of concrete behavior is assumed to be caused by two different physical mechanisms. The first is a dependence of the fracture process on the rate of crack opening, and the second is the viscoelastic deformation of the intact (unfractured) cement paste. In this study, the first mechanism is described by the activation energy theory applied to the ruptures occurring along the crack surfaces, whereas the second mechanism is modeled by the Microprestress-Solidification theory. The developed model is calibrated and validated on the basis of experimental data gathered from the literature.     

Microstructure and fracture mechanical response of wood

Investigations on the fracture properties of wood in relation to its microstructure are reported. The inhomogeneous and hierarchical structure of wood is addressed. Wood species, the influence of orientation, the role of structural features, like rays are considered and discussed. Likewise the mode of loading, which determines the mode of fracturing, and the influence of humidity have been studied by using new fracture mechanical techniques and ways of evaluation. The specific fracture energy has been determined under crack opening conditions. In-situ loading in an environmental scanning electron microscope (ESEM), which allows observation in moistured condition, has been performed in order to investigate the mechanisms of fracturing of wood on a sub-microscopic scale. In the nanometer range, especially the influence of the microfibril angle on deformation and fracture behaviour has been studied.     

A comparison of high temperature defect assessment methods

Cracked high temperature components which are subjected to creep or creep-fatigue loading may fail by crack growth, net section rupture or a combination of both processes. In this paper, models are presented for describing these modes of failure in terms of fracture mechanics concepts, limit analysis methods and cumulative damage laws. It is shown that these models form the basis of a number of high temperature defect assessment procedures that are available for plant. These procedures are then applied to a semi-elliptical defect in a plate which is subjected to creep-fatigue loading. It is found that the predictions are sensitive to the crack initiation criteria assumed and the limit analysis solutions adopted.     

Fracture toughness of exponential layer-by-layer polyurethane/poly(acrylic acid) nanocomposite films

This paper characterizes the fracture toughness of layer-by-layer (LBL) manufactured thin films with elastic polyurethane, a tough polymer, and poly(acrylic acid) as a stiffening agent. A single-edge-notch tension (SENT) specimen is used to study mode I crack propagation as a function of applied loading. Experimental results for the full-field time histories of the strain maps in the fracturing film have been analyzed to obtain R-curve parameters for the nanocomposite. In particular, by using the strain maps, details of the traction law are measured. A validated finite strain phenomenological visco-plastic constitutive model is used to characterize the nanocomposite film while a discrete cohesive zone model (DCZM) is implemented to model the fracture behavior. The LBL manufactured nanocomposite is found to display a higher fracture toughness than the unstiffened base polymer.     

On the nature of “dynamic” strength of brittle bodies

The enhanced strength observed for bodies under dynamic loading has been attributed to fracturing duration, i.e. the time interval between the commencement of fracture at the most heavily loaded point of the body and the instant of completion of fracture which is recorded by a sensor. The “dynamic” strength of brittle bodies is shown to be related to the crack formation process.     

混凝土Ⅲ型裂缝动静力损伤断裂分析

本文把混凝土的动静力损伤本构模型通过应变等价性假设引入到线弹性断裂力学中，推出了混凝土Ⅲ型裂缝在动静力荷载作用下缝端附近的损伤场，给出了相应的判据，实现了损伤与断裂的耦合分析，为把损伤理论应用到混凝土断裂分析中提供了一条可行的途径。文末还给出了混凝土Ⅲ型裂缝允许损伤尺度的估计值。     

Formulation and implementation of strain rate-dependent fracture toughness in context of the phase-field method

The phase-field approach is a promising technique for the realistic simulation of brittle fracture processes, both in quasi-static and transient analysis. Considering fast loading, experimental evidence indicates a strong relationship between the rate of strain and the material's resistance against fracture, which can be considered by a dynamic increase factor for the strength of the material. The paper at hand presents a novel approach within the framework of phase-field models for brittle fracture. A rate-dependent fracture toughness is formulated as a function of the rate of crack driving strain components, which results in higher strength for faster loading. Beside the increased amount of energy necessary to evolve a crack at a high strain rate loading situation, the model incorporates quasi-viscous stress-type quantities that are not directly related to the formation of the crack and exist only in the phase-field transition zone between broken and sound material. The governing strong form equations for a transient simulation are derived and the relevant information for an implementation of the model into a finite element code is outlined in detail. The performance of the model is demonstrated for static and dynamic benchmark simulations and for a comparison to experimental findings.     

Constraint issues in cleavage fracture and matters arising from practical application

In the past 10 years much research has been carried out to deal with the question of how crack‐tip constraint effects can be described and, moreover, how crack‐tip constraint (stress tri‐axiality in the vicinity of the crack tip) contributes to matters arising from practical application of structures and components containing postulated or real cracks and made of ferritic steel. In fracture mechanics, application crack‐tip constraint can be influenced by loading (out of plane or multi‐axial loading) or by the crack shape and crack depth to ligament ratio. Temperature loading of a crack in a structure is different compared to the loading condition of a fracture toughness specimen and a deep crack behaves differently compared to a short crack. In this paper a model for the prediction of cleavage fracture of ferritic steel is briefly described and summarized and the issues for practical application are pointed out. It turns out that crack‐tip constraint, induced either by loading or geometry, can be described quantitatively by local approach models, but there is still a need to understand the micro structural features behind.     

Analysis of a dynamically loaded three-point-bend ductile fracture specimen

The three-point-bend bar is a common specimen configuration used in experimental fracture studies. It is essentially a two-dimensional configuration in the form of a simply supported beam with an initial edge crack on the cross-section at mid-span. The specimen is loaded to fracture initiation by means of a concentrated transverse force applied at mid-span on the uncracked surface of the beam. In the present case, it is assumed that the material of interest is ductile, and that fracture initiation occurs after substantial plastic deformation, which develops in the uncracked ligament under the applied load. Furthermore, it is assumed that the loading is rapidly applied so that material inertia must be taken into account in relating applied loads to crack tip fields. It is supposed that the crack tip conditions are such that the J-integral may be adopted as a characterizing parameter. The main purpose of the present study is to determine conditions under which the value of J at initiation may be inferred from quantities that are directly measurable in an experiment. To this end, J is determined from computed field quantities by means of a crack tip integral that is suitable for finite element procedures. The value of J is simultaneously computed in terms of measurable quantities by means of an appropriate deep crack formula, and implications for fracture testing of tough materials at relatively high loading rates are discussed. The notion of a transition time, defined as the time beyond which a deep crack formula may be used to compute J in an experiment, is introduced on the basis of simple model studies. Calculations are performed for typical specimen dimensions and material properties representative of a high-strength structural steel in a ductile condition.     

Ductile crack propagation by plastic collapse of the intervoid ligaments

In the present study ductile crack initiation and propagation is investigated by means of a micro-mechanical model under small-scale yielding conditions. Voids are resolved discretely in the fracture process zone where steep gradients occur during the loading history and are taken into accounted by a homogenized porous plasticity law elsewhere. The size of the region of discrete voids is not set a priori but is determined consistently. The results show that effective crack growth occurs by plastic collapse, i.e. purely geometric softening of the intervoid ligaments without incorporating material separation. Due to this mechanism a limit load exists coinciding with the maximum fracture toughness. In addition, it turns out that the shielding due to the growth of voids around the crack plane has a considerable influence on the computed R-curves compared to models neglecting this effect. Depending on the void arrangement a diffuse softening zone or even crack branching is observed. A comparison with experimental data from literature indicates that plastic collapse and the formation of diffuse zones of void growth are realistic mechanisms of ductile crack propagation.     

A local approach to cleavage fracture in ferritic steels following warm pre-stressing

Quantification of the enhancement in cleavage fracture toughness of ferritic steels following warm pre‐stressing has received great interest in light of its significance in the integrity assessment of such structures as pressure vessels. A Beremin type probability distribution model, i.e., a local stress‐based approach to cleavage fracture, has been developed and used for estimating cleavage fracture following prior loading (or warm pre‐stressing, WPS) in two ferritic steels with different geometry configurations. Firstly, the Weibull parameters required to match the experimental scatter in lower shelf toughness of the candidate steels are identified. These parameters are then used in two‐ and three‐dimensional finite element simulations of prior loading on the upper shelf followed by unloading and cooling to lower shelf temperatures (WPS) to determine the probability of failure. Using both isotropic hardening and kinematic hardening material models, the effect of hardening response on the predictions obtained from the suggested approach has been examined. The predictions are consistent with experimental scatter in toughness following WPS and provide a means of determining the importance of the crack tip residual stresses. We demonstrate that for our steels the crack tip residual stress is the pivotal feature in improving the fracture toughness following WPS. Predictions are compared with the available experimental data. The paper finally discusses the results in the context of the non‐uniqueness of the Weibull parameters and investigates the sensitivity of predictions to the Weibull exponent, m, and the relevance of m to the stress triaxiality factor as suggested in the literature.     

A probabilistic fracture mechanics model including 3D ductile tearing of bi-axially loaded pipes with surface cracks

This paper presents a probabilistic fracture mechanics model established from three-dimensional FEM analyses of surface cracked pipes subjected to tension load in combination with internal pressure. The models are particularly interesting for offshore pipelines under operational conditions or during laying, where inelastic deformations may occur. In the numerical models, the plastic deformations, including ductile tearing effects, are accounted for by use of the Gurson-Tvergaard-Needleman model. This model is calibrated to represent a typical X65 pipeline steel behaviour under ductile crack growth and collapse. Several parameters are taken into account, such as crack depth, crack length and material hardening. Another important topic is the examination of the influence of bi-axial loading due to internal pressure on capacity. From the results of the deterministic analyses a probabilistic fracture mechanics model is established using the response surface methodology. Two failure criteria are examined to represent the structural capacity. Based on the established model, we illustrate the methodology by examples employing the two different failure criteria solved with first and second order reliability methods.