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.

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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.

     

An FEM study on crack tip blunting in ductile fracture initiation

Ductile fracture is initiated by void nucleation at a characteristic distance (I_c) from the crack tip and propagated by void growth followed by coalescence with the tip. The earlier concepts expressed I_c in terms of grain size or inter-particle distance because grain and particle boundaries form potential sites for void nucleation. However, Srinivas et al. (1994) observed nucleation of such voids even inside the crack tip grains in a nominally particle free Armco iron. In an attempt to achieve a unified understanding of these observations, typical crack-tip blunting prior to ductile fracture in a standard C(T) specimen (Mode I) was studied using a finite element method (FEM) supporting large elasto-plastic deformation and material rotation. Using a set of experimental data on Armco iron specimens of different grain sizes, it is shown that none of the locations of the maxima of the parameters stress, strain and strain energy density correspond to I_c. Nevertheless, the size of the zone of intense plastic deformation, as calculated from the strain energy density distribution ahead of the crack tip in the crack plane, compares well with the experimentally measured I_c. The integral of the strain energy density variation from the crack tip to the location of void nucleation is found to be linearly proportional to J_IC. Using this result, an expression is arrived at relating I_c to J_IC and further extended to CTOD_c.     

Steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modeling

This work employs high resolution, finite element computations to investigate key features of the elastic–plastic fields near a steadily advancing crack at quasi-static rates under three-dimensional, small-scale yielding conditions. The model represents a structurally thin component constructed of a material (e.g., Al and Ti alloys) with flow stress and fracture toughness properties that together limit the size of the in-plane plastic zone during steady-growth to no more than several multiples of the plate thickness. The computational approach generalizes the streamline integration procedure used previously for two-dimensional studies into three dimensions to represent steady-state growth on a fixed mesh in a boundary-layer framework. The plate thickness provides the only geometrical length scale. Crack extension occurs at the remotely applied, fixed loading without the need for a local growth criterion. In the first computations of this type, the present work considers a straight crack front advancing under local and global mode I loading with zero T-stress in a moderately hardening material. Applied remote loads at steady growth generate plastic zone sizes ahead of the advancing crack front ranging from 0.25 to 6.4 times the thickness. Key results include: (1) the crack-front fields exhibit a self-similar scaling characterized by a non-dimensional loading parameter; (2) three-dimensional effects extend to distances of approximately 1.5–2.5 times the thickness ahead of the advancing crack front for key values of this loading parameter, beyond which the fields (elastic–plastic then linear-elastic at greater distances) become uniform over the thickness; and (3) crack opening profiles on the outside surface reveal a “wedge-like”, opening shape which simplifies the definition of a crack-tip opening angle.     

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.     

The effect of loading pattern on crack growth instability under small-scale yielding conditions

A theoretical analysis shows that if a plane strain crack is stable up to a certain $\text{J}$ level ($\text{J}{}_{\mathrm{ST}}$) under small-scale yielding conditions and the displacement control loading conditions that are characteristic of a machine loaded specimen, then unstable fracture under load control conditions cannot occur at $\text{J}$ levels that are appreciably less than $\text{J}{}_{\mathrm{ST}}$. The results, therefore, give added confidence to the use of linear elastic fracture mechanics data, for assessing the integrity of cracked engineering structures with regard to catastrophic fracture.     

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.

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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].     

Failure initiation at a blunt V-notch tip under mixed mode loading

A criterion to predict crack onset at a sharp V-notch tip in homogeneous brittle materials under a mixed-mode loading was presented and validated by experimental observations in a previous paper by the authors. This criterion slightly underestimates the experimental loads causing failure which is attributed to a small notch tip radius that blunts the sharp corner. This discrepancy is rigorously analyzed mathematically in this paper by means of matched asymptotics involving 2 small parameters: a micro-crack increment length and the notch tip radius. A correction is brought to the initial prediction and a better agreement is obtained with experiments on PMMA notched specimens.     

Determination of crack tip parameters for ASCB specimen under mixed mode loading using finite element method

In this paper, a new asymmetric semicircular bend specimen (ASCB) is presented. Having more geometric parameters in asymmetric bend elements gives the opportunity of covering a wider range of K_?, K_?? and T-stress in comparison with classical SCB specimens. Finite element method is used to obtain these parameters from pure mode I to pure mode II. Extensive numerical calculations are made to get a wide range data for crack tip parameters of this specimen. It is observed that for ASCB specimens with specified geometries under pure mode II loading, one of the bottom supports can move horizontally without significant variation in Y_I. The complete sets of numerical results are obtained and can be used for verification and interpretation of future experimental results.     

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.     

Strain and stress fields near the blunted tip of a crack under mixed mode loading and the implications for fracture

The strain distribution in the vacinity of a blunted crack-tip is analysed by slip line theory under the conditions of plane-strain, small-scale yielding, and mixed-mode loading of Modes I and II. A generalized crack-tip opening displacement is introduced by which the strain and stress fields near the blunted crack-tip are determined uniquely over a wide range of Mode I and II combinations. Also, coupled experimental and finite-element analyses under the condition of large-scale yielding reveal that the initiation of stable crack growth occurs when the generalized crack-tip opening displacement attains a critical value which is constant for the material tested. The finite-element analysis is based on the finite deformation theory of elastic-plastic materials. The generalized crack-tip opening displacement criterion is found to be superior to the J-integral and the usual COD for the characterization of the initiation of stable crack growth. The plastic work in a small circular region at the crack-tip is found to be equivalent to the generalized crack-tip opening displacement, as a fracture criterion.     

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.     

Evaluation of crack tip stress and strain fields under nonproportional loading in a viscoelastic material

The crack tip strain and stress fields in a viscoelastic material under nonproportional loading conditions are evaluated. In order to evaluate the strain field, the crack tip displacement field is measured by using the digital image correlation (DIC) technique. This displacement field is then approximated by using the theoretically obtained crack tip displacement field in viscoelastic materials. The result shows that the approximation method can smoothly reconstruct the experimentally obtained displacement field. From the approximated displacement field, the crack tip strain field can be precisely obtained by using the differential form of the theoretical displacement. On the other hand, the crack tip stress field is analyzed by using the stress function. This suggests that the strain and stress fields can be independently evaluated. In addition, different time dependencies between stress and strain fields near the crack tip are observed. Based on this experiment, we can discuss the several criteria for the crack propagation directions in viscoelastic materials.     

A crack tip blunting analysis in the ductile-to-brittle transition of a structural steel

The J -integral versus crack growth resistance curves ( J – R curves) of an AE-460 structural steel were evaluated at different temperatures in the ductile-to-brittle transition region. This curve as well as the critical J _I value at the initiation of crack growth was found to be independent of temperature, being dependent only on the stress triaxiality state of the specimen.
The blunting expressions described in the ASTM and ESIS standards were evaluated using fractographic techniques. The ESIS blunting line fits the linear regression of the experimental results very well, while the blunting line predicted by the ASTM standard is clearly located below the experimental points.     

An analysis of the crack tip fields in a ductile three-point bend specimen subjected to impact loading

Analyses of an impact fracture test of a precracked, three-point beam of HY100 steel were performed to determine the dynamic fracture toughness. During impact, the crack tip opening displacement (CTOD) 100 μm behind the crack tip was measured using an optical measuring device called the interferometric strain/displacement gage. Since fracture initiates when stress wave effects dominate, a numerical simulation of the fracture event was conducted to obtain relevant near crack tip field parameters. The specimen was modeled by a plane stress finite element simulation using a rate sensitive elastoplastic material law. Since the simulated CTOD was to be compared with the measured CTOD in a region of residual strains due to crack closure, this effect was included in the model. The simulation produces a CTOD versus time response within 10% of the observed response, indicating that the other field quantities (such as the J-integral) should also be reliable. The loading rate /.K₁ was approximately 8 × 10⁶MPa√m/sec. If the fracture initiation time is assumed to coincide with the time at which the simulated and observed CTOD curves diverge, then the impact fracture toughness is 56% higher than the static fracture toughness.     

T-stress effects on steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modeling

The non-singular T-stress provides a first-order estimate of geometry and loading mode, e.g. tension vs. bending, effects on elastic–plastic, crack-front fields under mode I conditions. The T-stress has a pronounced effect on measured crack growth resistance curves for ductile metals – trends most computational models confirm using a two-dimensional setting. This work examines T-stress effects on three-dimensional (3D), elastic–plastic fields surrounding a steadily advancing crack for a moderately hardening material in the framework of a 3D, small-scale yielding boundary-layer model. A flat, straight crack front advances at a constant quasi-static rate under near invariant local and global mode I loading. The boundary-layer model has thickness B that defines the only geometric length-scale. The material flow properties and (local) toughness combine to limit the in-plane plastic-zone size during steady growth to at most a few multiples of the thickness (conditions obtainable, for example, in large, thin aluminum components). The computational model requires no crack growth criterion; rather, the crack front extends steadily at constant values of the plane-stress displacements imposed on the remote boundary for the specified far-field stress intensity factor and T-stress. The specific numerical results presented demonstrate similarity scaling of the 3D near-front stresses in terms of two non-dimensional loading parameters. The analyses reveal a strong effect of T-stress on key stress and strain quantities for low loading levels and less effect for higher loading levels, where much of the plastic zone experiences plane-stress conditions. To understand the combined effects of T-stress on stresses and plastic strain levels, normalized values from a simple void-growth model, computed over the crack plane for low loading, clearly reveal the tendency for crack-front tunneling, shear-lip formation near the outside surfaces, and a minimum steady-state fracture toughness for T = 0 loading.     

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.     

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.