Expressions For Stress Intensity Factors Research Articles

Stress Intensity Factors of External Surface Cracks in a Vertical Cylindrical Steel Tank

The present research deals with external surface cracks in vertical steel tanks. Such defects occur very often. The analytical expression of stress intensity factor (SIF) is need, for prediction residual life of tanks with cracks. There are solutions for the SIF of surface cracks in hollow cylinders. However, all of them can not be used for steel tank. The main purpose of this research is to present analytical expression SIF for longitudinal external surface cracks in vertical cylindrical steel tanks. The finite element method is used for the SIF calculation of the crack in the tank. All SIF values are determined based on the submodel technique. Firstly, the whole shell model of tank without defects is generated. Real service load and tank size take into account. After that the solid submodel with crack is generated. A wide range of cracks geometries and filling level of oil are researched. The finite element models have a good agreement with wellknown analytical decisions. All SIF results are expressed with polynomial functions, that the fracture criteria can be used easily in the estimation of the residual life of tanks. SIF polynomial function should used in Paris-Erdogan equation. Results of this work could be useful for engineers in oil storage field.

A general methodology for calculating mixed mode stress intensity factors and fracture toughness of solder joints with interfacial cracks

Solder joints in electronic packages undergo thermo-mechanical cycling, resulting in nucleation of micro-cracks, especially at the solder/bond-pad interface, which may lead to fracture of the joints. The fracture toughness of a solder joint depends on material properties, process conditions and service history, as well as strain rate and mode-mixity. This paper reports on a methodology for determining the mixed-mode fracture toughness of solder joints with an interfacial starter-crack, using a modified compact mixed mode (CMM) specimen containing an adhesive joint. Expressions for stress intensity factor (K) and strain energy release rate (G) are developed, using a combination of experiments and finite element (FE) analysis. In this methodology, crack length dependent geometry factors to convert for the modified CMM sample are first obtained via the crack-tip opening displacement (CTOD)-based linear extrapolation method to calculate the under far-field mode I and II conditions (f1a and f2a), (ii) generation of a master-plot to determine ac, and (iii) computation of K and G to analyze the fracture behavior of joints. The developed methodology was verified using J-integral calculations, and was also used to calculate experimental fracture toughness values of a few lead-free solder-Cu joints.

Fracture Toughness Determined by the Centrally Cracked Brazilian-Disc Test: Some Critical Issues in the Light of an Alternative Analytic Solution

Abstract The centrally cracked Brazilian-disc test is among the configurations proposed for the experimental determination of fracture toughness of brittle rock-like materials. In spite of its wide use and the fact that the test is already standardized by the International Society for Rock Mechanics (ISRM), quite a few issues are still open related especially to the exact expressions of the respective stress intensity factors (SIFs) and the actual shape of the crack during loading. The latter is crucial since for specific crack inclinations, the crack lips are coming in contact and contact stresses appear which alter the boundary conditions, rendering analytic solutions erroneous. In this context, an alternative analytic approach is here introduced assuming that the cracked disc is under uniform pressure along two finite symmetric arcs of its periphery. The main advantage of the present approach is that it provides physically acceptable results independently of whether the crack is under opening- or closing-mode. The results of the analysis are validated against experimental data obtained using the method of Caustics and the Digital Image Correlation (DIC) technique. Based on the analytic solution, some critical practical aspects of the cracked Brazilian-disc test are explored including the influence of the loading type and the length of the loaded arc, the role of the specimen's thickness, the range of crack inclination angles for which generation of contact stress is unavoidable, and the rotation of the crack axis. Moreover, a redefinition of the SIFs in case of Mixed Fundamental problems is proposed. The limitations of the analysis are critically considered and finally some thoughts about the future steps concerning the fracture toughness determination using the cracked Brazilian-disc test are discussed.

Stress Singularities near Interface Crack Tip for Mode II of Orthotropic Bi-Material

Using the method of composite material complex and constructing new stress functions with complex singularity exponents, the problem of singularities near interface crack tip for mode II of orthotropic bi-material is studied. Boundary value problems of generalized bi-harmonic equations can be solved with the help of boundary conditions, then four kinds of stress singularities are deduced, respectively, such as the constant singularity at λ=-1/2, the non-constant singularity at λ=-1/2+ε , the constant oscillation singularity at λ=-1/2+iε, and non-constant oscillation singularity at λ=-1/2+c+iε. For each case, the analytic expressions for stress intensity factors near the central-penetrated interface crack tip for mode II of orthotropic bi-material are obtained.

The Singularity of Stress near the Tip of a Crack Perpendicular to an Elastically Mismatched Bimaterial Interface

This Based on the elastic theory of a crack perpendicular to and terminating at bimaterial interface, a generalized expression of the stress intensity factor is provided for a crack in single material and a crack perpendicular to bimaterial interface, finite element methods are used to calculate the stress intensity factors. The influences of the material combination and crack length on the the stress intensity factors were investigated. Results show that when the crack terminates at bimaterial interface, singular order ofKIis different from that of single material, and the values ofKIincrease with increasingE1/E2andμ1/μ2.

Crack Front Shapes and Stress Intensity Factors in Plates under a Pure Bending Loading that Induces Partial Closure of the Crack Faces

This paper investigates, both experimentally and numerically, stress intensity factors and evolving geometry of through edge cracks in plates loaded under pure transverse bending; the loading induces partial closure of the crack faces. After performing experiments on steel specimens, it was found that a kink in the crack front forms as the crack grows, changing the initial straight front into an L-shape. To understand the distribution of stress intensity factors along such crack fronts, the observed crack shapes were reproduced in a three-dimensional fracture code (FRANC3D) coupled with a finite element analysis program (ABAQUS). With this coupled system, linear elastic stress analyses were performed considering the nonlinear effects caused by the crack face contact in the compressed region. The proposed methodology computes stress intensity factors along the crack front based on observed crack front shapes obtained from the experimental tests. A method is proposed to treat numerical noise around the crack front kink and at crack front end. In addition, normalized empirical expressions for stress intensity factors are proposed.

Determination of Stress Intensity Factors in Low Pressure Turbine Rotor Discs

An attention in this paper is focused on the stress analysis and the determination of fracture mechanics parameters in low pressure (LP) turbine rotor discs and on developing analytic expressions for stress intensity factors at the critical location of LP steam turbine disc. Critical locations such as keyway and dovetail area experienced stress concentration leading to crack initiation. Major concerns for the power industry are determining the critical locations with one side and fracture mechanics parameters with the other side. For determination of the critical locations in LP turbine rotor disc conventional finite elements are used here. For this initial crack length and during crack growth it is necessary to determine SIFs. In fatigue crack growth process it is necessary to have analytic formulas for the stress intensity factor. To determine analytic formula for stress intensity factor (SIF) of cracked turbine rotor disc special singular finite elements are used. Using discrete values of SIFs which correspond to various crack lengths analytic formula of SIF in polynomial forms is derived here. For determination of SIF in this paper, combinedJ-integral approach and singular finite elements are used. The interaction of mechanical and thermal effects was correlated in terms of the fracture toughness.

Determination of Stress Intensity Factor for Cracked Brazilian Disc Using Weight Function

To reduce stress concentration near the loading point, Brazilian test, normally used for the indirect determination of tensile strength for rock or concrete quasi-brittle materials, is usually subjected to uniformly distributed pressure over finite width (or arc) of the Brazilian disc. As an extension, the same geometry, in which a central notch is introduced in the symmetrical plane of applied forces, can be used to determine fracture toughness of the materials. For this purpose, accurate expression for stress intensity factor (SIF) for central cracked Brazilian disc should be provided first. To this end, weight function method is used in this paper where weight function for central cracked disc is available in general SIF handbooks. Meanwhile, the exact stress solution in the intact Brazilian disc subjected to uniformly distributed pressure is derived by use of numerical method. In the end, some finite element analysis is carried out using the program package ABAQUS on the Brazilian disc, and the comparison results regarding the SIF between the weight function method and numerical method prove the validity of the proposed method.

Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip

The anti-plane fracture problem for a finite crack in a one-dimensional hexagonal quasicrystal strip is analyzed. By using Fourier transforms, the mixed boundary value problems are reduced to the dual integral equations. The solution of the dual integral equations is then expressed by the complete elliptic integrals of the first and the third kinds. The expressions for stress, strains, displacements and field intensity factors of the phonon and phason fields near the crack tip are obtained exactly. The path-independent integral derived by a conservation law equals the energy release rate, which can be used as a fracture criterion for a mode III fracture problem.

Analytical expressions for the notch stress intensity factors of periodic V-notches under tension by using the strain energy density approach

This article presents the notch stress intensity factors of a wide range of finite-width flat plates with periodic edge notches. A broad series of notch configurations, varying from shallow to very deep notches, complete range of notch spacing and wide range of notch opening angles, is considered. The significance of sharp notch approximation for the case of blunt notches with a very small notch radius is also demonstrated. Overall, more than 1200 models are carried out. Due to the desirability of analytical expressions for notch stress intensity factors evaluation, the numerical results, obtained from the strain energy density approach, are used to find some simple analytical expressions for the prediction of notch stress intensity factors of periodic sharp notches. The use of a coarse mesh in the finite element models and multiscaling are the promising advantages of such a method underlined in this study.

Analysis of Fatigue Life and Influence Factors of Tubing with Initial Crack

Three-dimensional stresses are analyzed in the tubing working based on the theory of thick-walled cylinder, and the stress intensity factor expression is derived on longitudinal and circumferential extending cracks in the tubing. Tubing fatigue life is calculated according to given condition through the preparation of the corresponding program. Influence of initial crack length, change of the pressure and the tubing wall thickness on the fatigue life are discussed.

Transient thermal cracking associated with non-classical heat conduction in cylindrical coordinate system

This paper studies the fracture behavior of a thermoelastic cylinder subjected to a sudden temperature change on its outer surface within the framework of non-classical heat conduction. The heat conduction equation is solved by separation of variable technique. Closed form solution for the temperature field and the associated thermal stress are established. The critical parameter governing the level of the transient thermal stress is identified. Exact expression for the transient stress intensity factor is obtained for a crack in the cylinder. The difference between the non-classical solutions and the classical solution are discussed. It is found that the traditional classical heat conduction considerably underestimates the transient thermal stress and thermal stress intensity factor.

Influence of cracks on dynamic characteristics and stress intensity factor of gears

Dynamic and three-dimensional finite element analytical models of cracked gears are established, and the dynamic characteristics (natural frequency and vibration shape) of the gear body are investigated when tooth cracks in the gear appear. Further, the influence of crack position and length on the dynamic characteristics of the gear structure are simulated and discussed. From this it was possible to determine that the crack position and crack length had an influence on the gear’s natural frequency and vibration shape. When the crack appears, the natural frequencies of the gear drop, and the vibration shapes also change. The influence of crack position on natural frequency and vibration shape are bigger than that of crack length. The natural frequencies’ decreases are bigger when the crack in the gear occurs at the tooth root rather than at the top of the tooth. Moreover, when the crack appears, the vibration shapes of the gear body are obviously different from without the crack, and the dynamic stresses of the gear body also change. On this basis, using the theory of fracture mechanics and the finite element method by ANSYS, the stress intensity factors were analyzed when the gear crack appeared. The variations in stress intensity factors are discussed related to changes in crack length, gear module, number of teeth, and load, when gear cracks occur at the pitch circle and tooth root. The pragmatic expressions of stress intensity factors are given by combining with the results obtained from the finite element method and the influence laws of multiple parameters on stress intensity factors are gained. These establish a good foundation for the fracture analysis and fault diagnosis of gears.

A semi-inverse method of a Griffith crack in one-dimensional hexagonal quasicrystals

A semi-inverse method of a Griffith crack in a one-dimensional hexagonal quasicrystal is proposed in this work. By introducing Westerggaard’s stress functions, the expressions for stress, strains, displacements and field intensity factors of the phonon and the phason fields in the vicinity of a crack tip are obtained in closed-form. The path-independent integral derived by a conservation law equals the energy release rate for a mode III fracture problem in a one-dimensional hexagonal quasicrystal. It is found that the energy release rate is dependent on the phonon fields as well as the phason fields, which can be used as a fracture criterion. In the absence of the phason fields, the obtainable results in this paper match with the classical ones.

Research of effects of bar geometric parameters on initial external force in precision cropping

In precise cropping of the metal bars with different geometrical parameters, it is very important to obtain the reasonable initial external forces required by the bars rapidly and accurately. Stress concentration factor and stress intensity factor are proposed to calculate the initial external force accurately in this paper according to geometrical sizes of metal bars. By means of numerical simulation and linear fitting method, the correctional expressions of stress concentration factor and stress intensity factor of V-shaped notch tip, which include the geometric parameters of the bar, are obtained respectively and the error analyses are also performed. The ratio of the depth of V-shaped notch to the diameter of the bar is presented to be a condition parameter. The numerical analytical results show that the method of stress concentration factor can be used to calculate the initial external force accurately when the ratio of the depth of V-shaped notch to the diameter of the bar is more than 0.05, and the method of the stress intensity factor can be used when the ratio is less than 0.05. The calculating formulas of reasonable initial external forces for the bars with different geometrical parameters are also given. The cropping experimental results have confirmed that it is feasible that the reasonable initial external forces required by the bars with different geometrical parameters are calculated effectively by using the stress concentration factor or the stress intensity factor in precision cropping.

Transient dislocation emission from a crack tip in an anisotropic elastic material

The emission of a dislocation with a general Burgers vector from the tip of a stationary semi-infinite crack in an anisotropic elastic material is examined. The dislocation is assumed to leave the crack tip along the crack extension plane at constant speed. Explicit expressions for the transient shielding stress intensity factors at the crack tip and the drag forces on the dislocations are derived. Numerical results for a class of cubic materials and two hexagonal crystals, zinc and cobalt, are given. Dislocation emission under plane stress wave loading is discussed.

Research of fracture problem of composite plate with curved crack by conformal mapping

In this paper, the fracture problem of an anisotropic composite plate with a curved crack is solved by conformal mapping method. Using the conformal mapping method, the problem with boundary conditions is transformed into boundary value problem of partial differential equations, which can be solved analytically by complex function method. Based on the quoted conformal mapping function, the crack curve is mapped into a unit circle. The fracture analysis near the curve crack tip for both Mode I and Mode II of anisotropic composite materials in a state of plane stress is made. The stress functions are determined under the stress boundary conditions. Then the analytical expressions of stress intensity factors, stress fields and displacement fields near the curved crack tip for Mode I and Mode II in anisotropic composite materials are obtained.

Analysis of Mode I Periodic Parallel Cracks-Tip Stress Field in an Infinite Orthotropic Plate

The mechanical behavior near crack tip for periodic parallel cracks in an orthotropic composite plate subjected to the uniformly distributed load within the cracks surface is studied. The mechanical problem is turned into the boundary value problem of partial differential equation. By using the periodicity of the hyperbolic function in the complex domain and constructing proper Westergaard stress function, the periodicity of parallel cracks can be removed. Using the complex variable function method and the undetermined coefficients method, the boundary value problem of partial differential equation can be solved with the help of boundary conditions. The analytic expressions for stress intensity factor, stress, and displacement near the crack tip of periodical parallel cracks are obtained. When the vertical distance of cracks tends to infinity, the stress intensity factor degenerates into a single central crack situation. The stress intensity factor around the crack tip of periodic parallel cracks in an orthotropic composite plate depends on the shape factor. The interaction happens between the cracks. Finally, a numerical analysis of the stress and displacement changed with the polar angle is done.

Original Crack and Adhesion Development of Nano Diamond Coating

A theoretical original microvoid model which describes the micro structural aspects of tiny original crack was developed based on fracture mechanics to explain the interface strength between nano diamond coating (NDC) and its substrate. The expression of stress fields, displacement and stress intensity factor ( SIF ) K at interface crack tip were deduced. Based on the original microvoid model, the technique to improve the adhesion strength between the NDC and its substrate was obtained by improving the diamond nucleation density and adjusting the coating residue stress.

Dynamic characteristics of cracked gear and three-dimensional crack propagation analysis

The dynamic model and three-dimensional finite element analytical model of cracked gear structure are established respectively according to the cracked beam theory, and the dynamic characteristics (natural frequency, vibration shape) of cracked gear body are investigated. Further the influences of crack position and crack length on the dynamic characteristics of gear structure are simulated and discussed. On this basis, the fracture and damage of gear structure are investigated according to the theory of fracture mechanics. Using FRANC3D software, the three-dimensional (3D) propagation of crack at tooth root for involute gear is simulated, and stress intensity factor (SIF)s of semi-circular crack at tooth root including three types are analyzed, their variation laws are gained, then the expressions of SIFs are obtained by numerical fitting FEM results. Based on this, the 3D crack propagation path at tooth root is simulated and discussed, then, it is verified by comparing to experimental results, according to the mutation of the maximum SIF at crack tip, the fracture and damage of gear tooth are judged, and its work life also is predicted. These have very important value for damage monitoring and diagnosis of gear.