Closed Crack Tip Research Articles

Analytical solution of the stress field and plastic zone at the tip of a closed crack

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.

Transient in-plane analysis of cracked general orthotropic planes

An edge dislocation with a time-dependent Burgers vector in an orthotropic plane is analyzed. The stress field caused by a dynamic point force in the intact plane is determined. The displacement discontinuity of dislocation and direction of point load are at arbitrary angles with the Material Principal Axes (MPA), the general orthotropic plane. The density of dislocations is employed to construct integral equations for orthotropic planes weakened by several interacting cracks, making arbitrary angles with the MPA. The numerical solution of integral equations results in the density of dislocations on a crack surface. The results are employed to determine time-dependent Stress Intensity Factors (SIFs) at the tips of a crack and Crack Opening Displacement (COD). In the problem of general orthotropic planes containing cracks subjected to dynamic loads, crack orientations may cause their partial/total closing. Under the hypothesis of frictionless contact, we devise an iterative procedure to handle crack closing, as well. By contrast to cracks in the directions of MPA, expressions for the two modes of SIFs contain the density functions of both the climb and glide of the edge dislocation. Therefore, mode I SIF at an open tip of a crack may not be positive. Conversely, positive mode I SIF may occur at a closed crack tip. Numerical results of the SIFs and COD for several examples of orthotropic planes under dynamic point loads, while weakened by single and double parallel cracks, are presented. Moreover, in double cracks cases, the interaction between cracks is studied.

Theoretical and Experimental Study considering the Influence of T-Stress on the Fracture Behavior of Compression-Shear Crack

The influence of the nonsingular stress term (T-stress), existing in the Williams expansion of the crack tip stress field, on the fracture behavior of the compressive-shear crack is comprehensively considered in this paper. According to the stress boundary conditions on the crack surfaces, the theoretical solution of the stress field around the closed crack tip is established using the complex potential theory, which includes not only the singular term containing the stress intensity factor KII but also three nonsingular terms (T-stress: Tx, Ty, and Txy) without KII. Then, the differences between tangential stresses and shear stresses at the crack tip obtained in the cases with and without T-stress are compared, respectively. Outcome indicates that there are tangential tensile stress and compressive stress zones at the crack tip when considering T-stress, while only the former exists when ignoring T-stress. In the case of considering T-stress, an improved crack initiation criterion is proposed, where the maximum tangential tensile stress and maximum shear stress are utilized simultaneously to predict the crack initiation and failure mode under the multiaxial compression stress state. Besides, the triaxial compression tests were carried out on the prismatic rock-like material samples with a central crack, and the effectiveness and validity of the proposed criterion are verified by comparing the experimental results and theoretical predictions. It reveals that the proposed criterion can reliably predict the crack initiation angles and failure modes with higher accuracy than traditional fracture criterion.

Nonlinear ultrasonic phased array with fixed-voltage fundamental wave amplitude difference for high-selectivity imaging of closed cracks.

The nondestructive evaluation of closed cracks is a challenging subject in ultrasonic testing. Recently, nonlinear ultrasonic phased array with fixed-voltage fundamental wave amplitude difference (fixed-voltage FAD) has been proposed as a practical approach. In this study, the maximum incident wave amplitude, which is one of the most critical parameters in closed-crack imaging, was investigated. First, a theoretical model was formulated to explicitly show the essence of the fundamental principle of FAD and the advantage of fixed-voltage FAD over different-voltage FAD. In experiments, the authors imaged a closed fatigue crack using a nonlinear ultrasonic phased array with fixed-voltage FAD while varying the incident wave amplitude. It was found that when the incident wave amplitude was sufficiently high, the nonlinear image visualized the closed crack tip, which could not be visualized in linear images. In addition, the incident-wave-amplitude dependence of the nonlinear responses was quantified. It was found that different parts within a single fatigue crack showed different nonlinear behaviors. This suggests that fixed-voltage FAD is useful not only for practical application of closed crack imaging but also for examining the nonlinear dynamics at various parts of closed cracks with a high spatial resolution.

Opening and contact zones of an interface crack in a piezoelectric bimaterial under combined compressive-shear loading

A plane problem for a tunnel electrically permeable interface crack between two semi-infinite piezoelectric spaces is studied. A remote mechanical and electrical loading is applied. Elastic displacements and potential jumps as well as stresses and electrical displacement along the interface are presented using a sectionally holomorphic vector function. It is assumed that the interface crack includes zones of crack opening and frictionless contact. The problem is reduced to a combined Dirichlet–Riemann boundary value problem which is solved analytically. From the obtained solution, simple analytical expressions are derived for all mechanical and electrical characteristics at the interface. A quite simple transcendental equation, which determines the point of separation of open and close sections of the crack, is found. For the analysis of the obtained results, the main attention is devoted to the case of compressive-shear loading. The analytical analysis and numerical results show that, even if the applied normal stress is compressive, a certain crack opening zone exists for all considered loading values provided the shear field is present. It is found that the shear stress intensity factor at the closed crack tip and the energy release rates at the both crack tips depend very slightly on the magnitude of compressive loading.

OS0708 大変位超音波送信のための積層素子の高周波数化

Nonlinear ultrasound is expected to detect the closed crack tip using a large displacements ultrasonic measurement system. As the more advanced technique we have developed the laminated transducer system up to 700 kHz in frequency. In this study, a new high frequency laminated transducer of MHz range was developed for the use of industrial steel structures. New fabricate procedures relate to the thin element thickness due to the high frequency ultrasound were applied. As the result we can fabricate the 2.9 MHz laminated transducer using the 5 MHz elements.

OS0710 サブハーモニック超音波フェーズドアレイによる閉口き裂評価の高度化

The novel imaging method, subharmonic phased array for crack evaluation (SPACE) is a practical solution for imaging closed cracks. SPACE uses a LiNbO3 transmitter and array receiver based on the subharmonic waves and phased array. In this paper, we developed SPACE with a single array transducer applicable to small objects. We imaged a stress corrosion crack (SCC) in a heat affected zone (HAZ) of type 304 stainless steel specimen, formed in high temperature pressurized water to simulate the actual cracks formed in nuclear power plants. As a result, we succeeded in visualizing the closed crack tip in the subharmonic image with high selectivity, although the specimen led to strong linear scattering at the grain boundaries. This result will significantly contribute to improving the safety and reliability of nuclear power plants.

A closed crack tip model for interface cracks inthermopiezoelectric materials

The interface crack problem of a bimaterial thermopiezoelectric solid was treated byapplying the extended version of Strohs formalism and singular integral equation approach. Theinterface crack considered is subjected to combined thermal, mechanical and electric loads.Under the applied loading, the interface crack is assumed to be partially opened. Formulation ofthe problem results in a set of singular integral equations which are solved numerically. Thestudy shows that the contact zone is extremely small in comparison with the crack length. Basedon the formulation, some physically meaningful quantities of interest such as stress intensityfactors and size of contact zone for a particular material group are analyzed.

A frictional interfacial crack under combined shear and compression

For a center frictional interfacial crack in a bimaterial medium under remote shear loading, one crack tip is `open' and the other closed. Near the closed crack tip, the crack surfaces are in contact over a sizeable region, and the stress singularity is weaker than r −1/2. At the `open' tip, the crack-tip stress field is well approximated by the classical oscillatory solution obtained by assuming traction-free crack surfaces although theoretically there is still an extremely small contact zone near the tip. Under a combined shear and compression loading, the open crack zone may become small, and the near-tip stress field approaches the characteristic of a field with a singularity stronger than r −1/2. The strong singularity is obtained from a closed-form solution for a bimaterial medium with a frictional interfacial crack under combined shear and compression loading based on the assumption of a completely closed crack. It is found that the classical definition of strain-energy release rate becomes zero (or unbounded) if the singularity is weaker (or stronger) than r −1/2 and, thus, cannot be used as a fracture criterion. The finite crack-extension strain-energy release rate, which is uniquely related to the generalized mode-II stress-intensity factor, is proposed to establish a fracture criterion for frictional interfacial cracks with weak as well as strong singularities. Finally, a numerical example simulating a fiber pull-out problem is presented to demonstrate the fiber/matrix debonding crack tip behavior during pull-out.

An Arc-Shaped Debonding Crack Opened by Internal Pressure

AbstractIn this paper, the problem of a debonding crack at the interface between a circular fiber and an infinite matrix opened by internal pressure is discussed. We concentrated on the effect of contact near the crack tips within the content of linear elastic fracture mechanics. The Muskhelishvili complex variable method is used in this analysis. The frictionless contact crack tip condition is adopted in this study in order to avoid the oscillatory stress singularity at the crack tip as shown in the classical solution. By using the crack opening displacement gradient as the primary variable, the problem is then reduced to two coupled singular integral equations, and the final discretization of the equations employs the method given by Erdogan and Gupta (1972). The comprehensive numerical results of stress fields and the mode II SIF at the closed crack tip will be given in the paper. It is also found from the numerical evidences that the contact length at the crack tip is independent of one of the Dundurs parameters, α.

Finite element analysis of a subsurface crack

The problem of a subsurface crack parallel to the surface of a half space was studied by the finite element method. Without using the interface or gap elements over the crack faces, the crack faces would penetrate into each other for the traction-free boundary condition under shear loading, which is physically impossible. Using the gap elements, this problem was avoided, and a contact zone was observed near one crack tip. The size of the contact zone decreases but the maximum contact pressure at the closed crack tip increases as the crack approaches the surface. For tensile and shear loadings, both K I (mode I stress intensity factor) and K II (mode II stress intensity factor) increase as the crack approaches the surface. For shear loading there is no K I at the closed tip and the K I and K II at the open tip are comparable as the crack approaches the surface.

A hybrid finite element analysis of interface cracks

AbstractA versatile hybrid finite element scheme consisting of special crack‐tip elements and crack face contact elements is developed to analyse a partially closed interface crack between two dissimilar anisotropic elastic materials. The crack‐tip element incorporates higher‐order asymptotic solutions for an interfacial crack tip. These solutions are obtained from complex variable methods in Stroh formalism. For a closed interfacial crack tip, a generalized contact model in which the crack‐tip oscillation is eliminated is adopted in the calculation. The hybrid finite element modelling allows the stress singularity at an open and closed crack tip to be accurately treated. The accuracy and convergence of the developed scheme are tested with respect to the known interface crack solutions. Utilizing this numerical scheme, the stress intensity factors and contact zone are calculated for a finite interface crack between a laminated composite material.

Frictional contact of an arc crack in an isotropic homogeneous material due to uniaxial loading

In this paper we continue the work of Chao and Laws [ Q. Jl Mech. appl. Math. 45(4), 629–640 (1992)] and study the partial contact problem of an arc crack in an infinite isotropic elastic solid under uniaxial loading at infinity. We extend the analysis to include the effect of friction on the crack surfaces and consider the contact crack surfaces in slip and no-slip contact conditions. Formulation of the problem is based on the integral equation. The solutions are compared with the Muskhelishvili [ Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordhoff Ltd., Groningen] open-crack solution and the earlier Chao and Laws frictionless contact solution. We find that the stress intensity factors at the open crack tip do not change very much for the frictional contact case when compared with the frictionless contact solution. We also find that K II at the closed crack tip changes significantly as the friction coefficient varies. The size of contact zone is also affected by both the loading orientation and the coefficient of friction.

Elastoplastic crack analysis for pressure-sensitive dilatant materials-Part II: Interface cracks

The problem of a plane strain crack lying along an interface between a rigid substrate and an elastic-plastic material has been studied. The elastic-plastic material exhibits pressure-sensitive yielding and plastic volumetric deformation. Two-term expansions of the asymptotic solutions for both closed frictionless and open crack-tip models have been obtained. The Mises effective stress in the interfacial crack-tip fields is a decreasing function of the pressure-sensitivity in both open and closed-crack tip models. The variable-separable solution exists for most pressure-sensitive materials and the limit values for existence of the variable-separable solution vary with the strain-hardening exponents. The governing equations become singular as the pressure-sensitivity limit is approached. Strength of the leading stress singularity is equal 1/(n+1) for both crack-tip models, regardless of the pressure-sensitivity. The second-order fields have been solved as an additional eigenvalue problem and the elasticity terms do not enter the second-order solutions as n≥2. The second exponents for the closed crack model are negative for the weak strain hardening, whereas the negative second-order eigenvalue in the open crack model slightly grows with the pressure-sensitivity. The second-order solutions are of significance in characterising the crack-tip fields. The leading-order solution contains the dominant mode I components for both open and closed crack-tip models when the materials do not have substantial strain hardening. The second-order solutions are generally mode-mixed and depend significantly on the pressure-sensitivity.

An elastoplastic analysis of the interface crack with contact zones

A n elastoplastic solution for the interface crack with contact zones is presented in this paper. The elastic asymptotic solution is discussed first and approximate expressions for the Comninou stress intensity factors of a Griffith interface crack are obtained. It is shown that the elastic asymptotic solutions based on the assumption of a closed crack tip predict material interpenetration: material elements near the crack tip are “turned inside out” and cross the bimaterial interface. The problem of a crack along the interface of an elastoplastic material and a rigid substrate is analyzed in detail. Any contact between the crack surface and the substrate is assumed to be frictionless. An elastoplastic asymptotic solution for a closed crack tip under plane strain conditions is presented. It is shown that the asymptotic solution is separable in r and θ, where ( r, θ) are polar coordinates at the crack tip. A boundary layer approach is used to study the near tip elastoplastic fields under small-scale yielding conditions. The finite element method is used for the solution of the small-scale yielding problem and the validity of the developed plastic asymptotic solution is verified numerically. Large-scale yielding solutions for a Griffith interface crack with contact zones are also presented.

A Closed Crack Tip Terminating at an Interface

Considering a crack that lies in one solid, but has a tip that touches an interface with a second solid, Bogy found that the assumption of a traction-free crack tip leads to oscillating singularities for certain combinations of materials. Such oscillating singularities lead to overlapping of the crack faces, and the purpose of this article is to eliminate the oscillating singularities by allowing the crack tip to be closed. Using an asymptotic analysis and incorporating the effect of friction, the nature of the singularity at the closed crack tip is established and the various consequences explored.