The motion of a brittle crack

The acceleration phase and the subsequent motion of a brittle crack of finite length extending in an infinite sheet loaded in the extensional mode are determined. The energy release is assumed to depend on crack length and on crack-tip velocity but not on higher time-derivatives of crack length. This energy release should equal the fracture energy, which, for a given material is supposed to be dependent only on the crack-tip velocity. For polymethylmethacrylate this velocity dependence has been determined from recent experiments on the long strip configuration. The results obtained show good agreement with earlier experimental investigations of crack motion in large sheets of polymethyl-methacrylate. Since the validity of existing exact analytical solutions of the accelerating crack problem seems to be severely restricted from a practical point of view, the approximate method in the present paper could offer a promising alternative approach.     

Crack propagation in an elastic solid subjected to general loading—III. Stress wave loading

The stress intensity factor of a half-plane crack extending non-uniformly in an isotropic elastic solid subjected to stress wave loading is determined. A plane stress pulse strikes the crack at time t = 0, the wavefront being parallel to the plane of the crack. At some arbitrary later time t = τ, the crack begins to extend at a non-uniform rate. It is found that the stress intensity factor is a universal function of instantaneous crack-tip velocity times the stress intensity factor for an equivalent stationary crack. An energy rate balance fracture criterion is applied to obtain an equation of motion for the crack tip. The delay time between the arrival of the incident pulse and the onset of fracture is also calculated for this fracture criterion.     

Transient crack growth under creep conditions

Transient crack growth in an elastic/power-law creeping material is investigated under antiplane shear loading and small-scale-yielding conditions. At time t = 0 the solid is suddenly loaded far from the crack by tractions that correspond to the elastic crack-tip stress distribution. At that time the crack begins to propagate at a constant velocity. The stress fields evolve in a complex manner as the crack propagates due to the competing effects of stress relaxation due to constrained creep and stress elevation due to the instantaneous elastic material response to crack growth. From detailed finite element calculations it is shown that these fields can be approximated by a simple matching of three asymptotic singular crack-tip solutions. A characteristic stress, distance and time are defined for this problem which provide a normalization that accounts for any crack velocity, loading and all material properties for a given creep exponent n. Results are presented for crack-tip stresses, strains, crack opening displacements and creep zones.     

Role of elasticity in elastic–plastic fracture: Analytical bi-linear crack-tip fields and finite element analysis

A solution for Model-I plane strain crack tip fields in a bi-linear elastic–plastic material is presented. The elastic–plastic Poisson's ratio is introduced to characterize the influence of elastic deformation on the near tip constraint. Attention is focused on the distribution of elastic/plastic strain energy in the sensitive region of the forward sector ahead of a crack tip. The present study shows that the elastic strain energy can be higher than the plastic strain energy in this sensitive sector while large amount of the plastic strain energy develops outside this sector around the crack tip. The effect of elastic deformation in this sensitive region on the structure of crack-tip fields is considerable and the assumption in some important solutions for crack-tip fields reported in literature that the elastic deformation is small and can be ignored is therefore not physically reasonable. Besides, finite element analysis is carried out to validate the analytical solution and good agreement between them is found. It is seen that the present solution with T-stress can properly describe the crack-tip fields under various constraints for different specimens and an analytical relation is established between the critical value of J-integral, J_c, and T-stress for elastic–plastic fracture.     

Combination of fracture and damage mechanics for numerical failure analysis

A new approach for the analysis of crack propagation in brittle materials is proposed, which is based on a combination of fracture mechanics and continuum damage mechanics within the context of the finite element method. The approach combines the accuracy of singular crack-tip elements from fracture mechanics theories with the flexibility of crack representation by softening zones in damage mechanics formulations. A super element is constructed in which the typical elements are joined together. The crack propagation is decided on either of two fracture criteria; one criterion is based on the energy release rate or the J-integral, the other on the largest principal stress in the crack-tip region. Contrary to many damage mechanics methods, the combined fracture⧹damage approach is not sensitive to variations in the finite element division. Applications to situations of mixed-mode crack propagation in both two- and three-dimensional problems reveal that the calculated crack paths are independent of the element size and the element orientation and are accurate within one element from the theoretical (curvilinear) crack paths.     

Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture

Analyses of the stress and strain fields around smoothly-blunting crack tips in both non-hardening and hardening elastic-plastic materials, under contained plane-strain yielding and subject to mode I opening loads, have been carried out by use of a finite element method suitably formulated to admit large geometry changes. The results include the crack-tip shape and near-tip deformation field, and the crack-tip opening displacement has been related to a parameter of the applied load, the J-integral. The hydrostatic stresses near the crack tip are limited due to the lack of constraint on the blunted tip, limiting achievable stress levels except in a very small region around the crack tip in power-law hardening materials. The J-integral is found to be path-independent except very close to the crack tip in the region affected by the blunted tip. Models for fracture are discussed in the light of these results including one based on the growth of voids. The rate of void-growth near the tip in hardening materials seems to be little different from the rate in non-hardening ones when measured in terms of crack-tip opening displacement, which leads to a prediction of higher toughness in hardening materials. It is suggested that improvement of this model would follow from better understanding of void-void and void-crack coalescence and void nucleation, and some criteria and models for these effects are discussed. The implications of the finite element results for fracture criteria based on critical stress or strain, or both, is discussed with respect to transition of fracture mode and the angle of initial crack-growth. Localization of flow is discussed as a possible fracture model and as a model for void-crack coalescence.     

An integral equation method in dynamic fracture mechanics

This paper contains a study of the problems of crack propagation under static stresses and under transient stress waves. Within the assumption of linear elastic fracture mechanics, an integral-equation method has been developed for the analysis of these problems. The method has been applied to: (a) the determination of the stress intensity factor at a given loading and a crack-tip velocity and (b) the determination of the crack-tip motion under a given transient loading.     

基于新型裂尖杂交元的压电材料断裂力学研究

提出了一种裂尖邻域杂交元模型,将其与标准杂交应力元结合来求解压电材料裂纹尖 端的奇性电弹场和断裂参数的数值解．裂纹尖端杂交元的建立步骤为：1) 利用高次内插有限元特征法求解特征问题,得到反映裂尖奇异性电弹场状况的特 征值和特征角分布函数;2) 利用广义Hellinger-Reissner变分泛函以及特征问题的解来建立裂尖邻域杂交元模型．该 方法求解电弹场时,摒弃了传统有限元方法中裂尖奇异性场需要借助解析解的做法,也避免 了单纯有限元方法中需要在裂尖端部进行高密度单元划分．采用PZT5板中心裂纹问题 作为考核例,数值结果显示了良好的精确性．作为进一步应用,求解了含中心界面裂纹 的PZT4-PZT5两相压电材料的应力强度因子和电位移强度因子．所有的算例都考虑 了3种裂纹面电边界条件．     

Intersonic crack propagation along interfaces: Experimental observations and analysis

The isochromatic fringe patterns surrounding an intersonically propagating interface crack are developed and characterized using the recently developed stress field equations. A parametric investigation is conducted to study the influence of various parameters such as the crack-tip velocity and the contact coefficient on the isochromatic fringe patterns. It has been observed that the crack-tip velocity has a significant effect on the size and shape of isochromatic fringe patterns. The contact coefficient, on the other hand, does not affect the fringe pattern significantly. The paper also presents a numerical scheme to extract various parameters of interest such as the series coefficients of the stress field, the contact coefficient and the dissipation energy. The results show that the crack growth is highly unstable in the intersonic regime, and the energy dissipation decreases monotonically with increasing crack-tip velocity. The experimental data fit well with the recently proposed fracture criterion for intersonic interfacial fracture.     

A micromechanical model for predicting the fracture toughness of functionally graded foams

In this paper, finite element method based micromechanical analysis is used to understand the fracture behavior of functionally graded foams. The finite element analysis uses a micro-mechanical model in conjunction with a macro-mechanical model in order to relate the stress intensity factor to the stresses in the struts of the foam. The stress intensity factor at the crack tip of the macro-mechanical model can be evaluated using either the J-contour integral or the stresses in the singularity-dominated zone. The fracture toughness is evaluated for various crack positions and length within the functionally graded foam. Then the relationship between the fracture toughness of the graded foam and the local density at the crack tip is studied. Convergence tests for both macro-mechanical and micro-mechanical model analysis were conducted in order to maintain adequate accuracy with reasonable computational time. Fracture toughness of homogenous foams and functionally graded foams for various cases are presented as a function of relative density. This study indicates that the fracture toughness of functionally graded foams mainly depends on the relative density at the crack-tip.     

BRITTLE-DAMAGE ANALYSIS OF CRACKS UNDER CONDITIONS OF PLANE-STRAIN TENSILE LOADING

An abrupt damage model, taking full account of finite geometrychanges, is used to study both the shapes of damage zones and the stress strain fieldsfor a plane-strain tensile crack under the small-scale yielding condition. Two typicalcrack-tip damage profiles are simulated by the element vanish technique. The fracturetoughness increment due to damage dissipation is evaluated in terms of the energyrelease rate.     

裂纹面摩擦接触引起的断裂韧性增长的研究

采用弹黏塑性的材料本构关系, 建立了压、剪混合型裂纹常速准静 态扩展的力学模型, 求得了裂纹面摩擦接触条件下裂纹尖端场的数值解, 并基于数 值结果讨论了扩展裂纹的摩擦效应. 计算和分析表明, 裂纹面的摩擦效应主要表现 在两个方面. 第一方面是摩擦会导致裂纹尖端区材料的断裂韧性增高, 并且裂纹面间的摩擦作用越强, 增韧效果越显著. 摩擦增韧的机制可以解释为裂纹 面间的摩擦作用导致裂纹尖端塑性区尺寸变大, 使裂纹尖端场的塑性变形能增加, 从而使得裂纹尖端区材料增韧. 摩擦生热并不是导致材料断裂韧性增长的根本机制. 第二方面是摩擦会导致``断裂延缓'. 利用裂纹面的摩擦来提高构件的承载能力和延长构件的服役寿命具有较大的工程实用价值.     

平面应变脆性损伤裂纹场分析

根据Ｂｕｉ的脆性损伤理论，本文研究Ｉ型拉伸裂纹的平面应变过程区及其力学场。计及裂尖材料的有限变形，细致的有限元分析给出了小范围屈服条件下两种典型的损伤区形貌。     

On stable crack growth

Before catastrophic fracture occurs, certain changes in the distribution of the plastic strain near the crack-tip seem to take place. These changes make crack growth possible but contribute to a gradually increasing screening of the energy flow through the plastic region towards the crack-tip. Stable crack growth results. This phenomenon appears to be a feature common to almost all fractures. Therefore, a unified theory may be developed to describe fracture at both small-scale and large-scale yielding as well as stable crack growth which is accompanied by plastic collapse rather than being superseded by catastrophic fracture. In the present paper, an attempt is made to show how such a unified theory would work.     

Effects of near-tip rotation on pre-buckle crack growth of compressed beams bonded to a rigid substrate

The macroscopic pre-cracked line scratch test (MPLST), in which a debonded edge of a film is loaded in in-plane compression, has been modeled as a generic, coupled fracture–buckle problem using simple beam theory. Near crack-tip beam rotation (also called root rotation in literature), which always exists due to the eccentric loading in this type of test, has been incorporated into the governing equations. An analytical solution to the augmented problem has been derived. It is found that the near-tip rotation can introduce pre-buckle bending in the film. One important consequence of this pre-buckle bending is that it leads to the reduction of the critical buckling condition. This agrees well with the results of [Int. J. Fract. 113 (2002) 39] obtained by solving the full elastic field near the crack-tip. Furthermore, the pre-buckle bending moment at crack-tip remains negative (leading to crack closure) as long as the pre-buckle crack length is small, but it becomes positive (leading to crack opening) at larger pre-buckle crack length. The negative bending moment causes the crack-tip energy release rate to decrease as the crack propagates, which results in a stable pre-buckle crack growth. Once it becomes positive, however, the bending moment causes crack-tip energy release rate to increase rapidly as crack length increases and hence leads to an unstable (pre-buckle) crack growth. Further, the nominal phase angle is initially larger than the classic prediction of 52.1° owing to the existence of the negative crack-tip bending moment, but it drops quickly upon approaching the buckle point. All these results are confirmed by a rigorous 2D FEM calculation using cohesive zone modeling (CZM) approach. Finally the derived analytical solution has been used to analyze a set of PLST data reported in the literature. It has been demonstrated that plasticity in the adhesive layer and in the bonded film is responsible for the strong R-curve toughening characteristics in the deduced interface toughness data. It has also been shown that, once the deduced interface toughness is incorporated into a CZM simulation, both the axial loading and buckling point can be accurately predicted.     

Interaction between an incompressible flow and elastic cantilevers of circular cross-section

The purpose of this work is to study the deformation of elastic cantilevers due to hydrodynamic forces by coupled fluid–structure interaction simulations. The cantilever is placed in a rectangular duct and the Reynolds number based on bulk velocity and cantilever diameter is 400. Reduced velocities in the range π/4 to 2π are studied, which covers both un-synchronised motion and the initial branch of synchronisation. The cantilever surface is represented by a virtual boundary method which replaces a solid object in flow by additional force distribution to satisfy local boundary condition. The flow field is solved using a Cartesian finite difference code and the deformation of the cylinder a finite element approach using one-dimensional beam elements is used.     

Mixed mode crack initiation in piezoelectric ceramic strip

When piezoelectric ceramics are subjected to mechanical and electrical load, they can fracture prematurely due to their brittle behavior. Hence, it is important to know the electro–elastic interaction and fracture behavior of piezoelectric materials. The problem of a through crack in a piezoelectric strip of finite thickness is studied in this paper. Fourier transforms are used to reduce the problem to the solution of singular integral equations. The model technique can solve for polarization in an arbitrary direction and material anisotropy. Numerical values of the crack-tip field amplification for a piezoelectric strip under in-plane electromechanical loading are obtained. Energy density factor criterion is applied to obtain the maximum of the minimum energy density and direction of crack initiation. The influence of crack length and crack position on stress intensity and energy density factors is discussed.     

Effect of T-stress on branch angle of moving cracks

The effects of the T-stress on Yoffe crack propagation are analyzed. Using a maximum k_I fracture criterion near the kink of a moving crack tip, a branch angle is determined via asymptotic crack-tip field containing two fracture parameters related to singular and constant terms. Results indicate that crack speeds decrease the T-stress. The crack-tip field and the branch angle depend on the T-stress, especially for higher crack velocities. The critical speed for crack bifurcation is independent of remote transverse loading if neglecting the T-stress. Otherwise, the crack branch speed is reduced or raised, depending on positive or negative transverse loading, respectively.     

Field structure at mode III dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.