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.     

The effect of overloads on threshold and crack closure

The effect of tensile and compressive overloads on the threshold stress intensity level and crack closure behaviour of one aluminium alloy and three steels has been investigated. A few tensile overloads significantly decreased the crack propagation rate and increased the threshold stress intensity. An initially decreased and then increased opening stress was mostly responsible for the delayed retardation and crack arrest. Intermittant compressive overloads significantly accelerated the crack propagation and decreased the threshold stress intensity which was a function of the frequency of overloading. The opening stress was decreased to below zero after a large compressive peak load, and it took >10⁵ cycles for the opening stress to return to its stable level. During this period an initially high crack propagation rate also gradually decreased to the stable value.     

Stress intensity factors of an inclined elliptical crack near a bimaterial interface

In this paper the stress intensity factors are discussed for an inclined elliptical crack near a bimaterial interface. The solution utilizes the body force method and requires Green’s functions for perfectly bonded semi-infinite bodies. The formulation leads to a system of hypersingular integral equation whose unknowns are three modes of crack opening displacements. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of stress intensity factors along the crack front accurately. Distributions of stress intensity factors are presented in tables and figures with varying the shape of crack, distance from the interface, and elastic modulus ratio. It is found that the inclined crack can be evaluated by the models of vertical and parallel cracks within the error of 24% even for the cracks very close to the interface.     

Mixed-mode crack opening displacement measurements for small fatigue cracks growing in Astroloy at 20 °C

Crack opening displacements were measured for small fatigue cracks in Astroloy being grown with uniaxial stress application under high-cycle fatigue conditions. Four cracks were investigated including one that grew from 27 to 74 μm in three increments. Most of the cracks grew at an angle to the loading axis and all opened bimodally. Crack opening scaled with distance from the crack tip similar to an elastic crack, which allowed the calculation of a local stress intensity factor for both mode I and mode II. The proportion of mode II stress intensity factor was relatively large, varying as 0.06 < Δ K _II /Δ K _I < 0.42, with an average of ~0.3. Thus, uniaxial loading remote to the cracks resulted in a bimodal opening response on the scale of the cracks.     

External surface cracks in shells under cyclic internal pressure

The stress analysis and fatigue crack growth behaviour of a part‐through‐cracked double‐curvature thin‐walled shell is examined. An external surface crack is assumed to lie in one of the principal curvature planes of the shell, and to present a semi‐elliptical shape. The stress intensity factors (SIFs) along the crack front for different elementary opening stresses acting on the crack faces are determined through a three‐dimensional finite element analysis. Then approximate values of SIF in the case of a cracked pressure vessel are computed by employing the above results together with the superposition principle and the power series expansion of the actual opening stress. Finally, a numerical simulation procedure is carried out to predict the crack growth under cyclic internal pressure. Some results are compared with those of other authors.     

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.     

A weight function method for mixed modes hole‐edge cracks

A weight function approach is proposed to calculate the stress intensity factor and crack opening displacement for cracks emanating from a circular hole in an infinite sheet subjected to mixed modes load. The weight function for a pure mode II hole‐edge crack is given in this paper. The stress intensity factors for a mixed modes hole‐edge crack are obtained by using the present mode II weight function and existing mode I Green (weight) function for a hole‐edge crack. Without complex derivation, the weight functions for a single hole‐edge crack and a centre crack in infinite sheets are used to study 2 unequal‐length hole‐edge cracks. The stress intensity factor and crack opening displacement obtained from the present weight function method are compared well with available results from literature and finite element analysis. Compared with the alternative methods, the present weight function approach is simple, accurate, efficient, and versatile in calculating the stress intensity factor and crack opening displacement.     

FGH95粉末高温合金裂纹闭合数值模拟

夹杂、温度和远场应力是影响粉末高温合金材料断裂性能的重要因素.本文建立了考虑裂纹面接触的含夹杂裂纹体有限元模型,计算了夹杂、温度和远场应力对裂纹闭合的影响.计算结果分析表明:当夹杂与基体未脱开时,夹杂对裂纹的张开应力几乎没有影响,而裂开的夹杂可显著提高裂纹的张开应力;均匀的温度场能显著降低裂纹的张开应力;远场应力越大,裂纹张开应力越大.     

Dynamic stress intensity factors due to scattering of harmonic waves by a crack in an infinite medium

An analytical method for calculating dynamic stress intensity factors in the mixed mode (combination of opening and sliding modes) using complex functions theory is presented. The crack is in infinite medium and subjected to the plane harmonic waves. The basis of the method is grounded on solving the two‐dimensional wave equations in the frequency domain and complex plane using mapping technique. In this domain, solution of the resulting partial differential equations is found in the series of the Hankel functions with unknown coefficients. Applying the boundary conditions of the crack, these coefficients are calculated. After solving the wave equations, the stress and displacement fields, also the J‐integrals are obtained. Finally using the J‐integrals, dynamic stress intensity factors are calculated. Numerical results including the values of dynamic stress intensity factors for a crack in an infinite medium subjected to the dilatation and shear harmonic waves are presented.     

Stress intensity factors and crack opening displacements for slanted axial through-wall cracks in pressurized pipes

This paper proposes elastic stress intensity factors and crack opening displacements (CODs) for a slanted axial through-wall cracked cylinder under an internal pressure based on detailed three-dimensional (3D) elastic finite element (FE) analyses. The FE model and analysis procedure were validated against existing solutions for both elastic stress intensity factor and COD of an idealized axial through-wall cracked cylinder. To cover a practical range, four different values of the ratio of the mean radius of cylinder to the thickness ( R _m/ t ) were selected. Furthermore, four different values of the normalized crack length and five different values of the ratio of the crack length at the inner surface to the crack length at the outer surface representing the slant angle were selected. Based on the elastic FE results, the stress intensity factors along the crack front and CODs through the thickness at the centre of the crack were provided. These values were also tabulated for three selected points, that is, the inner and outer surfaces and at the mid-thickness. The present results can be used to evaluate the crack growth rate and leak rate of a slanted axial through-wall crack due to stress corrosion cracking and fatigue. Moreover, the present results can be used to perform a detailed leak-before-break analysis considering more realistic crack shape development.     

Variation of mixed modes stress intensity factors of an inclined semi-elliptical surface crack

In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D inclined semi-elliptical surface crack in a semi-infinite body under tension. The stress field induced by displacement discontinuities in a semi-infinite body is used as the fundamental solution. Then, the problem is formulated as a system of integral equations with singularities of the form r ^–3. In the numerical calculation, the unknown body force doublets are approximated by the product of fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately for various geometrical conditions. The effects of inclination angle, elliptical shape, and Poisson's ratio are considered in the analysis. Crack mouth opening displacements are shown in figures to predict the crack depth and inclination angle. When the inclination angle is 60 degree, the mode I stress intensity factor F _I has negative value in the limited region near free surface. Therefore, the actual crack surface seems to contact each other near the surface.     

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.     

Dislocation pile-up and cleavage: effects of strain gradient plasticity on micro-crack initiation in ferritic steel

The paper is devoted to the problem of slow crack growth in heterogeneous media. The crack is subjected to arbitrary pressure distribution on the crack surface. The problem relates to construction of the so-called equilibrium crack. For such a crack, stress intensity factors are equal to the material fracture toughness at each point of the crack contour. The crack shape and size depend on spatial distributions of the elastic properties and fracture toughness of the medium, and the type of loading. In the paper, attention is focused on the case of layered elastic media when a planar crack propagates orthogonally to the layers. The problem is reduced to a system of surface integral equations for the crack opening vector and volume integral equations for stresses in the medium. For discretization of these equations, a regular node grid and Gaussian approximating functions are used. For iterative solution of the discretized equations, fast Fourier transform technique is employed. An iteration process is proposed for the construction of the crack shape in the process of crack growth. Examples of crack evolution for various properties of medium and types of loading are presented.     

Multiscale continuum modeling of a crack in elastic media with microstructures

Cosserat type continuum theories have been employed by many authors to study cracks in elastic solids with microstructures. Depending on which theory was used, different crack tip stress singularities have been obtained. In this paper, a microstructure continuum theory is used to model a layered elastic medium containing a crack parallel to the layers. The crack problem is solved by means of the Fourier transform. The resulting integrodifferential equations are discretized using the Chebyshev polynomial expansion method for numerical solutions. By using the present theory, the explicit internal length effects upon the crack opening displacement and stress field can be observed. It is found that the stress field near the crack tip is not singular according to the microstructure continuum solution although the level of the opening stress shows an increasing trend until it gets very close to the crack tip. The rising portion of the near tip opening stress is used to project the stress intensity factor which agrees fairly well with that obtained using the FEM to perform stress analyses of the cracked layered medium with the exact geometry. The numerical solutions also indicate that treating the layered medium as an equivalent homogeneous classical elastic solid is not adequate if cracks are present and accurate stress intensity factors in the original layered medium is desired.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

Weight function for an elliptic crack in an infinite medium. I. Normal loading

A recently developed integral equation method has been used to derive the crack opening displacement of an elliptic crack in an infinite elastic medium subjected to a concentrated pair of point force loading at an arbitrary location on the crack faces. These results have been used to obtain the stress intensity factor along the elliptic crack front which corresponds to the weight function for an elliptic crack under normal loading. Analytical expression of the weight function can be used to derive the stress intensity factor for both polynomial loading as well as non-polynomial loading.     

Cylindrical crack <Emphasis Type="Italic">vis a vis</Emphasis> parallel crack pair: shielding,energetics and asymptotics

Fibrous materials often contain cylindrical cracks due to delamination along the matrix-fiber interface. It is instructive to analyse a cylindrical crack of length 2a and diameter 2h in a homogeneous medium and compare the results with those for a pair of parallel cracks of length 2a and spacing 2h. The pair of parallel cracks mutually shielding each other is examined here with regard to the variation of stress intensity factors and energetics including the asymptotic limit of a pair of nearly coalescing parallel cracks. A unified formulation for parallel cracks/cylindrical crack based on crack opening displacement (COD) in terms of Chebyshev polynomials is developed. The characteristic variation of stress intensity factors as the cracks approach each other (h → 0) shows that the stress intensity factors vanish for the case of a vanishingly small cylindrical crack but not for the 2D parallel pair of cracks. The 2D case of a pair of collapsing parallel cracks ensures a finite energy release rate asymptoting to that of a single crack. Further research is needed to establish definitive asymptotic bounds for the case of extremely closely spaced cracks on the lines of Hutchinson and Suo (Adv Appl Mech 29:377–384, 1992), Kachanov (1993) and Gorbatikh et al. (Int J Fract (Lett Fract Micromech) 143:377–384, 2007). Results are presented for different values of Poisson’s ratio.     

Prediction of crack opening stress levels for 1045 quenched and tempered steel under service loading spectra

The opening stresses of a crack emanating from an edge notch in a 1045 quenched and tempered steel specimen were measured under two different Society of Automotive Engineers (SAE) standard service load histories having different average mean stress levels. The two spectra are the Grapple Skidder history (GSH), which has a positive average mean stress, and the Log Skidder history (LSH), which has a zero average mean stress. To capture the behaviour of the crack opening stress in the material, the crack opening stress levels were measured at 900X using an optical video microscope, at frequent intervals for each set of histories scaled to two different maximum stress ranges.A crack growth analysis based on a fracture mechanics approach was used to model the fatigue behaviour of the steel specimens for the given load spectra and stress ranges. Crack growth analysis was based on an effective strain‐based intensity factor, a crack growth rate curve obtained during closure‐free loading cycles and a local notch strain calculation based on Neuber's rule.The crack opening stress (S_op) was modelled and the model was implemented in a fatigue notch model, and the fatigue lives of the specimens under the two different spectra scaled to several maximum stress levels were estimated. The average measured crack opening stresses were between 6 and 12% of the average calculated crack opening stresses. In the interest of simplifying the use of S_op in design, the average S_op was correlated with the frequency of occurrence of the cycle reducing the S_op to the average crack opening stress level. The use of an S_op level corresponding to the cycle causing a reduction in S_op to a level reached once per 10 cycles gave a conservative estimate of average crack opening stress for all the histories.     

Ultrasonic Measurement of the Crack Depth and the Crack Opening Stress Intensity Factor Under a No Load Condition

This paper describes a new methodology for evaluating the crack depth and the crack opening stress intensity factor of small closed cracks using an ultrasonic technique. Surface connected back-wall cracks of depth ranging from 0.4 to 4.0 mm in steel specimens are considered. The crack corner echo amplitude of an ultrasonic shear wave, SW, beam of 50° incidence in material is used. First, the ultrasonic echo response of an open crack is determined as a function of crack depth. Next, based on changing the crack closure stress, an empirical relation between the crack closure stress and the crack-echo response is formulated. The crack depth and the crack closure stress of an unknown closed crack based on these relations are determined by inverse analysis of the ultrasonic response of the crack. From the evaluated crack depth and crack closure stress, the crack opening stress intensity factor is determined. The accuracy and reliability of this new nondestructive evaluation (NDE) method is verified by comparing the evaluated crack depth with the actual one. The latter is measured on the fractured surface obtained after carrying out ultrasonic testing. The ultrasonic method developed is proved to be a powerful tool for quantitative and nondestructive evaluation of the crack depth as well as the crack closure stress.     

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.