Griffith Crack Research Articles

Analysis of stress intensity factor for moving Griffith crack in a transversely isotropic strip under punch pressure

Analysis of stress intensity factor for moving Griffith crack in a transversely isotropic strip under punch pressure

Does the capacitor analogy model in fracture mechanics of elastic dielectrics constitute an appropriate approximation?

The analytical solution of an elliptic dielectric cavity in an infinite dielectric plate is taken as basis to investigate a Griffith crack problem which is obtained by letting the semi-minor axis tend towards zero. In the course of this, an erroneous conformal mapping, commonly employed in literature and correctly reproducing the electric field only in a part of the physical space, is rectified. Interpreting the elliptic interface as faces of a mechanically opened crack which is exposed to an oblique remote electric field, surface charges and electrostatic tractions are calculated. In contrast to the established capacitor analogy model, approximately yielding electric charge densities and Coulombic tractions from displacements and electric potentials in the undeformed crack configuration, the work at hand provides exact solutions accounting for different implications of the crack curvature and for the inclination of the electric field. Crack weight functions are finally used to calculate stress and electric displacement intensity factors. As turns out, the surface charges of the capacitor analogy represent an excellent substitute for the exact electric boundary conditions within a relevant range of parameters, whereas inaccurate Coulombic tractions in the vicinity of crack tips may lead to a significantly overestimated mode I stress intensity factor.

A study of Griffith crack in nonlocal infinite magneto-elastic media

The main objective of this article is to study how a finite crack interacts with horizontally polarized shear (SH) waves in a nonlocal isotropic medium when a magnetic field is present. The study aims to understand the effects of various parameters, including the magnetic field, non-locality, frequency of SH waves, on the behavior of the crack. Here, we are concerned with Griffith’s theory of a brittle fracture crack mechanism. The article starts by formulating the governing equations for the propagation of SH waves in the given medium. These equations are converted into dual integral equations using Fourier transformation. Subsequently, an Abel transformation is applied to obtain a Fredholm integral equation of the second kind. Daubechies wavelet family has been used to find an approximate solution for the integral equation. The article aims to determine specific physical quantities including stress intensity factor (SIF), shear stress, and crack energy related to the behavior of the crack. These parameters are essential in understanding how the crack responds to the external conditions. The application of Daubechies wavelets in this context is a relatively new and innovative approach. The study performed a comprehensive analysis of the effects of various factors on the crack behavior. The findings are presented in graphical form to help illustrate the conclusions. To assess the accuracy of the proposed method, the obtained results are compared with known results from the literature for local media. This is an important step in demonstrating the reliability and effectiveness of the proposed approach.

The Problem Solution on the Propagation of a Griffith Crack Based on the Equations of a Nonlinear Model

The Problem Solution on the Propagation of a Griffith Crack Based on the Equations of a Nonlinear Model

Bone crack inspired pair of Griffith crack opened by forces at crack faces

The mathematical theory of elasticity helps in the study of physical quantities in the problem of structures. The structures face the problem of crack’s presence, which makes the problem difficult but not impossible to deal. Integral equations are useful in a variety of problems. Integral equations are used to solve problems like fracture mechanics or fracture design. The physical interest in the fracture design criterion is due to stress and crack opening displacement components. We have an accurate form of stress and displacement components for a pair of longitudinal crack propagations in the bone fracture of the human body at the interface of an isotropic and orthotropic half-space that are bounded together in the proposed study. The expression was calculated using the Fourier transform approach near the crack tips, but these components were evaluated using Fredholm integral equations and subsequently reduced to coupled Fredholm integral equations. We employ the Lowengrub and Sneddon problem in this research and reduce it to triple integral equations. The Srivastava and Lowengrub method reduces the solution of these equations to a coupled Fredholm integral equation. The problem is further reduced to a decoupled Fredholm integral equation of the second kind. Triple integral equations are solved, and the problem is reduced to a coupled Fredholm integral equation. The Fredholm integral equation is solved and reduced to a decoupled Fredholm integral equation of the second kind. Stress and crack opening displacement components drive physical interest in fracture design criteria. Finally, the stress and displacement components may be simply calculated in their exact form.

The Problem Solution on the Propagation of a Griffith Crack Based on the Equations of a Nonlinear Model

On the basis of a nonlinear model of deformation of a crystalline medium with a complex lattice, the problem of the stationary propagation of a Griffith crack under the action of homogeneous expanding stresses is posed and solved. It is shown that the stressed and deformed states of the medium are determined both by external influences on the medium and by the gradients of the optical mode (mutual displacement of atoms). The contributions from these factors are separated. Finding the components of the stress tensor and macro-displacement vector is reduced to solving Riemann–Hilbert boundary value problems. Their exact analytical solutions are obtained.

On analytical study of Griffith crack propagation in a transversely isotropic dry sandy punch pressured strip

Abstract The present study focuses on the propagation of a Griffith crack in an infinitely extending finitely thick transversely isotropic dry sandy strip with moving parallel punch pressure acting at the bounding surface of the layer because of the propagation of plane waves under mechanical point loading. Furthermore, the present model is developed using coupled singular integral equations, Dirac delta function & Cauchy-type singularities; it is applied to investigate the point load at the moving crack edge, and Hilbert transformation properties are also used to derive the stress intensity factor (SIF) with constant point loading in closed form. Numerical computation & graphical demonstrations have been provided to examine the influence of the prevailing parameters, viz. length & speed of the crack, pressure of the punch, sandiness parameter and different positions of point load, on the SIF for transversely isotropic dry sandy and also for isotropic material strips. A comparative study of the SIF at the tip of moving crack has been carried out for the transversely isotropic dry sandy strip and isotropic materials strip with and without punch pressure and sandiness to highlight some of the important peculiarities of the problem.

Mode-I Plane Elasticity Problem of Two Asymmetrical Edge Cracks Emanating from an Elliptical Hole in Two-Dimensional Decagonal Quasicrystals

We consider the plane elasticity problem of two asymmetrical edge cracks emanating from an elliptical hole in two-dimensional decagonal quasicrystals (QCs) under remotely uniform tensile stress. A complex variation method of two-dimensional QCs is developed to solve the plane elasticity problem of two-dimensional decagonal QCs containing complex defects. The analytical solutions for the stress field and the stress intensity factors near the crack tip are expressed by using a conformal mapping technique and complex potential theory. Some special cases of the results are also obtained, such as the T-type crack, cross crack, and Griffith crack. The effects of geometrical parameters of crack configuration on the stress intensity factors are presented graphically.

Mathematical analysis on the propagation of Griffith crack in an initially stressed strip subjected to punch pressure

The purpose of this study is to analyze the features of a moving Griffith crack in an initially stressed infinitely long and finitely thick isotropic strip with moving parallel punches of constant load acting on its boundaries owing to plane wave propagation under point loading. Coupled singular integral equations and singularities of the Cauchy-type are used to formulate the present model, Dirac delta function is employed to analyze point load located at the moving crack edge, and Hilbert transformation properties are used for obtaining stress intensity factor (SIF) with constant point loading. Numerical simulations and graphical illustrations are performed to analyze the influences of the prevalent parameters, viz. initial stresses, punch pressure, distinct positions of point load, length and speed of the crack on the SIF for the considered isotropic material strip.

A Griffith crack model in a generalized nonhomogeneous interlayer of bonded dissimilar half-planes

A Griffith crack model in a generalized nonhomogeneous interlayer of bonded dissimilar half-planes

Non-Fourier thermoelastic interaction of two collinear cracks in a functionally graded layer

Designed with a predetermined composition profile, functionally graded materials receive much attention in serving as the thermal barrier coating to resist severe thermal loading. By means of the dual-phase-lag theory, the non-Fourier effect is considered in establishing the theoretical thermoelastic model of the cracked layer under sudden thermal shocks. The novelty of the present work lies in revealing the interaction effects of two collinear Griffith cracks embedded in a functionally graded layer on the transient temperature and thermal stress fields under harsh conditions. Via employing the Fourier sine and cosine transforms, coupled with the Laplace transform, the dynamic mixed mode crack problem is transformed into a group of Cauchy-type singular integral equations. Numerical calculations are implemented to evaluate the transient temperature and stress intensity factors. The influences of the crack spacing, the nonhomogeneous parameters, and the thermal lags on the thermal and stress concentrations are explored. The results indicate that both mode I and II thermal stress intensity factors of the outer crack tips are noticeably higher than those of the inner crack tips, even up to 71% and 128%, respectively. A larger fracture risk may occur with the closer two adjacent cracks.

Mechanical analysis of the interaction between a semi-infinite Griffith crack and its coplanar plastic zone

This paper presents the complete analytical solution of the equilibrium conditions between a dislocation pile-up and a loaded crack. Contrary to previous models, the image distributions (forces) are included in this calculation. Their influence on the effective stress intensity factor, on the dislocation nucleation conditions and on the criteria for crack propagation are underlined. In semi-brittle materials the size of the dislocation free zone is controlled by material characteristics, as expected from the literature. However, in ductile materials its size is controlled by a mathematical limitation resulting from introduction of image forces. If plastic deformation takes place, the Griffith criteria must be modified by adding additional work required to separate dislocations from their images to the surface energy and the plastic work done to develop the plastic zone. A detailed analysis of the stress distribution inside and outside the pile-up is presented. Within the plastic zone the total stress is not constant while outside and close to the limits a square root variation with distance is noticed. RésuméCet article présente la solution analytique complète des conditions d’équilibre d’un empilement de dislocations en présence d’une fissure chargée. Les effets des distributions (forces) image sont inclus, contrairement aux modèles disponibles dans la littérature. Leur influence sur le facteur d’intensité des contraintes effectif, sur les conditions de nucléation des dislocations et sur le critère de propagation de la fissure est soulignée. L’hypothèse d’une enclave élastique dont la taille est fixée par les caractéristiques physiques de la matière n’est valable que pour les matériaux semifragiles ; elle est limitée par une condition mathématique induite par les forces images dans le cas des matériaux ductiles. En présence de déformation plastique, le critère de Griffith doit être modifié par l’adjonction à l’énergie de surface et au travail plastique d’un travail additionnel effectué lors de la séparation des dislocations et de leurs images. Une analyse détaillée de la répartition des contraintes dans et hors de l’empilement est présentée. A l’intérieur de la zone plastique la contrainte totale n’est pas constante. A l’extérieur, à proximité de ses limites, elle varie comme la racine carrée de la distance

Ligament size effect in largely cracked tensile structures

The influence of the ligament size on the brittle failure behaviour of largely cracked elements is analysed. To this aim, the following tensile infinite geometries are taken into account: (i) a double edge cracked plate; (ii) a cylindrical bar with an external circular crack. These configurations can be considered as complementary to the Griffith crack and the Penny shaped crack, respectively. The failure size effect is investigated through a semi-analytical approach by two different methods: the coupled Finite Fracture Mechanics (FFM) criterion and the Cohesive Crack Model (CCM) implementing a Dugdale type cohesive law. Theoretical predictions are compared with each other and with experimental data obtained by tensile testing PMMA samples containing two collinear cracks. Strength estimations reveal accurate.

STATISTICAL NATURE OF STRENGTH AND SCALE EFFECT IN MOUNTAIN ROCKS

The theories and concepts of strength considered above are based on the model of a body either as a homogeneous structureless medium, or as a material having a structure, but uniform throughout its volume. Rocks are obviously not such bodies. They are composed of mineral grains of different properties, contain macrodefects in the form of pores and various inclusions, as well as objects of various aggregate states (gases, liquids). Under these conditions, deterministic theories of strength turn out to be clearly untenable. In particular, the use of the classical theory of Griffith cracks is complicated by the following circumstance. Since the rock is an aggregate of mineral grains, a microcrack developing inside the grain inevitably reaches its boundary and, consequently, the radius r of the crack mouth increases abruptly. Therefore, for the transition of a crack to another grain and its further development, a stress greater than that follows from Griffith's theory is required. Thus, there is some "barrier" stress, at which only the development of a crack in a real rock is possible. In addition, the development of cracks in the rock occurs mainly along the contact of mineral grains, i.e., along the cementing material, often of a clay composition. For such a material, the theory of brittle fracture is applicable. The destruction of the rock (from the standpoint of any theory of strength) is determined by the stresses acting in it. But due to the heterogeneous structure of the rocks, the local stress concentration centers are randomly distributed in its volume. Therefore, the strength and destruction of rocks must be considered from a statistical standpoint. This approach is justified for most other materials used by humans. The idea of the statistical nature of strength was first put forward in scientific terms by A.P. Aleksandrov and N.S. Zhurkov in 1933. Keywords: rock strength, constant material, scale effect, rocks, destruction probability, fractured, robustness theory, microcrack, mineral, experience constant, density of defects, displacement, compression.

Griffith crack analysis in nonlocal magneto-elastic strip using Daubechies wavelets

The main focus of this work is to employ the Daubechies wavelet approximation to analyze the Griffith crack in nonlocal magneto-elastic horizontally shear (SH) wave propagation. It is assumed that the crack is located in a homogeneous isotropic nonlocal magneto-elastic strip of finite thickness and infinite length. The governing mixed boundary value problem is converted to a dual integral equation by employing Fourier transformation. This dual integral equation is further converted to a Fredholmn integral equation of the second kind by means of Abel transformation. A wavelet-based approximation scheme is employed to solve the integral equation as it is not solvable analytically. Griffith crack analysis using Daubechies wavelets has not been studied previously in the literature. The stress intensity factor (SIF) at the crack tip is determined numerically after solving the integral equation. In the end, the effect of the magnetic field, strip breath, incident angle of the shear wave and the nonlocal parameter on stress intensity factor (SIF) at the crack tip is explained through various graphical representations.

SIF and DF of a Griffith crack problem bounded together isotropic and orthotropic medium

The motive why most materials break suddenly is that they have tiny internal and surface fissures that spread when tension is applied. The development and presentation of reinforced composite materials' properties has received a lot of effort. Composite materials have a very promising future. We will continue to see broader use of composite materials for advanced constructions. In order to properly analyse the cracking mechanisms of these structures, which are being used in engineering buildings using composite materials at an ever-increasing rate. The Griffith crack is the most intricate and peeling is the simplest of the numerous forms in which cracks can spread. After the First World War, Alan Griffith created it as an engineer at the Royal Aircraft Establishment in Farnborough. This paper explores the terms “stress intensity factor,” and “displacement factor” of this particular type of fracture at the intersection of two medium half-space bound together, there is a Griffith Crack.

MATHEMATICAL AND MECHANICAL RELATIONSHIP OF CRACKS GRIFFITH AND A NEW TYPE IN SOME NANOTECHNOLOGIY

In this paper, for the first time, mathematically strictly, models of Griffiths cracks and cracks of a new type are constructed simultaneously. Accurate models are constructed as a result of the convergence of the ends of two semi-infinite deformable stamps located on a deformable multilayer base. The mechanism of destruction of the medium by cracks of a new type is radically different from the mechanism of destruction of the medium by Griffiths cracks, and has so far been poorly studied. Griffiths formed his cracks with a smooth border as a result of compression from the sides of an elliptical cavity in the plate. Cracks of a new type have a piecewise smooth border, resulting from the replacement of an ellipse with a rectangle compressed from the sides. The construction of models in the work is based precisely on these descriptions of the origin of cracks formed by the ends of approaching deformable stamps. When creating models, the results of previously performed studies on the construction of exact solutions to a number of integral equations of contact problems, published in this journal, are used. The paper found that the destruction of the medium, in the case of cracks of a new type, occurs earlier than in the case of Griffith cracks, which, according to the authors, explains the reason for the destruction of Griffith cracks in the experiment earlier than the theory predicts. Griffiths explained this by the presence of microcracks on the border of the main crack. The authors tend to believe that the cause is the transformation of one type of crack into another, under certain conditions. The results obtained make it possible to compare and quantify the behavior of the medium in the presence of both one and another type of cracks. It is applicable both for the study of mutual transitions of Griffith cracks and cracks of a new type, and for the construction of models of adhesion of wetted nanoparticles.

Mathematical study on the propagation of Griffith crack in a dry sandy strip subjected to punch pressure

The present study investigates the features of a moving Griffith crack in a dry sandy infinitely long and finitely thick isotropic strip with moving parallel punches of constant load acting on both of its boundaries owing to plane wave propagation under point loading. Coupled singular integral equations and Cauchy-type singularities are used to formulate the present model, the Dirac delta function is employed to analyse the point load located at the moving crack edge, and the properties of Hilbert transformation are employed to obtain the closed-form expression of the stress intensity factor with constant point loading. For the sake of validation, the obtained results are matched with pre-established and standard results. Numerical simulations and graphical illustrations are performed to analyse the influences of the prevalent parameters, viz., sandiness parameter, punch pressure, distinct point load positions, length and speed of the crack corresponding to the plane wave on the stress intensity factor for the considered dry sandy isotropic material strip.

The Griffith Crack and the Interaction between Screw Dislocation and Semi-Infinite Crack in Cubic Quasicrystal Piezoelectric Materials

The Griffith crack problem and the interaction between screw dislocation and semi-infinite crack in cubic quasicrystal piezoelectric materials are studied by using the complex variable function method. The stress intensity factors and electric displacement intensity factors are obtained. The effects of the linear force and coupling elastic coefficient on the stress intensity factor of phonon field and phason fields are discussed in detail. By numerical examples, it is found that the linear force and the coupling elastic constant have a significant effect on the stress intensity factor.

A Griffith Crack at the Interface of an Isotropic and Orthotropic Half Space Bonded Together

In the past 53 years, many efforts have been contributed to develop and demonstrate the properties of reinforced composite materials. The ever-increasing use of composite materials through engineering structures needs the proper analysis of the mechanical response of these structures. In the proposed work, we have an exact form of Stress components and Displacement components to a Griffith crack at the interface of an Isotropic and Orthotropic half-space bounded together. The expression was evaluated in the vicinity of crack tips by using Fourier transform method but here these components have been evaluated with the help of Fredholm integral equations and then reduce to the coupled Fredholm integral equations. In this paper, we use the problem of Lowengrub and Sneddon and reduce it to dual integral equations. Solution of these equations through the use of the method of Srivastava and Lowengrub is reduced to coupled Fredholm integral equation. Further reduces the problem to decoupled Fredholm integral equation of 2nd kind. We get the solution of dual integral equations and the problem is reduced to coupled Fredholm integral equation. We find the solution of the Fredholm integral equation and reduce it to decoupled Fredholm integral equation of 2nd kind. The Physical interest in fracture design criterion is due to Stress and crack opening Displacement components. In the end, we can easily calculate the Stress components and Displacement components in the exact form.