Stress Crack Opening Displacement Research Articles

Analytical solution of two non‐identical edge cracks in an infinite strip under anti‐plane shear wave

AbstractThis article presents an extensive analytical solution addressing the interaction between two non‐identical edge cracks in an infinite orthotropic strip under anti‐plane shear waves. Most studies assume identical cracks or single edge crack in a strip, but this research breaks new ground by considering cracks of different sizes. By incorporating mixed‐type boundary conditions, the study derives dual integral equations. These equations are then transformed into a singular integral equation of Cauchy type with the aid of a trial solution and contour integration technique. The singular integral equation is further converted into a system of integral equations, which are solved numerically utilizing Jacobi polynomials. The obtained solutions are utilized to derive expressions for the stress intensity factor (SIF) and crack opening displacement (COD) at the crack tip using Krenk's interpolation formulae. The derived results are presented graphically and compared against existing solutions for single edge crack and symmetric edge cracks in static scenario.

Elastic fields of double branched and Kalthoff–Winkler cracks in a half-plane

We demonstrate in this paper a combination of the Schwarz–Christoffel mapping and Muskhelishvili’s approach with fractional function series in solving the elastic fields of a cracked half-plane, and zoom in on two typical problems, a double branched crack with two rays emanating from one point on the edge and two edge cracks spaced by a certain distance. Typical loading conditions are considered, including far-field uniform tensile stress and concentrated loads along either the tangential or the normal direction of the free surface. We supply a semi-analytic solution to those boundary-value problems in the cracked half-plane, and validate the theory by comparing the theoretical results in terms of stress fields, stress intensity factors (SIFs) and crack opening displacement (COD) with those from finite-element simulations. The theoretical approach shows how two edge cracks may shield the stress intensity factors of each other in a quantitative manner. For the typical Kalthoff–Winkler cracks of length a and being spaced by a distance d, their SIFs KI decay with decreasing d, and KI=KI0−KI1[1−exp(−a/d)]. It converges to KI0—the SIF of a single edge crack when d approaches to infinity. Those observations and the theory approach itself provide a general way to analyze the mechanical consequence of edge cracks in engineering practice.

On the dynamic mode-III crack in the elastic continuum consisting of sandy properties

The aim of the present investigation is to come up with a mathematical model for the dynamic mode III crack in an elastic continuum consisting of dry sandy properties. For the analysis of the propagating mode-III crack due to the propagating Love-type wave in an elastic continuum consisting of dry sandy properties, the results for the stress intensity factor (SIF) and crack opening displacement (COD) have been developed in closed form. The complex variable integral transform mathematical methods consisting of the two-sided Fourier integral transform, along with the Weiner-Hopf technique, serve as a salient feature of the adopted mathematical treatment for determining the SIF and COD in the closed form. Moreover, the expressions for the SIF and COD for the limiting static mode-III crack have also been obtained and deduced for special cases of elastic medium. The effects of existing parameters, viz. dry sandiness parameter, depth of crack, crack velocity on the SIF and COD . The effect of physical materials parameters has been shown numerically as well as graphically.

Analytical fracture parameters of two unequal collinear interface cracks in an orthotropic bimaterial

This paper presents an analytical solution for the fracture parameters of two unequal collinear interface cracks (UCIC) in an infinite orthotropic or isotropic bimaterial subject to in-plane loading. The analytical expressions for the stress intensity factors (SIF), Jk-integrals and crack opening displacements of two UCIC in an infinite orthotropic bimaterial plate are derived using the complex variable method and evaluated in conjunction with the Jacobian elliptic functions and elliptic integrals. The presented theory is verified using the boundary element method. The results show that the fracture parameters are greatly influenced by mismatch of the engineering constants and proximity of the cracks. The inner tip of the longer crack always experiences the maximum SIFs and Jk values. These fracture parameters will increase exponentially when the distance between the cracks inner tips decreases. This study provides a baseline for the verification of future numerical and experimental studies of two UCIC which is not currently available in the SIFs handbooks.

Moving semi-infinite crack between dissimilarorthotropic strips

In this paper, we conceive a semi-infinite moving crack between two bonded dissimilar orthotropic strips. Fourier transform has been adopted to reduce the mixed boundary value problem to a standard Wiener–Hopf equation. To determine the stress intensity factor (SIF) and crack opening displacement (COD), the Wiener–Hopf equation is solved in asymptotic sense for stress free boundary. Finally, the variation of SIF and COD is demonstrated graphically with respect to several orthotropic material parameters.

Mathematical modelling for multiple straight cracks with coalesced yield zones

Solution for a normally loaded infinite isotropic plate containing five cracks with coalesced yield zones is obtained using a complex variable method. Influence of coalesced yield zones on the load bearing capability of the infinite plate is analyzed. Analytical expressions for stress intensity factors, displacement components and crack opening displacement (COD) are obtained. Numerical study is carried out to determine the yield zone length, applied load ratio and COD. The numerical results are reported graphically between some important parameters.

Semi-infinite moving crack between two orthotropic strips

The problem of a semi-infinite moving crack at the interface of two bonded dissimilar strips of finite width has been solved. The crack is subjected to two constant normal displacements applied on the boundaries. The problem has been converted to one that is suitable for applying the Wiener–Hopf technique. Fourier transform technique has been used to reduce the mixed boundary value problem to the standard form of the Wiener–Hopf equation which has been solved in asymptotic form to obtain the analytical expressions of stress intensity factor and crack opening displacement. Finally the numerical values of stress intensity factor and crack opening displacement have been plotted graphically against various parameters to show the effects of these parameters on stress intensity factor and crack opening displacement.

Impact response of a finite crack in the presence of magnetic field

The present article studies the problem concerning impact load on a finite crack in the presence of a magnetic field on the crack surface in an infinite isotropic medium. Laplace and Abel’s transform are used to reduce the boundary value problem to a Fredholm integral equation of the 2nd kind. The integral equation is solved by the asymptotic series expansion method for low frequency. The expression of stress intensity factor (SIF) and crack opening displacement (COD) have been derived in the Laplace transform domain, and finally switched to the time domain with the help of Zakian’s Laplace inversion algorithm. The effect of magnetic field on SIF and COD has been discussed employing graphs. We found that a ferromagnetic substance such as Iron, Cobalt, Nickel, etc. is more capable to prevent the crack growth due to normal impact. Magnetic field can not prevent the crack growth in case of sliding mode.

Semi-infinite crack between two dissimilar orthotropic strips

The problem of a semi-infinite crack between two dissimilar orthotropic strips has been solved. Using the Fourier transform technique the problem has been converted to a standard Wiener-Hopf equation which has been solved for asymptotic cases only to deduce the expressions for the stress intensity factor and crack opening displacement. Finally the values of the stress intensity factor and crack opening displacement have been plotted graphically to show the effects of strip width and material orthotropy.

Riemann–Hilbert approach-based analytical solutions for strip saturated two unequal collinear cracks in piezoelectric media

The objective of the work is to derive analytical solutions based on the Riemann–Hilbert (R–H) approach for semipermeable strip saturated two unequal collinear cracks in arbitrary polarized piezoelectric media. We particularly consider the influence of far field electromechanical loadings, poling direction and different crack-face boundary conditions. The problem is mathematically formulated into a set of non-homogeneous R–H problems in terms of complex potential functions (related to field components) using complex variable and extended Stroh formalism approach. After solving these equations, explicit solutions are obtained for the involved unknown complex potential functions and hence, the stress and electric displacement components at any point within the domain. Furthermore, after employing standard limiting conditions, explicit expressions for some conventional fracture parameters such as saturated zone lengths (in terms of nonlinear equations), local stress intensity factors and crack opening displacement are obtained. Numerical studies are presented for the PZT-4H material to analyze the effects of prescribed electromechanical loadings, inter-cracks distance, crack-face conditions and poling direction on the defined fracture parameters.

Moving Three Collinear Griffith Cracks at Orthotropic Interface

This work deals with the interaction of P-waves between a moving central crack and a pair of outer cracks situated at the interface of an orthotropic layer and an elastic half-space. Initially, we considered a two-dimensional elastic wave equation in orthotropic medium. The Fourier transform has been applied to convert the basic problem to solve the set of four integral equations. These set of integral equations have been solved to to get the analytical expressions for the stress intensity factor (SIF) and crack opening displacements (COD) by using the finite Hilbert transform technique and Cooke’s result. The main objective of this work is to investigate the dynamic stress intensity factors and crack opening displacement at the tips of the cracks. The aims of the study of these physical quantities (SIF, COD) is the prediction of possible arrest of the cracks within a certain range of crack velocity by monitoring applied load. SIF and COD have been depicted graphically for various types of orthotropic materials. We presented a parametric study to explore the influence of crack growing and propagation. This result is very much applicable in bridges, roads, and buildings fractures.

Weight function method and its application for orthotropic single edge notched specimens

The crack opening displacement-based weight function method (WFM) for isotropic material is extended to analyze crack in orthotropic material for the first time. Explicit weight function for an edge crack in an orthotropic strip is provided in this paper. After thorough verifications against results from finite element analyses (FEA) and existing literature data, the weight function is used to determine the stress intensity factors (SIF) and crack opening displacements (COD) for three-point bending, eccentrical point load and segment bridging stress. The results agree very well with those obtained from extensive FEAs and existing literature. The dependences of the SIF and crack mouth opening displacement (CMOD) on the material orthotropy are quantitatively determined. By comparisons to the SIF and CMOD for the edge cracked strip in isotropic material, it is found that the material orthotropy should not be ignored for some cases. The present weight function is highly accurate, efficient and versatile for calculating SIFs and CODs for edge cracks in orthotropic materials.

Study of Semi-Infinite Crack in a Sandwiched Orthotropic Strip

ABSTRACTThe problem of semi-infinite crack situated in an orthotropic strip sandwiched between two identical half planes has been considered. The considered boundary value problem has been solved to convert it into the standard Wiener-Hopf equation by using Fourier transform technique, which has been solved to obtain analytical expressions for stress intensity factor and crack opening displacement. The variations of stress intensity factors and crack opening displacement are displayed graphically for different pair of orthortropic materials and for various depth of the strip of the composite media.

Semi‐infinite moving crack in an orthotropic strip sandwiched between two identical half planes

AbstractThe problem of semi‐infinite moving crack situated in a strip of a linear elastic orthotropic material sandwiched between two identical half planes has been considered. The assumed boundary value problem has been solved to convert it into the standard Wiener‐Hopf equation by using Fourier transform technique. Further the Wiener‐Hopf equation has been solved to obtain analytical expression for stress intensity factor and crack opening displacement at the tip of the crack for plane stress and plane strain cases. The effects of elastic material constants and propagation of the crack tip on stress intensity factor and crack opening displacement at the tip of crack have been displayed graphically for various particular cases for a combination of orthotropic materials.

A review and verification of analytical weight function methods in fracture mechanics

AbstractThe weight function method is a powerful method for the determination of key parameters of stress intensity factors and crack opening displacements that are fundamental for crack analyses in fracture mechanics. The method has a remarkable computational efficiency without compromising solution accuracy and is especially suited for cracks in real‐world components with complex stress gradients. A critical issue for successful development and reliable application of the weight function method is how to achieve verifiable and high solution accuracy. This paper provides several reflections on the current state of the art in weight function methods, including a brief historical perspective and description of several analytical and numerical weight methods, with main focus on the two‐dimensional (2D) analytical approaches. A variety of typical application examples are presented. A platform for point‐by‐point accuracy assessment by using the Green's functions is established on a completely independent numerical approach with validated accuracy. Rigorous and systematic verification of the accuracy levels of different 2D analytical weight function approaches are conducted using several benchmark crack geometries. The sources of inaccuracy of different analytical weight function methods are analysed. A brief account on weight function methods for analysing three‐dimensional (3D) crack problems with arbitrary univariant and bivariant stress distributions is provided.

Semi-infinite moving crack between two bonded dissimilar isotropic strips

The problem of a moving semi-infinite crack between two bonded dissimilar isotropic strips has been considered. The mixed boundary value problem has been reduced to a standard Wiener–Hopf equation by the use of Fourier transform technique. The Wiener–Hopf equation has been solved in asymptotic cases to obtain the required expressions of the stress intensity factor and crack opening displacement. The effects of various parameters on the stress intensity factor and crack opening displacement have been shown by virtue of the graphs.

Weight functions and stress intensity factors for two unequal-length collinear cracks in an infinite sheet

The configuration of two unequal-length collinear cracks is common in aircraft structures containing multiple site damage. In this paper, the weight functions for this crack configuration are derived from the exact crack opening displacements under remote tension, and verified by reducing to the existing exact weight functions of two equal-length collinear cracks and by good agreement with finite element analysis. These weight functions are used to calculate the stress intensity factors and crack opening displacements for different load cases. The results agree very well with those from literature and finite element analysis. The weight functions for two unequal-length cracks provide a highly efficient and accurate tool for damage tolerance analysis of structure with multiple site damage.

Hybrid effects of steel fiber and carbon nanotube on self-sensing capability of ultra-high-performance concrete

This study aims to examine the feasibility of using steel fibers and carbon nanotubes (CNTs) on developing self-sensing ultra-high-performance concrete (UHPC). For this, four steel fiber types with two different shapes (straight vs. twisted) and three different aspect ratios from 65 to 100 were considered at a fiber volume fraction of 2%. 0.5% by volume of CNTs were simultaneously added. A plain UHPC with only 0.5% CNTs was also fabricated and tested as a control specimen. Test results indicated that the addition of 2% steel fibers was effective in enhancing compressive strength, elastic modulus, tensile strength, and strain capacity of the plain UHPC. The compressive behaviors of UHPC and ultra-high-performance fiber-reinforced concrete (UHPFRC) with CNTs were not predicted based on a fractional change in resistance (FCR) measurement, whereas the total tensile behaviors in terms of both stress-strain and stress-crack opening displacement (COD) curves were quite well simulated based on the FCR measurement and curve-fitting equations suggested. The unintended noise in the FCR of the plain UHPC was largely mitigated by adding the steel fibers, while the highest gauge factor (GF) under tensile load was found for the plain UHPC with CNTs. Using micro steel fibers was more effective in increasing the GF than using macro steel fibers, and better predictive results were obtained for the UHPFRC with straight steel fibers as compared to that with twisted steel fibers.

Review and evaluation of weight functions and stress intensity factors for edge-cracked finite-width plate

Various analytical and numerical weight functions (WFs) for the edge-cracked finite-width plate (ECFWP) configuration have been developed in the literature for efficient fracture mechanics analyses involving complicated/arbitrary load conditions. The objective of this paper is to make a comprehensive review and critical evaluation of the existing WFs determined by using various approaches, e.g. singular integral equation methods; one/two reference load case(s) based analytical approaches; fitting and interpolation to handbook solutions; numerical methods such as the complex Taylor series expansion and finite element analysis. The accuracy levels of existing WFs are assessed in rigorous manner by benchmarking the corresponding Green’s functions (GFs) against well-recognized accurate solutions. A highly accurate and wide-ranging analytical ECFWP WF is presented in closed form. Stress intensity factor and crack opening displacement solutions for various load cases of practical interest are easily determined with high accuracy by using the well-verified analytical ECFWP WF for this crack configuration to demonstrate the methods.

Evaluation of various analytical weight function methods base on exact K-solutions of an edge-cracked circular disc

Weight function method is a powerful tool for the determination of key parameters such as stress intensity factors and crack opening displacements for cracks in complicated stress fields. Accurate determination of the weight functions is of primary importance for successful applications of the method. This paper makes detailed accuracy evaluation of two kinds of widely used approximate (2D) weight function approaches, one being based on crack opening displacement for one reference stress, the other on two reference stresses coupled with a geometric condition. To eliminate possible sources of error, the edge-cracked circular disc with exactK-solutions is used for deriving the two kinds of weight functions and for benchmarking against the two weight function approaches. A highly accurate numerical weight function method using the complex Taylor series expansion and complex finite element computation is also used for assessing the related Green’s functions. Results from the evaluation will be instructive for accurate determination of weight functions for other finite edge-crack geometries.