Extrapolation methods for calculating the contour J 2 ‐integral at the mixed mode of fracture

Experimental and numerical study on the effect of friction between crack faces on fracture parameters of hot mix asphalt concrete

The investigation of stress field and plastic zone distribution at the closed crack tip provides a fundamental basis for failure analysis and life prediction of geotechnical materials. Closed crack is a common crack in geotechnical materials. Studying the distribution of stress field and plastic zone at the tip of closed crack can provide theoretical basis for stability evaluation of geotechnical structures. In this study, we employ the superposition principle to obtain complex function solutions for the stress field and displacement field at the crack tip. Furthermore, we analyze the plastic zone distribution at the crack tip based on the Mohr Coulombs criterion. We investigate how factors such as crack angle, confining pressure, and material properties influence the stress field, displacement field, plastic zone size, and crack propagation direction. Our results demonstrate that this method effectively characterizes the distribution of stress field and displacement field at closed crack tips. Moreover, we elucidate that wing cracks are primarily formed due to tension-shear coupling effects. The solutions for the stress field and displacement field at the crack tip are obtained using the superposition principle. The distribution of the plastic zone at the crack tip is analyzed based on the M-C (Mohr-Coulomb) criterion. Subsequently, an analysis is conducted to investigate the influence of crack angle, confining pressure, and material properties on stress field, displacement field, plastic zone, and crack propagation direction. Lower crack angles and higher confining pressures effectively suppress slip between crack surfaces by reducing tension-shear coupling effects and inhibiting wing foil crack development. The results further indicate that the rock cohesion and internal friction angle exert negligible influence on the stress field, displacement field, plastic zone shape at the crack tip, as well as the growth direction of new cracks. The results demonstrate the effective representation of stress field and displacement field at the closed crack tip using this method. The stress distribution at the crack tip reveals that the tension-shear coupling effect primarily contributes to wing crack formation. Lower crack angles and higher confining pressures effectively suppress surface slip, reduce tension-shear coupling effects, and inhibit wing crack propagation. Furthermore, material properties do not influence the crack propagation angle, stress field, or displacement field.

In this study, the transient response of a surface crack in an elastic solid subjected to dynamic anti-plane concentrated loadings is investigated. The angles of the surface crack and the half-plane are 60° and 90°. In analyzing this problem, an infinite number of diffracted and reflected waves generated by the crack tip and edge boundaries must be taken into account and it will make the analysis extremely difficult. The solutions are determined by superposition of the proposed fundamental solution in the Laplace transform domain and by using the method of image. The fundamental solution to be used is the problem for applying exponentially distributed traction on the crack faces. The exact transient solutions of dynamic stress intensity factor are obtained and expressed in formulations of series form. The solutions are valid for an infinite length of time and have accounted for the contribution of an infinite number of diffracted waves. The explicit value of the dynamic overshot for the perpendicular surface crack is obtained from the analysis. Numerical results are evaluated which indicate that the dynamic stress intensity factors will oscillate near the correspondent static values after the first three or six waves have passed the crack tip.

This paper investigates a numerical solution for multiple crack problem in an infinite plate under remote compression. The influence of friction is taken into account. In the first step of the solution, we make a full contact assumption on the crack faces. The full contact assumption means that one component of the dislocation distribution vanishes, and the first mode stress intensity factors (K 1) at the crack tips become zero. On the above-mentioned assumption, the problem can be solved by using integral equation method, and the second mode stress intensity factors (K 2) at the crack tips can be evaluated. Meantime, after solving the integral equation the normal contact stress on the crack faces can be evaluated. The next step is to examine the full contact assumption. If the contact stresses on the crack faces are definitely negative, the solution is true. Otherwise, the obtained solution is not true. It is found from present study that in most cases the full contact condition is satisfied, and only in a few cases the full contact condition is violated. Numerical examples are given. It is found that the friction can lower the stress intensity factors at crack tips in general.

We examine the contribution of crack bridging and surface elasticity to the elastic interaction between a mode III finite crack and a screw dislocation. The surface effect on the crack faces is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. The crack faces are subjected to a bridging force which is assumed to be proportional to the crack opening displacement, whereas the bridging stiffness is allowed to vary arbitrarily along the crack. By considering a continuous distribution of both screw dislocations and line forces on the crack, the boundary value problem is reduced to two decoupled first-order Cauchy singular integro-differential equations. After the expansion of the unknown line dislocation and line force densities and the known variable bridging stiffness into Chebyshev polynomials, these singular integro-differential equations are solved numerically using the collocation method. Owing to the incorporation of surface elasticity, the stresses at the crack tips only exhibit the weak logarithmic singularity when the dislocation is located on the real axis where the crack is located, whereas in the case when the dislocation is not on the real axis, the stresses at the crack tips exhibit both the weak logarithmic and the strong square-root singularities. The two densities, the crack opening displacement across the crack faces and the image force acting on the screw dislocation are specifically calculated. We note that crack bridging only exerts an effect on the line dislocation density but has no influence on the line force density. In addition, we demonstrate that both surface elasticity and crack bridging can reduce the strengths of the logarithmic stress singularity at the crack tips and the magnitude of the crack opening displacement across the crack faces. Our results also clearly show that both crack bridging and surface elasticity exert a significant influence on the magnitude and direction of the image force acting on the screw dislocation.

This paper presents a simple method based on the strain energy density factor ΔS to study the fatigue characteristics of rhombic plates with induced angled flaws under biaxial stress field. The paper discusses in detail the procedures followed to predict the fracture crack initiation angle, θo, as a function of induced crack angle, β, the path of the crack trajectory at the initial stage of fracture and develop an expression for the crack growth rate. This method assumes that the crack extends in a radial direction and that the initial fracture crack angle, θo, is obtained by maximizing the hoop stress along a circumference of a radius r. Expressions for the stress-state near the crack tip were developed for computing the crack trajectory and the strain energy density factor. The crack trajectory path was estimated by computing the new values of the crack angle and a fictitious crack length. These computed values were in turn used to determine the strain energy density factor. The developed method revealed two important observations: i) The crack trajectory was in close agreement with the experimental data for the first 20% of the lifetime to failure, ii) the crack propagation rate is dependent on the crack angle using the stress intensity factor and exhibited no variation with respect to the crack angle when the strain energy density factor is used.

A computational method is outlined for modelling the three-dimensional development of hydraulic fractures due to the injection of a non-Newtonian fluid at the well bore. The rock formation is modelled as an infinite, homogeneous, isotropic, elastic solid with in situ stresses that vary with depth. The three dimensional problem is made two-dimensional by assuming that the velocity profile through the thickness of the crack opening is the same as for flow between parallel plates and by reducing the elasticity problem to an integral equation that relates pressure on the crack faces to crack openings. Crack openings for a given crack geometry and pressure distribution are obtained by using properties of two-dimensional Chebyshev polynomials to properties of two-dimensional Chebyshev polynomials to facilitate inversion of the integral equation. Two-dimensional fluid flow between the crack faces is analyzed using a finite element method. Introduction Current computational procedures for predicting vertical hydraulic fractures are based on an assumed height of the fracture and on fluid flow in the horizontal direction only. These assumptions, while necessary and useful in many cases, are clearly not fully satisfactory since the height of the fracture is an important quantity that one would like to predict from the computations. Furthermore, knowledge of the two-dimensional flow should be helpful in predicting proppant transport. In addition, knowledge of the proppant transport. In addition, knowledge of the pressure-time history predicted at the well bore in a well pressure-time history predicted at the well bore in a well formulated simulation of hydraulic fracturing should allow field pressure-time records to be interpreted with greater insight and confidence. Experience with computations of one-dimensional hydraulic fractures suggests that fluid flow and elastic stiffness characteristics are of primary importance whereas fracture mechanics considerations affect only the length of a small cracked region between the fluid front and the crack tip. This situation results because the weight function that relates pressure on the crack face to the stress intensity factor at the crack tip has a square root singularity at the crack tip. Consequently, the in situ compressive stress tending to close the crack can, when not nullified by fluid pressure on the crack faces near the crack tip, offset major changes in loading at positions distant from the crack front. These observations suggest that in two-dimensional hydraulic fracturing the height-length ratio of the crack may be determined primarily by considerations of fluid flow and elastic stiffness. Thus, the analysis outlined herein gives primary attention to these aspects of hydraulic fracturing. ELASTICITY OF THE FORMATION Consider the rock formation to be an infinite, isotropic elastic body under an initial stress fieldo ij. Let this body be subjected to an additional stress field 1 ij corresponding to reducing the initial stress o zz (x,y,O) on a region B bounded by(x,y) = 0 to a pressure p(x,y). (See Figure 1.) Then, the stress field 1 ij is obtained as the solution to the problem of an infinite medium with pressurep {p(x,y) - ozz (x,y,0)} acting on the planar region B. The other boundary conditions on B are assumed to be 1 xz = 1 yz = 0, corresponding to being a surface on which o xz = o yz = 0.The pressure p(x,y) can be related to the crack opening w(x,y) by making use of the fundamental solution for the stress field due to an infinitesimal segment of a dislocation line. For this, consider a dislocation segment with Burger's vector b = bh k and~ z ~ length dx' at position (x',y') as shown in-Figure 2. The normal stress on the plane z = 0 due to the dislocation segment is (Hirth and Lothe, p. 125) (1) where (2) P. 307

The crack in the rock is a system that coexists with multiscale and interaction cracks, and it is necessary to evaluate the deformation, stability, and strength of rock mass in many engineering projects and the failure of rock are related to the distribution of cracks in the rock. Cracks of different lengths were set in rock-like materials, and uniaxial loading experiments were carried out under 30 test conditions by changing the length of the rock bridge and the short crack as well as the crack angle. During the experiment, a high-speed camera system was used to record the failure process of the specimens. The following conclusions were obtained: when the crack angle is in the range of 30°to 60° with the loading direction, the shorter cracks are more likely to propagate and coalesce. Most of the specimens initiate cracking from the outer tip of the longer crack and the key point of crack instability is the outer tip of the longer crack. The coalescence mode of the cracks is mainly related to the length of the rock bridge length and the crack angle.

When the dimension of a structure falls to the micro-/nanoscale, surface effect is significant and plays a key role in affecting the mechanical behavior. This article studies the influence of surface elasticity on the stress intensity factor of an antiplane shear crack embedded in an elastic strip made of functionally graded materials. Surface elasticity is applied on the strip surfaces and crack faces, and classic elasticity is invoked for the strip interior. An antiplane shear crack problem is solved for a symmetric FGM with a crack parallel to the strip surfaces. The associated problem is converted to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces through the Fourier transform and then to a singular integro-differential equation with Cauchy kernel. The Galerkin method is applied to expand the crack face displacement as a Chebyshev series, and the singular integro-differential equation reduces to a system of algebraic linear equations. Stress intensity factors at the crack tips and the out-of-plane displacement on the crack faces are calculated numerically. It is found that surface elasticity and gradient index strongly alter the bulk stress and its intensity factors near the crack tips. Positive surface shear modulus decreases the mode III stress intensity factors and negative surface shear modulus has an opposite behavior. The influence of the variation of material gradient on the mode III stress intensity factors is expounded in graph.

Based on the work by Eshelby, the path‐independent Jk‐, M‐, L‐ and interaction‐ or Ik‐integrals were introduced and applied to cracks for the accurate calculation of crack tip loading quantities. Applying the FE‐method to solve boundary value problems with cracks, numerically inaccurate values are observed within the crack tip region affecting the accuracy of local approaches. Simulating crack paths, local approaches face further problems as cracks are running towards interfaces, internal boundaries or other crack faces. Within global approaches, path‐independent integrals are calculated along remote contours far from the crack tip, essentially exploiting numerically reliable data requiring special treatment only for the near‐tip crack faces. To provide path‐independence, additional integrals along interfaces, internal boundaries and crack faces are necessary.In this paper, new global approaches of path‐independent integrals are presented and applied to the calculation of crack paths at two‐cracks systems. A second focus is directed to the accurate loading analysis and crack path prediction considering anisotropic properties and material interfaces. The numerical model provides crack paths which are in good agreement with those obtained from crack growth experiments. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Wiener-Hopf technique in elastodynamic crack problems with characteristic lengths in loading

There is normally the occurrence of pre-existing cracks and holes in coal mass to influence its mechanical behaviours. And the crack initiation and propagation around the tip of pre-existing cracks can be observed to induce the overall failure of coal mass finally. In this study, two groups of hole with the radius of 10 mm connecting one crack with length and width of 20 mm and 1 mm, respectively, were pre-existed in sample to explore the influence of crack angle (from 0 to 90o) on the unconfined compressive strength (UCS), crack initiation and propagation, and failure modes of coal mass with combined faults by using RFPA2D. The results showed that the stress-strain curves of specimen with double-hole-crack exhibit multiple stress drop compared to that of intact coal sample, especially in the post-peak stage. Moreover, UCS decreased firstly with the crack angle increasing to 30o and then increased until the crack angle reaching to 75o following by decreasing with the continuous increase of crack angle to 90o. In addition, the failure mode of double-hole-crack specimen with the crack angle of 0-30o can be regards as the dominated tensile failure combined with shear failure, which was consist with the failure pattern of intact specimen. On the other hand, the failure mode of double-hole-crack specimen with the crack angle of 45-90o is the dominated shear failure combined with tensile failure. Meanwhile, the distribution characteristics of acoustic emission energy can be used to better reflect the deformation and failure process of coal mass with combined defects.

The near-tip asymptotic field and full-field solution are obtained for a mode III crack in an elastic material with strain gradient effects. The asymptotic analysis shows that, even though the near-tip field is governed by a single parameter B (similar to the mode III stress intensity factor), the near-tip field is very different from the classical KIII field; stresses have r -3/2 singularity near the crack tip, and are significantly larger than the classical K III field within a zone of size l to the crack tip, where l is an intrinsic material length, depending on microstructures in the material. This high-order stress singularity, however, does not violate the boundness of strain energy around a crack tip. The parameter B of the near-tip asymptotic field has been determined for two anti-plane shear loadings: the remotely imposed classical K III field, and the arbitrary shear stress tractions on crack faces. The mode III full-field solution is obtained analytically for an elastic material with strain gradient effects subjected to remotely imposed classical K III field. It shows that the near-tip asymptotic field dominates within a zone of size 0.5 l to the crack tip, while strain gradient effects are clearly observed within 5l. It is also shown that the conventional way to evaluate the crack tip energy release rate would lead to an incorrect, infinite value. A new evaluation gives a finite crack tip energy release rate, and is identical to the J-integral.

The analysis of the interaction of a main crack with an array of microcracks, placed near the tip of the main crack, has been performed considering 2-D polycarbonate flat specimens. Analytical results, based on the elastic potential theory for the stress intensity factor K, have been well compared with experimental results obtained through the application of the caustic method. Two different specimen configurations have been analysed involving one and two microcracks. INTRODUCTION The problem of the interaction of a main crack with an array of microcracks placed near its tip, of strong relevance in the prediction of the reliability of structural materials, can be experimentally analyzed by applying the caustic method on 2-D specimens. In fact, the method of caustics can be applied to the determination of the stress intensity factors at crack tips approaching other crack tips or boundaries of the specimen. Several problems of interaction of cracks with other cracks or boundaries have been reported [1,2]. In these problems two special cases can be distinguished: in the first case, the crack tip at which the stress intensity factor is determined does not lie very near another crack or boundary [3]. In the second case the crack tip lies very near another crack tip (or boundary) and the shape of the caustic around this crack tip is influenced by the other crack (or boundary). On the other hand, the elastic interaction between both types of cracks can be analytically studied by applying a method based on the combination of the double layer potential technique and the Willis polynomial conservation theorem stating that the COD of a crack embedded into a polynomial stress field of degree N has the form (ellipse)x(polynomial) [4]. Following this approach and Transactions on Modelling and Simulation vol 10, © 1995 WIT Press, www.witpress.com, ISSN 1743-355X 620 Computational Methods and Experimental Measurements expressing the displacement field as well as the stress field by an integral of the double layer potential type, the problem can be reduced to the one of finding vectorial functions-microcrack CODs b(x) and the stress intensity factor K, at the macrocrack tip from the system obtained by solving the system of integral equations which express the boundary conditions on the crack faces. The microcrack CODs are represented in the form (ellipse)x(polynomial) where the first multiplier corresponds to the crack embedded into a uniform stress field and the second multiplier accounts for crack interactions. Thus, the system of singular integral equations is reduced to a system of linear algebraic equations which must be solved to obtain the polynomial coefficients. POTENTIAL REPRESENTATION THEORY The schematic representation of a main crack (-L, i) which can interact with one microcrack is shown in Fig. La and with two microcracks in Fig.l.b. In order to describe the elastic interaction between both cracks, plane stress conditions and Mode I are assumed. Moreover, the case where the microcrack is embedded into the stress field of the main crack tip is considered. Using the constant approximation the stress field within the microcrack line (c-l, c+l) may be approximated by a constant equal to the value of the field at the microcrack centre (x-c) and, following symmetry, a* = â fcj is the only stress component acting along the microcrack line. Fig.l .a: Main crack collinear with one microcrack. Transactions on Modelling and Simulation vol 10, © 1995 WIT Press, www.witpress.com, ISSN 1743-355X Computational Methods and Experimental Measurements 621 Thus, the overall stress field in the vicinity of the main crack tip is given by the superposition:

The presence of micro-cracks in solar cells hinders the movement of charges leading to charge accumulation around the crack surfaces. Cracks grow during real-time operation and also new cracks will be formed, leading to further charge accumulation. In this study, the influence of cracks on the movement of charges and hence the current–voltage characteristics of silicon based solar cells is investigated through molecular dynamics simulations. Simulations are performed considering a domain of dimensions 260.64 Å × 222.63 Å × 43.44 Å with five different cases: (1) without any initial defects, (2) an edge crack, (3) a center crack, (4) two angled edge cracks and (5) two oblique cracks from a circular crack at the center of the domain, considering a time step of 1 fs. Charges of the atoms at a given time instant are estimated after charge equilibration. The electric current, voltage, and power are estimated based on the charges. As the crack starts propagating, the charge fluctuations of a group of atoms around the crack tip are observed to be in the range of −9×10−3 e to −6×10−3 e, where the highest fluctuation is noticed in case of angled edge cracks. The electric current for the same atoms is found to be fluctuating between −0.02 nA and −0.098 nA, peak fluctuations observed in the case of the edge crack. Similar ranges of charge and current without any initial crack are found to be: 0 × 10−3 e and 3 × 10−3 e, and 0 nA and 0.01 nA, respectively. This confirms that the presence of cracks can hinder the charge flow. Furthermore, the peak voltage estimated between two groups of charges: near and far away from the crack tip is found to be 1.24 ×10−22 V. Whereas, the peak voltage between two groups considered away from the crack tip is equal to 0.1×10−22 V, indicating that the potential difference is significant due to charge accumulation around the damaged areas. Moreover, the electric power associated with the group of atoms around the crack tip is observed to be fluctuating rapidly. The peak electrical power of the intact cell is estimated to be 2.14 nW, whereas it is dropped down to 31.93% in the presence of two inclined edge cracks. The electric power associated with a group of charges lying within the included region between two inclined cracks with a hole is observed to converge to zero, in agreement with the experimental observations in the literature.