Expressions For Stress Intensity Factors Research Articles

Analytical study on the propagation of rectilinear semi-infinite crack due to Love-type wave propagation in a structure with two dissimilar transversely isotropic layers

The present study deals with the propagation characteristics of rectilinear semi-infinite non-centrally located interfacial crack associated with progressing Love-type wave in the structure comprised of two dissimilar transversely isotropic layers. The closed form expressions of stress intensity factor for different quasi-static conditions (viz. diverse loading and stress-free conditions on the edges of crack subjected to non-harmonic and harmonic loading) and also for static condition concerned with non-harmonic and harmonic loading have been derived by using Weiner-Hopf technique along with two-sided Fourier transform. The substantial influence of thickness ratio, inhomogeneity, stress amplitude ratio, displacement amplitude ratio, and different material combinations of layers on stress intensity factor at the crack tip has also been examined for the considered structure. Some notable characteristics are also highlighted and revealed by means of numerical results and graphs.

Crack interaction with nanoscale twinning near a second-phase particle in fine-grained magnesium alloy

A theoretical model is established to address the effect of nanoscale twinning near a second-phase particle on crack growth in fine-grained magnesium alloys. The numerical solutions of singular integral equations are obtained by the considering complex variable method of Muskhelishvili, the superposition principle of elasticity, and the distributed dislocation technique. The expressions of stress intensity factors near the left crack tip are derived, and the energy release rate (ERR) characterizing the condition for crack propagation is also calculated. The influences of relevant parameters such as the location of nanoscale twin, the size of particle, and the relative distance between the inhomogeneity and the left crack tip on the ERR are examined in detail. The results indicate that the ERR is strongly influenced by the nanoscale twinning and the second-phase particle. The hard inhomogeneity decreases the ERR while the soft inhomogeneity increases the ERR when nanoscale twinning is not taken into account. At the same time, there is competition between the effects of the hard particle and nanoscale twinning on the ERR, and the nanoscale twin band with an optimum size displays the best toughening effect.

Combined effects of cooperative grain boundary sliding and migration and reinforced particles on crack growth in fine-grained Mg alloys

A theoretical model is developed to describe the combined influences of cooperative grain boundary (GB) sliding and migration and reinforced particles on crack propagation in fine-grained Mg alloys. The numerical solutions of singular integral equations are derived on the basis of complex variable method of Muskhelishvili, the superposition principle of elasticity and distributed dislocation technique. The expressions of stress intensity factors (SIFs) near the right tip of basal crack are obtained and the energy release rate (ERR) characterizing the condition for crack growth is calculated. The influences of important parameters such as the location of cooperative GB sliding and migration, the size of reinforced particle and the misorientation angle of higher angle grain boundaries on the ERR are discussed in detail. The results show that the fracture toughness of fine-grained Mg alloys can be improved by cooperative GB sliding and migration and particle hardening and refining. There exists an optimum particle size that makes the fracture toughness of fine-grained Mg alloys best. Besides, the GB migration remarkably contributes to the fracture toughness of fine-grained Mg alloys and there is an optimum migration distance making the fracture toughness of fine-grained Mg alloys best.

Analysis of cracks in one-dimensional hexagonal quasicrystals with the heat effect

The extended displacement discontinuity (EDD) method is proposed to analyze cracks in the periodical plane of one-dimensional (1D) hexagonal quasicrystals with the heat effect. Based on the operator theory and the Fourier transform, the fundamental solutions for EDDs are derived, where the EDDs include phonon and phason displacement discontinuities and the temperature discontinuity. The EDD boundary integral equation method is used to analyze the singularities of the near-crack tip fields, and the extended stress intensity factor (ESIF) expressions are obtained in terms of the EDDs across the crack faces. The EDD boundary element method is proposed to calculate the ESIFs of cracks in 1D hexagonal quasicrystals. COMSOL software is used to validate the developed method. The influences of applied mechanical and heat loads on cracks in a finite plate are investigated.

Assessment of the interaction between three collinear unequal straightcracks with unified yield zones

The Dugdale model has been modified in the paper to include the effect of linearly varying yield stress distribution. An isotropic infinite plate is considered with three collinear unequal straight cracks with coalesced yield zones. Muskhelishvili’s complex variable approach is used to solve the problem. Closed form analytical expressions for stress intensity factor and crack tip opening displacement at each crack tip are obtained when boundary of the plate is subjected to uniform stress distribution and developed yield zones are assumed variable stress distribution. Different yield zone lengths and crack tip opening displacements are observed at each crack tip. A comparative case with the solution of two equal cracks is studied to show that the problem considered in this paper is the predecessor of the two equal cracks problem.

Analysis of dynamic stress intensity factors for interfacial crack near shallow circular inclusion in bi-material half-space to SH wave

An analytical approach is used to investigate the SH-wave scattering problems of the bi-material half-space including a circular inclusion embedded and an interfacial crack. Considering the multiple scattering phenomena between the boundaries and the circular inclusion, the “image” principle is used to establish the expressions of the wave field. With the aid of interfacial conjunction, the interfacial crack is constructed. According to the displacement and stress continuity conditions, the Fredholm integral equations of the first kind are obtained to determine the unknown forces attached to the linking sections. Finally, analytical expressions for dynamic stress intensity factors are obtained. The numerical results for dynamic stress intensity factors are presented graphically considering the factors of the crack length, the incident wave number, the inclusion defect and the different media parameters.

Influence of anisotropy, porosity and initial stresses on crack propagation due to Love‐type wave in a poroelastic medium

AbstractAn analytical solution has been attained to establish the closed form expression of stress intensity factor at the tip of a semi‐infinite crack, dynamically propagating in an initially stressed transversely isotropic poroelastic strip due to Love‐type wave for the case of concentrated force of constant intensity as well as for the case of constant load. The study presents the sound effect of various affecting parameters viz. speed of the crack, length of the crack, horizontal compressive/tensile initial stress, vertical compressive/tensile initial stress, porosity parameter and anisotropy parameter on the stress intensity factor. In order to delineate the effects of these aforementioned parameters on the stress intensity factor graphically, numerical simulations have been accomplished. One of the major highlight of the paper is the comparative study carried out for horizontal compressive/tensile initial stress, vertical compressive/tensile initial stress, porosity parameter and anisotropy parameter with the case when the strip is isotropic, non‐porous and free from initial stresses. Wiener–Hopf technique and the Fourier integral transform has been effectuated for the procurement of the closed form expression (exact solution) of stress intensity factor.

Variation of Local Flexibility Coefficients of Functionally Graded Cracked Shaft

The present paper deals with the analytical determination of local flexibility coefficients (LFCs) for a functionally graded (FG) cracked shaft. Finite element (FE) analysis has been performed using two nodded beam element having six degrees of freedom (DOF) at each node and Timoshenko beam theory (TBT) has been used. Material properties are assumed to be graded in radial direction following linear, power law and exponential law gradation respectively. In the present analysis, the mixture of Aluminum Oxide (Al2O3) and Stainless Steel (SUS304) is considered as FG material where metal content is considered decreasing towards the outer diameter of shaft. The LFCs of a cracked shaft is derived using Castigliano's theorem and Paris's equations in conjunction with the expression for stress intensity factors. A complete code has been developed using MATLAB program and validated with the existing results available in literatures. The LFCs are calculated for different crack depths and for different crack closer lines for a cracked shaft and the effect of crack depth and crack closer line position on the LFCs has been studied. Numerical results reveal that for a given crack configuration, LFCs values are maximum for metallic, minimum for ceramic shaft and the LFCs for FG shaft in between, and also the LFCs values increase with the increasing power law gradient indices.

New Development to Measure Mode I. Fracture Toughness in Rock

Tensile fracture toughness is one of the dominant characteristics of rocks that play an important role in fracture mechanics of rock structures. In spite of the substantial amount of work that has been conducted as suggested methods to determine the fracture toughness of rock in Mode I., research on new methods are still demanded. The compact tension (CT) specimen is widely used to determine the fracture toughness of metals and is a standard method in accordance with ASTM standard. To conduct similar direct tests in rock, preparation of rock specimen a difficult task and needs some special tools to create notched sample. To address these issues, experimental techniques based direct tensile apparatus have been developed. Developed experimental procedure involves the use of the specimens having a central hole and two notches or cracks at both inner side of proposed specimen. The stress intensity factor formula at the crack tip of a CT specimen can be availed through the literature. However a new expression of stress intensity factor is needed for the developed test in order to account geometry and loading configuration. To develop a new stress intensity factor expression, the numerical models of suggested method were worked out. The focus of the current work is on development of the experimental technique, which involves determining the optimum sample size and shape, loading procedure, fracture toughness for different rock types, and verification of suggested method based on CT test which has been carefully considered in this investigations.

Effect of Specimen Geometry and Press-Fit on the Stress Intensity Factor Solution for Scaled Wheelset Axles under Bending

One problem in wheelset axles are the failures due to the propagation of fatigue cracks. To avoid those failures with sometimes catastrophic consequences analytical crack propagation simulations are used. For these simulations the knowledge of the crack propagation path and the corresponding stress intensity factor (SIF) is required. To calculate the SIF a numerical simulation, such as finite element method (FEM), is generally needed. To this end numerical simulations for shouldered solid shafts with elliptical surface cracks are done. In this work different influencing factors of the stress intensity factor expressions have been identified. Therefore, two different specimen types with three different stress concentration factors each were designed. The results represent an overview for the interaction of stress concentration factor, crack depth, crack aspect ratio, bending and press-fit load in shouldered solid shafts. To represent the rotation, different characteristic angular positions for one selected characteristic crack geometry have been considered.

Propagation of crack in a pre-stressed inhomogeneous poroelastic medium influenced by shear wave

A mathematical model has been developed for the analytical study of propagation of crack due to horizontally polarized shear wave (SH-wave) in an initially stressed inhomogeneous poroelastic strip. The expression of stress intensity factor for the force of constant intensity has been determined in closed form. The emphatic impact of various affecting parameters viz. speed of the crack, length of the crack, horizontal compressive or tensile initial stress, vertical compressive or tensile initial stress, heterogeneity parameter, porosity parameter and anisotropy parameter on the stress intensity factor in an initially stressed inhomogeneous poroelastic strip has been depicted by means of graphs. In order to solve the problem, Wiener–Hopf technique specified by Matczynski has been employed. Moreover, some special cases have been deduced from the procured expression of stress intensity factor for the force of constant intensity.

Dynamic stress intensity factor of 2A12T4 Al compact tension specimen with loading point displacement and spring mass model

Loading point displacement and spring mass model were introduced into the calculation of the dynamic stress intensity factor for a fatigue pre-cracked 2A12T4 aluminium alloy compact tension specimen. A simple formula of dynamic stress intensity factor for a fatigue pre-cracked compact tension specimen is derived using the vibration analysis method. By determining equivalent mass and equivalent stiffness of 2A12T4 aluminium alloy compact tension specimen, the dynamic stress intensity factor expression has been given. Three examples of simple dynamic loading were calculated, the results were in good agreement with their finite element model values.

Application of modified Dugdale model to two pairs of collinear cracks with coalesced yield zones

In this paper, a modified Dugdale’s approach has been presented to arrest four straight collinear quasi-static cracks with coalesced yield zones. An infinite elastic perfectly plastic plate, containing four cracks, is subjected to uniform stresses applied at infinite boundary of the plate. As a result, yield zones develop at each crack tip. On increasing the stresses, yield zones between two pairs of cracks get coalesced and after that at each internal crack tip. In order to detain propagation of cracks, a quadratic yield stress distribution is applied on the rims of the yield zones. This distribution enables to investigate the residual strength of an infinite plate, when the plate fails at a stress which is well below the yield stress of the plate. Muskhelishvili’s complex variable technique is use to solve the stated problem. Analytical expressions for stress intensity factor and crack-tip opening displacements are established. The results validate with previously published works.

Effect of initial T-stress on stress intensity factor for a crack in a thin pre-stressed layer

Fracture of a pre-stressed thin elastic layer is studied and modelled as bending of double clamped beams with axial force. Using the beam theory, the strain energy in bending state of the beam with an axial force is derived. Explicit expressions for energy release rate and stress intensity factor (SIF) for a crack with initial transverse stress (T-stress) are obtained. Results show that tensile (compressive) initial T-stress decreases (increases) the SIF and impedes (promotes) crack growth. The influence of compressive initial stress is greater than that of tensile initial stress in the same magnitude.

Thermal stress intensity factor expressions for functionally graded cylinders with internal circumferential cracks using the weight function method

In this paper, the weight function method is used to derive mathematical expressions in terms of the Gauss hypergeometric function for the mode-I thermal stress intensity factor of functionally graded cylinders with internal circumferential cracks. To determine the weight function coefficients, a unique function is fitted to reference stress intensity factors obtained from finite element analysis. Effects of the internal convection cooling coefficient and the material power law index are investigated, as well. It is shown that the thermal stress intensity factors predicted by the developed mathematical expression are in good agreement with those directly obtained from finite element analysis. Results of this study may be used in material selection, design optimization, safety assessment against fracture, and fatigue life evaluation of functionally graded cylinders.

Stress Intensity Factor and Elastic Crack Opening Displacement Solutions of Complex Cracks in Pipe Using Elastic Finite-Element Analyses

In the present paper, the closed-form expressions for the stress intensity factors (SIFs) and the elastic crack opening displacements (CODs) of complex-cracked pipes are derived based on the systematic three-dimensional (3D) elastic finite-element (FE) analyses. The loading conditions that are evaluated include global bending moment, axial tension, and internal pressure. In terms of geometries, the geometric variables affecting the SIFs and the elastic CODs of complex-cracked pipes, i.e., the crack angle of through-wall cracks (TWCs), the crack depth of fully circumferential, internal surface cracks in the inner surface of pipe, and the ratio of pipe mean radius to thickness, are systematically considered in the present FE analyses. The FE analysis procedure employed in the present study has been validated against the existing solutions for the circumferential TWC pipes. Using the present FE results, the shape factors of SIF and elastic COD for complex-cracked pipes are tabulated as a function of geometric variables. The results are applied for closed-form expressions of SIF and elastic COD when the pipe is subjected to simple loading conditions of bending, axial tension, or internal pressure. The proposed closed-form expressions can estimate SIF and elastic COD of complex-cracked pipes within maximum differences of 2.4% and 5.9% with FE results, respectively.

Effect of Confining Pressure on Stress Intensity Factors for Cracked Brazilian Disk

An analytical model, verified by the finite element method, is developed to study the effect of confining pressure on stress intensity factors for the cracked Brazilian disk. The closed-form expressions for stress intensity factors under both confining pressure and diametric forces are obtained based on the weight function method. The results show that the confining pressure has no effect on the mode II stress intensity factor; however, the mode I stress intensity factor decreases with the increase of confining pressure and the change may be above 100% for a large confining pressure. In addition, the effect of confining pressure on the loading condition of pure mode II crack is also investigated. It is shown that the critical loading angle for pure mode II crack decreases as the confining pressure increases. Depending on the magnitude of confining pressure, the failure problem of a disk may be no longer a pure fracture problem. These results have established the theoretical foundation to measure the fracture toughness of materials under confining pressure.

Fracture analysis and remaining life prediction of aluminium alloy 2014A plate panels with concentric stiffeners under fatigue loading

Fracture analysis and remaining life prediction has been carried out for aluminium alloy (Al 2014A) plate panels with concentric stiffener by varying sizes and positions under fatigue loading. Tension coupon tests and compact tension tests on 2014A have been carried out to evaluate mechanical properties and crack growth constants. Domain integral technique has been used to compute the Stress intensity factor (SIF) for various cases. Generalized empirical expressions for SIF have been derived for various positions of stiffener and size. From the study, it can be concluded that the remaining life for stiffened panel for particular size and position can be estimated by knowing the remaining life of corresponding unstiffened panel.

Determination of the Stress Intensity Factor of an elliptical breathing crack in a rotating shaft

The failures due to the propagation of fatigue cracks are one of the most frequent problems in rotating machines. Those failures sometimes are catastrophic and are sufficient to provoke the loss of the complete machine with high risks for people and other equipments. When a cracked shaft rotates, the breathing mechanism appears. The crack passes from an open state to a close state with a transition in which a partial opening or closing of the crack is produced. In this work, a new general expression that gives the Stress Intensity Factor (SIF) along the crack front of an elliptical crack in a rotating shaft in terms of the crack depth ratio, the crack aspect ratio, the relative position on the front and the angle of rotation has been developed for linear elastic materials. By the moment, no expressions of the SIF in term of these variables have been found in the literature. To this end, a quasi-static 3D numerical model of a cracked shaft with straight and elliptical cracks subjected to rotary bending using the Finite Element Method (FEM) has been made. To simulate the rotation of the shaft, different angular positions have been considered. The SIF in mode I along the crack front has been calculated for each angular position of the cracked shaft and for different crack geometries. The expression results have been compared with solutions obtained from the literature. It has been found that they are in good agreement. The model has been applied to other crack geometries with good results. The obtained SIF expression allows studying the dynamic behavior of cracked shafts and can be used to analyze the crack propagation.

Antiplane Problem of Periodically Stacked Parallel Cracks in an Infinite Orthotropic Plate

The antiplane problem of the periodic parallel cracks in an infinite linear elastic orthotropic composite plate is studied in this paper. The antiplane problem is turned into the boundary value problem of partial differential equation. By constructing proper Westergaard stress function and using the periodicity of the hyperbolic function, the antiplane problem of the periodic parallel cracks degenerates into an algebra problem. Using the complex variable function method and the undetermined coefficients method, as well as with the help of boundary conditions, the boundary value problem of partial differential equation can be solved, and the analytic expressions for stress intensity factor, stress, and displacement near the periodical parallel cracks tip are obtained. When the cracks spacing tends to infinity, the antiplane problem of the periodic parallel cracks degenerates into the case of the antiplane problem of a single central crack.