Crack-tip Model Research Articles

Effects and recommendations of crack tip modelling on compliance solutions

Elastic unloading compliance is a common technique to determine instantaneous crack size in fracture mechanics properties such as resistance curves (R-curves) and fatigue crack growth. This technique uses a polynomial, developed using finite elements models, to correlate compliance with crack depth. In these models, crack tip is usually modeled with a blunt mesh (with a crack tip radius ρ) or a sharp mesh (with element size in the width direction l), and neither are standardized in the literature. This paper shows that both affect compliance, therefore potentially impacting results of fracture mechanics properties acquired from simulations. This paper focuses on how those parameters affects compliance in SE(B) models. Finally, a parameter φ that normalizes ρ or l with the width was developed and recommended values based on compliance convergence are suggested. Controlling φ may help to ensure reliable results in fracture mechanics models where compliance needs to be determined.

A pure complex variable enrichment method for modeling progressive fracture of orthotropic functionally gradient materials

Reasonably reducing degrees of freedom (DOFs) to achieve higher numerical efficiency and robustness is a long-standing challenge in all aspects of computational mechanics and this is especially true in the case of computational fracture analysis. In this work, a novel technique pursuing the further reduction of the number of the general singular enriched terms for fracture modelling is developed by means of the complex variable basis system inspirited by the Euler’s identity. By introducing the complex solution, the desirable enrichment to capture the crack-tip field can be constructed as a complex variable form with fewer terms. Thereby, the necessary number of nodes, as well as the DOFs, can be reasonably reduced for crack-tip modelling. The proposed novel pure complex variable enriched basis system, which is jointed with a standard complex variable meshless scheme, is applied to analyze the crack propagation problems in orthotropic functional gradient materials (FGM). The numerical results, of both the stress fields and the crack path, demonstrated that the proposed novel pure complex variable enrichment can effectively model fractures in orthotropic FGM with fewer DOFs compared with the element-free Galerkin method and the extended finite element method.

Insights from electrochemical crack tip modeling of atmospheric stress corrosion cracking

Crack tip electrochemical conditions are explored utilizing a reactive transport Finite Element Method model for stainless steel 304 exposed to 3 M NaCl. Under simulated full immersion conditions, a steady state pH of 3 and metal chloride concentration of 1.4 M were calculated. It was determined that an increase in cathode length and a decrease in water layer thickness increases total cathodic current per unit thickness, metal chloride concentrations, and decreases crack tip pH. The presented results and discussion call into question whether electrochemical similitude is achieved between different specimens, crack lengths, and between laboratory specimen and field relevant samples.

Development of Automatic Crack Growth Simulation Program Based on Finite Element Analysis

A crack growth simulation program based on the advanced iterative-finite element method (AI-FEM) was developed to predict realistic crack growth of structures. The developed program was suggested to calculate the exact stress intensity factor for arbitrary structures by regenerating the crack tip mesh as the crack grows. The main advantages of the developed program are to estimate each different crack growths along the crack tip line and to simulate the cracking transition from a surface crack to a through-wall crack under a complex stress field. For these purposes, the sensitivity analyses were performed for various influence variables on stress intensity factors, such as element types and crack dimensions. Based on the results of sensitivity analyses, the appropriate criteria for crack tip modeling to be used in AI-FEM were suggested to calculate sufficient converged SIF. The program developed in this research was validated through stress corrosion crack growth and natural crack growth examples including cracking transition, and it was confirmed that the program simulates crack growth well and has reasonable methods for cracking transition.

Characterization of non‐planar crack tip displacement fields using a differential geometry approach in combination with 3D digital image correlation

AbstractThis paper describes a novel differential geometry method that is used in combination with 3D digital image correlation (3D‐DIC) for crack tip field characterization on non‐planar (curved) surfaces. The proposed approach allows any of the two‐dimensional crack tip field models currently available in the literature to be extended to the analysis of a 3D developable surface with zero Gaussian curvature. The method was validated by analyzing the crack tip displacement fields on hollow thin‐walled cylindrical specimens, manufactured from either 304L or 2024‐T3 alloy that contained a central circumferential crack. The proposed approach was checked via a comparison between experimentally measured displacement fields (3D‐DIC) and those reconstructed from a modified 2D crack tip model (utilizing either 2, 3, or 4 terms of the William's expansion series) and implementing a 3D geometrical correction. Further validation was provided by comparing model‐derived stress intensity factors with values provided by empirical correlations.

Lichtenberg optimization algorithm applied to crack tip identification in thin plate-like structures

PurposeThis study aims to develop a numerical identification and characterization of crack propagation through the use of a new optimization metaheuristics called Lichtenberg optimization.Design/methodology/approachThe damage-identification problem is treated as an inverse problem, which combines finite element methods with intelligent computational methods to obtain the best possible response. To optimize the objectives, the Lichtenberg algorithm is applied, which includes concepts of random cluster growth in nature.FindingsThe simulations show that it is possible to determine the Lichtenberg spectrum algorithm a part of the structure to be removed and replaced in this case to stop the propagation.Originality/valueThe results show a very good crack identification in plates-like structures using the Lichtenberg algorithm (LA) based only in strain fields. Although many studies have reported on damage-identification-based optimization methods, very few have focused on the crack tip modeling and LA as the main solver.

The effect of global viscoelasticity on rubber friction with Schallamach waves

A mathematical model developed previously for predicting adhesive sliding friction of rubber on a hard surface in the presence of Schallamach waves is extended here to include global effects of viscoelasticity, besides the previously included viscoelastic effects in the neighborhood of the debond (shear crack) tips; the crack tip model is unchanged. We find that global viscoelasticity limits the maximum sliding speed and minimum wavelength for which a wave train can exist and, as expected, increases the friction force. The model is shown to improve upon the agreement with experimental data on wave speed and wavelength in the previous paper.

A new asymptotic crack tip model to predict failure processes

A two dimensional finite element asymptotic crack tip model (ABAQUS® code) to simulate crack tip opening displacement (CTOD) and damage in ductile materials has been developed. Pure Mode I is considered with asymptotic displacements applied on the outer boundary of the finite element mesh constructed using plane strain solid and cohesive elements. CTOD profiles and damage variation along cohesive element path are clearly observed. Evolution of CTOD for various ratios of solid/cohesive Young´s Modulus are also presented. The evolution of damage was based on a power law fracture criterion. In an effort to extend isotropic hardening plasticity theory to small scales, the research was focused on room-temperature crack growth in polycrystalline Aluminium alloys. In a future work, stress enhancement will be considered in a framework of cohesive zone modeling and strain gradient plasticity based on the micromechanics of dislocations. An asymptotic crack tip micro structured mesh will be also accounted into the finite element analysis.

Experimental methodology for the quantification of crack tip plastic zone and shape from the analysis of displacement fields

The current work presents a novel methodology for the experimental quantification of the crack tip plastic zone during fatigue crack growth. This methodology is based on the application of yield criteria to estimate the area and the shape of the plastic zone at the crack tip. The implementation of the proposed methodology requires the use of strain maps calculated from the differentiation of the displacement fields obtained by digital image correlation (DIC). Stress maps can subsequently be inferred from both von Mises and Tresca yield criteria. Fatigue tests and associated measurements of plastic zone size and shape were conducted on a compacttension specimen made from commercially pure titanium at R ratio of 0.6. In addition, the ability to predict the shape and size of the experimentally observed crack tip plastic zone has been explored using three different analytical elastic crack tip models [Westergaard, Williams and Christopher- James-Patterson (CJP)]. This analysis indicated that the CJP model provided the most accurate prediction of the plastic zone and shape.

Crack-Tip Singularity Growth Modeling without Remeshing

On the basis of fracture mechanics, a numerical study has been considered in order to model the crack growth without re-meshing. Firstly, we have carried out a numerical comparative investigation on the finite elements for capturing crack-tip singularity. The aim is to numerically assess the values of the stress intensity factors by using different crack-tip finite elements (QPE, DQPE and IQPE). Secondly, we display a new numerical scheme to model the crack propagation without re-meshing. The fundamental idea is to couple the level set method (LSM) with several kinds of singular finite elements to solve the cracks problem: the special elements are employed to evaluate the crack-tip displacement fields, whereas the level sets are utilized to represent the crack location. By the end, a physical problem of crack growth simulations is supplied to show the robustness and versatility of the crack-tip finite element modelling.

Dislocation shielding of a cohesive crack

Dislocation interaction with a cohesive crack is of increasing importance to computational modelling of crack nucleation/growth and related toughening mechanisms in confined structures and under cyclic fatigue conditions. Here, dislocation shielding of a Dugdale cohesive crack described by a rectangular traction-separation law is studied. The shielding is completely characterized by three non-dimensional parameters representing the effective fracture toughness, the cohesive strength, and the distance between the dislocations and the crack tip. A closed form analytical solution shows that, while the classical singular crack model predicts that a dislocation can shield or anti-shield a crack depending on the sign of its Burgers vector, at low cohesive strengths a dislocation always shields the cohesive crack irrespective of the Burgers vector. A numerical study shows the transition in shielding from the classical solution of Lin and Thomson (1986) in the high strength limit to the solution in the low strength limit. An asymptotic analysis yields an approximate analytical model for the shielding over the full range of cohesive strengths. A discrete dislocation (DD) simulation of a large (>10 3) number of edge dislocations interacting with a cohesive crack described by a trapezoidal traction-separation law confirms the transition in shielding, showing that the cohesive crack does behave like a singular crack at very high cohesive strengths (∼7 GPa), but that significant deviations in shielding between singular and cohesive crack predictions arise at cohesive strengths around 1GPa, consistent with the analytic models. Both analytical and numerical studies indicate that an appropriate crack tip model is essential for accurately quantifying dislocation shielding for cohesive strengths in the GPa range.

Estimation of Crack Growth Life in Rail with a Squat Defect

Rolling contact fatigue initiated defects such as surface corrugation, head check, squat, are one of growing problems in high speed railway line. A squat is generally developed below the rail surface and grows parallel to surface until it turns down into rail. Estimation of critical crack size and crack growth rate is an essential to prevent rail from failure and develop cost effective railway maintenance strategy. In this study, we predict crack growth rate of a rail with a squat defect. For this purpose, a rail model with a squat defect is developed. Timoshenko’s beam theory is applied to calculate the global bending stress at the crack tip and Hertzian contact model is applied to calculate the local contact stresses at the surface of rail by simulating rolling over a railway wheel on a rail. Stress intensity factors are calculated from the total stress at the crack tip. Fatigue crack growth curve of 60kg rail steel is applied to calculated crack growth rate. Software to predict crack growth life through whole life cycle is developed. We expect that we can make a more cost effective rail maintenance strategy by considering the crack growth analysis for a defective rail.

Time-dependent cohesive zone modelling for discrete fracture simulation

The cohesive crack tip model became very popular in fracture and failure mechanics, starting with the original publications of (Dugdale, 1960) [1] and (Barenblatt, 1962) [2] and first practical applications in the early 1980s. A compact representation of the fracture mechanical basis, kinematic and constitutive issues as well as some special characteristics of the related finite element formulation are given in the first part of this contribution. The main part is dedicated to the presentation and discussion of recent developments of cohesive finite element methods for time-dependent fracture. On the basis of a rheological model assumption, a novel viscoelastic extension for cohesive traction separation laws is presented and the resultant characteristic behaviour is depicted and compared for different loading conditions. Adopting an industrial application of a peel foil specimen, the time-dependent characteristics as well as some aspects of parameter identification and application of the material model are shown.

Fatigue crack shielding and deflection in plain bearings under large-scale yielding

Multi-layered bearing systems used in the automotive industry show shielding and anti-shielding effects that reduce or amplify the crack driving force under large-scale yielding conditions. Using finite element analysis, it is shown that shielding in such systems results in path deflection and bifurcation despite the absence of mixed-mode loading. As the crack approaches a stiff layer, the tangential strains measured around a blunted crack tip model show a maximum corresponding to the direction of crack propagation. The distribution of such strains indicates the effect of shielding and the likelihood of the tip to deflect or bifurcate. The suitability of bi-layer and tri-layer bearing architectures is assessed through crack path and respective crack driving force predictions.

Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K H IC for a single hydride platelet at the crack tip has been also estimated.

Electro-elastic stress analysis for a Zener–Stroh crack at the metal/piezoelectric bi-material interface

A Zener–Stroh crack may nucleate at the interface of metal/piezoelectric bi-materials, when piled-up dislocations along a slip plane in metal material are stopped by the interface which works as an obstacle. As a kind of microcrack, Zener–Stroh crack controls the initial phase of crack nucleation. In the first step of our current research, the stress and electric displacement fields for a single interfacial dislocation in general piezoelectric/metal bimaterials are obtained in an explicit close form. With this solution as the Green function, the interfacial Zener–Stroh crack problem is formulated into a set of singular integral equations with distributed dislocation technique. These singular integral equations are then solved by numerical method based on the singularity nature at the crack tips. To deal with the oscillation behaviors near the interfacial crack tips, both open crack tip model and contact zone model are considered and compared each other in the paper. Solutions on the stress field, stress and electric displacement intensity factors and contact zone length are obtained. A numerical example of a Zener–Stroh crack at the Pt/PZT-5H bimaterial interface is given. Some useful electro-elastic characteristics related to the crack are found and discussed.

Atomistic process on hydrogen embrittlement of a single crystal of nickel by the embedded atom method

A molecular dynamics (MD) simulation by the embedded atom method (EAM) was conducted on hydrogen embrittlement (HE) at a crack tip of a single crystal of nickel composed of {1 0 0} planes under uniaxial tension along the [1 0 0] direction perpendicular to the crack plane at room temperature. A three-dimensional (3D) crack-tip model with 16,632 nickel atoms and from 2 to 653 hydrogen atoms segregated in the notched area was designed and the EAM potentials proposed by Baskes and co-workers were used for the simulation. The hydrogen-free specimen showed good ductility associated with a significant blunting of the crack tip. In the hydrogen-charged specimen, the elongation at the fracture decreased with increasing hydrogen content due to the suppressed and localized plastic deformation. Fracture occurred macroscopically on the (1 0 0) plane perpendicular to the tensile direction. Twin deformation associated with the fracture was formed in the specimen with 653 hydrogen atoms, corresponding to a thin layer of hydride at the notched area, while no twin deformation was formed in the other hydrogen-charged specimens or the hydrogen-free specimen. The simulation results were in good agreement with published experimental results.

Process zone size effects on naturally curving cracks

In a fuselage crack turning application, a crack may grow under stable tearing conditions for some length, achieving a somewhat steady-state condition, then encounter a region of high tensile T-stress as it nears a stiffener, causing the crack to turn. While a T-stress related crack path instability has been described by Cotterell and Rice [Int J Fract 16 (1980) 155], recent experimental evidence indicates that crack turning is also influenced by the size of the process zone. In order to study this problem, the perturbed crack path solution of Cotterell and Rice is extended to include the effects of a process zone represented by cohesive tractions on the crack flanks trailing the crack tip. The characteristic length, r c, over which the tractions are prescribed, represents a strain-localization zone which precedes the physical crack tip, and through which the crack presumably must pass. The strain-localization zone is assumed to be smaller than common measures of plastic zone size, and other than the strain-localization zone, plasticity is not explicitly modeled. The solution, which uses cohesive tractions analogous to the Dugdale–Barenblatt crack tip model [J Mech Phys Solids 8 (1960) 100; Advances in Applied Mechanics, vol. VII, Academic Press, 1962. p. 55], is accurate to first-order deviations from a straight crack path in an infinite medium for small strain-localization zones. A correction factor is provided to the process zone parameter to approximate the solution for larger process zones, but loses accuracy as the strain-localization zone approaches the plastic zone size. Increased process zone size is shown to result in an increase in the perturbation sensitivity of a crack in a positive T-stress environment, causing more rapid turning of the crack, as has also been experimentally observed.

A closed crack tip model for interface cracks inthermopiezoelectric materials

The interface crack problem of a bimaterial thermopiezoelectric solid was treated byapplying the extended version of Strohs formalism and singular integral equation approach. Theinterface crack considered is subjected to combined thermal, mechanical and electric loads.Under the applied loading, the interface crack is assumed to be partially opened. Formulation ofthe problem results in a set of singular integral equations which are solved numerically. Thestudy shows that the contact zone is extremely small in comparison with the crack length. Basedon the formulation, some physically meaningful quantities of interest such as stress intensityfactors and size of contact zone for a particular material group are analyzed.