A general model for simulating the effect of the intercommunication between various elements of a structure on the extension of a crack

The characteristics of the intercommunication between the various elements of a structure have a major effect on the structure's resistance to failure by the extension of a crack. Two extremes are: (a) where the intercommunication is very strong when the structure is particularly vulnerable to crack extension, and (b) where the elements are almost totally isolated when the structure is highly resistant to crack extension. This paper presents a very simple and general model that quantifies the effect of the intercommunication on a structure's crack extension resistance, and unifies the various types of intercommunication within a single model analysis.     

Displacement control crack-growth instability in an elastic-softening material

The criterion for crack growth instability in an elastic-softening material that is subjected to displacement control loading conditions is examined. A theoretical analysis of the model of a solid containing two symmetrically situated deep cracks and with tensile loading of the remaining ligament, defines the criterion for crack growth instability. The criterion is expressed in terms of the material's softening characteristics and the solid's geometrical parameters. The analysis covers the complete spectrum of material behaviour from the case where the softening zone is very small to the case where instability does not occur until the softening zone traverses the ligament between the crack tips.     

The value of the J-integral at the onset of crack extension from a weld defect: weld material is harder than parent material

The paper addresses the problem of crack extension in a weld in an engineering structure for the situation where the crack is parallel to the plane of the weld. An earlier analysis, for the case where the weld material is softer than the parent material, has demonstrated the extent to which the value of the J-integral at the onset of crack extension depends on the flow properties of the weld and parent materials, the crack size and the weld thickness. The present paper extends these earlier considerations to the case where the weld material is harder than the parent material, and again demonstrates the non-uniqueness of the value of J at the onset of crack extension.     

On quarter-point three-dimensional finite elements in linear elastic fracture mechanics

Use of the finite element method for calculating stress intensity factors of two-dimensional cracked bodies has become commonplace. In this study, the more difficult task of applying finite elements to three-dimensional cracked bodies is investigated. Since linear elastic material is considered, square root singular stresses exist along the edge of an embedded crack. To deal with this numerical difficulty, twenty noded, isoparametric, serendipity, quarter-point, singular, solid elements are employed. Examination of these elements is carried out in order to determine the extent of the singular behavior.In addition, the stiffness derivative technique is explored, together with quarter-point elements, to determine an accurate procedure for computing stress intensity factors in three-dimensions. The problem of chosing a proper virtual crack extension is addressed. To this end, the disturbance in the square root singular stresses is examined and compared with a similar disturbance which occurs in two-dimensions. As a numerical example, a pennyshaped crack in a finite height cylinder is considered with various meshes. It is found that stress intensity factors can be calculated to an accuracy within 1 percent when quarter-point cylindrical elements are employed with the stiffness derivative technique such that the crack extension is one in which one corner node is not moved, the other corner node is moved a small distance, and the midside node is moved one-half that distance. This crack extension is analogous to that of a straight crack advance for a brick element. Both of these crack advances disturb the square root singular stresses in a manner similar to that which occurs with the two-dimensional eight noded element in which the crack has been advanced a small distance.     

Analysis of Mode I and Mode II Crack Growth Arrest Mechanism with Z-Fibre Pins in Composite Laminated Joint

This paper presents the numerical study of the mode I and mode II interlaminar crack growth arrest in hybrid laminated curved composite stiffened joint with Z-fibre reinforcement. A FE model of hybrid laminated skin-stiffener joint reinforced with Z-pins is developed to investigate the effect of Z- fibre pins on mode I and mode II crack growth where the delamination is embedded inbetween the skin and stiffener interface. A finite element model was developed using S4R element of a 4-node doubly curved thick shell elements to model the composite laminates and non linear interface elements to simulate the reinforcements. The numerical analyses revealed that Z-fibre pinning were effective in suppressing the delamination growth when propagated due to applied loads. Therefore, the Z-fibre technique effectively improves the crack growth resistance and hence arrests or delays crack growth extension.     

A discussion of results from models simulating crack tip reinforcement: Fully bridged situation

Against the background of the need to understand the mechanics of toughening for various types of advanced material, several recent papers have analysed crack models in which the crack faces are bridged by a continuous distribution of springs. This paper addresses the full bridging situation, and shows how results for the linear elastic (hardening) case can, after appropriate modification, be used for the case where the springs have a linear softening law.     

Characterization of stable crack extension in aluminium sheet material using the crack tip opening angle determined optically and by the δ5 clip gauge technique

Tests standards aimed at deriving fracture toughness data and crack resistance curves under low constraint condition have recently been finished by ASTM and ISO. These standards cover various experimental methods for determining critical crack tip opening angles, CTOA, for characterising stable crack extension in sheet material. In this paper, some key items of these standard methods are validated, namely the experimental determination of the crack tip opening angle by optical observation and using the δ₅ clip gauge method. When applying such standard methods to material characterization it is of particular interest to know how CTOA-data derived by different methods compare with each other. This paper compares CTOA-data as derived by the optical method with those derived by using the δ₅ clip gauge method. In order to study possible specimen size and geometry effects the methods have been applied to a wide range of specimen geometries. The results demonstrate that CTOA-data derived by the optical method are well suited to provide a specimen size and geometry independent characterization of stable crack extension where the thus obtained CTOA-data are constant over a large amount of stable crack extension. In contrast to this result, CTOA-data obtained from the δ₅ clip gauge method revealed a complex pattern of size and geometry effects, and only in case of compact specimens with a selected size the two CTOA-methods provide nearly identical CTOA-data over a large amount of crack extension.     

Comparative assessment of fatigue and fracture behaviour of cast and forged railway wheels

A comparative evaluation of fatigue and fracture behaviour of commercially produced cast and forged rail wheels has been made using specimens extracted from various locations of the wheel quadrant. A systematic investigation in the web and rim regions of the wheel quadrant with various notch orientations showed that the forged material exhibited a better intrinsic resistance to fatigue crack growth than the cast material. Since linear elastic fracture mechanics (LEFM) based fracture toughness could not be validated for both the cast and forged wheel material, elastic plastic fracture mechanics (EPFM) based characteristic fracture toughness was used. Results showed that fracture resistance of the forged material is superior to that of the cast material. Cast wheel specimens exhibited unstable crack extension in comparison to substantial stable tearing in forged specimens. Microstructural and fractographic analyses showed that the cast wheel material contained large amounts of inclusions. The poor fracture resistance of cast wheel material is therefore attributed to the inferior material quality.     

Crack growth resistance behavior of a functionally graded material: computational studies

This paper describes crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) using a cohesive zone ahead of the crack front. The plasticity in the background (bulk) material follows J₂ flow theory with the flow properties determined by a volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model). A phenomenological, cohesive zone model with six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) describes the constitutive response of the cohesive zone. Crack growth occurs when the complete separation of the cohesive surfaces takes place. The crack growth resistance of the FGM is characterized by a rising J-integral with crack extension (averaged over the specimen thickness) computed using a domain integral (DI) formulation. The 3-D analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. The paper describes applications of the cohesive zone model and the DI method to compute the J resistance curves for both single-edge notch bend, SE(B), and single-edge notch tension, SE(T), specimens having properties of a TiB/Ti FGM. The numerical results show that the TiB/Ti FGM exhibits significant crack growth resistance behavior when the crack grows from the ceramic-rich region into the metal-rich region. Under these conditions, the J-integral is generally higher than the cohesive energy density at the crack tip even when the background material response remains linearly elastic, which contrasts with the case for homogeneous materials wherein the J-integral equals the cohesive energy density for a quasi-statically growing crack.     

A cohesive crack model to study the fracture behaviour of fiber-reinforced brittle-matrix composites

A cohesive crack model analysis of the fracture resistance and ductility of fiber-reinforced brittle-matrix composites was performed. The bulk material is assumed to behave as a linear elastic solid. The constitutive equation for the cohesive crack is obtained through micromechanical considerations based on the shear lag model of fiber-matrix interaction and on the statistical nature of fiber failure. A parametrical study of the influence of fiber volume fraction, fiber strength and flaw distribution, interfacial shear stress and specimen geometry on the fracture resistance of these composites was carried out. The results illustrate the role of each of these factors in overall composite toughening, and suggest various strategies to improve the composites' performance.     

Lattice modeling of aggregate interlocking in concrete

In this paper, we study a mixed-mode fracture process using a conventional two dimensional lattice model with incorporated meso-level internal material structure. Simple elasto-brittle elements of the network are divided into three phases according to a projected grain layout. The stiffness of any element that fulfils a failure criterion is removed. As a new feature of the otherwise standard lattice approach, we added the recovery of normal stiffness when a severed element enters the compressive regime. This enhancement enables capture of the shear resistance of an existing crack caused by crack roughness, i.e. what is termed aggregate interlocking. We demonstrate this enhancement via the simulation of mixed-mode experiments on concrete performed at a laboratory at the Technical University of Denmark. Double notched concrete specimens were initially pre-cracked in tension. Then, various combinations of tensile and shear load (normal and tangential to the crack plane) were applied. Simulated crack patterns and load–displacement curves are compared to the experimental observations.     

An adaptive singular finite element in nonlinear fracture mechanics

In the case of nonlinear fracture mechanics the type of singularity induced by the crack tip is commonly not known. This results in a poor approximation of the near crack tip fields in a finite element setting and induces so called spurious—or residual—discrete material forces in the vicinity of the crack tip. Thus the numerical calculation of the crack driving material force in nonlinear fracture is often not that precise as in linear elasticity where we can use special crack tip elements and/or path independency. To overcome this problem we propose an adaptive singular element, which adapts automatically to the type of singularity. The adaption is based on an optimisation procedure using a variational principle.     

The value of the J-integral at the onset of crack extension from a weld defect: Weld material is softer than parent material

The paper addresses the problem of crack extension in a weld in an engineering structure for the case where the weld crack is parallel to the plane of the weld, a situation for which the J-integral is path independent with regard to any contour surrounding a crack tip. Assuming that crack extension is associated with the attainment of a critical crack tip opening displacement _w, a theoretical analysis based on the strip yield representation of plastic deformation shows, for the case where the weld material is softer than the parent material, how the relation between the value of J at the onset of crack extension and _w depends on the flow properties of the weld and parent materials, the crack size and the weld thickness.     

Fatigue analysis and testing of adhesive joints

The fatigue resistance of a single-lap aluminium adhesive joint to cyclic loading in combined shear and bending mode is investigated by nonlinear finite element analysis and crack propagation experiments. The epoxy adhesive is modelled by an elasto-plastic overlay material model. The initial cycles build up a residual stress state, leading to nearly linear material behaviour in the following cycles. Fatigue crack propagation is modelled by removing adhesive elements. Two series of experiments with one-sided cyclic load were carried out. The crack length was monitored by measuring the bending compliance around the end of the overlap with clip-gauges. The crack length is determined as a simple linear function of the measured compliance. The experiments show nearly constant rate crack growth until failure, with no appreciable crack initiation period. The rate of crack growth is proportional to the stress level to the power m = 6.2. Fatigue life results are given in the form of S---N curves for adhesive thickness of 0.1 and 0.3 mm. There is no systematic influence of the thickness of the adhesive on the fatigue life. This supports the use of a crack propagation and fatigue life criterion formulated in terms of the energy release rate.     

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.     

Virtual crack closure technique for an interface crack between two transversely isotropic materials

The virtual crack closure technique makes use of the forces ahead of the crack tip and the displacement jumps on the crack faces directly behind the crack tip to obtain the energy release rates \({{\mathcal {G}}}_I\) and \({\mathcal {G}}_{II}\). The method was initially developed for cracks in linear elastic, homogeneous and isotropic material and for four noded elements. The method was extended to eight noded and quarter-point elements, as well as bimaterial cracks. For bimaterial cracks, it was shown that \({\mathcal {G}}_I\) and \({\mathcal {G}}_{II}\) depend upon the virtual crack extension \(\varDelta a\). Recently, equations were redeveloped for a crack along an interface between two dissimilar linear elastic, homogeneous and isotropic materials. The stress intensity factors were shown to be independent of \(\varDelta a\). For a better approximation of the Irwin crack closure integral, use of many small elements as part of the virtual crack extension was suggested. In this investigation, the equations for an interface crack between two dissimilar linear elastic, homogeneous and transversely isotropic materials are derived. Auxiliary parameters are used to prescribe an optimal number of elements to be included in the virtual crack extension. In addition, in previous papers, use of elements smaller than the interpenetration zone were rejected. In this study, it is shown that these elements may, indeed, be used.     

Mixed mode I/II fracture of 2024-T3 friction stir welds

For the first time, a series of mixed mode I/II fracture experiments have been performed on both base material and three families of friction stir welds (FSWs) in 6.4 mm thick, 2024-T351 aluminum plate; the FSW joints are designated hot, medium and cold due to the level of nominal weld energy input per unit weld length (specific weld energy) during the joining process.Results from the fracture tests indicate that the measured critical crack opening displacement (COD) at a fixed distance behind the crack tip properly correlates both load-crack extension response and microstructural fracture surface features for both the base metal and all FSWs, providing measure of a quantitative fracture toughness. The COD values also indicate that transition from mode I to mode II dominant crack growth occurs at lower loading angles for FSW joints having higher specific weld energy input, with a truly mixed mode I/II COD measured during crack growth in the medium FSW joint. Using results from recent detailed FSW metallographic studies, specific features in the fracture process are correlated to the FSW microstructure. Finally, the observed ductile crack growth path in all three welds tends to exit the under-matched FSW weld region as the far-field applied shear loading is increased, with the medium FSW being the only case where the flaw remained within the FSW region for all combinations of shear and tensile loading.     

Some problems involving distributions of screw dislocations in a finite slab

This paper considers two problems involving distributions of screw dislocations in a finite slab. The first problem is concerned with the spread of plasticity from a crack ¦x₁¦c, x₂ = 0 in the finite slab ¦x₂¦ h/2 subject to an externally applied shear stress p₂₃ = σ, and is treated in terms of the theory of continuous distributions of dislocations. The extent to which the dislocations describing the plastic relaxation spread from a crack tip is determined, together with the relative displacement of the crack surfaces at its tips. Thus the criterion for crack extension is calculated, particular attention being given to the case when fracture occurs at low applied stresses. The second problem studied is the special case of the first problem that arises when no plastic relaxation is allowed; the criterion for crack extension is then determined from energetic considerations; the similarity between the two sets of results for low stress fractures is emphasized.     

On the physical meaning of the J-δa-curves

In elastic-plastic fracture mechanics often so-called J-R -curves are recorded to describe a material's resistance against crack extension. In this paper the physical meaning of these curves is worked out. The starting point is the energy balance during quasi-static crack-growth. From this the difference between crack-initiation toughness and crack-growth toughness is derived. The J-Δa-curves are no resistance curves in the Griffith sense. The real crack-growth resistance, R, is reflected by the slope of the curves. A formula is given to estimate R for deeply notched bend or CT-specimens.     

Dynamic steady-state crack propagation in a transversely isotropic viscoelastic body

The problem considered herein is the dynamic, subsonic, steady-state propagation of a semi-infinite, generalized plane strain crack in an infinite, transversely isotropic, linear viscoelastic body. The corresponding boundary value problem is considered initially for a general anisotropic, linear viscoelastic body and reduced via transform methods to a matrix Riemann–Hilbert problem. The general problem does not readily yield explicit closed form solutions, so attention is addressed to the special case of a transversely isotropic viscoelastic body whose principal axis of material symmetry is parallel to the crack edge. For this special case, the out-of-plane shear (Mode III), in-plane shear (Mode II) and in-plane opening (Mode I) modes uncouple. Explicit expressions are then constructed for all three Stress Intensity Factors (SIF). The analysis is valid for quite general forms for the relevant viscoelastic relaxation functions subject only to the thermodynamic restriction that work done in closed cycles be non-negative. As a special case, an analytical solution of the Mode I problem for a general isotropic linear viscoelastic material is obtained without the usual assumption of a constant Poissons ratio or exponential decay of the bulk and shear relaxation functions. The Mode I SIF is then calculated for a generalized standard linear solid with unequal mean relaxation times in bulk and shear leading to a non-constant Poissons ratio. Numerical simulations are performed for both point loading on the crack faces and for a uniform traction applied to a compact portion of the crack faces. In both cases, it is observed that the SIF can vanish for crack speeds well below the glassy Rayleigh wave speed. This phenomenon is not seen for Mode I cracks in elastic material or for Mode III cracks in viscoelastic material.