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.     

Effect of curvature at the crack tip on the stress intensity factor for curved cracks

In this paper, the numerical solution of the hypersingular integral equation using the body force method in curved crack problems is presented. In the body force method, the stress fields induced by two kinds of standard set of force doublets are used as fundamental solutions. Then, the problem is formulated as a system of integral equations with the singularity of the form r ^–2. In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density functions and power series. The calculation shows that the present method gives rapidly converging numerical results for curved cracks under various geometrical conditions. In addition, a method of evaluation of the stress intensity factors for arbitrary shaped curved cracks is proposed using the approximate replacement to a simple straight crack.     

Anti-plane transient analysis of planes with multiple cracks

In this paper, the transient dynamic stress intensity factor is determined for multiple curved cracks under impact loading. The dislocation method has rarely been applied to the problems involving dynamic loading. The transient response of Volterra-type dislocation in a plane is obtained by means of the Cagniard-de Hoop method. The distributed dislocation technique is used to construct integral equations for an infinite isotropic plane weakened by cracks. These equations are of Cauchy singular type at the location of dislocation which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determine stress intensity factors for multiple smooth cracks. Numerical results are obtained to validate the formulation and illustrate its capabilities.     

An accurate scheme for mixed‐mode fracture analysis of functionally graded materials using the interaction integral and micromechanics models

The interaction integral is a conservation integral that relies on two admissible mechanical states for evaluating mixed‐mode stress intensity factors (SIFs). The present paper extends this integral to functionally graded materials in which the material properties are determined by means of either continuum functions (e.g. exponentially graded materials) or micromechanics models (e.g. self‐consistent, Mori–Tanaka, or three‐phase model). In the latter case, there is no closed‐form expression for the material‐property variation, and thus several quantities, such as the explicit derivative of the strain energy density, need to be evaluated numerically (this leads to several implications in the numerical implementation). The SIFs are determined using conservation integrals involving known auxiliary solutions. The choice of such auxiliary fields and their implications on the solution procedure are discussed in detail. The computational implementation is done using the finite element method and thus the interaction energy contour integral is converted to an equivalent domain integral over a finite region surrounding the crack tip. Several examples are given which show that the proposed method is convenient, accurate, and computationally efficient. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary elements for cracks and notches in three dimensions

Plane and curved cracks are modelled by boundary elements, of geometry defined by conforming quadratic and hybrid quadratic‐Hermitian cubic shape functions. Displacement and traction are interpolated by the quadratic functions, supplemented by singular functions by which are multiplied stress intensity factors corresponding to each of the three modes of crack opening displacement, for the first three eigenvalues of the Williams eigenfunction expansion and its equivalent for antiplane strain. Singular and hypersingular boundary integral equations are taken at nodes of elements and auxiliary collocation points. Singular and hypersingular components of integrals are evaluated by consideration of trial displacement fields (simple solutions) for subdomains lying to either side of the crack. Examples are shown of buried and surface cracks, and computed results compared with those obtained by other methods. For surface cracks, the computed results reveal the cause of significant discrepancies between values given by well established empirical and other formulae. The modelling of notches is demonstrated by an analysis of stress in rock near a tunnel intersection. Computational efficiency is discussed, and improvements and extensions of the analysis are proposed. Copyright © 2005 John Wiley & Sons, Ltd.     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

Partition of unity enrichment for bimaterial interface cracks

Partition of unity enrichment techniques are developed for bimaterial interface cracks. A discontinuous function and the two‐dimensional near‐tip asymptotic displacement functions are added to the finite element approximation using the framework of partition of unity. This enables the domain to be modelled by finite elements without explicitly meshing the crack surfaces. The crack‐tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The concept of partition of unity facilitates the incorporation of the oscillatory nature of the singularity within a conforming finite element approximation. The mixed‐mode (complex) stress intensity factors for bimaterial interfacial cracks are numerically evaluated using the domain form of the interaction integral. Good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized. Copyright © 2004 John Wiley & Sons, Ltd.     

Weight functions for the determination of stress intensity factor and T‐stress for semi‐elliptical cracks in finite thickness plate

This paper presents the application of weight function method for the calculation of stress intensity factors (K) and T‐stress for surface semi‐elliptical crack in finite thickness plates subjected to arbitrary two‐dimensional stress fields. New general mathematical forms of point load weight functions for K and T have been formulated by taking advantage of the knowledge of a few specific weight functions for two‐dimensional planar cracks available in the literature and certain properties of weight function in general. The existence of the generalised forms of the weight functions simplifies the determination of specific weight functions for specific crack configurations. The determination of a specific weight function is reduced to the determination of the parameters of the generalised weight function expression. These unknown parameters can be determined from reference stress intensity factor and T‐stress solutions. This method is used to derive the weight functions for both K and T for semi‐elliptical surface cracks in finite thickness plates, covering a wide range of crack aspect ratio (a/c) and relative depth (a/t) at any point along the crack front. The derived weight functions are then validated against stress intensity factor and T‐stress solutions for several linear and nonlinear two‐dimensional stress distributions. These derived weight functions are particularly useful for the development of two‐parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

Non‐oscillatory and non‐singular asymptotic solutions to stress fields at interface cracks

This work concerns the complex oscillatory singularities revealed in Williams's asymptotic solutions to stress fields around arbitrary interface cracks, which are the foundation of phenomenological interface fracture mechanics. First, we highlight the fatal discrepancy between the asymptotic stress fields for cracks in a homogeneous material obtained by assigning an identical material on both regions embracing an interface crack, and the solutions directly derived from cracks in a single material. Next, following a brief introduction to Williams's formulation process, we adopt the method of repeatedly eliminating variables instead of solving the determinant equation for the coefficient matrix to reformulate the asymptotic analysis of stress fields at arbitrary interface cracks. The resultant stresses get rid of oscillatory character. Further, under two specific loading conditions, namely, remotely uniaxial tension or shear, non‐oscillatory and non‐singular asymptotic solutions to stress fields around interface cracks are obtained.     

Incremental‐secant modulus iteration scheme and stress recovery for simulating cracking process in quasi‐brittle materials using XFEM

In this paper, an incremental‐secant modulus iteration scheme using the extended/generalized finite element method (XFEM) is proposed for the simulation of cracking process in quasi‐brittle materials described by cohesive crack models whose softening law is composed of linear segments. The leading term of the displacement asymptotic field at the tip of a cohesive crack (which ensures a displacement discontinuity normal to the cohesive crack face) is used as the enrichment function in the XFEM. The opening component of the same field is also used as the initial guess opening profile of a newly extended cohesive segment in the simulation of cohesive crack propagation. A statically admissible stress recovery (SAR) technique is extended to cohesive cracks with special treatment of non‐homogeneous boundary tractions. The application of locally normalized co‐ordinates to eliminate possible ill‐conditioning of SAR, and the influence of different weight functions on SAR are also studied. Several mode I cracking problems in quasi‐brittle materials with linear and bilinear softening laws are analysed to demonstrate the usefulness of the proposed scheme, as well as the characteristics of global responses and local fields obtained numerically by the XFEM. Copyright © 2006 John Wiley & Sons, Ltd.     

Frobenius’ method for curved cracks

The distribution of stresses produced by an undulated crack in a plane elastic solid, and in particular, at its tips where stresses approach infinity, requires the solution of two coupled singular integral equations. Except for simple crack geometries such as rectilinear and circular arcs in infinite plates, for which explicit analytic solutions have been obtained, the integral equations require numerical solutions. We propose a treatment of the integral equations by Frobenius’ method, which is particularly suitable for evaluating the stress intensity factors of slightly curved cracks.     

Estimating the importance of cyclic thermal loads in thermo-mechanical fatigue

A method for evaluating the effect of cyclic thermal loading on crack tip stress fields is developed. In its development, advantage is taken of the periodic nature of fatigue loading and only harmonic loadings are evaluated. Formulating the problem in this way permits the extraction of time as an explicit variable and replaces its role with a dependence on the frequency of the thermal loading. The means for evaluating the effect of periodic loadings on crack tip stress fields is the stress intensity factor which is calculated from numerically defined stress and displacement fields using a path independent integral. Results obtained indicate that stress intensity factors of cracked components exposed to thermal fatigue conditions have a significant dependence on the frequency of the thermal cycle and the crack geometry. Numerical estimates for mode I thermal stress intensity have been obtained using thermal fatigue test data for a titanium alloy and can be as high as 25 percent of the critical mode I mechanical stress intensity.     

Dynamic stress intensity factors around two parallel cracks in an infinite elastic plate

Summary Dynamic stresses around two parallel cracks in an infinite elastic plate are obtained. An incoming shock stress wave impinges on the cracks at right angles to their faces. The Fourier-Laplace transform technique is utilized to reduce the problem to dual integral equations. To solve these equations, the differences in the crack surface displacements are expanded in a series of functions which are zero outside the cracks. The unknown coefficients occurring in those series are solved using the Schmidt method. The stress intensity factors defined in the Laplace transform domain are inverted numerically, in the physical space.     

Mixed mode fatigue growth of curved cracks emanating from fastener holes in aircraft lap joints

The finite element alternating method is extended further for analyzing multiple arbitrarily curved cracks in an isotropic plate under plane stress loading. The required analytical solution for an arbitrarily curved crack in an infinite isotropic plate is obtained by solving the integral equations formulated by Cheung and Chen (1987a, b). With the proposed method several example problems are solved in order to check the accuracy and efficiency of the method. Curved cracks emanating from loaded fastener holes, due to mixed mode fatigue crack growth, are also analyzed. Uniform far field plane stress loading on the plate and sinusoidally distributed pin loading on the fastener hole periphery are assumed to be applied. Small cracks emanating from fastener holes are assumed as initial cracks, and the subsequent fatigue crack growth behavior is examined until long arbitrarily curved cracks are formed near the fastener holes under mixed mode loading conditions.     

Weight function,stress intensity factor and crack opening displacement solutions to periodic collinear edge hole cracks

Periodic collinear edge hole cracks and arbitrary small cracks emanating from collinear holes, which are two typical multiple site damages occurred in the aircraft structures, are studied by using the weigh function method. An explicit closed form weight function for periodic edge hole cracks in an infinite sheet is obtained and further used to calculate the stress intensity factor and crack opening displacement for various loading cases. Compared to finite element method, the present weight function is accurate and highly efficient. The interactions of the holes and cracks on the stress intensity factor and crack opening displacement are quantitatively determined by using the present weight function. An approximate weight function method is also proposed for arbitrary small cracks emanating from multiple collinear holes. This method is very useful for calculating the stress intensity factor for arbitrary small cracks.     

Intrinsic material length,Theory of Critical Distances and Gradient Mechanics: analogies and differences in processing linear‐elastic crack tip stress fields

The Theory of Critical Distances (TCD) is a bi‐parametrical approach suitable for predicting, under both static and high‐cycle fatigue loading, the non‐propagation of cracks by directly post‐processing the linear‐elastic stress fields, calculated according to continuum mechanics, acting on the material in the vicinity of the geometrical features being assessed. In other words, the TCD estimates static and high‐cycle fatigue strength of cracked bodies by making use of a critical distance and a reference strength which are assumed to be material constants whose values change as the material microstructural features vary. Similarly, Gradient Mechanics postulates that the relevant stress fields in the vicinity of crack tips have to be determined by directly incorporating into the material constitutive law an intrinsic scale length. The main advantage of such a method is that stress fields become non‐singular also in the presence of cracks and sharp notches. The above idea can be formalized in different ways allowing, under both static and high‐cycle fatigue loading, the static and high‐cycle fatigue assessment of cracked/notched components to be performed without the need for defining the position of the failure locations a priori. The present paper investigates the existing analogies and differences between the TCD and Gradient Mechanics, the latter formalized according to the so‐called Implicit Gradient Method, when such theories are used to process linear‐elastic crack tip stress fields.