About the Dugdale crack under mixed mode loading

The plane problem of a Dugdale crack under mixed mode loading is investigated. An exact closed form solution is given and the corresponding crack displacements are discussed.

---

Résumé On étudie le problème plan d'une fissure de Dugdale soumise à une sollicitation selcn un mode mixte. Une solution exacte de forme fermée est fournie et les déplacementsde fissure correspondants sont discutés.

     

Fatigue crack propagation under non-proportional mixed mode loading

Under non-proportional mixed I+II loading, two kinds of stable crack propagation may be distinguished. An existing precrack will either kink, mode I controlled (tensile mode), or will propagate, coplanarly mode II controlled (shear mode). Shear mode growth will occur if the effective mode II range exceeds the material-specific threshold ΔK_{II th sm} and in addition to that, the ΔK_II-value on the starter crack is larger than the ΔK¹_I (Δϕ)-range on the infinitesimally short additional crack. Examination under the scanning electron microscope showed that flaws are not the reason for the mode II controlled crack propagation and support the criteria introduced. If the crack opening is large enough, the crack propagation rate is higher for shear-stress controlled crack growth than for normal-stress controlled crack extension, the deviation angle of which is well predictable via the MTS criterion due to Erdogan and Sih [On the crack extension in plates under plane loading and transverse shear. J Basic Engng 1963;85:519–25].     

Fracture analyses and experimental results of crack growth under general mixed mode loading conditions

In this paper detailed results of 3D finite element (FE) and mixed mode analyses of different fracture specimens are presented and discussed. Special interest is taken in 3D and mode coupling effects to be found in strain energy release rate (SERR) results along crack fronts, in particular adjacent to corners, where a crack front intersects a free surface of a specimen. It will be shown that these effects stay small if they are related to Poisson’s ratio but that they can also be considerably pronounced if they are related to the global deformation behaviour of the specimen. The computational fracture analysis is based on the calculation of separated energy release rates (SERRs) by the aid of the modified virtual crack closure integral (MVCCI)-method in order to calculate the local SERR-distributions along the crack front. Furthermore some qualitative experimental results will show the influence of these variable mixed mode I, II and III loading conditions along the crack front on crack initiation and on the further development of 3D crack growth in the specimens.     

Incipient yielding at a debond crack tip under mixed mode loading

Plastic relaxation by a single slip band of a debond crack at the interface between a rigid and an incompressible elastic material is studied. It is shown that for mixed mode loading there is a unique angle, at which the slip band meets the interface crack tip, such that the stress intensity factor is zero. A simple model is discussed which gives a good approximation to the crack opening displacement and slip band angle.

---

Résumé On étudie la relaxation plastique que produit une simple bande de glissement associée à une fissure de décollement à l'interface entre un matériau rigide et un matériau élastique incompressible. On montre que, pour une mise en charge de mode mixte, il existe un angle unique pour lequel la bande de glissement rencontre l'extrémité de la fissure d'interface, de telle manière que le facteur d'intensité des contraintes s'annule. On discute d'un modèle simple fournissant une bonne approximation pour le COD et pour l'angle de la bande de glissement.

     

Vapor pressure assisted crack growth at interfaces under mixed mode loading

Moisture diffuses into the numerous pores and cavities formed in polymeric molding compounds, at the filler particle–polymer matrix interfaces and at polymer–silicon interfaces of IC packages. During reflow soldering, the rapidly expanding moisture generates high internal pressures within the voids which are comparable to yield strengths of the molding compounds at glass transition temperatures. The combined action of thermal stresses and high vapor pressure accelerates void growth, and ultimately leads to interface delamination and package cracking. In this study, the molding compound is taken to be an elastic–plastic material while the silicon substrate is treated as an elastic material. The extended Gurson model which incorporates vapor pressure as an internal variable is used to characterize the void growth and coalescence process at the interface. When the mode II loading is dominant, high vapor pressure can cause several-fold reduction in the interface fracture toughness.     

Dislocation emission criterion for a wedge crack under mixed mode loading

Dislocation emission criterion for a wedge crack under mixed mode loading was investigated using Airy stress function. The order of singularity at the wedge crack tip due to remote loading was found to vary with the loading mode. The plastic zones for plane stress and plane strain were studied based on von Mises' and Tresca criteria. The dislocation emission criterion was examined for both loading modes. The mechanism of crack propagation was believed to be controlled by dislocation emission. Under an action of Mode I loading, the wedge tip movement occurred when a pair of edge dislocations of Burgers vectors be ⁱ and –be ^–i were emitted from the wedge tip where b and were the magnitude of Burgers vector and the angle between the positive x axis and the line connecting from the tip to dislocation. Similarly, under an action of Mode II loading, the wedge crack tip moved as soon as either an edge dislocation of Burgers vector along the x direction was emitted from its tip or a pair of edge dislocations of Burgers vectors be ⁱ and be ^–i were emitted from the wedge tip. The conventional mechanism of crack propagation based on the energy release rate was not expected to occur. The calculated results for a few special cases were presented and compared with those reported in the literature.     

Linear micropolar elastic crack-tip fields under mixed mode loading conditions

Micropolar elasticity laws provide a possibility to describe constitutive properties of materials for which internal length scales may become important. They are characterized by the presence of couple stresses and nonsymmetric Cauchy stress tensor. Beyond the classical displacement field, the kinematical variables are augmented by a so-called microrotation field and its gradient, the latter introducing an internal length scale in the theory. For an isotropic, linear micropolar elastic material, the near-tip asymptotic field solutions for mode I and mode II cracks are derived. It is shown that these solutions behave similar to those according to the so-called couple stress theory, which has been investigated by Huang et al. (1997a), or similar to those derived for cellular materials by Chen et al. (1998). In particular, the singular fields have an order of singularity r ^–1/2 and are governed by some amplitude factors, having the meaning of stress intensity factors as in the classical linear elastic theory. The effect of material parameters on the stress intensity factors is studied by applying the finite element method to calculate the values of the stress intensity factors for an edge-cracked specimen of finite width.     

Elastic-plastic analysis of a stationary crack under cyclic loading and effect of overload

A recently proposed elastoplastic constitutive model has been implemented in a finite element code to study crack front behaviour under variable loading. The importance of proper modelling of a material's behaviour becomes evident when a variable loading condition is considered. We present stress, strain and displacement distribution along a stationary crack front for constant amplitude cyclic loading with an overload cycle. The analysis predicts a decreased tensile stress and damage accumulation following an overload.     

Weld root crack propagation under mixed mode I and III cyclic loading

This study examined fatigue propagation behaviour and fatigue life of weld root cracks under mixed mode I and III loading. Fatigue tests were performed on butt-welded joints with a continuous lack-of-penetration (LOP) inclined at angles of 0°, 15°, 30° or 45° to the normal direction of the uniaxial cyclic load. Branch and/or co-planar crack propagation was observed, depending on the initial mode I stress intensity factor (SIF) range. Co-planar crack propagation predominated when the SIF range was large. The fatigue crack propagation mode affected fatigue life; the life of branch crack propagation was longer than that of co-planar crack propagation. Using an initial equivalent SIF range based on a maximum strain energy release rate criterion, the results obtained from the 0°, 15°, 30°, and 45° specimens indicated almost the same fatigue lives, despite the different inclination angles.     

Role of plastic zone in crack growth direction criterion under mixed mode loading

An approach to determine the crack growth direction under mixed-mode loading conditions is presented. The plastic zone shape around the crack tip is applied for evaluating angle of crack propagation. It is proposed that a mixed-mode crack will extend along the plastic zone radius with a minimum value. The prediction of the proposed criterion is compared with the experimental data and other models. The agreement is fairly good.     

Crack paths under mixed mode loading

Long fatigue cracks that initially experience mixed mode displacements usually change direction in response to cyclic elastic stresses. Eventually the cracks tend to orient themselves into a pure mode I condition, but the path that they take can be complex and chaotic. In this paper, we report on recent developments in techniques for tracking the crack path as it grows and evaluating the strength of the mixed mode crack tip stress field.     

Two parameter fracture mechanics: Fatigue crack behavior under mixed mode conditions

The fatigue crack path has been studied on a tensile specimen with holes. The experimental crack path trajectories were compared with those calculated numerically. To incorporate the influence of constraint on the crack curving, we predicted the fatigue crack path by using the two-parameter modification of the maximum tensile stress (MTS) criterion. The values of the mixed-mode stress intensity factors K_I, and K_II as well as the corresponding constraint level characterized by T-stress were calculated for the obtained curvilinear and reference crack path trajectories. It is shown that in the studied configuration the effect of T-stress on the crack path is not significant. On the other hand the effect of constraint on the fatigue crack propagation rate is more pronounced.     

Solute distribution in crack tips under mixed mode I/II conditions

Summary Concentration solutions of crack problems under steady state conditions are expressed in terms of the Westergaard stress function. The problem of a central crack in an infinite plate subjected to a biaxial stress field at any angle of inclination with respect to the crack axis is considered in detail. The concentration distribution in the vicinity of the crack tip is obtained and is expressed in terms of the opening-mode and sliding-mode stress intensity factors. Constant terms, usually omitted, are incorporated into the concentration solution. The crack growth criterion based on the maximum concentration of diffusing species in front of the crack tip is reformulated by incorporating the constant terms of the concentration solution. It is shown that the omission of the constant terms may result in a significant error in the prediction of the critical quantities for crack growth.     

Fracture from a straight crack subjected to mixed mode loading

Fracture from a straight crack under mixed mode loading conditions and small scale of yielding is studied. It is assumed that crack growth occurs in either mode I or mode II. Comparison of theoretically obtained values of the mode I stress intensity factor at incipient kinking with experimental results indicates that mode I is preferred to mode II when the loading is such that the crack surfaces are traction-free, i.e. in the absence of a confining pressure. Mode II would be preferred only if the ratio K _{II c}/K _{I c} between the stress intensity factors is very low, between 0.38–0.81, depending on the load situation. Since there are reasons to believe that most materials are characterized by higher values of K _{II c}/K _{I c}, the conclusion is that mode II hardly occurs in the absence of a high confining pressure.

---

Résumé On etudie la rupture entraînée par une fissure droite sujette à des sollicitations selon un mode mixte et comportant une plastification sur une petite échelle. On suppose que la croissance de la fissure se produit sous un mode I ou un mode II.En comparant les valeurs théoriques du facteur d'intensité de contraintes selon le mode I correspondant à l'évasement initial et les résultats expérimentaux, il s'indique que le mode I est privilégié par rapport au mode II lorsque la sollicitation est telle que les surfaces de la fissure sont libres de toutes contraintes, c'est-à-dire en l'absence de toute pression de confinement.Le mode II ne serait privilégié que si le rapport K _{II c}/K _{I c} des facteurs d'intensité de contraintes est très faible, à savoir entre 0,38 et 0,81, selon les conditions de sollicitation.Comme il y a de bonnes raisons de croire que la plupart des matériaux sont caractérisés par des rapports K _{II c}/K _{I c} plus élevés, on en conclut que, en l'absence de hautes pressions de confinement, le mode II a peu de chances de se manifester.

     

Elastic-plastic analysis of an arbitrarily-oriented mode III crack touching an interface

In this paper we investigate a semi-infinite crack terminating at an arbitrarily oriented interface between two elastic-plastic materials under an anti-plane shear loading. An analytical solution is first developed for general power-law hardening materials under a mode III loading. If both materials have the same hardening exponent, the formulation results in a nonlinear eigenequation which can be solved numerically. The stress singularities are determined as a function of two material constants: the hardening exponent n and parameter G which represents the relative resistance of the two materials. In addition to the power of the singularity, the stress, strain and displacement asymptotic fields are also determined. If the hardening exponents are not the same, the leading order terms of an expansion model ensure the stress continuity across the interface. The results show that the stress singularity mainly depends upon the material having the larger hardening exponent, with the highest stresses in the material having the smaller hardening exponent. By taking the hardening exponent n , the perfectly plastic bimaterial problem is studied. It has been found that if the crack lies in the less stiff material, the entirely plastic asymptotic fields around the crack tip can be determined. On the other hand, if the crack lies in the stiffer material, the crack-tip fields are partially elastic and partially plastic. For both cases, unique asymptotic fields can be determined explicitly. For those cases when the materials present a strain hardening property, different mathematical models are established.     

Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading

Disc-type specimens are among favorite test samples for determining mode I and mixed mode fracture toughness in brittle materials like rocks, brittle polymers, ceramics, etc. In this research, the finite element method is used to analyze two disc-type specimens: a semi-circular disc specimen containing an edge crack and subjected to three-point-bend loading (SCB specimen), and a centrally cracked circular disc subjected to diametral compressive loading, often called the Brazilian disc specimen. The crack parameters K_I, K_II and T are calculated for different mode mixities from pure mode I to pure mode II. Although the stress intensity factors K_I and K_II are presented mainly for validation of the analyses, they are also used for determining the crack angle corresponding to pure mode II for each specimen. It is shown that in general the T-stress increases for larger crack angles. While the T-stress in the Brazilian disc specimen is always negative for any combinations of mode I and mode II, the sign of T-stress in the SCB specimen depends on the mode mixity. A very good agreement is shown to exist between the calculated results for T and those very limited data presented in other papers.     

Numerical analysis of blunting of a crack tip in a ductile material under small-scale yielding and mixed mode loading

The blunting of the tip of a crack in a ductile material is analysed under the conditions of plane strain, small-scale yielding, and mixed mode loading of Modes I and II. The material is assumed to be an elastic-perfectly plastic solid with Poisson's ratio being 1/2. The stress and strain fields for a sharp crack under mixed mode loading are first determined by means of elastic-plastic finite element analysis. It is shown that only one elastic sector exists around the crack tip, in contrast with the possibility of existence of two elastic sectors as discussed by Gao. The results obtained for a sharp crack are used as the boundary conditions for the subsequent numerical analysis of crack tip blunting under mixed mode loading, based on slip line theory. The characteristic shapes of the blunted crack tip are obtained for a wide range of Mode I and Mode II combinations, and found to resemble the tip of Japanese sword. Also the stress field around the blunted crack tip is determined.     

Crack trajectories under mixed mode and non-proportional loading

A novel testing procedure for mixed mode crack propagation is presented; it offers the possibility of non-proportional loading and of changing the crack trajectory during testing, features which allow non-traditional mixed mode tests which, in turn, may help in discriminating mixed mode fracture criteria. Detailed experimental stable crack trajectories and corresponding load-CMOD (or load-displacement) curves from these non-traditional tests are proposed as benchmarks for numerical programs of mixed mode crack propagation.     

Atomic simulation of cracks under mixed mode loading

A discrete atomic model of a crack tip in iron under mixed mode loads is examined. The results indicate that the behavior of the crack at the atomic scale as a function of the ratio of mode I to mode II component of load is quite complex. In general, crack tip plasticity appears to increase as the mode II component of load increases.

---

Résumé On examine un modèle atomique discret de l'extrémité d'une fissure dans du fer soumis à des charges de mode mixte. Les résultats indiquent que le comportement de la fissure à l'échelle atomique en fonction du rapport des composantes de charge du mode I sur celles du mode II est totalement complexe. En général, la plasticité à l'extrémité d'une fissure parait s'accroitre lorsque la composante de mode II de la contrainte s'accroit.