Calculation of mixed mode (I/II) stress intensities at sharp V‐notches using finite element notch opening and sliding displacements

In this paper, a simple, robust, and an efficient technique has been proposed for accurate estimation of mixed mode (I/II) notch stress intensity factors (NSIFs) of sharp V‐notched configurations using finite element notch opening and sliding displacements at the selected number of nodes along the notch flanks. Unlike the crack problems, displacement field is rarely employed in the notch problems due to complexities introduced by the presence of rigid body displacements. One of the main emphasis of the present work is to neatly bypass these rigid body displacements and develop a simple approach for accurate computation of the NSIFs so that it can be easily incorporated in the existing code. Several benchmark problems have been analyzed. The results obtained using the present method show excellent agreement with the solutions available in the literature. Some new results have also been reported in the present work.     

Solution of flat crack problems in shear mode

Solution of the flat crack problems in shear mode is presented. The least potential energy principle is used to solve the problems. In the solution a family of the crack opening displacements (COD) with some undermined coefficients is assumed. The strain energy stored in body is expressed in the form of a quadratic form with some undetermined coefficients. In the formulation of the quadratic form, the differential–integral equation for the flat crack problem is used. After using the least potential energy principle, the coefficients in the family of COD can be determined and the crack opening displacements can be evaluated immediately. The stress intensity factors along the crack border can be obtained from the known crack opening displacements. A particular feature of the present method is that no singular integral is involved in computation. Several numerical examples are given with the calculated stress intensity factors along the crack border. Copyright © 2005 John Wiley & Sons, Ltd.     

A numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems

A computational model based on the numerical Green's function (NGF) and the dual reciprocity boundary element method (DR-BEM) is presented for the study of elastodynamic fracture mechanics problems. The numerical Green's function, corresponding to an embedded crack within the infinite medium, is introduced into a boundary element formulation, as the fundamental solution, to calculate the unknown external boundary displacements and tractions and in post-processing determine the crack opening displacements (COD). The domain inertial integral present in the elastodynamic equation is transformed into a boundary integral one by the use of the dual reciprocity technique. The dynamic stress intensity factors (SIF), computed through crack opening displacement values, are obtained for several numerical examples, indicating a good agreement with existing solutions.     

Wedging of an edge crack in the elastic wedge

We study the problem of wedging of an elastic wedge realized along the edge crack with the help of a hard wedge. The analytic solution of the problem is obtained by the Wiener–Hopf method. We determine the stress intensity factors, the distributions of stresses on the continuation of the crack and in the contact zone, and the circular displacements of the crack lips.     

The arbitrarily loaded single-edge cracked circular disc; accurate weight function solutions

A general solution for the stresses and displacements of collinear cracks in an infinite homogeneous anisotropic medium subjected to uniform loading at infinity has been given in this paper by using the Stroh's formulation. The solutions are valid not only for plane problems but also for antiplane problems and the problems whose inplane and antiplane deformations couple each other. Two special collinear crack problems are solved explicitly: (1) two collinear cracks, (2) an infinite row of evenly spaced collinear cracks. A closed form solution of the stresses and displacements in the entire domain is obtained. Through the use of identities developed in the literature, the stress intensity factors, crack opening displacements and energy release rate are expressed in real form, which are valid for any kind of anisotropic materials including the degenerate materials such as isotropic materials. The simple explicit form solutions for the crack opening displacements and energy release rate reveal that the effect of anisotropy is totally determined by the fundamental elasticity matrix L. The relation between the stress intensity factors and energy release rate is obtained in quadratic form and related to L.     

Investigation of fatigue crack closure using multiscale image correlation experiments

Two full-field macroscale methods are introduced for estimating fatigue crack opening levels based on digital image correlation (DIC) displacement measurements near the crack tip. Crack opening levels from these two full-field methods are compared to results from a third (microscale) method that directly measures opening of the crack flanks immediately behind the crack tip using two-point DIC displacement gages. Of the two full-field methods, the first one measures effective stress intensity factors through the displacement field (over a wide region behind and ahead of the crack tip). This method reveals crack opening levels comparable to the limiting values (crack opening levels far from the crack tip) from the third method (microscale). The second full-field method involves a compliance offset measurement based on displacements obtained near the crack tip. This method delivers results comparable to crack tip opening levels from the microscale two-point method. The results of these experiments point to a normalized crack tip opening level of 0.35 for R ∼ 0 loading in grade 2 titanium. This opening level was found at low and intermediate ΔK levels. It is shown that the second full-field macroscale method indicates crack opening levels comparable to surface crack tip opening levels (corresponding to unzipping of the entire crack). This indicates that effective stress intensity factors determined from full-field displacements could be used to predict crack opening levels.     

An analytical method for mixed-mode propagation of pressurized fractures in remotely compressed rocks

An analytical method for mixed-mode (mode I and mode II) propagation of pressurized fractures in remotely compressed rocks is presented in this paper. Stress intensity factors for such fractured rocks subjected to two-dimensional stress system are formulated approximately. A sequential crack tip propagation algorithm is developed in conjunction with the maximum tensile stress criterion for crack extension. For updating stress intensity factors during crack tip propagation, a dynamic fictitious fracture plane is used. Based on the displacement correlation technique, which is usually used in boundary element/finite element analyses, for computing stress intensity factors in terms of nodal displacements, further simplification in the estimation of crack opening and sliding displacements is suggested. The proposed method is verified comparing results (stress intensity factors, propagation paths and crack opening and sliding displacements) with that obtained from a boundary element based program and available in literatures. Results are found in good agreements for all the verification examples, while the proposed method requires a trivial computing time.     

Weight functions and strip yield solution for two equal-length collinear cracks in an infinite sheet

The problem of two equal-length collinear cracks in an infinite sheet is treated using the weight function method. Exact weight functions for the inner and outer crack tips are derived based on the crack opening displacement solution for a reference load case. These weight functions are used to calculate stress intensity factors for different load cases, plastic zone sizes and crack tip opening displacements of the strip yield model. The approach is validated by the perfect agreement between the present strip yield model solutions and Collins and Cartwright’s analytical results based on the direct complex stress function formulation.     

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.     

Hygrothermal stresses for a plane crack in a generally anisotropic plate

The solutions are presented for the hygrothermal stress field of a generally anisotropic plate under uniform heat flux and moisture concentration transfer obstructed by a hygrothermally insulated crack. For uncoupled diffusion of temperature and moisture, the solutions of both temperature and moisture are obtained directly from the Hilbert problem approach, and are treated as the particular solutions to a pair of nonhomogeneous partial differential equations for an uncoupled hygrothermoelastic system. The associated homogeneous solutions are expressed in terms of three stress functions based on the complex variable approach of Lekhnitskii. With some identities concerning the eigenvalues and eigenvectors, the general expressions of the stress and displacement fields can then be found in an explicit form. The stress intensity factors, crack opening displacements and energy release rate are expressed in terms of the heat flow, moisture concentration, material geometry, elastic and hygrothermal anisotropy. The simultaneous existence of mode I, II and III fracture is found to be due to material inherent anisotropy. Special cases for isotropic and orthotropic materials are also discussed.     

A comparison of direct and iterative methods for determining traction-separation relations

Traction-separation relations have been used to represent the adhesive interactions at bimaterial interfaces for contact and fracture analyses. There are a variety of methods for determining these relations, which are broadly sorted into iterative and direct methods. Here we compare the traction-separation relations for a silicon/epoxy interface extracted by two such methods. Interferometric measurements of the normal crack opening displacements near the crack front in a double-cantilever beam specimen were exploited along with an augmented analytical solution for J-integral as an illustration of the direct method. As an example of the iterative method, we relied on comparisons of measured crack length and normal crack opening displacements with numerical simulations obtained from two types of candidate traction-separation relations. It was found that the shape of the traction-separation relation, in addition to the interfacial toughness and strength, was needed to bring the numerical solutions into optimal registration with the measurements. On the other hand, the direct method lived up to its name in terms of ease of parameter extraction while providing a reasonable set of parameters.     

Method of translations for elliptic mode I cracks in infinite bodies. Part 1. Polynomial loads

We propose a method for evaluation of the displacement of the lips of an elliptic crack in an infinite body and, hence, for the determination of the stress intensity factors under the action of a polynomial load. The method is based on the Rice integral formula for the fields of stresses and displacements in two different states of the body, the Dyson theorem for the form of the field of displacements for a given polynomial load, the author's theory of translations of an elliptic crack in a nonuniform stress field, and the use of the existing solution for the case of uniform loading. The structure and complexity of the proposed method are similar to those of the well-known methods of weight functions based on the application of known particular solutions of the problem for a given body in constructing new more general solutions. For the special case of elliptic cracks in infinite bodies, the proposed method realizes the prediction of Fett concerning integration of the method of weight functions with general approaches of the theory of elasticity.     

The improved D-M model solutions for single edge cracked plates by considering the yielding at the back side

The method of calculating the stress intensity factor (SIF) and the crack opening displacement (COD) for double edge cracks in plates under arbitrary loadings that results in solving a system of Cauchy-type singular integral equations is presented. The improved D-M model is then constructed for edge cracked plates by considering the yielding at the back side. For the cases of tension and bending, the plastic zone sizes and the crack opening displacements are calculated from the improved model solution, and the envelopes for the beginning of backside yielding and ligament yielding are obtained. The numerical results are compared with known solutions which take no account of the yielding at the back side and with experimental results.     

A solution method for finding dynamic stress intensity factors for arbitrarily oriented cracks in transversely isotropic materials

The transient elastodynamic response of a transversely isotropic material containing a semi-infinite crack under uniform impact loading on the faces is examined. The crack lies in a principle plane of the material, but the crack front does not coincide with a principle direction. Rather, the crack front is at an angle to a principle direction and thus the problem becomes more three-dimensional in nature. Three loading modes are considered, i.e., opening, in-plane shear and anti-plane shear. The solutions for the stress intensity factor history around the crack tip are found. Laplace and Fourier transforms together with the Wiener-Hopf technique are employed to solve the equations of motion directly. The asymptotic expression of stress near the crack tip leads to a closed-form solution for the dynamic stress intensity factor for each loading mode. It is found that the stress intensity factors are proportional to the square root of time as expected. Results given here converge to known solutions in transversely isotropic materials with a crack oriented along a principle direction and isotropic materials as special cases. The results of this analysis are used to find approximate strain energy release rates for dynamically loaded penny shaped cracks.     

Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading

Disc-type specimens are among favorite test samples for determining mode I and mixed mode fracture toughness in brittle materials like rocks, brittle polymers, ceramics, etc. In this research, the finite element method is used to analyze two disc-type specimens: a semi-circular disc specimen containing an edge crack and subjected to three-point-bend loading (SCB specimen), and a centrally cracked circular disc subjected to diametral compressive loading, often called the Brazilian disc specimen. The crack parameters K_I, K_II and T are calculated for different mode mixities from pure mode I to pure mode II. Although the stress intensity factors K_I and K_II are presented mainly for validation of the analyses, they are also used for determining the crack angle corresponding to pure mode II for each specimen. It is shown that in general the T-stress increases for larger crack angles. While the T-stress in the Brazilian disc specimen is always negative for any combinations of mode I and mode II, the sign of T-stress in the SCB specimen depends on the mode mixity. A very good agreement is shown to exist between the calculated results for T and those very limited data presented in other papers.     

Fatigue crack growth in Haynes 230 single crystals: an analysis with digital image correlation

Fatigue crack growth was investigated in Haynes 230, a nickel‐based superalloy. Anisotropic stress intensity factors were calculated with a least squares algorithm using the displacements obtained from digital image correlation. Crack opening/sliding levels were measured by analysing the relative displacement of crack flanks. Reversed crack tip plastic zones were calculated adopting an anisotropic yield criterion. The strains measured in the reversed plastic zone by digital image correlation showed a dependence on crystallographic orientation. Finally, a finite element model was adopted to examine plasticity around the crack tip. Results were compared with the experimentally observed strains.     

Method of translations for elliptic mode I cracks in infinite bodies. Part 1. Polynomial loads

We propose a method for evaluation of the displacement of the lips of an elliptic crack in an infinite body and, hence, for the determination of the stress intensity factors under the action of a polynomial load. The method is based on the Rice integral formula for the fields of stresses and displacements in two different states of the body, the Dyson theorem for the form of the field of displacements for a given polynomial load, the author's theory of translations of an elliptic crack in a nonuniform stress field, and the use of the existing solution for the case of uniform loading. The structure and complexity of the proposed method are similar to those of the well-known methods of weight functions based on the application of known particular solutions of the problem for a given body in constructing new more general solutions. For the special case of elliptic cracks in infinite bodies, the proposed method realizes the prediction of Fett concerning integration of the method of weight functions with general approaches of the theory of elasticity. Translated from Problemy Prochnosti, No. 6, pp. 102–121, November–December, 1997.     

Element free galerkin study of steady quasi-static crack growth in plane strain tension in elastic-plastic materials

Employing an extension of the convective mesh technique, an Element Free Galerkin (EFG) based formulation for steady quasi-static crack growth in elastic-plastic materials undergoing small scale yielding is established. In accordance with the steady state condition, a parallel path constitutive law integration scheme is implemented into the formulation and a body force type term is introduced for the contribution of crack tip plasticity. A detailed numerical analysis is performed for steady quasi-static growth of a mode I crack under plane strain conditions in elastic-perfectly plastic materials. The numerical solutions to this problem confirm the existence of an elastic unloading wedge. The computed near-tip stress distributions and the crack opening displacements for this problem provide good agreements with the corresponding asymptotic solutions and demonstrate the validity of the EFG based formulation. A rough estimate of the range of validity of the asymptotic solutions is also made based on the numerical solutions. In addition, the effect of strain hardening on the steady quasi-static crack growth is investigated. Both power hardening and linear hardening models are considered.     

Crack opening displacement in plate with bonded repair patch

Mathematical techniques are extended to compute crack opening displacements in a cracked plate with an adhesively bonded composite patch. The plate and the patch are considered as orthotropic materials. The problem is reduced to the solution of integral equations. A software program is written to compute shear stresses in adhesive, stress intensity factors in the plate and the crack openings at the centreline of the crack. The effects of adhesive thickness, adhesive modulus, patch thickness and plate thickness on crack openings are investigated. A test program is carried out to obtain crack opening displacements in plate with bonded patch. A good agreement with analytical predictions is obtained. The effects of patches bonded on one or both sides of a plate on stress intensity factors are evaluated.