Fracture of solids containing arrays of cracks

This survey is largely a collection of the author's recent results on the fracture characteristics of elastic solids containing inhomogeneities in the form of slitlike cracks. The crack tip opening displacement is used as a measure of the fracture behaviour of the solid under three different deformation situations: (a) Opening Mode I; (b) Sliding Mode II; and (c) Tearing Mode III. Various crack geometries have been considered for which closed-form analytical solutions do not seem feasible. To this category belong (a) A stack of cracks (cracks with constant distance of vertical separation) and (b) A doubly-periodic array of cracks forming a rectangular or diamond-shaped pattern. Besides summarizing various results which have been published elsewhere in Scientific Literature, some new results, and necessary amplification of the previous results, are included herein to make the survey as self-contained as possible. It is hoped that it will be found useful by many research workers engaged in crack interaction and propagation problems.The survey is divided into five sections. Section 1 gives a general introduction to the problem at hand and brings out the importance of studying the subject. Sufficient reference is made to the available literature on the subject, without being unduly overemphatic. It is likely that many otherwise good papers have been omitted either through oversight or because they were thought to be peripheral to the subject matter of the survey. Interested readers should, however, be able to find sufficient cross references in the literature cited herein. This section also touches upon the necessity of using the dislocation formalism in solving the problems at hand. For obvious reasons, no attempt has been made to dwell upon this equivalence of slitlike cracks and straight dislocations, the interested reader being again referred to relevant literature.Section 2 formulates the problem in mathematical terms as one consisting in the solution of a singular integral equation. The latter results from the traction-free conditions on the crack faces. The equation is suitably non-dimensionalized and its kernel decomposed into a singular and a non-singular part as dictated by the method of solution.Section 3 presents a perturbation solution for widely spaced cracks. The solution is restricted to a stack of cracks under plane strain conditions.Section 4 deals with an approximate method used to solve the singular integral equation. It is based on an expansion of the non-singular part of the kernel in a series of orthogonal polynomials. The solution of the singular integral equation allows us to calculate the crack tip opening displacement as a function of the externally applied stress and the crack geometry.The results for various loading modes and crack configurations are presented graphically in Section 5 and discussed from the point of fracture initiation from multiple cracks. Where possible, the results are compared with that for an isolated relaxed crack for a better understanding of the change brought about by an array of interacting cracks.In order not to interrupt the text all complicated mathematical expressions have been grouped together and listed at the end of the relevant section.     

Stability of arrays of multiple edge cracks

The creation and subsequent shedding of arrays of edge cracks is a natural phenomenon which occurs in heat-checked gun tubes, rapidly cooled pressure vessels and rock, dried-out mud flats, paint and concrete and in ceramic coatings and permafrost. The phenomenon covers five orders of magnitude in crack spacing and the driving mechanisms may include fast fracture, environmental cracking and fatigue crack growth. A simple model is developed which indicates that the shedding behaviour is governed by the behavior of individual cracks rather than global energy changes. The model predicts that all cracks will deepen until a crack-spacing/crack depth ratio (2h/a) of 3.0 is achieved, at which stage crack-shedding will commence. Two out of every three cracks will be shed, leading to a new (higher) crack spacing/crack depth ratio at which stage growth of all currently active cracks will be dominant. An approach based upon rapid, approximate methods for determining stress intensity provides good indications of behaviour provided near-surface stress gradients are not excessive. In cases where stress gradients are high it is shown that it is necessary to employ numerical techniques in calculating stress intensity. Two specific examples are presented, the first at very small scale (heat-check cracking in a gun tube, typical crack spacing 1 mm) and the second at very large scale (permafrost cracking, typical crack spacing 20 m). The predicted ratios for the proportion of cracks shed and for crack spacing/crack depth are in agreement with experimental evidence for gun tubes, concrete and permafrost. The ratios also appear to match experimental observations of “island delamination” in ceramic coatings and paint films.     

Bifurcation predictions for moving cracks by the T-criterion

The directional instability of moving cracks was studied within the context of the maximum dilatational energy density, or T-criterion. To make bifurcation predictions according to the T-criterion theory it suffices to consider only the singular stress field. In this paper the stress field for a non-uniformly propagating crack under mode-I deformation was adopted. Predictions concerning both the critical crack velocity and the bifurcation angle was given. In this way, not only totally brittle fractures described previously by the S-criterion were correctly encountered, but also the results were extended to initially ductile materials where the T-criterion conserves its total efficiency.

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Résumé On a étudié l'instabilité directionnelle de fissures en cours de propagation, en recourant au critère d'énergie de dilatation maximum, ou critère T. Pour prédire les bifurcations de la fissure suivant le critère T, il suffit de considérer le champ de contraintes singulières. On a adopté dans la présente étude le champ de contraintes correspondant à une fissure se propageant de manière non uniforme sous une déformation de mode I. On fournit les prédictions de la vitesse critique de propagation et de l'angle de bifurcation de la fissure. De la sorte, on a non seulement retrouvé de manière correcte les résultats de rupture totalement fragiles décrits précédemment par le critère S, mais on a pu également élargir les résultats à des matériaux initialement ductiles, pour lesquels le critère T s'avère être parfaitement adapté.

     

Semiconducting two-dimensional graphene nanoconstriction arrays

The fabrication and characterization of two-dimensional nanoconstriction arrays consisting of sub-20-nm graphene constrictions and interconnecting graphene islands are reported. The arrays are fabricated in a scalable top-down fashion using self-assembled close-packed polystyrene nanospheres as lithographic templates and characterized using electron microscopy, Raman spectroscopy, and charge transport measurements. At room temperature, the arrays behave as semiconductors with a field-effect conductance modulation of up to 450 and charge mobilities of ~1 cm(2) V(-1) s(-1) . The effective bandgap of the arrays scales inversely with the nanoconstriction width, indicating that its magnitude is determined by quantum confinement in the constrictions. At low temperatures, the arrays act as semiconductors, with increasing ON/OFF conductance modulation up to ～1000, and simultaneously act as 2D arrays of coupled Coulomb islands affected by single-electron charging events. The high conductance modulation of these nanopatterned graphene materials, combined with the scalability of the patterning approach is expected to impact thin film, flexible, and transparent semiconductor electronics.     

Bifurcation and stability of structures with interacting propagating cracks

A general method to calculate the tangential stiffness matrix of a structure with a system of interacting propagating cracks is presented. With the help of this matrix, the conditions of bifurcation, stability of state and stability of post-bifurcation path are formulated and the need to distinguish between stability of state and stability path is emphasized. The formulation is applied to symmetric bodies with interacting cracks and to a halfspace with parallel equidistant cooling cracks or shrinkage cracks. As examples, specimens with two interacting crack tips are solved numerically. It is found that in all the specimens that exhibit a softening load-displacement diagram and have a constant fracture toughness, the response path corresponding to symmetric propagation of both cracks is unstable and the propagation tends to localize into a single crack tip. This is also true for hardening response if the fracture toughness increases as described by an R-curve. For hardening response and constant fracture toughness, on the other hand, the response path with both cracks propagating symmetrically is stable up to a certain critical crack length, after which snapback occurs. A system of parallel cooling cracks in a halfspace is found to exhibit a bifurcation similar to that in plastic column buckling.     

Size distribution of surface cracks and crack pattern in austenitic SUS316 steel plates fatigued by cyclic bending

The size distribution of surface cracks and the crack pattern were examined on the specimens of the SUS316 steel plates fatigued by cyclic bending. The size distribution of the cracks could be approximated to a logarithmic normal distribution, irrespective of the maximum total strain range or the number of fatigue cycles. The number of the cracks (Nu) of the length (x) equal to or larger than a given size (X) could be approximated to a power law, Nu X^–a , with a scaling exponent a at the larger crack sizes in the fatigued specimens of the SUS316 steel. The value of a decreased with increasing the number of fatigue cycles because of the increase in the number and size of fatigue cracks, and was larger in the specimens tested at the smaller total strain range. Effects of experimental variables on the scaling exponent (a) were also shown in this study. The fractal dimension of spatial crack distribution (the fractal dimension of crack pattern) (D) increased in the range from about 0.9 to about 1.2 with increasing the number of fatigue cycles, and was larger in the specimens fatigued at the larger total strain range. There was a negative correlation between the value of a and the value of D on fatigue cracks, although there was no unique relationship between these two values.     

Nucleation of cracks in two-dimensional periodic cellular materials

This paper describes the nucleation and propagation of cracks in brittle cellular material. Four basic patterns with triangular, square, hexagonal and kagome-type cells are considered. The cracks propagate by sequential failure of critical elements. The analysis technique hinges on the combined use of the structural variation method and the representative cell method. While the latter allows for the analysis of periodic structures under arbitrary loads, by means of the discrete Fourier transform, the former analyzes modified structures (the cracked lattices) on the basis of analysis of the pristine structure (the periodic lattices). Within the assumptions of Bernoulli–Euler beam theory the suggested method for the analysis of infinite cracked lattices is exact. Although most cracks follow intuitive paths it was found that the microstructure of cellular materials has a significant influence on the crack pattern.     

The control of crack arrays in thin films

Thin-film fracture can be used as a nano-fabrication technique, but generally, it is a stochastic process that results in nonuniform patterns. Crack spacings depend on the interaction between intrinsic flaw populations and the fracture mechanics of crack channeling. Geometrical features can be used to trigger cracks at specific locations to generate controlled crack patterns. However, while this basic idea is intuitive, it is not so obvious how to realize the concept in practice, nor what the limitations are. The control of crack arrays depends on the nature of the intrinsic flaw population. If there is a relatively large density of long flaws, as commonly assumed in fracture mechanics analyses, reliable crack patterns can be obtained fairly robustly using relatively blunt geometrical features to initiate cracks, provided the applied strain is carefully matched to the properties of the system and the desired crack spacing. This process is analyzed both for cracks confined to the thickness of a film and for cracks growing into a substrate. The latter analysis is complicated by the fact that increases in strain can either drive cracks deeper into the substrate or generate new cracks at shallower depths. If the intrinsic flaws are all very short, the geometrical features need to be very sharp to achieve the desired patterns. While careful control of the applied strain is not required, the strain needs to be relatively large compared to that which would be required to propagate a large flaw across the film. This results in an approach that is not robust against the introduction of accidental damage or a few large flaws.     

Surface crack and cracks networks in biaxial fatigue

Semi-elliptical fatigue crack growth in 304 L stainless steel, under biaxial loading, was investigated. Compared to those of through-cracks under uniaxial loading, the growth rate of surface cracks is increased by a non-singular compressive stress and reduced by a tensile stress, when R = 0. Plasticity-induced crack closure under biaxial loading was investigated through 3D finite element simulations with node release. Roughness and phase-transformation-induced closure effects were also discussed. The interactions in two-directional crack networks under biaxial tension were investigated numerically. It appears that the presence of orthogonal cracks should not be ignored. The beneficial influence of interaction-induced mode-mixities was highlighted.     

Experimental investigation of crack growth direction in multiple cracks

In this paper, the interaction between multiple cracks in crack growth direction is studied in an aluminium alloy under static and fatigue loading. Self similar as well as non‐self‐similar crack growth has been observed which depends on the relative crack positions defined by crack offset distance and crack tip distance. On the basis of experimental observations, the criterion for crack coalescence and crack growth direction are expressed in terms of the crack positions defined by crack offset and crack tip distances. The criterion presented in this study can be used to determine the limiting value of crack tip and crack offset distance and to determine the mode in which cracks coalesce during their growth process. Experimental results and crack interaction criterion presented under various crack positions and size conditions could be used to derive a new evaluation method of crack growth in multiple crack geometry.     

Solving two-dimensional cases for cracks in piecewise-homogeneous bodies

Summary A survey is made on papers concerned with a general approach to solving planar problems in the theory of elasticity for finite and infinite piecewise-homogeneous bodies containing curvilinear cracks, where singular integral equations are used. Cases are considered for an infinite plane and half-plane containing arbitrary and periodic elastic-inclusion systems together with curvilinear cracks, as well as a curvilinear two-component ring having internal and edge cracks.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 4, pp. 80–89, July–August, 1989.     

Functional DNAzymes organized into two-dimensional arrays

DNAzymes are catalytically active DNA molecules, which have previously been described in solution. Here, we organize these molecules into a series of two-dimensional (2D) arrays using a periodic arrangement of DNA structures based on the DNA double crossover motif. We demonstrate by means of atomic force microscopy that the DNAzymes are organized according to the design and that they retain their activity when attached in linear strings within the context of the 2D array.     

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.     

Automatic crack growth tracking of bimaterial interface cracks

The propagation process of an interfacial crack in composite material is studied using the modified maximum dilatational strain energy density criterion, NT-criterion. Some necessary assumptions have been adopted to facilitate the use of the NT-criterion in this case. The stress intensity factors at the crack tip are extracted from the complex displacement field and finite element results. A simple algorithm for automatic crack propagation is presented with an illustrative example.

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Résumé On étudie le processus de propagation d'une fissure d'interface dans un matériau composite en utilisant le critère NT, version modifiée du critère de densité d'énergie maximale de dilatation. On a adopté diverses hypothèses pour faciliter l'application du critère NT à ce cas. Les facteurs d'intensité de contraintes à l'extrémité de la fissure sont déduits du champ de déplacement complexe et des résultats de l'analyse par éléments finis. Un algorithme simple relatif à une propagation automatique de la fissure est présenté avec un exemple à la clé.

     

Bifurcation and propagation of a mixed-mode crack in a ductile material

The bifurcation and the propagation of a 2-D mixed-mode crack in a ductile material under static and cyclic loading were investigated in this work. A general methodology to study the crack bifurcation and the crack propagation was established. First, for a mixed-mode crack under static loading, a procedure was developed in order to evaluate the fracture type, the beginning of the crack growth, the crack growth angle and the crack growth path. This procedure was established on the basis of a set of criteria developed in the recent studies carried out by the authors [Li J, Zhang XB, Recho N. J-M^p based criteria for bifurcation assessment of a crack in elastic-plastic materials under mixed mode I-II loading. Engng Fract Mech 2004;71:329-43; Recho N, Ma S, Zhang XB, Pirodi A, Dalle Donne C. Criteria for mixed-mode fracture prediction in ductile material. In: 15th European conference on fracture, Stockholm, Sweden, August 2004]. A new criterion, by combining experimentation and numerical calculation, was developed in this work in order to predict the beginning of the crack growth. Second, in the case of cyclic loading, the crack growth path and crack grow rate are studied. A series of mixed-mode experiments on aluminium and steel specimens were carried out to analyse the effect of the mixed mode on the crack growth angle and the crack growth rate. On the basis of these experimental results, a fatigue crack growth model was proposed. The effect of the mixed mode on the crack growth rate is considered in this model. The numerical results of this model are in good agreement with the experimental results.     

Modeling of fatigue crack closure in inclined and deflected cracks

A 2-dimensional, elastic-plastic finite element model has been developed to simulate plasticity induced crack closure in slanted and deflected cracks growing outside the small scale yielding (SSY) regime. The finite element model allows for contact between deformable surfaces to capture the complex contact interaction between the crack faces. Coulomb's friction law has been used to model friction between the crack faces and has been incorporated in the finite element model. This paper examines the mode I and mode II behavior of slanted cracks subjected to remote mode I, constant amplitude cyclic loading. Two possible types of mode II crack face interaction have been identified: (a) complete slip in mode II before mode I opening and, (b) mode I crack opening before the crack faces undergo mode II displacements. Both types of interactions were observed in slanted cracks. The finite element study also reveals a clear dependence of mode I and mode II crack opening levels for a slanted crack on R ratio and maximum stress, S_max/₀. The crack opening levels for a slanted crack are found to be significantly higher than the stable opening values for a straight crack growing in pure mode I. The mode I and mode II crack opening levels are also found to depend on the friction between the crack faces. A four-fold increase in friction coefficient resulted in almost 50% increase in normalized mode I and mode II opening values. This paper also describes the effect of crack deflection on closure. Deflection of a fatigue crack from 45° inclination to pure mode I caused a decrease in mode I opening level, but, an increase in mode II opening level. This difference in opening behavior is attributed to the transition of the nature of crack interaction from complete slip before opening to opening in mode I before mode II shear offset. Final stable opening levels for a deflected crack are found to be close to the stable value for straight cracks.     

Effect of structural geometry and crack location on crack driving forces for cracks in welds

This paper quantifies the effect of geometry (planar or cylindrical) and crack location (internal or edge cracks; weld center or interface cracks) on crack driving force for welded joints, via systematic elastic-creep and elastic-plastic finite element (FE) analyses for welded joints. For engineering estimates of crack driving forces for mismatched welded joints, the equivalent material approach is employed. It is found that the equivalent material concept works very well only for a planar geometry with an internal crack, such as the middle cracked tension specimen. For a planar geometry with an edge crack, it works reasonably well, but tends to provide conservative results for under-matching and for interface cracks. For a cylindrical geometry with an edge crack, the results are similar to those for a planar geometry with an edge crack, but caution should be exercised for over-matching, as non-conservative estimates are possible due to gross-section yielding. For a cylindrical geometry with an internal crack, excessively conservative estimates for under-matching are found, and thus an improved estimation method is desired.     

Interaction of a crack with certain microcrack arrays

Interaction of a crack with microcracks (modelling “damage”) can significantly alter the stress concentration at the crack tip. Certain important effects of interaction—shielding effect (“toughening by microcracking”), amplification effect, influence of the orientations of microcracking and irregularities in its patterns, change of the character of interaction (shielding to amplification and vice versa) with change of the mode of loading, etc.—are demonstrated on several microcrack systems. It appears that these relatively simple systems exhibit the essential features of the crack-damage interactions. Consideration is based on the method of analysis of elastic solids with many cracks proposed recently [Kachanov, Int. J. Fracture 28, R11–R19 (1985); Kachanov, Int. J. Solids Structures (in press)] and briefly presented here.