mode-III Stress Intensity Factors Research Articles

Mode-III external circular crack in an infinite orthotropic medium under torsional loading

This paper contains an investigation of interaction of torsional wave with a mode-III outer circular crack in an infinite orthotropic elastic medium. Using mixed boundary condition, the problem reduced into a dual integral equation. After some manipulation and use of Abel’s singular integral equation, the dual integral equation transformed into the Fredholm second kind integral equation, which is solved numerically. The obtained solution has been adopted to interpret the physical quantities like Crack Torsional Displacement (CTD) and mode-III Stress Intensity Factor (SIF) which are shown graphically.

Closed form solution for a semi-infinite crack moving in an infinite orthotropic material with a circular crack breaker under antiplane strain

This study investigates the influence of a circular crack breaker on mode-III deformation behavior of a semi-infinite crack in a homogeneous, elastic orthotropic material subjected to longitudinal shear loads. The Galilean transformation is employed to convert the governing wave equation to Laplace’s equation which is time independent, rendering the problem amenable to analysis within the realm of the classical theory of two-dimensional elasticity. Considering the geometrical configuration of the problem, the analytical solution of the problem is possible if the problem is transformed using the appropriate mapping function. Our construction of a holomorphic function that maps the circular hole into a straight line with the edge terminating at the origin is a novelty which enables the use of integral transform method to obtain an analytic solution of the displacement, leading to closed-form expression for mode-III stress intensity factor, K_{111}. The asymptotic values of the fields are obtained and shown to depend on the radius of the crack breaker. A parametric study shows that, for a fixed loading interval, a crack breaker of larger radius leads to increased stress intensity factor.

Three-dimensional mixed-mode stress intensity factors for deflected external surface cracks in thin and midsize-thick-walled spherical pressure vessels

Normalized three-dimensional mixed-mode stress intensity factor (SIF) distributions are presented for deflected external surface cracks in thin- and midsize-thick-walled spherical pressure vessels. With an aim to build a library of SIF solutions, values of the governing parameters of the problem, namely crack deflection angle, normalized crack depth and crack shape aspect ratio, are changed within their practical ranges. The obtained results show that for a given case, increasing the deflection angle yields a decrease in mode-I SIFs and increases in mode-II and mode-III SIFs. It is also observed that as the normalized crack depth increases, SIFs generally increase along the crack front. The results also show that magnitudes of the normalized mixed mode stress intensity factors at the depth location for midsize-thick-walled vessels are lower than those of thin-walled spherical vessels. The crack deflection angle has the highest effect among all parameters, emphasizing the importance of this study. It is concluded that the presented SIF solutions can be used to assess damage tolerance of a spherical vessel with a deflected external surface crack.

Failure Analysis of a Polygonal Void with an Oxide Layer in a Cracked Matrix

An oxide layer can commonly be found between a void in and the matrix of metal materials subjected to remote mechanical loading. This paper presents an evaluation of the failure behavior of a polygonal nano-void surrounded by an oxide layer in an infinite cracked matrix under a remote anti-plane load. Based on the complex variable, conformal mapping and analytical continuation with alternating methods, the stress functions are obtained in series form and the mode-III stress intensity factors (SIFs) are derived. The results indicate that a softer oxide layer (zirconia) would cause a higher SIF compared to a stiffer oxide layer (alumina). The results of this investigation provide a quantitative description of the effects of an oxide layer and inform an optimum design method for metal forming.

Mode-I, Mode-II, and Mode-III Stress Intensity Factor Estimation of Regular- and Irregular-Shaped Surface Cracks in Circular Pipes

Stress intensity factor (SIF) of surface cracks (semi-circular, semi-elliptical, and irregular shapes) located on a circular pipe (hollow) has been determined using numerical method. The irregular-shaped crack geometry was considered in the present work in order to simulate the cracks that are generally developed during service condition. The interaction behavior of multiple cracks on SIF calculation has also been investigated. Crack aspect ratios [a/c; “a” is the crack depth and where “c” is semi-major axis of elliptical crack] ranging between 0.6 and 1.0 and crack depth ratios [(a/d); where “d” diameter of the pipe] ranging between 0.1 and 0.4 were considered. Irregular-shaped cracks were modeled by keeping the crack depth (a) constant. For a regular profile semi-elliptic crack [(a/c) = 0.6], higher SIF was observed at crack middle region [(S/S0) = 0]. In contrast, higher SIF was observed at crack surface region [(S/S0) = ± 1] for semi-circular crack. Asymmetric spreading of SIF was noticed because of added effect of mode-II and mode-III fracture. In addition, the leading mode of fracture at short crack depth [(a/d) ≤ 0.1] was mode-II (in-plane shear) and mode-III (out of plane shear) fractures whereas at deep crack depths [(a/d) ≥ 0.4], the leading mode of fracture was mode-I fracture which is crack opening mode. Marginal influence of crack aspect ratio was noticed at crack surface region [(S/S0) = ± 1] and the effect is significant at crack middle region [(S/S0) = 0]. For a constant crack depth, higher SIF was obtained at the middle region [(S/S0) = 0] compared to regular profile crack. When comparing the observations of regular semi-elliptic crack, higher influence of mode-II and mode-III fracture was observed for irregular-shaped semi-circular crack which may leads to additional crack growth. Thus, approximating the standard crack geometry may over/under predict the SIF values. It is also inferred that the presence of adjacent cracks significantly affects the SIF of middle crack due to interaction effect. The interaction behavior was observed to be more pronounced when multiple cracks are located at closer interval. The interaction effect reduces gradually when the distance between crack centers increases. The improved SIF solutions obtained with consideration of additional fracture modes will be useful during the residual life determination.

Three-dimensional mixed-mode stress intensity factors for deflected internal surface cracks in thin and midsize-thick-walled spherical pressure vessels

In this paper, three-dimensional mixed-mode stress intensity factor solutions are obtained for deflected inner surface cracks contained in thin- and midsize-thick-walled spherical pressure vessels. In an effort to build a library of stress intensity factor solutions for different realistic cases, main parameters affecting the problem, which are crack deflection angle, normalized crack depth and crack shape aspect ratio, are changed systematically to cover all combinations. The results show that as the deflection angle increases, mode-I stress intensity factors decrease and mode-II and mode-III stress intensity factors increase for a given case. It is also observed that as the normalized crack depth increases, stress intensity factors increase along the crack front. The results show that magnitudes of the normalized mixed mode stress intensity factors for midsize-thick-walled vessels are higher than those of thin-walled spherical vessels. Crack deflection angle is shown to have the highest influencing effect among all parameters. It is also observed that especially for higher crack deflection angles, which have higher mode-mixity, crack surfaces may come into contact partially or fully.

Mode-III stress intensity factors for two circular inclusions subject to a remote uniform shear load

ABSTRACTAn analytical solution to the antiplane elasticity problem associated with two circular inclusions interacting with a line crack is provided in this article. A series solution for the stress field is derived in an elegant form by using complex variable theory in conjunction with the alternation method. Based on the superposition method, a singular integral equation (SIE) is established from the traction-free condition along the crack surface. After solving the SIE, the mode-III stress intensity factors (SIFs) can be obtained to quantify the singular behavior of the stress field ahead of the crack tips. Numerical results of the SIFs, when a crack is embedded either in the inclusion or in the matrix, are discussed in detail and displayed in graphic form.

Complex Potential Methods for a Crack and Three-phase Circular Composite in Anti-plane Elasticity

An interaction between an anti-plane crack with a three-phase circular composite by using complex potential methods is considered in this paper. The solution procedures for solving this problem consist of two parts. In the first part, based on complex potential methods in conjunction with analytical continuation theorem and alternating technique, the complex potential functions of a screw dislocation interacting with three-phase circular composites are obtained. The second part consists of the derivation of logarithmic singular integral equations by introducing the complex potential functions of screw dislocation along the crack border together with superposition technique. The logarithmic singular integral equations are then solved numerically by modeling a crack in place of several segments. Linear interpolation formulae with undetermined coefficients are applied to approximate the dislocation distribution along the elements, except at vicinity of the crack tip where the dislocation distribution preserves a square-root singularity. The mode-III stress intensity factors are then obtained numerically in terms of the values of the dislocation density functions of the logarithmic singular integral equations.

Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear

We investigate the anti-plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. The problem is first reduced to a non-homogeneous Riemann–Hilbert problem, the solution of which can be obtained by evaluating the associated Cauchy integral. A set of linear algebraic equations is obtained from the compatibility condition imposed on the resulting analytic function defined in the inhomogeneity and its Faber series expansion. Each of the unknown coefficients in the corresponding analytic functions can then be uniquely determined by solving the linear algebraic equations, which are written concisely in matrix form. The resulting analytical solution is then used to quantify the displacement jump across the debonded section of the interface as well as the traction distribution along the bonded section of the interface. In addition, our solution allows us to obtain mode-III stress intensity factors at the two crack tips. The solution to the anti-plane problem of a partially debonded elliptical inhomogeneity containing a confocal crack is also derived using a similar method.

A Shape-Topological Control Problem for Nonlinear Crack-Defect Interaction: The Antiplane Variational Model

We consider the shape-topological control of a singularly perturbed variational inequality. The geometry-dependent state problem that we address in this paper concerns a heterogeneous medium with a micro-object (defect) and a macro-object (crack) modeled in two dimensions. The corresponding nonlinear optimization problem subject to inequality constraints at the crack is considered within a general variational framework. For the reason of asymptotic analysis, singular perturbation theory is applied, resulting in the topological sensitivity of an objective function representing the release rate of the strain energy. In the vicinity of the nonlinear crack, the antiplane strain energy release rate is expressed by means of the mode-III stress intensity factor that is examined with respect to small defects such as microcracks, holes, and inclusions of varying stiffness. The result of shape-topological control is useful either for arrests or rise of crack growth.

Finite element modeling and experimental studies on mixed mode-I/III fracture specimens

In this study, finite element modeling and experimental studies on a mode-I/III specimen similar to the compact tension specimen are presented. By using bolts, the specimen is attached to two loading apparatus that allow different levels of mode-I/III loading by changing the loading holes. Specimens having two different thicknesses are analyzed and tested. Modeling, meshing and the solution of the problem involving the whole assembly, i.e., loading devices, bolts and the specimen, with contact mechanics are performed using ANSYSTM. Then, the mode-I/III specimen is analyzed separately using a submodeling approach, in which threedimensional enriched finite elements are used in FRAC3D solver to calculate the resulting stress intensity factors along the crack front. In all of the analyses, it is clearly shown that although the loading is in the mode-I and III directions, mode-II stress intensity factors coupled with mode-III are also generated due to rotational relative deformations of crack surfaces. The results show that the mode-II stress intensity factors change sign along the crack front and their magnitudes are close to the mode-III stress intensity factors. It is also seen that magnitudes of the mode-III stress intensity factors do not vary much along the crack front. Fracture experiments also performed and, using the stress intensity factors from the analyses and crack paths and surfaces are shown.

Interaction between a Crack and an Isotropic Tri-Material Media in Anti-Plane Elasticity

Solution of a crack interacting with a tri-material under a remote shear load for anti-plane elasticity problem is considered in this paper. The main purpose of this work is to study the interaction between a crack and a tri-material for anti-plane elasticity problem. This can be achieved by determination of the stress intensity factors that allow the characterization of this interaction from the point of view of linear elastic fracture mechanics. The proposed method is based on complex variable solution of a screw dislocation together with logarithmic singular integral equations. The singular integral equation is then solved numerically by modeling a crack in place of several segments. Some numerical results are performed to show the effects of material property combinations and geometric parameters on the normalized mode-III stress intensity factors. The results show that the stiffer materials may always give retardation effect on stress intensity factors when a crack approaching interfaces. On the other hand, the softer materials may always give enhancement effect on stress intensity factors.

Mode-III Stress Intensity Factors of an Arbitrarily Oriented Crack Crossing Interface in a Layered Structure

ABSTRACTThe problem of a layered structure containing an arbitrarily oriented crack crossing the interface in anti-plane elasticity is considered in this paper. The fundamental solution of displacements and stresses is obtained in a series form via the method of analytical continuation in conjunction with the alternating technique. A dislocation distribution along the prospective site of a crack is used to model a crack crossing the interface and the singular integral equations with logarithmic singular kernels for a line crack are then established. The crack is approximated by several line segments and the linear interpolation equation with undetermined coefficients was applied for the dislocation function along line segments. Once the undetermined dislocation coefficients are solved, the mode-III stress intensity factors KIII at two crack tips can be obtained for various crack inclinations with different material property combinations. All the numerical results are checked to achieve a good approximation that demonstrates the accuracy and the efficiency of the proposed method.

Anti-Plane Interaction between a Crack and an Elliptically Cylindrical Layered Media

AbstractAnti-plane interaction of an elliptically cylindrical layered media with an arbitrarily oriented crack embedded in an infinite matrix, intermediate layer, or inner inclusion under a remote uniform shear load is considered in this paper. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the solution for a screw dislocation in the inclusions and the matrix is first derived in a series form. The integral equations with logarithmic singular kernels for a line crack can easily be obtained by using the screw dislocation solutions as the Green's function together with the principle superposition. The stress intensity factors, which can properly reflect the interaction between a crack and an elliptically cylindrical layered media, are then obtained numerically in terms of the values of the dislocation density functions of the integral equations. The effects of material property combinations and geometric parameters on the normalized mode-III stress intensity factors are discussed in detail and shown in graphic form.

A note on stress intensity factors for a crack emanating from a sharp V-notch

A concise and precise formula for mode-I stress intensity factor of a short crack emanating from the bisector of a V-notch has been derived by Philipps et al. The crack tip stress intensity factor is directly evaluated via the mode-I generalized stress intensity factor of an uncracked notch. This note further supplements an accurate formula to calculate mode-III stress intensity factor for a short crack emanating from a V-notch.

Mode-III Stress Intensity Factors of a Three-Phase Composite with an Eccentric Circular Inclusion

Mode-III Stress Intensity Factors of a Three-Phase Composite with an Eccentric Circular Inclusion

Mode-III Stress Intensity Factor by Williams Element with Generalized Degrees of Freedom

Williams series are developed for mode III cracks, based on which the displacement field is defined in the singular region around the crack tip. The Williams element with generalized degrees of freedom (GDOFs) is proposed for analysis of stress intensity factor (SIF) of mode III crack. The SIF at the crack tip can be evaluated analytically by one of the undetermined constants of the Williams element. The influence of the relative length of the crack on the SIF is investigated. Three important parameters for the Williams element, including the radial scale factor, the number of subelements and the terms of the Williams series, are discussed in detail. Numerical example shows that the Williams element is of accuracy and efficiency.

Higher-Order Terms for the Mode-III Stationary Crack-Tip Fields in a Functionally Graded Material

This paper provides full asymptotic crack-tip field solutions for an antiplane (mode-III) stationary crack in a functionally graded material. We use the complex variable approach and an asymptotic scaling factor to provide an efficient procedure for solving standard and perturbed Laplace equations associated with antiplane fracture in a graded material. We present the out-of-plane displacement and the shear stress solutions for a crack in exponentially and linearly graded materials by considering the gradation of the shear modulus either parallel or perpendicular to the crack. We discuss the characteristics of the asymptotic solutions for a graded material in comparison with the homogeneous solutions. We address the effects of the mode-III stress intensity factor and the antiplane T-stress onto crack-tip field solutions. Finally, engineering significance of the present work is discussed.

Anti-plane interaction of a crack and reinforced elliptic hole in an infinite matrix

Anti-plane interaction of a crack with a coated elliptical hole embedded in an infinite matrix under a remote uniform shear load is considered in this paper. Analytical treatment of the present problem is laborious due to the presence of material inhomogeneities and geometric discontinuities. Nevertheless, based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, general expressions for displacements and stresses in the coated layer and the matrix are derived explicitly in closed form. By applying the existing complex function solutions for a dislocation, the integral equations for a line crack are formulated and mode-III stress intensity factors are obtained numerically. Some numerical examples are given to demonstrate the effects of material inhomogeneity and geometric discontinuities on mode-III stress intensity factors.

On a semi-infinite crack penetrating a piezoelectric circular inhomogeneity with a viscous interface

We investigate a semi-infinite crack penetrating a piezoelectric circular inhomogeneity bonded to an infinite piezoelectric matrix through a linear viscous interface. The tip of the crack is at the center of the circular inhomogeneity. By means of the complex variable and conformal mapping methods, exact closed-form solutions in terms of elementary functions are derived for the following three loading cases: (i) nominal Mode-III stress and electric displacement intensity factors at infinity; (ii) a piezoelectric screw dislocation located in the unbounded matrix; and (iii) a piezoelectric screw dislocation located in the inhomogeneity. The time-dependent electroelastic field in the cracked composite system is obtained. Particularly the time-dependent stress and electric displacement intensity factors at the crack tip, jumps in the displacement and electric potential across the crack surfaces, displacement jump across the viscous interface, and image force acting on the piezoelectric screw dislocation are all derived. It is found that the value of the relaxation (or characteristic) time for this cracked composite system is just twice as that for the same fibrous composite system without crack. Finally, we extend the methods to the more general scenario where a semi-infinite wedge crack is within the inhomogeneity/matrix composite system with a viscous interface.