The crack-modelling technique: optimization of parameters

The crack-modelling technique is a method for prediction of fatigue in components using finite element (FE) analysis. The technique, which is based on the estimation of equivalent K factors for stress-concentrators, has had some initial success in analysing components of complex shape, but this has raised a number of questions about the potential accuracy of the method and its sensitivity to the choice of operating parameters. The present paper reports on a systematic study using four different specimen types and one component geometry. Accurate estimates of equivalent K values are shown to be possible for both sharp notches and blunt notches, loaded in uniaxial tension or bending, using a very simple approach in which the stress distribution from the notch is compared to that from a standard cracked body. The method shows some sensitivity to the optimization routines used, and to some extent to the choice of the standard cracked body. It is relatively insensitive to mesh refinement and can be used with simple, elastic FE models.     

Failure of a Porous Solid from a Deep Notch

A finite strain finite element method is used to examine the stress state near the tip of a deep notch in an elastic-plastic porous solid. The notch is loaded in mode I plane strain tension and small scale yielding is assumed. Two rate independent strain hardening material models are used: a version of the Gurson model (1977) and the more recent FKM model developed by Fleck, Kuhn and McMeeking (1992). Under increasing K _I, void growth is initially stable and independent of mesh dimension. Localization of plastic flow sets in at a finite value K _i, and the deformation field is mesh-size dependent thereafter. The initiation of crack growth at the notch root is assumed to occur when a critical level of porosity is attained. The results show that the shape of the plastic zone for both the Gurson and the FKM material is highly dependent on the initial porosity. In the case of low initial porosity, the plastic zone shape is similar to that of a fully dense material; at higher initial porosities the plastic zone is concentrated ahead of the notch tip. The effect of the initial void volume fraction on the porosity field and the critical stress intensity factor is studied, and the mesh-size dependence of the results is discussed. The analysis is useful for prediction of the notched strength of porous metals. This revised version was published online in August 2006 with corrections to the Cover Date.     

Multi‐level hp‐adaptivity for cohesive fracture modeling

Discretization‐induced oscillations in the load–displacement curve are a well‐known problem for simulations of cohesive crack growth with finite elements. The problem results from an insufficient resolution of the complex stress state within the cohesive zone ahead of the crack tip. This work demonstrates that the hp‐version of the finite element method is ideally suited to resolve this complex and localized solution characteristic with high accuracy and low computational effort. To this end, we formulate a local and hierarchic mesh refinement scheme that follows dynamically the propagating crack tip. In this way, the usually applied static a priori mesh refinement along the complete potential crack path is avoided, which significantly reduces the size of the numerical problem. Studying systematically the influence of h‐refinement, p‐refinement, and hp‐refinement, we demonstrate why the suggested hp‐formulation allows to capture accurately the complex stress state at the crack front preventing artificial snap‐through and snap‐back effects. This allows to decrease significantly the number of degrees of freedom and the simulation runtime. Furthermore, we show that by combining this idea with the finite cell method, the crack propagation within complex domains can be simulated efficiently without resolving the geometry by the mesh. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical evaluation of the quarter-point crack tip element

This paper attempts to answer two commonly raised questions during the preparation of a finite element mesh, for the linear elastic fracture analysis of cracked structure: how to set up the finite element mesh around the crack tip, and what level of accuracy is to be expected from such a modelling. Two test problems, with known analytical expressions for their stress intensity factors, are analysed by the finite element method using the isoparametric quadratic singular element. The modified parameters were the order of integration, aspect ratio, number of elements surrounding the crack tip, use of transition elements, the singular element length over the total crack length, the symmetry of the mesh around the crack tip. Based on these analyses, a data base is created and various plots produced. The results are interpreted, the accuracy evaluated and recommendations drawn. Contrary to previous reports, it is found that the computed stress intensity factor (SIF) remains within engineering accuracy (10 per cent) throughout a large range of l/a (singular element length over crack length) for problems with a uniform non-singular stress distribution ahead of the crack tip (i.e. double edge notch), and l/a should be less than 0·1 for problems with a non-singular stress gradient (i.e three-point bend). Also, it is found that the best results are achieved by using at least four singular elements around the crack tip, with their internal angles around 45 degrees, and a reduced (2 × 2) numerical integration.     

Assessment of discretized errors and adaptive refinement with quadrilateral finite elements

This paper studies discretized errors, and their estimation in conjunction with quadrilateral finite element meshes which are generated by the intelligent mesh generator XFORMQ.¹ The exact energy error is used to evaluate the distortion effect of the quadrilateral mesh. The Zienkiewicz–Zhu² error estimate and actaptive procedure are applied to the short cantilever and the square plate problems using the quadrilateral mesh generator XFORMQ. It is shown that the multistage quadrilateral element refinement produces results superior to the triangular element refinement in the test cases.     

Singular and non-singular approaches for predicting fatigue crack growth behavior

In this work, three classes of mechanisms that can cause load sequence effects on fatigue crack growth are discussed: mechanisms acting before, at or after the crack tip. After reviewing the crack closure idea, which is based on what happens behind the crack tip, quantitative models are proposed to predict the effects at the crack tip due to crack bifurcation. To predict the behavior ahead of the crack tip, a damage accumulation model is proposed. In this model, fatigue cracking is assumed caused by the sequential failure of volume elements or tiny εN specimens in front of the crack tip, calculated by damage accumulation concepts. The crack is treated as a sharp notch with a small, but not zero radius, avoiding the physically unrealistic singularity at its tip. The crack stress concentration factor and a strain concentration rule are used to calculate the notch root strain and to shift the origin of a modified HRR field, resulting in a non-singular model of the strain distribution ahead of the crack tip. In this way, the damage caused by each load cycle, including the effects of residual stresses, can be calculated at each element ahead of the crack tip using the correct hysteresis loops caused by the loading. The proposed approach is experimentally validated and extended to predict fatigue crack growth under variable amplitude loading, assuming that the width of the volume element broken at each cycle is equal to the region ahead of the crack tip that suffers damage beyond its critical value. The reasonable predictions of the measured fatigue crack growth behavior in steel specimens under service loads corroborate this simple and clear way to correlate da/dN and εN properties.     

Analysis of elastoplastic sharp notches

In this paper the elastoplastic solutions with higher-order terms for apex V-notches in power-law hardening materials have been discussed. Two-term expansions of the plane strain and the plane stress solutions have been obtained. It has been shown that the leading-order singularity approaches the value for a crack when the notch angle is not too large. In plane strain cases the elasticity does not enter the second-order solutions when the notch opening angle is too small. For a large notch angle, the two-term expansions of the plane strain near-tip fields are described by a single amplitude parameter. The plane stress solutions generally contain the elasticity terms. The boundary layer formulations based on the small-strain plasticity theory confirm that a dominance zone exists ahead of the notch tip. Finite element results give good agreement to the asymptotic solutions under both plane strain and plane stress conditions. The second-order terms cannot improve the predictions significantly. The near-tip fields are dominated by a single parameter. Finite element calculations under the finite strain J ₂-flow plasticity theory revealed that the finite strains can only affect local characterization of the asymptotic solution. The asymptotic solution has a large dominance zone around the notch tip. For an apex notch bounded to a rigid substrate the leading-order singularity falls with the notch angle significantly more slowly than in the homogeneous material. It vanishes at the notch angle about 135° for all power-hardening exponents. The elasticity effects enter the second-order solutions when the notch angle becomes large enough. The tip fields are characterized by the hydrostatic stress and the shear stress ahead of the notch.     

Fracture of anodic-bonded silicon-thin film glass-silicon triple stacks

In this study we focus on the fracture behavior of two types silicon-thin film glass-silicon (Si-Glass-Si) triple stacks specimens with a sharp corner. We determine the notch stress intensity factor Kⁿ for both specimens using a combination of the Williams eigenfunction expansion method, Stroh’s sextic formalism, finite element analysis, and the path-independent H-integral. Empirical solutions of dimensionless stress intensity factors are proposed for two typical specimens, and the dependence of geometry is analyzed. Furthermore, the effect of glass thickness on stress intensity is explored for anodic-bonded Si-Glass-Si triple stacks. We discuss the feasibility of using a critical value of Kⁿ to correlate the failure results for both specimens with various bond area and glass thickness.     

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.     

The growth of short fatigue cracks ahead of a notch in high strength steel

This paper is concerned with the short crack growth behaviour in notched specimens of high strength steels mainly used for automotive parts. Attention is focused on the appropriate estimation of the stress intensity range ΔK, when the short cracks grow in the stress field of a notch and attempts are made to investigate whether this is appropriate or whether further allowance may have to be made for short crack effects at low ΔK.The stress distribution for the notch in bending was taken into account for the estimation of ΔK. This enabled us to produce a more precise evaluation of short crack growth in the stress field of the notch. Our findings are that the short cracks, which propagate in the notch field, grow faster at low ΔK when their lengths are extremely short compared with the notch root radius. The short crack effect at low ΔK in the notch stress field is analysed by expressing the crack growth data in terms of the parameter, ΔK_s, in which stress gradient ahead of the notch is taken into account.     

Analysis of the effects of rigid and elastic boundary conditions in the elastic-plastic finite element analysis of rolling contact

The effects of boundary conditions and mesh refinement of finite element models of elasto-plastic 2-dimensional pure rolling contact are examined. The pure rolling of a cylinder is simulated by incrementally translating a semi-elliptical Hertzian pressure distribution over a finite element model of a semi-infinite half space. The half space is treated as an elastic-linear-kinematic-hardening-plastic (ELKP) material. The calculations evaluate the effects of rigid and elastic boundary conditions and two degrees of mesh refinement on the steady state residual stresses, the continuing cyclic plasticity, and the residual displacements. The results show that the boundary conditions influence the circumferential residual stresses and residual displacements, while they do not affect the axial residual stresses and the continuing cyclic plasticity. Careful choice of mesh refinement is shown to improve results and decrease computation time.List of symbols E elastic modulus - G shear modulus - _K kinematic tensile yield stress: - k _K kinematic shear yield strength - M plastic modulus - P normal contact load per unit length - p _o peak pressure - N number of cycles - w semi-contact width (macrocontact) - w semi-contact width (microcontact) - cyclic equivalent strain amplitude - _x circumferential stress - _z axial stress - _x ^R circumferential residual stress - _z ^R axial residual stress     

An h-p- multigrid method for finite element analysis

The finite element method generates solutions to partial differential equations by minimizing a strain energy based functional. Strain energy based techniques for adaptive mesh refinements are not always effective, however. The adaptive refinement technique proposed in this paper uses strain energy but also incorporates advantages from the h- and p- finite element methods, the multigrid method and a Delaunay based mesh generation method. The refinement technique converged rapidly and was numerically efficient when applied to determining stress concentrations around the circular hole of a thick plate under tension.     

A p-adaptive scaled boundary finite element method based on maximization of the error decrease rate

This study enhances the classical energy norm based adaptive procedure by introducing new refinement criteria, based on the projection-based interpolation technique and the steepest descent method, to drive mesh refinement for the scaled boundary finite element method. The technique is applied to p-adaptivity in this paper, but extension to h- and hp-adaptivity is straightforward. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh, is used to represent the unknown exact solution. In the new adaptive approach, a projection-based interpolation technique is developed for the 2D scaled boundary finite element method. New refinement criteria are proposed. The optimum mesh is assumed to be obtained by maximizing the decrease rate of the projection-based interpolation error appearing in the current solution. This refinement strategy can be interpreted as applying the minimisation steepest descent method. Numerical studies show the new approach out-performs the conventional approach.     

T‐stress corrected notch stress intensity factors with application to welded lap joints

The definition, content and application of the notch stress intensity factors (NSIFs) characterizing the stress field at rounded slit tips (keyholes) is discussed. The same is done in respect of the T‐stress transferred from the corresponding pointed slit tips. A T‐stress based correction of the NSIF K_1,ρ is found to be necessary. The applicability of the T‐stress term supplemented by higher‐order terms in Williams’ solution to the slit tip stresses in tensile‐shear loaded lap joints is discussed in more detail. The role of the T‐stress in constituting the near‐field stresses of rounded slit tips is shown to cause a difference between internal and external slit tip notches. The notch stress equations for lap joints proposed by Radaj based on structural stress and by Lazzarin based on a finite element model of the rounded notch are reconsidered and amended based on the derivations above.     

CORRELATION BETWEEN HYDROGEN INDUCED CRACKING INITIATION SITES AND CRITICAL STRESS INTENSITY FACTORS

Hydrogen induced cracking (HIC) initiation sites and their correlation with the critical stress intensity factors of hydrogen charged specimens were studied under combined I/II mode loading. Two series of tests, is. constant load (CL) tests and slow strain rate (SSR) tests, were carried out. Experimental results showed that in CL tests, irrespective of the ratio KIJKl, the HIC initiation sites always correspond to the point of maximum hydrostatic stress; which is located some distance ahead of the notch tip. However, for SSRT tests, when K₁₁/K₁> 1, HIC started at the notch tip which corresponds to the point of maximum equivalent plastic strain. When K₁₁/K₁<1 in SSR tests, HIC occurred initially some distance ahead of the notch tip. The relationship between the critical stress intensity factor for HIC and K₁₁/K₁ was shown to be different for the two types of test. Multiple effects of stress and strain on hydrogen redistribution and hence on HIC initiation sites, as well as critical stress intensity factors, are discussed.     

Significance of the elastic peak stress evaluated by FE analyses at the point of singularity of sharp V-notched components

The paper presents an expression useful to estimate the notch stress intensity factor (NSIF) from finite element analyses carried out by using a mesh pattern with a constant element size. The evaluation of the NSIF from a numerical analysis of the local stress field usually requires very refined meshes and then large computational effort. The usefulness of the presented expression is that (i) only the elastic peak stress numerically evaluated at the V‐notch tip is needed and no longer the whole stress–distance set of data; (ii) the adopted meshes are rather coarse if compared to those necessary for the evaluation of the whole local stress field. The proposed expression needs the evaluation of a virtual V‐notch tip radius, i.e. the radius which would produce the same elastic peak stress than that calculated by FEM at the sharp V‐notch tip by means of a given mesh pattern. Once such a radius has been theoretically determined for a given geometry, the expression can be applied in a wide range of notch depths and opening angles.     

A BENCHMARK COMPUTATIONAL STUDY OF FINITE ELEMENT ERROR ESTIMATION

Benchmark solutions are presented for a simple linear elastic boundary value problem, as analysed using a range of finite element mesh configurations. For each configuration, various estimates of local (i.e. element) and global discretization error have been computed. These show that the optimal mesh corresponds not only to minimization of global energy (or L₂) norms of the error, but also to equalization of element errors as well. Hence, this demonstrates why element error equalization proves successful as a criterion for guiding the process of mesh refinement in mesh adaptivity. The results also demonstrate the effectiveness of the stress projection method for smoothing discontinuous stress fields which, for this investigation, are more extreme as a consequence of the assumption of nearly incompressible material behaviour. In this case, lower order smoothing produces a continuous stress field which is in close agreement with the exact solution.     

The Mode I elastic stress distribution in the immediate vicinity of an intrusion-type notch

The representation of the elastic stress distribution in the immediate vicinity of a notch root is a key element in the development of a fracture mechanics methodology for blunt notches. Earlier work has shown that, for Mode III deformation, the stress at a distance x ≪ ρ (notch root radius of curvature) ahead of the root is unique in that it only depends on x, ρ and the peak stress σ_p irrespective of the notch shape and the loading characteristics. However with Mode I deformation, it has been shown that the local stress does not exhibit this uniqueness characteristic. This conclusion is underpinned by this paper’s quantification of the Mode I elastic stress distribution in the immediate vicinity of an intrusion-type notch.     

Mesh refinement and iterative solution methods for finite element computations

Global and element residuals are introduced to determine a posteriori, computable, error bounds for finite element computations on a given mesh. The element residuals provide a criterion for determining where a finite element mesh requires refinement. This indicator is implemented in an algorithm in a finite element research program. There it is utilized to automatically refine the mesh for sample two-point problems exhibiting boundary layer and interior layer solutions. Results for both linear and nonlinear problems are presented. An important aspect of this investigation concerns the use of adaptive refinement in conjunction with iterative methods for system solution. As the mesh is being enriched through the refinement process, the solution on a given mesh provides an accurate starting iterate for the next mesh, and so on. A wide range of iterative methods are examined in a feasibility study and strategies for interweaving refinement and iteration are compared.