On the assessment of cracks within the interface of thin layers on a substrate

Although bimaterial wedge or notch configurations are identified as potential weak locations, the assessment of the degree of criticality of cracks in such regions is still a demanding problem. The singular character of the stress field at cracks or at bimaterial notches can be calculated analytically or numerically. The angle of the direction of potential crack initiation may also be determined, but the decisive question is whether a hypothetical crack will be initiated or not. An essential question in the context of crack assessment is to find a criterion for crack nucleation. For that aim, the hypothesis of Leguillon is modified. Herein, the crack is assumed to be critical when and only when both the released energy and the local stress reach critical values along a hypothetical crack of finite length. This concept can be transferred to a bimaterial interface configuration of a thin layer on a substrate. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling crack initiation by means of finite fracture mechanics

In this study, a generally applicable hybrid criterion for brittle crack initiation overcoming deficiencies of classical approaches is applied to regular and singular stress concentrations. Possible formulations of the resulting optimisation problem for the failure load are discussed. Examples demonstrate the capability of the model to cover relevant effects as well as a versatile finite element analysis procedure for notch problems. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of crack–propagation using embedded discontinuities

The transition from microscale damage phenomena to crack initiation and growth at the macroscale is an important mechanism which constrains the lifetime of concrete structures. Analysing crack growth using the finite element method without enhancement of the shape functions is possible only by continuously updating the corresponding meshes, which constitutes a significant computational effort. But even then the results can be substantially mesh–dependent and hard to interpret. The extended Finite Element Method (XFEM) uses additional discontinuous shape–functions and is one possibility to overcome these problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A closed‐form analysis of material and geometry effects on stress singularities at unsymmetric bimaterial notches

An important issue in the mechanics of adhesive bonds is the knowledge of local mechanical fields. In the present study, an asymptotic analysis of the stress fields near an unsymmetric bimaterial notch with arbitrary opening angle is performed. Using the complex potential method, the order of the singularity of the stress fields at a notch tip can be determined in closed‐form analytical manner, so that the dependency of the occurring singularity exponents on geometry and material properties can be studied systematically.     

Elastic stress analysis of blunt V-notches under mixed mode loading by considering higher order terms

This paper presents the in-plane asymptotic displacement and stress fields for blunt V-notched components based on Kolosov–Muskhelishvili's approach. In the first part, the displacement and stress components in the polar coordinate system are determined by choosing appropriate complex potential functions. In order to construct the notch geometry, the Neuber's mapping relation is utilized. Then, the notch boundary conditions are imposed to calculate the free parameters of the stress distribution. Eventually, the stress and displacement components are calculated in the Cartesian and polar coordinates in the forms of series expansion. In the second part, the coefficients of series expansions are computed by using the least square method (LSM). The blunt V-notched Brazilian disk (BV-BD) specimen under mixed mode loading is used as an example to verify the proposed procedure. The stress components in arbitrary distances and directions are determined for different blunt V-notches in order to evaluate the accuracy of the calculated stress series solutions and their associated coefficients. The numerical results indicate that a single-term solution can lead to considerable errors, and to achieve good accuracy in the stress field calculation, one should take account of at least three terms in the stress series solution.     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape sensitivity of a plane crack front

The 3D‐elasticity model of a solid with a plane crack under the stress‐free boundary conditions at the crack is considered. We investigate variations of a solution and of energy functionals with respect to perturbations of the crack front in the plane. The corresponding expansions at least up to the second‐order terms are obtained. The strong derivatives of the solution are constructed as an iterative solution of the same elasticity problem with specified right‐hand sides. Using the expansion of the potential and surface energy, we consider an approximate quadratic form for local shape optimization of the crack front defined by the Griffith criterion. To specify its properties, a procedure of discrete optimization is proposed, which reduces to a matrix variational inequality. At least for a small load we prove its solvability and find a quasi‐static model of the crack growth depending on the loading parameter. Copyright © 2003 John Wiley & Sons, Ltd.     

Adaptive extended isogeometric analysis based on PHT-splines for thin cracked plates and shells with Kirchhoff–Love theory

In this paper, a posteriori error estimation and mesh adaptation approach for thin plate and shell structures of through-the-thickness crack is presented. This method uses the extended isogeometric analysis (XIGA) based on PHT-splines (Polynomial splines over Hierarchical T-meshes), which is abbreviated as XIGA-PHT. In XIGA-PHT, the isogeometric displacement approximation is locally enriched with enrichment functions, which efficiently capture the displacement discontinuity across the crack face as well as the stress singularity in the vicinity of the crack tip. On the one hand, the rotational degrees of freedom (RDOFs) are not required in Kirchhoff–Love theory, which drastically reduces the complexity of enrichment mode and computational scale for crack analysis. On the other hand, the PHT-splines basis functions can automatically satisfy the requirement of C¹-continuity for the Kirchhoff–Love theory. Moreover, the PHT-splines facilitate the local refinement, which is the deficiency of NURBS-based isogeometric formulations. The local refinement is highly suitable for adaptive analysis. The stress recovery-based posteriori error estimator combined with the superconvergent patch recovery (SPR) technique is used to evaluate the approximate local discretization error. A new strategy for selecting enriched recovered functions in the enriched areas was proposed. Special functions extracted from the asymptotic stress solutions are applied to obtain the recovered stress field in the enriched area. The results of stress intensity factors or J-integral values obtained by the adaptive XIGA-PHT are compared with reference solutions. Several thin plate and shell illustrative examples demonstrate the effectiveness and accuracy of the proposed adaptive XIGA-PHT.     

蠕变损伤材料中缺口尖端稳定扩展的应力场

研究了在拉伸载荷和反平面载荷作用下蠕变损伤材料缺口尖端稳定扩展的应力场．假设材料的应力及位移势函数,得到了缺口尖端场的各参数分量,进而在小范围蠕变条件下,建立了缺口尖端稳定扩展的蠕变损伤控制方程,并考虑缺口尖端蠕变钝化作用和问题的边界条件,对控制方程进行了数值分析,得到了缺口尖端的应力场,并讨论了缺口尖端应力场随各影响参数的变化规律．结果表明,缺口尖端的应力具有r1/(1-n)的奇异性,应力率具有rn/(1-n)的奇异性,n是蠕变指数．     

Fast algorithms for designing variable FIR notch filters

The paper presents fast algorithms for designing finite impulse response (FIR) notch filters. The aim is to design a digital FIR notch filter so that the magnitude of the filter has a deep notch at a specified frequency, and as the notch frequency changes, the filter coefficients should be able to track the notch fast in real time. The filter design problem is first converted into a convex optimization problem in the autocorrelation domain. The frequency response of the autocorrelation of the filter impulse response is compared with the desired filter response and the integral square error is minimized with respect to the unknown autocorrelation coefficients. Spectral factorization is used to calculate the coefficients of the filter. In the optimization process, the computational advantage is obtained by exploiting the structure of the Hessian matrix which consists of a Toeplitz plus a Hankel matrix. Two methods have been used for solving the Toeplitz‐plus‐Hankel system of equations. In the first method, the computational time is reduced by using Block–Levinson's recursion for solving the Toeplitz‐plus‐Hankel system of matrices. In the second method, the conjugate gradient method with different preconditioners is used to solve the system. Comparative studies demonstrate the computational advantages of the latter. Both these algorithms have been used to obtain the autocorrelation coefficients of notch filters with different orders. The original filter coefficients are found by spectral factorization and each of these filters have been tested for filtering synthetic as well as real‐life signals. Copyright © 2007 John Wiley & Sons, Ltd.     

An experimental-analytical hybrid model for the elasto-plastic stresses at a crack

The primary obstacle preventing the analytical determination of physically sensible stresses at a crack tip is the presence of a mathematical singularity there. This singularity is best known in its elastic form; however it persists even in elasto-plastic crack-tip stresses. To overcome the difficulty we adopt the following strategy: we attempt to capture initial elastic stresses experimentally, than track subsequent elasto-plastic stress distributions analytically.We infer a finite stress at a crack tip from the experimental behaviour of cracked specimens at fracture when the specimens are made of a truly brittle material. Given a size-independent result, we argue that the crack-tip stress at fracture must equal the ultimate stress for such a material; thus dividing by the applied stress at the same point gives a measure of the stress concentration factor, K_T. The approach is checked for size independence and against hole configurations with known theoretical, yet physically reasonable, K_T. Then the effective experimental KT are taken as inputs for the second phase of the study in which we model the crack as being a smooth notch having the same stress concentration factor as found experimentally. In this way our configuration initially shares the same stresses at the crack tip as we inferred physically. Next we track effects of incremental plastic flow on a set of finite element grids. Satisfactory resolution in return for modest computational effort is obtained by employing a substructuring method. The accuracy in both the elastic and the elasto-plastic regime is checked against trial problems with exact solutions. Thereafter, physically interpretable stress distributions ahead of the crack are determined for a range of materials and for varying load levels.     

Mixed antiplane boundary‐value problem for a piecewise‐homogeneous elastic body with a semi‐infinite interfacial crack

Antiplane stress state of a piecewise‐homogeneous elastic body with a semi‐infinite crack along the interface is considered. The longitudinal displacements along one of the crack edges on a finite interval, adjacent to the crack tip, are known. Shear stresses are applied to the body along the crack edges and at infinity. The problem reduces to a Riemann–Hilbert boundary‐value matrix problem with a piecewise‐constant coefficient for a complex potential in the class of symmetric functions. The complex potential is found explicitly using a Gaussian hypergeometric function. The stress state of the body close to the singular points is investigated. The stress intensity factors are determined. Copyright © 2016 John Wiley & Sons, Ltd.     

Conditions for Cracking and Debond Development at Partially Bonded Bimaterial Half-Planes

Fracture phenomena at the debond tip of partially bonded bimaterial half-planes subjected to concentrated normal forces, couples, and uniform tension are considered. The crack initiation conditions are described by the stress distribution before the initiation and the energy release rate of the crack immediately after the initiation. The debond development conditions are described by the stress distribution and the energy release rate of the debond before its initiation. When both the crack and the debond have chances to occur, or when cracks can arise in both the materials, the fracture phenomena are predicted by comparing the ratio of energy release rates and the ratio of fracture toughnesses.     

带k条径向边裂纹的圆形孔口问题的应力分析

利用复变函数方法,通过构造保角映射研究了带k条径向边裂纹的圆形孔口的平面弹性问题,给出了裂纹尖端Ⅰ型与Ⅱ型问题应力强度因子的精确分析解.在极限情况下,不仅可以还原出星形裂纹,Griffith裂纹,十字裂纹等经典的裂纹问题的结果,而且当k取任意正整数值时,可以模拟出更多的、复杂的带裂纹的圆形孔口问题.     

A coupled stress and energy failure model for crack initiation in arbitrarily shaped adhesive lap joints

In this work, crack initiation in adhesive lap joints of arbitrary joint configuration is studied by means of a finite fracture mechanics approach. The analysis is based on a general stress solution for adhesive joints combined with a coupled stress and energy criterion. The instantaneous formation of a crack of finite size is predicted if a stress and energy criterion are satisfied simultaneously. The closed-form analytical solution of the stress field allows for an efficient evaluation of the crack initiation load and corresponding finite crack length. A comparison to experimental results from literature and to numerical results obtained with a cohesive zone model approach shows a good agreement. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inclusion Size Distribution and Endurance Limit of a Hard Steel

High cycle fatigue failure of hard steels is dominated by a long crack initiation phase and a short crack propagation phase. The crack initiation sites are inclusions or surface defects. Due to this, the endurace probability of a hard steel part depends on its crack initiating inclusion size or surface defect size distribution. These consideration lead to the weakest-link concept which allows to calculate local and total endurance probabilities of the cycled parts. From these probabilities, the endurance limit and the probable crack initiation sites can be predicted. Base of the prediction is the knowledge of the endurance limits under three different load conditions. From these data and fracture-mechanic relations for the different inclusion types, the distributions of the crack initiating inclusion sizes can be predicted.     

Thermal Analysis of Free‐Corner Effects in Cross‐Ply Plates by a Higher‐Order Theory Approach

Stresses in the vicinity of free edges and corners of composite laminates exhibit a distinct localized three‐dimensional behaviour [1] and thus represent an important technical situation. Since numerical analyses of stress concentration phenomena in layered structures are computationally expensive, the present contribution is devoted to a simple closed‐form higher‐order theory approach for the calculation of displacements, strains and stresses in the vicinity of a rectangular corner of symmetric cross‐ply laminates under uniform thermal load ΔT. An appropriate representation for the displacement field in the form of a single‐layer theory with unknown inplane components and appropriately assumed functions through the plate thickness yields closed‐form expressions for the strains and stresses throughout the whole laminate. Equilibrium and boundary conditions are fulfilled in an integral sense. The present approach is easily applied, is of a completely closed‐form analytic nature and requires only little computational effort. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Static and extending interface cracks with contact zones in anisotropic as well as in piezoelectric bimaterials

An interface crack with an electrically permeable and mechanically frictionless contact zone in a piezoelectric bimaterial under the action of a remote mixed mode mechanical loading as well as thermal and electrical fields is considered in the first part of this paper. By use of the matrix‐vector representations of thermal, mechanical and electrical fields via sectionally‐holomorphic functions the problems of linear relationships are formulated and solved exactly both for an electrically permeable and an electrically impermeable interface crack. For these cases the transcendental equations and clear analytical formulas are derived for the determination of the contact zone lengths and the associated fracture mechanical parameters. A plane strain problem for a crack with a frictionless contact zone at the leading crack tip extending stationary along an interface of two semi‐infinite anisotropic spaces with a subsonic speed under the action of various loading is considered in the second part of this paper. By introducing of a moving coordinate system connected with the crack tip and by using the formal similarity of static and propagating crack problems the combined Dirichlet‐Riemann boundary value problem is formulated and solved exactly for this case as well and a transcendental equation is obtained for the determination of the real contact zone length. It is found that the increase of the crack speed leads to an increase of the real contact zone length and the correspondent stress intensity factors which increase significantly for a quasi‐Rayleigh wave speed.     

Investigation of the local behavior of different cohesive zone models

Cohesive interface elements are well suited for three-dimensional crack propagation analyses as long as the crack path is known. This is the case e.g. in delamination analyses of laminated composite structures or failure analyses of adhesively bonded joints. Actually, they are widely used in such applications for both brittle and ductile systems. As long as the strength and fracture toughness of the material are accurately captured it is generally accepted that the shape of the cohesive law has little to no influence on the mechanical behavior of the investigated structures. However, when having a look on the local behavior of different cohesive zone models, such as stress distribution in the fracture process zone, the results exhibit certain differences. These will be studied in the present contribution. Especially the local stress distribution will be investigated and the effect on the computational efficiency will be pointed out. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Crack initiation in elliptically notched plates

In this work, crack initiation in elliptically notched plates under uniaxial tension is studied by means of a Finite Fracture Mechanics analysis. The stress concentration factor changes with the aspect ratio of the half axes: from unity for slender ellipses in loading direction to infinity for crack-like ellipses normal to the loading. The closed-form solutions of the stress field and an approximate formulae of the stress intensity factor of mode I cracks allows for an efficient Finite Fracture Mechanics analysis of plates with elliptical notches with arbitrary aspect ratios. The results show that the transition from vanishing stress concentration to infinite (crack-like) stress concentration is rendered continuously by the present approach. A transition from strength of materials to Linear Elastic Fracture Mechanics is obtained that shows distinct size effects. A comparison to experimental results on size effects of notched plates shows good agreement. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)