Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high‐order enrichment functions

This paper presents a comprehensive study on the use of Irwin's crack closure integral for direct evaluation of mixed‐mode stress intensity factors (SIFs) in curved crack problems, within the extended finite element method. The approach employs high‐order enrichment functions derived from the standard Williams asymptotic solution, and SIFs are computed in closed form without any special post‐processing requirements. Linear triangular elements are used to discretize the domain, and the crack curvature within an element is represented explicitly. An improved quadrature scheme using high‐order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. Furthermore, because the Williams asymptotic solution is derived for straight cracks, an appropriate definition of the angle in the enrichment functions is presented and discussed. This contribution is an important extension of our previous work on straight cracks and illustrates the applicability of the SIF extraction method to curved cracks. The performance of the method is studied on several circular and parabolic arc crack benchmark examples. With two layers of elements enriched in the vicinity of the crack tip, striking accuracy, even on relatively coarse meshes, is obtained, and the method converges to the reference SIFs for the circular arc crack problem with mesh refinement. Furthermore, while the popular interaction integral (a variant of the J‐integral method) requires special auxiliary fields for curved cracks and also needs cracks to be sufficiently apart from each other in multicracks systems, the proposed approach shows none of those limitations. Copyright © 2017 John Wiley & Sons, Ltd.     

Disc analogy for approaching circular arc cracks

The limiting situation of a pair of approaching circular arc crack tips in a homogeneous medium is examined to draw analogies with circular disc problems. As the crack tips approach each other, the narrow material ligament bridging the crack tips controls the stresses, stress intensity factors and energy release rates. In particular, this ligament sets up the length scale for singularity analysis for a given radius of the arc crack. Following a detailed analytical examination of approaching crack tips, a photoelastic visualisation of the stress field is presented. Experimental isochromatics are also compared with theoretical predictions for some specific cases. Finally, based on the ideas developed in this paper, existing notions on interacting cracks and cavities treated in the literature are reinterpreted.     

Two cracks with coalesced interior plastic zones — The generalised Dugdale model approach

The problem investigated is of an elastic-perfectly plastic infinite plate containing two equal collinear and symmetrically situated straight cracks. The plate is subjected to loads at infinity inducing mode I type deformations at the rims of the cracks. Consequently, plastic zones are formed ahead of the tips of the cracks. The loads at infinity are increased to a limit such that the plastic zones formed at the neighbouring interior tips of the cracks get coalesced. The plastic zones developed at the tips of the cracks are closed by applying normal cohesive quadratically varying stress distribution over their rims. The opening of the cracks is consequently arrested. Complex variable technique is used in conjugation with Dugdale’s hypothesis to obtain analytical solutions. Closed form analytical expressions are derived for calculating plastic zone size and crack opening displacement. An illustrative numerical example is discussed to study the qualitative behaviour of the loads required to arrest the cracks from opening with respect to parameters viz. crack length, plastic zone length and inter-crack distance. Crack opening displacement at the tip of the crack is also studied against these parameters.     

A weight function for 2D subsurface cracks under general loading conditions

A procedure for evaluating the fracture mechanics parameters of a subsurface two-dimensional crack parallel to the boundary in an elastic half plane is presented. A Weight Function (WF) with a matrix structure is proposed, to account for the coupling effects between modes I and II, typical in non-symmetrical problems. In order to face any loading condition, the WF was formulated by symmetric and anti-symmetric components and the ‘multiple reference loading’ approach was used to derive their analytical expressions. To this purpose, a parametric Finite Element (FE) analysis was set up and the Stress Intensity Factors (SIFs) were determined for several independent loading conditions. The analysis was carried out for different ratios between crack length and in-depth position and, consequently, the dependence of the WF on this parameter was studied. The WF accuracy was assessed by considering different loading and the method applied for evaluating the SIFs produced by a point-like load travelling on the semi-plane surface. The results indicated that the correct fracture mechanics analysis requires crack closure (either complete or partial) to be taken into account. Consequently, the crack opening displacement (COD) components under general loading conditions have to be evaluated. On the basis of the WF, the related Green Function (GF) was also derived by which the COD components can be efficiently evaluated for any applied load including the contact due to crack closure.     

A set of interface cracks with contact zones in a combined tension-shear field

Summary. A set of cracks lying along the interface of two dissimilar isotropic materials under a mixed-mode loading is considered. The interface cracks are assumed to be fully open, partially closed with frictionless contact zones and fully closed. The problem is reduced to a homogeneous combined Dirichlet-Riemann boundary value problem, which is solved in closed form. A set of transcendental equations for the determination of the contact zone lengths for an arbitrary number of cracks and the closed-form expressions for the stresses and the displacement jumps on the material interface are obtained. A single crack with one and two contact zones has been considered in details. An explicit set of two transcendental equations for the relative contact zone length and closed-form expressions for the stress intensity factors at the crack tips are obtained for both cases. The contact zone lengths and the stress intensity factors are investigated numerically for different material pairs under different values of the loading, and a comparison of the results for a crack with one and two contact zones is carried out.     

Cracked Elastic Bi-Material Layer Under Compressive Loads

The boundary value problem of an elastic bi-material layer containing a finite length crack under compressive mechanical loadings has been studied. The crack is located on the bi-material interface and the contact between crack surfaces is frictionless. Based on Fourier integral transformation techniques the solution of the formulated problem is reduced to the solution of singular integral equation, then, with Chebyshev`s orthogonal polynomials, to infinite system of linear algebraic equations. The expressions for contact stresses in the elastic compound layer are presented. Based on the analytical solution it is found that in the case of frictionless contact the shear and normal stresses have inverse square root singularities at the crack tips. Numerical solutions have been obtained for a series of examples. The results of these examples are illustrated graphically, exposing some novel qualitative and quantitative knowledge about the stress field in the cracked layer and their dependence on geometric and applied loading parameters. It can be seen from this study that the crack tip stress field has a mixture of mode I and mode II type singularities. The numerical solutions show that an interfacial crack under compressive forces can become open in certain parts of the contacting crack surfaces, depending on the applied forces, material properties and geometry of the layers.     

Mode-III stress intensity factors for two circular inclusions subject to a remote uniform shear load

ABSTRACT

An analytical solution to the antiplane elasticity problem associated with two circular inclusions interacting with a line crack is provided in this article. A series solution for the stress field is derived in an elegant form by using complex variable theory in conjunction with the alternation method. Based on the superposition method, a singular integral equation (SIE) is established from the traction-free condition along the crack surface. After solving the SIE, the mode-III stress intensity factors (SIFs) can be obtained to quantify the singular behavior of the stress field ahead of the crack tips. Numerical results of the SIFs, when a crack is embedded either in the inclusion or in the matrix, are discussed in detail and displayed in graphic form.     

Approximate weight function technique using single-layer potential for a cracked circular disc

An efficient weight function technique using the indirect boundary integral method was presented for cracked circular discs. The crack opening displacement field was presented by a single layer whose kernel was a modified form of the fundamental solution in elastostatics. The application of a single-layer potential to the weight function method leads to a unique closed-form SIF (stress intensity factor) solution. The solution can be applied to a cracked circular discs with or without an internal hole or opening. For these crack geometries over a wide range of crack ratios, the SIF solution can be applied without any modification.

The calculation procedure of SIFs for the various cracked circular discs using only one analytical solution is very simple and straightforward. The information necessary in the analysis includes only two or three reference load cases. In most cases the SIF solution using two reference SIFs gives reasonably accurate results while the SIF solution with three reference load cases may be used to improve the solution accuracy of the crack configurations, with an internal opening or hole, compared with the solutions of the available literature.     

Elastic compliance of a partially debonded circular inhomogeneity

In the present work, we predict contribution of a partially debonded circular inhomogeneity into the material overall elastic compliance. Debonding at the matrix/inclusion boundary is modeled as interfacial arc cracks. The change in the elastic compliance caused by interface cracking is estimated through the accompanying energy change that is related to the mode I and mode II stress intensity factors at the crack tips. The sum of the crack compliance and the inhomogeneity compliance (with perfect bonding) gives the total compliance of the debonded inhomogeneity. The latter is obtained in terms of the material properties and crack length. Additional analysis shows that the replacement of an interface crack with a crack in a homogenized medium is an inadequate approach when seeking approximate solutions. The paper also provides guidelines how to determine properties of a fictitious perfectly bonded orthotropic inhomogeneity that has the same contribution into the material compliance as the debonded isotropic one. This problem is of practical importance when modeling damage accumulation in composite materials by means of homogenization schemes.     

A fracture mechanics problem of a functionally graded layered structure with an arbitrarily oriented crack crossing the interface

In this paper, the fracture mechanics problem for an arbitrarily oriented crack crossing the interface in a functionally graded layered structure is investigated. The elastic modulus is assumed to be continuous at the interface, but its derivative may be discontinuous. Applying the superposition principle and Fourier integral transform, the stress fields and displacement fields are derived. A group of auxiliary functions defined in both layers are introduced and then the mixed-mode crack problem is turned into solving a group of singular integral equations. The mixed-mode stress intensity factors (SIFs) are obtained by solving the singular integral equations. The influences of the material nonhomogeneity parameter, normalized crack length and crack angle on the SIFs are investigated. It is found that the mixed-mode SIFs can be affected greatly by the crack angle. Moreover, the mixed-mode SIFs usually attain their extremum when the crack tips get to the interface during one crack moves from one layer into another layer. The present work may form the basic work for establishing a multi-layered fracture mechanics model of FGMs with an arbitrarily oriented crack and general mechanical properties.     

A solution for SIFs of part-elliptical cracks based on approximate crack surface displacement and energy method

A procedure has been developed to derive stress intensity factors (SIFs) for part-elliptical cracks based on an approximate crack surface displacement mode assumption for general configurations. The crack surface displacement mode is composed of available 2D crack surface displacement modes at intersections of the crack surface and boundaries, or in symmetry planes. Along with the obtained crack surface displacement mode, SIFs are determined by the magnitude of the crack surface displacement derived from energy release rate for virtual crack increments. The procedure was analytically verified with the exact solution for an embedded crack in an infinite body subjected to uniform crack surface pressure. Several examples show the obtained results in acceptable agreements with available solutions.     

Paradoxes in the deformation modes of crack flanks due to shear

Whereas the singular solution for an internal crack in an infinite plate induced to a biaxial loading at infinity defines that the crack tips remain unmoved during deformation, the twoterm approximate solution implies some movement of the tips. The exact solution based on Muskhelishvili's complex potentials gives a thorough and exact view of the form of the deformed crack. The paradoxes of the aspect of the exact solution are several and may be classified as follows, (i) While overall tension opens the crack lips and overall compression makes the lips overlap congruently, so that the crack resembles a completely closed line without any stress concentration at its tips, the contribution of shear is always to make the crack flanks overlap in a non-congruent manner, thus developing strongly variable friction between the closed lips, (ii) The deformed crack length under shear loading is always different than its initial length depending on the angle of obliquity and the loading step, (iii) The deformed crack axis is different from the initial axis, (iv) The shape of the deformed crack is always elliptic with the upper lip penetrating inside the lower lip and vice versa, (v) There is always a clockwise or anticlockwise displacement of the lips depending on the sign of shear in a carousel mode, (vi) In this way new points behind the crack tips are replacing them as points of the vertices of the elliptic crack during continued loading, (vii) The curvature of the vertices is decreasing as the load is increased, (viii) The elliptic shape of the crack for the exact solution becomes a double parabolic shape with corners behind the vertices of the parabolas for the singular and the two-term solution. These unrealistic shapes of the deformed crack overestimate its relative lip-displacements, as compared with the exact solution.     

集中载荷作用下的不同压电材料反平面应变状态的共圆弧界面裂纹问题

本文应用复变函数解析延展原理，得到了集中载荷作用下的不同压电材料反平面应变状态的共圆弧界面裂纹问题的一般解；对单个圆弧界面裂纹，给出了复函数封闭解和场强度因子。     

On the representation of solutions for the theory of generalized thermoelasticity with three phase-lags

An analytical investigation on the plastic zone size (PZS) of a crack near a circular inclusion has been carried out. Both the crack and the circular inclusion are embedded in an infinite matrix, with the crack oriented along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, two stripe plastic zones at both crack tips are introduced. Using the solution of a circular inclusion interacting with a single dislocation as the Green’s function, the physical problem is formulated as a set of singular integral equations. With the aid of Erdogan and Gupta’s method and iterative numerical procedures, the singular integral equations are solved numerically for the PZS and the crack tip opening displacement. The results obtained in the current work can be reduced to those simpler cases of the Dugdale model.     

T-stress in the Zener-Stroh arc crack problem in plane elasticity

A Zener-Stroh curved crack is defined such that the crack undergoes an initial displacement discontinuity. A singular integral equation is suggested to solve the Zener-Stroh curved crack problem. General formulation for evaluating the stress intensity factors and the T-stresses at the crack tips of a Zener-Stroh curved crack is carried out. For the Zener-Stroh arc crack, T-stresses at the crack tips can be evaluated in a closed form.     

FATIGUE CRACK GROWTH BEHAVIOUR OF SM45C STEEL UNDER CYCLIC MODE I WITH SUPERIMPOSED STATIC MODE II LOADINGS

Abstract— The practical applications of studies related to constant amplitude mode I loading are somewhat limited, since mode I crack growth is often influenced by mode II (sliding mode) or mode III (tearing mode) in industrial situations. For these cases, criteria, rules, and laws have to be worked out and verified by experiments. However, it is very difficult to evaluate mixed-mode fatigue cracking due to crack surface interference, crack closure, crack branching, etc. This paper, which defines the length of a branched crack as an effective slant crack with a length equal to the distance between the two crack tips, explains the influences of crack surface interference by introducing concepts of adhesive wear and scrutinizes some related researches on mixed-mode crack growth behaviour. Additionally an effective stress intensity factor range is described which considers crack closure and crack surface interference and is verified with crack growth tests under mode I fatigue loading and cyclic mode I with a superimposed static mode II loading.     

Two equal symmetric circular arc cracks in an elastic medium – the Dugdale approach

The Dugdale model for two equal, symmetrically situated coplanar circular arc cracks contained in an infinite elastic perfectly-plastic plate is proposed. Biaxial loads are applied at the infinite boundary of the plate. Consequently, the rims of the cracks open in Mode I and develop a plastic zone ahead of each of the cracks. These plastic zones are then closed by the distribution of uniform normal closing stresses over the rims of the plastic zones. Based on the complex-variable technique and the superposition principle, the solution for the above problem is obtained. Closed-form analytic expressions are obtained for the determination of the sizes of the plastic zones and the crack-opening displacement (COD) at the tip of the crack. Numerical studies are carried out to calculate the load ratio (load applied at infinity/yield point stress applied at the rims of the plastic zones) required for the closure of the plastic zones, for various radii of arc cracks and for various angles subtended by them at the centre. The crack-opening displacement is also investigated with respect to these parameters.     

求解双材料界面裂纹应力强度因子的扩展有限元法

基于双材料界面裂纹尖端的基本解,构造扩展有限元法(eXtended Finite Element Methods, XFEM)裂尖单元结点的改进函数。有限元网格剖分不遵从材料界面,考虑3种类型的结点改进函数:弱不连续改进函数、Heaviside改进函数和裂尖改进函数,建立XFEM的位移模式,给出计算双材料界面裂纹应力强度因子(Stress Intensity Factors, SIFs)的相互作用积分方法。数值结果表明:XFEM无需遵从材料界面剖分网格,该文的方法能够准确评价双材料界面裂纹尖端的SIFs。     

Zener–Stroh crack at the interface of multi-layered structures

A Zener–Stroh (Z–S) crack can be nucleated on the interface of a multi-layered structure when a dislocation pileup is stopped by the interface which works as an obstacle. During the entire fracture of a crack, Z–S crack mechanism controls the initial stage, or the first phase of crack initiation and propagation. In our current research, investigation on a Z–S crack at the interface of a multi-layered structure is carried out. The problem is formulated into a set of singular integral equations by applying the distributed dislocation based fracture mechanics. The obtained integral equations are then solved with numerical method after the singularities at crack tips are carefully checked. In the solution procedure, the contact zone model is adopted to cease the oscillation behavior. The contact zone length, the stress field near the crack tips and the stress intensity factors (SIFs) of the crack are discussed based on the numerical results of two typical structures. It was found that the contact zone length could be very large and was determined by the properties of all the three materials and loading conditions. Our analysis also shows that the thickness of the middle thin layer plays a critical role for the fracture behavior of the crack when it is comparable to the crack length.     

On the plastic zone size and CTOD study for a Zener�CStroh crack interacting with a circular inclusion

An analytical solution is given for plastic yielding of a Zener?CStroh crack near a circular inclusion embedded in an infinite matrix. The crack is orientated along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. Using the Dugdale model of small-scale yielding, plastic zones are introduced at both crack tips. Using the solution of a circular inclusion, interacting with a single dislocation as the Green??s function, the physical problem is formulated into a set of singular integral equations. With the aid of Erdogan??s method and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement. The results obtained in the current work are verified by reduction to simpler cases of the Dugdale model. Various parameters such as the distance, shear modulus ratio, Poisson??s ratio, and loading condition are studied.