The competition between the crack kinking away from the interface and crack propagation along the interface in elastic bicrystals

The problem analyzed is of the crack kinking away from the interface between the two different anisotropic materials. The attention is concentrated on the initiation of the crack kinking and the condition that the length of the crack segment that is leaving the interface is small in comparison to the crack segment that remains along the interface. The emphasis is placed to the application of the fracture mechanics concept for the interfacial crack that propagates dynamically between the two orthotropic materials. The simulations and calculations were done by application of the Mathematica ^® programming routine. The stress intensity factors and the energy release rate are obtained for the kinked crack, as functions of the corresponding values for the interfacial crack prior to kinking. The analysis was performed of the influence of anisotropy on the crack kinking versus crack propagating along the interface competition. Due to anisotropy the kinking is easier, i.e., it is easier for the crack to kink away from the interface into the “softer” of the two materials. The oscillatory index for the case of the dynamic crack growth along the interface between the two orthotropic materials increases with crack tip speed v and with increase of the difference in stiffnesses. The practical application of this analysis could be for the interface in the glued joints and protective coatings.     

单向静压下柱状炮孔端部爆生裂纹的扩展规律

高地应力对岩层地下工程爆破动态断裂过程有重要影响。采用数字激光动态焦散线测试系统,研究了不同单向静压下柱状炮孔端部爆生裂纹动态断裂行为,明确了柱状炮孔端部爆生裂纹的扩展规律。结果表明:单向静压越大,端部裂纹平均扩展长度越短,但单向静压下端部裂纹尖端积聚能量的快速释放会导致裂纹初始扩展速度提升;裂纹尖端应力强度因子基本随单向静压增加而递减,单向静压越大,应力强度因子随时间下降越剧烈,裂纹的止裂韧度越高,止裂时间越早;单向静压作用下的爆生裂纹在整个扩展阶段基本表现为I型裂纹,无静压作用下爆生裂纹在扩展初期表现为I型裂纹,中后期表现为复合型裂纹。研究结果对认识静压作用下的柱状炮孔端部破坏机理具有一定意义。     

Conditions for crack kinking under stress-wave loading

Kinking of a crack in a prestressed body under the influence of incident stress waves is investigated on the basis of the balance of rates of energies. It is assumed that the crack tip will choose to propagate at a time, in a direction, and at a speed for which the energy flux into the propagating crack tip attains a maximum value with respect to variation of the kinking angle. It is shown that the balance of rates of energies implies that the crack tip speed is zero at the onset of fracture. Consequently, the conditions for the onset of crack kinking and for the computation of the kinking time and kinking angle are completely defined by the elastodynamic field around the original crack tip. Examples of the incidence of step stress waves on a semiinfinite crack in a prestressed body have been investigated. It is shown that for an incident antiplane wave with Mode III fracture, kinking is generally not possible. For an incident inplane wave with mixed Mode I–II fracture, kinking may happen. For that case curves are presented which relate the kinking time and the kinking angle to the state of prestress and to the parameters of the incident wave.     

Crack kinking from an initially closed, ordinary or interface crack, in the presence of friction

One theoretically studies crack kinking from an ordinary crack (in some homogeneous solid) or an interface crack (between two dissimilar materials), in the situation where this crack is closed prior to kinking but open after it. This problem was recently considered by the authors with the simplifying, but physically quite unrealistic hypothesis of absence of friction between the crack lips. Their work is extended here to account for possible friction governed by Coulomb’s law. Problems of elastic fracture mechanics with unilateral contact and friction between the crack lips being not only non-linear, but incremental in nature, the theoretical treatment becomes notably more involved than without friction. It is still based, however, on the same basic ingredients, namely “homogeneity” properties of the type of problems considered, changes of scale and some reasonable hypotheses. It is shown that whatever the geometry of the body and the crack and whatever the loading, the asymptotic expression of the stress intensity factors (SIF) at the tip of a vanishingly small kinked crack extension depends solely upon the initial (mode II) SIF prior to kinking, the kink angle, Dundurs’s famous parameters α and β and the friction coefficient. The (history-independent) functions involved in the general formulae established are determined numerically through finite element computations. From there, using Goldstein and Salganik’s famous principle of local symmetry to predict the crack path, one derives a theoretical value for the kink angle. This value depends upon the loading only through the sign of the initial stress intensity factor; it also depends on the mismatch of elastic properties and the friction coefficient. However, its range of variation is numerically found to be rather narrow. Experiments conducted by various authors seem to confirm these theoretical predictions.     

Distributed cracks and kinked cracks in bonded dissimilar half planes with an interface crack

This paper is concerned with the interactions between an interface crack and other arbitrarily distributed cracks in two bonded dissimilar half planes. Special emphasis is placed on the cracks kinked at a tip of the interface crack, which remain unsolved as far as the authors are concerned. For the present, we pay attention to the stress intensity factors at the tips of the kinks or the distributed cracks, and not to those at the tips of the interface crack. The analysis is based on continuous distributions of the body forces along the cracks, and their densities are determined with a new procedure in order to get highly accurate results. The present analysis for distributed line cracks applies to kinked cracks, branched cracks and those piercing the interface just by joining some of the line cracks. Numerical calculations are performed for various important problems, and the effects of geometric and mechanical parameters on the stress intensity factors are examined.     

On crack kinking and curving

This paper attempts to provide a comprehensive summary of results on the following topics relating to crack kinking and curving: (a) Stress intensity factors for kinked cracks under combined loading; (b) applicability of Irwin's energy-release rate formula to kinked cracks; and (c) reasons behind crack kinking and curving. It is assumed throughout that the surfaces of the cracks do not come into contact.     

FRACTURE MECHANICS ASSESSMENT OF PLANAR BRANCHED CRACKS IN AN EQUIBIAXIAL STRESS FIELD

Abstract A simplified fracture mechanics assessment is presented of branched planar cracks in an equibiaxial stress state. In linear-elastic fracture mechanics the stress intensity factors which characterize the load at the crack tips depend, for a given external load, only on the crack geometry. The stress intensity factors of a large number of branched cracks were evaluated using the Boundary Element method, and correlations between the stress intensity factors and the crack geometry were investigated. Formulae are presented which assign an individual effective crack length to each crack tip of a branched crack and hence allow approximate stress intensity factors to be determined for very complicated crack geometries. An algorithm is used for the stochastic simulation of an irregular crack pattern formation in thermal fatigue.     

Stress intensity factors for a wide range of long-deep semi-elliptical surface cracks, partly through-wall cracks and fully through-wall cracks in tubular members

To calculate the rate of fatigue crack growth in tubular members, one approach is to make use of the fracture mechanics based Paris law. Stress intensity factors (SIF) of the cracked tubular members are prerequisite for such calculations. In this paper, stress intensity factors for circumferential deep semi-elliptical surface crack (a/t > 0.8), semi-elliptical partly through-wall crack and fully through-wall crack cracks in tubular members subjected to axial tension are presented. The work has produced a comprehensive set of equations for stress intensity factors as a function of a/T, c/πR and R/T for deep surface cracks. For the partly through-wall cracks and fully through-wall cracks, two sets of bounding stress intensity factor equations were produced based on which all stress intensity factors within the range of parameters can be obtained by interpolation.     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

A theory of caustics for mixed-mode fast running cracks

The method of caustics (shadow spot method) has proven to be a powerful optical method to measure stress intensity factors in static and dynamic fracture mechanics problems. In this paper, a theory of caustics was developed for elastodynamically propagating cracks under inplane mixed-mode conditions. Complex potentials for the general solutions of a near-tip field which have been previously derived by the authors were used in this theoretical development. Completely analytical expressions were derived for the caustic curves as well as for the initial curves for fast running cracks under inplane mixed-mode conditions. The effects of crack velocity and mixed-mode condition on the caustic pattern and the initial curve were investigated. New procedures were also proposed for the evaluation of the dynamic stress intensity factors K_I and K_II using the overall dimensions of the caustic pattern. The method of caustics developed here enables one to study quantitatively various mixed-mode dynamic fracture phenomena such as crack branching, crack curving, and crack kinking.     

Time-domain BEM analysis of a rapidly growing crack in a bi-material

Rapid propagation of a matrix crack in a bi-material system is studied with emphasis on the dynamic interaction between the crack and the interface by combining the traditional time-domain displacement boundary element method (BEM) and the non-hypersingular traction BEM. The crack growth is controlled by the fracture criterion based on the maximum circumferential stress, and is modeled by adding new elements to the moving crack tip. Detailed computation is performed for an unbounded bi-material with a crack subjected to incident impact waves and a bounded rectangular bi-material plate under dynamic wedged loading. Numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions are presented for various material combinations and geometries. The effects of the interface on the crack growth are discussed.     

A theoretical note on mode-I crack branching and kinking

An energy-based fracture mode has been derived for the mode-I crack branching and kinking. The classic J_i-integral has been further explored by a new partial integral path and the analytical solution of the energy release rate for crack branching and kinking from a mode-I crack tip has been established. The crack branching/kinking angle has also been analytically derived. It shows that the Griffith’s theorem and conservation law can be applied to both mode-I crack extension and mode-I crack branching and kinking. The branching mechanism for quasi-static mode-I crack has been theoretically investigated. The branching toughness and the K-based criterion for crack branching have been defined. The crack branching phenomena predicted by the present model are in well agreement with the experimental observations reported in the literatures.     

The interaction of two inclined cracks with dynamic stress wave loadings

To gain insight into the phenomenon of the interaction of stress waves with material defects and the linkage of two cracks, the transient response of two semi-infinite inclined cracks subjected to dynamic loading is examined. The solutions are obtained by the linear superposition of fundamental solutions in the Laplace transform domain. The fundamental solution is the exponentially distributed traction on crack faces proposed by Tsai and Ma [1]. The exact closed form solutions of stress intensity factor histories for these two inclined cracks subjected to incident plane waves and diffracted waves are obtained explicitly. These solutions are valid for the time interval from initial loading until the first wave scattered at one crack tip returns to the same crack tip after being diffracted by another crack tip. The result shows that the contribution of diffracted waves to stress intensity factors is much less than the incident waves. The probable crack propagation direction is predicted from the fracture criterion of maximum circumferential tensile stress. The linkage of these two cracks is also investigated in detail.     

Effects of T-stresses on fracture initiation for a closed crack in compression with frictional crack faces

This paper studies crack extension resulting from a closed crack in compression. The crack-tip field of such a crack contains a singular field relative to K _II and non-singular T-stresses T _x and T _y parallel and perpendicular to the crack plane, respectively. Using a modified maximum tensile stress criterion with the singular and non-singular terms, the kinking angle at the onset of crack growth is determined by a two parameter field involving the mode-II stress intensity factors and T-stresses, and at fracture initiation a wing crack may be created at an arbitrary angle from 0° to 90°. A compressive T _y increases the kinking angle and reinforces apparent mode-II fracture toughness, while a compressive T _x decreases the kinking angle and enhances apparent mode-II fracture toughness. The direction and resistance of fracture onset is strongly affected by T-stresses as well as frictional stress. The von Mises effective stress is determined for small-scale yielding near the crack tip. The effective stress contour shape exhibits a marked asymmetrical behavior unless 2T _x  = T _y  ≤ 0 for plane stress state. Coulomb friction between two crack faces generally increases the kinking angle, shrinks the size enclosed by the effective stress contour and enhances apparent fracture toughness. Field evidence and experimental observations of many phenomena involving the growth of closed cracks in compression agree well with theoretical predictions of the present model.     

Stress and Failure Analysis of Brittle Matrix Composites. Part II: Failure Analysis

A linear elastic fracture mechanics analysis of a cylindrical element of matrix with a single fiber and two matrix annular cracks perpendicular to the fiber direction under longitudinal tensile load was undertaken. The order of singularity and the angular dependence of the stress field in the neighborhood of the crack periphery were determined by using the stress function approach proposed by Zak and Williams. The stress intensity factor was evaluated by combining the results of the local stress solution with a finite element analysis. The case of fiber debonding originating from the periphery of the annular cracks was also studied. For that problem both opening-mode and sliding-mode stress intensity factors and the strain energy release rate were determined. These results help to understand the various failure mechanisms including matrix cracking, debonding along interfaces and kinking of interface cracks into fibers in brittle matrix composites. This revised version was published online in July 2006 with corrections to the Cover Date.     

Surface effects on mode-I crack tip fields: A numerical study

Surface energy often significantly influences the deformation and failure behavior of materials and devices at the nanoscale. However, how it alters the local deformation around a crack tip remains unclear. In the present paper, we investigate the surface effects on the near-tip fields of a mode-I blunt crack (or notch). The theory of surface elasticity is incorporated into the finite element method. It is found that when the curvature radius of the crack root shrinks to nanometers, surface effects considerably affect the local stress distributions near the crack tip. We also calculate the J-integral, which is almost independent of surface effects except when the integral path approaches the crack tip. This demonstrates that surface effects are localized in a small zone around the crack tip, where the classical fracture mechanics solutions neglecting surface effects should be modified.     

Time domain boundary element method for dynamic stress intensity factor computations

This paper presents a procedure for transient dynamic stress intensity factor computations using traction singular quarter-point boundary elements in combination with the direct time domain formulation of the Boundary Element Method. The stress intensity factors are computed directly from the traction nodal values at the crack tip. Several examples of finite cracks in finite domains under mode-I and mixed mode dynamic loading conditions are presented. The computed stress intensity factors are represented versus time and compared with those obtained by other authors using different methods. The agreement is very good. The results are reliable and little mesh dependent. These facts allow for the analysis of dynamic crack problems with simple boundary discretizations. The versatile procedure presented can be easily applied to problems with complex geometry which include one or several cracks.     

Experimental study of crack propagation in polymethyl methacrylate material with double holes under the directional controlled blasting

The crack propagation behaviours of polymethyl methacrylate material with double holes under the directional controlled blasting are studied using dynamic caustic method. A series of dynamic caustic patterns at the propagating crack tip under the explosive loading are recorded using the digital high‐speed camera. Some important dynamic fracture characterizations and parameters about two main cracks are determined including the crack path, the propagating speed and the crack tip stress intensity factors. The fracture mechanisms of the double blasting holes are analysed. The results provide the important experimental basis for evaluating and designing the directional controlled blasting for the rock tunnel and the rock face excavation.     

The T-stress solutions for through-wall circumferential cracks in cylinders subjected to general loading conditions

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stress solutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a circumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered; defined by the mean radius of the cylinder (R) to wall thickness (t) ratios: R/t = 5, 10, and 20. The T-stress was obtained at eight crack lengths (θ/π = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, θ is the crack half angle). Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubic crack face pressures and remote tension and bending. The results for constant and linear crack face pressure were used to derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validated against several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculations for circumferential cracks in cylinders under complex stress fields.     

A computational model and experimental validation of shielding and amplifying effects at a crack tip near perpendicular strength-mismatched interfaces

The stress field around the crack tip near an elastically matched but strength-mismatched interface body in a bimetallic system is influenced when the crack tip yield or cohesive zone spreads to the interface body. The concept of crack tip stress intensity parameter, K _tip, is therefore employed in fracture analysis of the bimetallic body. A computational model to determine K _tip is reviewed in this paper. The model, based upon i) Westergaard??s complex potentials coupled with Kolosov?CMuskhelishvili??s relations between a crack tip stress field and complex potentials and ii) Dugdale??s representation of the cohesive zone clearly indicates shielding or amplifying effects of strength mismatch across the interface, depending upon the direction of the strength gradient, over the crack tip. The model is successfully validated by conducting series of high cycle fatigue tests over Mode I cracks advancing towards various strength-mismatched interfaces in bimetallic compact tension specimens prepared by electron beam welding of elastically identical weak ASTM 4340 alloy and strong MDN 250 maraging steels.