Stress Trajectories for Mode I and Mode II Cracks

Maximum shear stress trajectories are obtained for the mode I and for the mode II crack in an isotropic elastic solid in plane strain.     

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.     

A mode II weight function for subsurface cracks in a two-dimensional half-space

The general properties of a mode II Weight Function for a subsurface crack in a two‐dimensional half‐space are discussed. A general form for the WF is proposed, and its analytical expression is deduced from the asymptotic properties of the displacements field near the crack tips and from some reference cases obtained by finite elements models. Although the WF has general validity, the main interest is on its application to the study of rolling contact fatigue: its properties are explored for a crack depth range within which the most common failure phenomena in rolling contact are experimentally observed, and for a crack length range within the field of short cracks. The accuracy is estimated by comparison with several results obtained by FEM models, and its validity in the crack depth range explored is shown.     

Error estimation of stress intensity factors for mixed‐mode cracks

This paper presents an a posteriori error estimator for mixed‐mode stress intensity factors in plane linear elasticity. A surface integral over an arbitrary crown is used for the separate calculation of the combined mode's stress intensity factors. The error in the quantity of interest is based on goal‐oriented error measures and estimated through an error in the constitutive relation. Copyright © 2006 John Wiley & Sons, Ltd.     

Mode I and mode II stress intensities for plates with cracks of finite opening

For the finite opening crack, there are two eigenvectors which give stress singularities at the crack tip. The Reciprocal Work Contour Integral Method is extended so as to yield the coefficients associated with these two eigenvectors. Several examples are given. In one, results using the algorithm are compared to a known solution.

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Résumé Dans le cas d'une fissure finie en cours d'ouverture, deux eigenvecteurs déterminent les singularités de contraintes à l'extrémité de la fissure. On procède àune extension de la méthode d'intégration sur un contour du travail réciproque de manière à déterminer les coefficients associés à ces deux eigenvecteurs. Divers exemples sont fournis et, dans l'un, on compare les résultats utilisant l'algoritme à ceux d'une solution connue.

     

Effect of compressive transverse normal stress on mode II fracture toughness in polymeric composites

Effect of transverse normal stress on mode II fracture toughness of unidirectional fiber reinforced composites was studied experimentally in conjunction with finite element analyses. Mode II fracture tests were conducted on the S2/8552 glass/epoxy composite using off-axis specimens with a through thickness crack. The finite element method was employed to perform stress analyses from which mode II fracture toughness was extracted. In the analysis, crack surface contact friction effect was considered. It was found that the transverse normal compressive stress has significant effect on mode II fracture toughness of the composite. Moreover, the fracture toughness measured using the off-axis specimen was found to be quite different from that evaluated using the conventional end notched flexural (ENF) specimen in three-point bending. It was found that mode II fracture toughness cannot be characterized by the crack tip singular shear stress alone; nonsingular stresses ahead of the crack tip appear to have substantial influence on the apparent mode II fracture toughness of the composite.     

Estimation of the plastic zone by finite element method under mixed mode (I and II) loading

Material fracture by opening (mode I) is not lonely responsible for fracture propagation. Many industrial examples show the presence of mode II and mixed mode I + II. The present work consists in the elaboration of a code to estimate the size of the plastic zone at the crack tip under mode I, mode II and mixed mode I + II loading. The computations are made according to Von Mises and Tresca criteria. The results obtained are compared to those measured by experiments.     

Simplified Greens functions for mode I and II cracks

When using integral equation techniques to solve fracture mechanics problems, a solution for the stress and displacement fields surrounding an arbitrarily loaded crack face within an infinite two-dimensional plane can be used to accurately represent the behavior of the crack. It is advantageous to have this solution in as simple a form as possible, since the accuracy and computation time for numerical solution of the integral equations is improved by using less complex kernels. Previous solutions for the Green's functions for loaded cracks in an infinite domain involve lengthy complex variable expressions or double integrals with Cauchy and crack tip singularities. The results presented here are a simplification of a previous double integral solution, and the simplified result is a single integral representation with a non-singular integrand which involves only real numbers. Both the mode I and mode II problems are reduced to this simplified form, and the two may be combined for mixed mode two-dimensional crack problems.

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Résumé Lorsqu'on utilise la technique de l'équation intégrale pour solutionner des problèmes de mécanique de rupture, on peut utiliser une solution pour les champs de contraintes et de déplacement environnant la surface d'une fissure arbitrairement mise en charge dans un plan infini à deux dimensions, pour représenter, de manière sûre, le comportement de la fissure. Il est avantageux que cette solution soit la plus simple possible en ce qui regarde sa forme, car la précision et le temps de calcul des solutions numériques des équations intégrales sont améliorés en utilisant des kernels moins complexes. Les solutions précédemment proposées pour les fonctions de Green relatives à des fissures chargées dans un domaine infini comportent de longues expressions de variables complexes ou de doubles intégrales à singularité de Cauchy, ainsi que des singularités d'extrémités de fissure. Les résultats présentés ici constituent une simplification de la solution précédente à double intégrale, et le résultat simplifié en est une représentation à simple intégrale avec un intégrant non singulier qui comporte seulement des nombres réels. Les problèmes de Mode I et de Mode II sont réduits à cette forme simple et ces deux problèmes peuvent être combinés pour des problèmes de fissures à deux dimensions, sollicitées selon un mode mixte.

     

Mode II stress intensity factors for central slant cracks with frictional surfaces in uniaxially compressed plates

The effect of crack surface friction on mode II stress intensity factor (SIF) of a central slant crack in a plate uniformly loaded in uniaxial compression is quantified. A previously developed two-dimensional finite element analysis was utilised after its modification to accommodate the friction between the crack surfaces. The plane strain state was assumed. A new numerical technique was devised to avoid the iteration procedures, which had to be employed due to the existence of frictional forces.

The crack inclination angle varied between zero and 75° measured from the horizontal direction. The coefficient of friction of the crack surfaces changed from zero to 1. In case of relatively sliding crack surfaces, mode II SIF existed. As is well known, the resulting mode II SIF decreased with increasing the coefficient of friction of the crack surfaces. Further, mode II SIF increased with increasing crack line inclination angle and then decreased after reaching a maximum value. The angle corresponding to that maximum SIF increased as the coefficient of friction of the crack surfaces increased.     

An improved stress recovery technique for low‐order 3D finite elements

In this paper, we propose a stress recovery procedure for low‐order finite elements in 3D. For each finite element, the recovered stress field is obtained by satisfying equilibrium in an average sense and by projecting the directly calculated stress field onto a conveniently chosen space. Compared with existing recovery techniques, the current procedure gives more accurate stress fields, is simpler to implement, and can be applied to different types of elements without further modification. We demonstrate, through a set of examples in linear elasticity, that the recovered stresses converge at a higher rate than that of directly calculated stresses and that, in some cases, the rate of convergence is the same as that of the displacement field.     

Numerical modelling of plasticity-induced crack closure for interfacial cracks in bi-material specimens

This paper reports the results of a fairly detailed finite element study which modelled the plasticity-induced crack closure (PICC) behaviour of interfacial cracks in various bi-material specimens. In particular, the fatigue crack-opening stress (S_op) level and the crack-tip deformation fields (Modes I and II) have been assessed for a number of different material combinations, chosen so as to throw some light on the effects of modulus of rigidity and strength level of the alloy on PICC. The material combinations included specimens based on aluminium alloy steel, medium strength-high strength steel, and aluminium or steel specimens coated with a rigid ceramic. Results obtained indicate that stabilised values of closure, S_op, can be interpreted as supporting the hypothesis that it is the elastic constraint on, and deformability of, the plastic zone surrounding a crack that are the major contributors to PICC, rather than any permanent ‘stretch’ associated with crack growth. Positive Mode II slip of the upper crack face over the lower face (i.e. the upper surface moving over the lower surface towards the crack-tip) can elevate S_op level, while a negative slip (i.e. the upper surface moving over the lower surface away from the crack-tip) causes a reduction in its value.     

A crack in a linear-elastic material under mode II loading, revisited

A sharp crack in a two-dimensional infinite linear-elastic material, under pure shear (mode II) loading is re-examined. Several criteria have been proposed for the prediction of the onset and direction of crack extension along a path emanating from the tip of the initial crack. These criteria date back some three decades and are well documented in the literature. All the predictions from the different criteria are close and indicate that the crack extension takes a direction at an angle of ≈ −70° measured counterclockwise from the positive x -axis, in the case of a remotely applied positive shear stress. However, the possibility seems to have been overlooked that the crack extension may initiate not from the crack tip itself, but instead may initiate on the free surface at an infinitesimal distance behind the crack tip. The effect of crack tip plasticity on the relevant stresses in the region of the crack tip is investigated by the application of an elastic–plastic finite element program.     

Determination of modes I and II stress intensity factors for oblique single edge cracks subjected to tension

A semi-theoretical and experimental approach is employed herein to investigate plates with 90, 75, 60, 45 and 30° oblique single edge cracks subjected to tensile loading. A series of tension tests were performed and the effect of the variation of the length of the cracks on the stress intensity factors were studied. Experimental measurements show that the mode one and two stress intensity factors are related mainly to the size of the caustic (optical singularity measured on a screen at a distance Z₀ from the screen), the length and the angle of the crack, and the width of the specimen. The calculated values of the semi-theoretical stress intensity factors were corroborated with the theoretical results; and then the method was further utilized to investigate more complicated oblique edge cracks subjected to tension. In particular, the stress intensity factors of plates with oblique 45° equal parallel edge cracks were studied by this method and the results were checked with the stress intensity factors of single oblique edge cracks.     

Measurement of mode I and mode II rock dynamic fracture toughness with cracked straight through flattened Brazilian disc impacted by split Hopkinson pressure bar

In order to find an effective and convincing method to measure rock dynamic fracture toughness for mode I and mode II, cracked straight through flattened Brazilian disc specimens of marble, which were geometrically similar for three size, were diametrically impacted by split Hopkinson pressure bar on the flat end of the specimen with three load angle respectively. History of stress intensity factors (K_I(t) for opening mode I, and K_II(t) for sliding mode II), mode mixture ratio (K_I(t)/K_II(t)), as well as mode I and mode II dynamic fracture toughness at crack initiation (K_Id and K_IId) were determined with the experimental–numerical method. It is found that there is a unique size effect for dynamic fracture test with the specimens, the mode mixture ratio is not solely determined by load angle (the angle between load direction and crack line) as in the static loading; the pure mode II load angle is 19° for the ?50 mm specimen, however it is 10° for the ?130 mm and ?200 mm specimens; the mode II load angle decreases with increment of specimen size. Realization of pure mode II is justified by the mode mixture ratio approaching zero, it can be realized under certain load angle and loading rate for the specimen of specified size. K_IId is generally greater than K_Id. Both K_Id and K_IId increase with increment of specimen size, and this trend for K_IId is more remarkable than that for K_Id.     

Prediction of crack growth direction for mode I/II loading using small-scale yielding and void initiation/growth concepts

Under the assumption that the processes which control the direction of crack growth in 2024-T3 aluminum are directly related to void initiation and growth, a theoretical framework is developed to predict the direction of crack growth. The basic premise of the framework is that, depending on the mode mixity of the remotely applied loading, either σ _m /σ_eff or σ_eff triggers the nucleation and growth of voids hence, fracture. The theoretical development uses linear elastic assumptions and two terms in the asymptotic expansion to describe the stress field in the vicinity of the crack tip for a mixed-mode I/II ARCAN specimen. Predictions based on the theory indicate that: (a) the transition from Mode~I type to Mode~II type crack propagation can be accurately quantified, and (b) the direction of crack growth is reasonably well predicted for both types of crack propagation. In addition, a qualitative, but microstructurally based, physical rationale for the observed phenomena is presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Three-dimensional asymptotic mode I/II stress fields at the front of interfacial crack/anticrack discontinuities in trimaterial bonded plates

An eigenfuntion expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of the front of a crack/anticrack discontinuity weakening/reinforcing an infinite pie-shaped trimaterial plate, of finite thickness, formed as a result of bimaterial (matrix/ARC plus reaction product/scatterer) deposit over a substrate (fiber/semiconductor). The wedge is subjected to mode I/II far field loading. Each material is isotropic and elastic, but with different material properties. The material 2 or the substrate is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying composition of the bimaterial deposit. Numerical results pertaining to the variation of the mode I/II eigenvalues (or stress singularities) with Young’s moduli ratio, as well as with the wedge aperture angle of the material 1 (reaction product/scatterer) are presented. Hitherto generally unavailable results, pertaining to the through-thickness variations of stress intensity factors or stress singularity coefficients for symmetric exponentially growing distributed load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the trimaterial plates under investigation, bridge a longstanding gap in the stress singularity/interfacial fracture mechanics literature.     

Threshold and growth mechanism of fatigue cracks under mode II and III loadings

ABSTRACT The fatigue crack growth behaviour of 0.47% carbon steel was studied under mode II and III loadings. Mode II fatigue crack growth tests were carried out using specially designed double cantilever (DC) type specimens in order to measure the mode II threshold stress intensity factor range, ΔK_IIth. The relationship ΔK_IIth > ΔK_Ith caused crack branching from mode II to I after a crack reached the mode II threshold. Torsion fatigue tests on circumferentially cracked specimens were carried out to study the mechanisms of both mode III crack growth and of the formation of the factory‐roof crack surface morphology. A change in microstructure occurred at a crack tip during crack growth in both mode II and mode III shear cracks. It is presumed that the crack growth mechanisms in mode II and in mode III are essentially the same. Detailed fractographic investigation showed that factory‐roofs were formed by crack branching into mode I. Crack branching started from small semi‐elliptical cracks nucleated by shear at the tip of the original circumferential crack.     

A new jig for mode II interlaminar fracture testing of composite materials under quasi-static and moderately high rates of loading

While it is still debated whether a pure mode II interlaminar fracture can physically exist in composites, several test methods have been proposed for its characterization. Lack of agreement between the results obtained with different test configurations has been attributed to the use of inconsistent data reduction schemes or inadequate correction factors used to correct for, e.g. the large deformations occurring with some tough modern materials.Aim of this work was to design a new jig that could provide an as pure as possible mode II crack initiation in unidirectional composites materials, that would allow a direct determination of fracture toughness, i.e. requiring almost no assumption for data reduction nor side effects correction and could be amenable to being used under impact as well as quasi-static loading conditions.The geometry of the system was designed in order to obtain great compactness, i.e. reduced masses and contained volume, making it usable with drop-weight testing machines, but at the same time enough stiffness to prevent flexural moments from closing or opening the crack faces, so granting the purity of the wanted mode, mode II, of loading. The compactness of the jig plus specimen system and the rigid confinement to which the composite specimen is subjected also grant that quite small displacements and overall deformations are reached at fracture.A static finite element analysis was conducted to optimize the jig geometry and is discussed here. Preliminary numerical and experimental results obtained with moderately high rate tests are also presented. The method employed for data reduction is based on the experimental calibration of the compliance and it is quite straightforward.     

On the calculation of derivatives of stress intensity factors for multiple cracks

In this paper, the work of Lin and Abel [Lin SC, Abel JF. Variational approach for a new direct-integration form of the virtual crack extension method. Int J Fract 1988;38:217-35] is further extended to the general case of multiple crack systems under mixed-mode loading. Analytical expressions are presented for stress intensity factors and their derivatives for a multiply cracked body using the mode decomposition technique. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system are computed in a single analysis. It is shown through two-dimensional numerical examples that the proposed method gives very accurate results for the stress intensity factors and their derivatives. It is also shown that the variation of mode I and II displacements at one crack-tip influence the mode I and II stress intensity factors at any other crack. The computed errors were about 0.4-3% for stress intensity factors, and 2-4% for their first order derivatives for the mesh density used in the examples.     

Crack-tip fields in elastic-plastic material under plane stress mode I loading

New results on the crack-tip fields in an elastic power-law hardening material under plane stress mode I loading are presented. Using a generalized asymptotic expansion of the stress function, higher-order terms are found which have newly-discovered characteristics. A series solution is obtained for the elastic-plastic crack-tip fields. The expansion of stress fields contains both the and terms where t_i is real and t_k is complex; the terms σ⁽ⁱ⁾ _pq(θt_i) and σ^(k) _rsθt_k) are real and complex functions of θ respectively. Comparing the results with that for the plane strain mode I loading shows that: (1) the effect of higher-order solutions on the crack-tip fields is much smaller; and (2) the path-independent integral J also controls the second-order or third-order term in the asymptotic solutions of the crack-tip fields for most of the engineering materials (1 < n < 11) in plane stress, while the J-integral does not control the second and the third-order terms for the plane strain mode I case for n > 3. These theoretical results imply that the crack-tip fields can be well characterized by the J-integral, and can be used as a criterion for fracture initiation under plane stress mode I loading. This is in agreement with existing full-field solutions and experimental data that J at crack growth initiation is essentially independent of in-plane specimen geometry. The comparison confirms the theoretical asymptotic solutions developed in this study. This revised version was published online in August 2006 with corrections to the Cover Date.