Crack–surface displacements for cracks emanating from a circular hole under various loading conditions

The purpose of this paper is to calculate and develop equations for crack–surface displacements for two‐symmetric cracks emanating from a circular hole in an infinite plate for use in strip‐yield crack‐closure models. In particular, the displacements were determined under two loading conditions: (1) remote applied stress and (2) uniform stress applied to a segment of the crack surface (partially loaded crack). The displacements were calculated by an integral‐equation method based on accurate stress–intensity factor equations for concentrated forces applied to the crack surfaces and those for remote applied stress or for a partially loaded crack surface. A boundary‐element code was also used to calculate crack–surface displacements for some selected cases. Comparisons made with crack–surface displacement equations previously developed for the same crack configuration and loading showed significant differences near the location where the crack intersected the hole surface. However, the previous equations were fairly accurate near the crack‐tip location. Herein an improved crack–surface displacement equation was developed for the case of remote applied stress. For the partially loaded crack case, only numerical comparisons were made between the previous equations and numerical integration. A rapid algorithm, based on the integral‐equation method, was developed to calculate these displacements. Because cracks emanating from a hole are quite common in the aerospace industry, accurate displacement solutions are crucial for improving life‐prediction methods based on the strip‐yield crack‐closure models.     

Fatigue crack retardation and closure in polymethylmethacrylate

The presented results are from an investigation of crack surface profiles and the influence of intermittent overloads on fatigue crack growth in polymethylmethacrylate, a transparent polymer. Fatigue cracks were grown in compact tension specimens under conditions of constant range in stress intensity factor. Tensile overloads were found to perturb subsequent crack growth in polymethylmethacrylate under certain conditions.

Examination of monochromatic light interference fringe patterns emanating from the fatigue cracks indicates that the crack perimeter was closed at zero load. The crack surfaces were displaced, however, in the interior of the specimen at zero load. Measurements of the crack opening displacements during loading revealed that a significant tensile load was required to displace the portions of the crack surfaces which Were initially closed. These observations are discussed in light of the Elber concept of crack closure.     

无限大平面中斜裂纹的压剪断裂分析

根据Muskhelishvili的复势理论,结合裂面边界条件和位移单值条件,将无限大平面受压应力作用的裂纹问题转化为对应的Hilbert问题,并运用复变函数法分别给出了在伪集中力作用下,不同裂面形态的基本解。对不同裂面形态的摩擦力大小和分布进行了详细分析,建立了新的摩擦力计算模型。采用“伪力法”和叠加原理,结合所求的基本解,给出了含中心斜裂纹的岩石类材料在压缩荷载作用下的应力强度因子(SIF)的解法。研究表明:裂面状态对KⅠ的大小没有影响,而对KⅡ的影响却很大,相同应力条件下,裂面状态会影响裂纹的开裂角和开裂方式。     

Three-dimensional stress and deformation fields around flat and slant cracks under remote Mode I loading conditions

This paper investigates the phenomenon of slant fracture observed in stable tearing tests of many ductile materials, where an initially flat crack, loaded under remote Mode I conditions, tends to grow into a slant crack and stay in the slant configuration until final fracture. In an effort to identify potential reasons why cracks prefer to grow in a slant manner, three-dimensional finite element analyses of crack-front stress and deformation fields in Arcan-type specimens containing a flat or slant crack are performed under elastic–plastic and remote Mode-I loading conditions. In particular, the crack-tip opening displacement (COD) at a position behind the crack tip, the mean stress, the effective stress, and a constraint factor (defined as the ratio of the mean stress and effective stress) are studied and compared for the two types of cracks. Analysis results reveal several stress/deformation field variations around flat and slant cracks under identical remote loading conditions. First, close to the crack front, the COD of a slant crack is greater than that of a flat crack. Second, at the specimen’s mid-plane, a flat crack leads to a higher constraint value ahead of the crack than a slant crack. Third, the effective stress ahead of a slant crack is greater than that ahead of a flat crack, especially close to the crack front. The above results seem to suggest that slant fracture may be preferred because a slant crack enhances the driving force in the form of a higher near-tip COD value and because a shearing type of failure is promoted in the case of a slant crack compared to a tensile type of failure in the case of a flat crack.     

Mixed-mode crack opening displacement measurements for small fatigue cracks growing in Astroloy at 20 °C

Crack opening displacements were measured for small fatigue cracks in Astroloy being grown with uniaxial stress application under high-cycle fatigue conditions. Four cracks were investigated including one that grew from 27 to 74 μm in three increments. Most of the cracks grew at an angle to the loading axis and all opened bimodally. Crack opening scaled with distance from the crack tip similar to an elastic crack, which allowed the calculation of a local stress intensity factor for both mode I and mode II. The proportion of mode II stress intensity factor was relatively large, varying as 0.06 < Δ K _II /Δ K _I < 0.42, with an average of ~0.3. Thus, uniaxial loading remote to the cracks resulted in a bimodal opening response on the scale of the cracks.     

A periodic array of cracks in functionally graded materials subjected to transient loading

A periodic array of cracks in an infinite functionally graded material under transient mechanical loading is investigated. In-plane normal (mode I) and shear (mode II) loading conditions are considered. For each individual loading mode, a singular integral equation is derived, in which the crack surface displacements are unknown functions. Numerical results are obtained to illustrate the variation of the stress intensity factors as a function of the crack periodicity for different values of material inhomogeneity, either at the transient state or steady state. The material inhomogeneity can increase or decrease the mode I and mode II stress intensity factors. Compared with the single crack solution, it is also shown that multiple cracking may decrease the mode I stress intensity factors, but enhance the mode II stress intensity factors significantly.     

In-situ observation of crack opening displacement (COD) in a metastable β titanium composite

The interfacial strength of β-Ti composites was increased via ageing, to investigate the effects of interfacial strength on the crack opening. Prior to crack opening displacement (COD) measurements, the number and location of fractured fibres were identified by acoustic emission and fibre probing techniques. This was later used to model the crack opening for number of tests conditions, such as different values of applied stress intensity factor, interfacial strengths and test temperatures. COD measurements were performed on bridged cracks, which were obtained by cycling the specimens under tension–tension loading. Test results have shown that crack opening is affected by both intrinsic factors such as interfacial debonding and frictional sliding stresses and extrinsic factors such as the maximum applied stress intensity factor and test temperature.     

Viscoelastic crack analysis using symplectic analytical singular element combining with precise time-domain algorithm

In this study, the crack problem in linear viscoelastic material is investigated numerically. The time dependent two-dimensional (2D) viscoelastic crack problem is treated by the precise time-domain expanding algorithm (PTDEA), such that the original problem is transformed into a series of quasi-elastic crack problems. The relationships among these quasi-elastic problems are expressed in terms of the time-domain expanding coefficients of displacement and stress in an improved recursive manner. Then a symplectic analytical singular element (SASE) which has been demonstrated to be effective and efficient for 2D elastic fracture problem is applied to solve the quasi-elastic crack problems obtained above. The SASE is constructed by using the symplectic eigen solutions with higher order expanding terms. An improved convergence criterion employing both displacement and stress for PTDEA is proposed. Taking advantage of the SASE, the stress intensity factors, crack opening and sliding displacements (COD and CSD) and strain energy release rate of the studied problem can be solved directly without any post-processing. Numerical examples show that the results of the present method can be solved accurately and effectively.     

Determination of Stress Intensity Factor for Cracks in Orthotropic Composite Materials using Digital Image Correlation

Abstract: This paper focuses on the application of the digital image correlation (DIC) technique to determine the stress intensity factor (SIF) for cracks in orthotropic composites. DIC is a full‐field technique for measuring the surface displacements of a deforming object and can be applied to any type of material. To determine the SIF from full‐field displacement data, the asymptotic expansion of the crack‐tip displacement field is required. In this paper the expansion of the crack tip displacement field is derived from an existing solution for strain fields. Unidirectional fibre composite panels with an edge crack aligned along the fibre were tested under remote tensile loading and the displacements were recorded using DIC. The SIF was calculated from the experimental data by fitting the theoretical displacement field using the least squares method. The SIF thus determined was in good agreement with theoretical results and therefore demonstrates the applicability of the derived displacement field and DIC technique for studying fracture in composites.     

Observing ideal “self-similar” crack growth in experiments

We here report on a detailed experimental study whose goal is to investigate spontaneous crack propagation in bonded and intact materials subjected to quasi-static far-field tensile loading. The cracks nucleate from a tiny circular hole and are triggered by an exploding wire. They subsequently propagate under the action of a constant far-field load. Dynamic photoelasticity in conjunction with high speed photography is used to capture the real-time photoelastic fringe patterns (isochromatics) associated with crack propagation. Dynamic stress intensity factors of propagating cracks are determined and the results are successfully compared with Broberg’s classical model of self-similar mode-I crack growth.     

Characterization of isochromatic fringe patterns for a dynamically propagating interface crack

The isochromatic fringes surrounding a crack propagating along a bimaterial interface have been developed and characterized. A parametric investigation has also been conducted to study the influence of various fracture parameters on this isochromatic fringe pattern. The relevant fracture parameters of interest were the crack-tip velocity, the mode mixity of loading and the non-singular stress field component. In all the cases the fringe pattern was compared with the more familiar patterns that are generated for the case of crack propagation in homogeneous media. It was found that both the crack tip velocity and the mode mixity of loading have a significant effect on the size and shape of the isochromatic fringe pattern surrounding a crack tip propagating along a bimaterial interface. However, the non-singular stress field component was found not to have a substantial effect on the fringe pattern. This is in contrast with the case of crack propagation in homogeneous media, where the non-singular stress field component determines the tilt of the fringe contours. The paper also presents an appropriate scheme to analyze experimental fringe contours to extract the various fracture parameters of interest. Finally, this scheme is employed to analyze actual experimental data from a typical bimaterial interface fracture experiment.     

Interactions of fatigue cracks with elastic obstacles

To quantify the growth behaviour of fatigue cracks growing towards microstructural barriers or elastic obstacles, parametric solutions are obtained for crack-tip opening displacement and plasticity-induced crack closure of a mode I fatigue crack growing towards elastic obstacles. Three common bi-material systems are analysed using the finite element method, in which both constituent materials have identical elastic properties but only the phase that contains the crack can deform plastically. It has been found that under monotonic loading the crack-tip opening displacement decreases as the crack-tip approaches the interface boundary, but reaching a non-zero value when the crack-tip terminates at the boundary. For a fatigue crack growing under constant amplitude loading, the crack-closure stress has been found to increase as the crack grows towards the barrier. Based on these results a mechanistic model is proposed to quantify the influence of stress level on the fatigue threshold of microstructurally small fatigue cracks, with predictions being in close agreement with experimental data.     

A Periodic Array of Cracks in a Functionally Graded Nonhomogeneous Medium Loaded under in-Plane Normal and Shear

In this paper, the problem of a periodic array of parallel cracks in a functionally graded medium is investigated based on the theory of plane elasticity for a nonhomogeneous continuum. Both the in-plane normal (mode I) and shear (mode II) loading conditions are considered. It is assumed that the material nonhomogeneity is represented as the spatial variation of the shear modulus in the form of an exponential function along the direction of cracks, and the Poisson's ratio is constant. For each of the individual loading modes, a hypersingular integral equation is derived, in a separate but parallel manner in which the crack surface displacements are the unknown functions. As the basic parameters in applying the linear elastic fracture mechanics criteria, the mode I and mode II stress intensity factors are defined from the stress fields with the square-root singularity ahead of the crack tips. Numerical results are obtained to illustrate the variations of the stress intensity factors as a function of the crack periodicity for different values of the material nonhomogeneity. The crack surface displacements are also presented for the prescribed loading, material, and geometric combinations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Estimations of creep fracture mechanics parameters for through-thickness cracked cylinders and finite element validation

This paper compares engineering estimation schemes of C* and creep crack opening displacement (COD) for cylinders with circumferential and axial through‐thickness cracks at elevated temperatures with detailed 3D elastic‐creep finite element results. Engineering estimation schemes include the GE/EPRI method; the reference stress (RS) method where the reference stress is defined based on the plastic limit load; and the enhanced reference stress (ERS) method where the reference stress is defined based on the optimised reference load, recently proposed by the authors. Systematic investigations are made not only on the effect of creep‐deformation behaviour on C* and creep COD, but also on effects of the crack location, the cylinder geometry, the crack length and the loading mode. Comparison of the finite element (FE) results with engineering estimations provides that for idealised power law creep, estimated C* and COD rate results from the GE/EPRI method agree best with FE results, suggesting that published plastic influence functions for plastic J and COD for through‐thickness cracked cylinders are reliable. For general creep‐deformation laws where either primary or tertiary creep is important and thus the GE/EPRI method is hard to apply, on the other hand, the ERS method provides more accurate and robust estimations for C* and COD rate than the reference stress method. As these two methods differ only in the definition of the reference stress, the ERS method maintains benefits of the reference stress method in terms of simplicity, but improves accuracy of the estimated J, C* and COD results.     

Elastic solutions for corner cracks undergoing uniform displacement-controlled loading

To understand the behavior of corner cracks involving displacement-controlled loading, the mode-I stress intensity factors and crack face displacements are determined for a quarter-elliptical corner crack undergoing uniform end displacement. A 3-D weight function method is used for the analysis. Various crack aspect ratios and crack-length-to-plate-height ratios are considered. The results show that the stress intensity factor and the crack opening profile for the uniform displacement-controlled loading can be significantly different from those for the uniform-stress controlled loading.     

Cyclic crystal plasticity analyses of stationary, microstructurally small surface cracks in ductile single phase polycrystals

ABSTRACT This paper explores the effects of microstructural heterogeneity on the cyclic crack tip opening and sliding displacements for stationary, microstructurally small transgranular surface cracks in a single phase metallic polycrystal using planar double slip crystal plasticity computations. Crack tip displacements are examined under plane strain conditions for stationary cracks of different lengths relative to grain size as a function of the applied nominal strain amplitude for tension-compression and cyclic shear. Nominal strain amplitudes range from well below to slightly above the nominal cyclic yield strength for each type of loading condition. Results indicate the complex nature of the crack tip sliding and opening displacements as functions of nominal strain amplitude and orientation of the nearest neighbour grains, the influence of the free surface in promoting the cyclic opening displacement even for cracks in the first surface grain, the rather restricted limits of applicability of linear elastic fracture mechanics, and very interesting crack tip plasticity effects which include crack tip displacement ratcheting or progressive accumulation, even for completely reversed, proportional applied loading. Results are compared for cases with and without crack face friction.     

Quasi-static and dynamic fracture of graphite/epoxy composites: An optical study of loading-rate effects

Strain-rate effects on fracture behavior of unidirectional composite materials are studied. Single-edge notched multi-layered unidirectional graphite composites (T800/3900-2) are investigated to examine fracture responses under static and dynamic loading conditions using a digital speckle correlation method. The fracture parameters for growing cracks are extracted as a function of fiber orientation. A 2D digital image correlation (DIC) method is used to obtain time-resolved full-field in-plane surface displacements when specimens are subjected to quasi-static and impact loading. Stress intensity factor and crack extension histories for pure mode-I and mixed mode cases are extracted from the full-field displacements. When compared to the dynamic stress intensity factors at crack initiation, the static values are found to be consistently lower. The stress intensity factor histories exhibit a monotonic reduction under dynamic loading conditions whereas an increasing trend is seen after crack initiation under quasi-static loading cases. This is potentially due to dominant crack face fiber bridging effects in the latter cases.     

An elastic-plastic finite element analysis of crack tip fields under biaxial loading conditions

A square plate containing a central crack and subjected to biaxial stresses has been studied by a finite element analysis. An elastic analysis shows that the crack opening displacement and stress of separation ahead of the crack tip are not affected by the mode of biaxial loading and therefore the stress intensity factor adequately describes the crack tip states in an elastic continuum.

An elastic-plastic analysis involving more than localized yielding at the crack tip provides different solutions of crack tip stress fields and crack face displacements for the different modes of biaxial loading.

The equi-biaxial loading mode causes the greatest separation stress but the smallest plastic shear ear and crack displacement. The shear loading system induces the maximum size of shear ear and crack displacement but the smallest value of crack tip separation stress.

     

The improved D-M model solutions for single edge cracked plates by considering the yielding at the back side

The method of calculating the stress intensity factor (SIF) and the crack opening displacement (COD) for double edge cracks in plates under arbitrary loadings that results in solving a system of Cauchy-type singular integral equations is presented. The improved D-M model is then constructed for edge cracked plates by considering the yielding at the back side. For the cases of tension and bending, the plastic zone sizes and the crack opening displacements are calculated from the improved model solution, and the envelopes for the beginning of backside yielding and ligament yielding are obtained. The numerical results are compared with known solutions which take no account of the yielding at the back side and with experimental results.     

Transformation zones, crack shielding, and crack-growth resistance of Ce-TZP/alumina composite in mode II and combined mode II and mode I loading

Crack-tip transformation zones, crack shielding and crack-growth-resistance (R-curve) behaviors of a transformation-toughened ceria-partially stabilized zirconia–alumina (Ce-TZP/alumina) composite were studied in mode II and combined mode I and mode II loading using compact-tension-shear (CTS) specimens. The mode II and mode I stress intensities for both the initial straight cracks and the subsequent kinked cracks were assessed by the method of caustics using geometrically equivalent specimens of polymethyl methacrylate (PMMA). The angle of formation of the transformation zones as well as of extension of the cracks increased systematically with increasing ratio of the mode II and the mode I stress intensities and approached a value of θ^*=−72° in pure mode II loading. This angle was close to the angle for maximum hoop tension in the stress field of a mode II crack (θ^*=−70.5°). A crack-initiation toughness envelope was constructed on a K_I–K_II diagram using the critical loads for incremental crack extension. The crack-initiation toughness in pure mode II loading was less than the corresponding toughness in mode I loading. This result was consistent with calculations that indicated no shielding from the asymmetric and elongated zones developed in mode II loading. The fracture toughness measured for the kinked cracks at long kink lengths approached the maximum fracture toughness measured for a mode I crack.