Analytical solution for orthotropic composite plate containing a mode I crack along principle axis

Analytical stress analyses are presented for orthotropic composite materials containing a through crack under uniaxial normal loads (mode I). A harmonic differential equation has been established for the orthotropic plates with axes normal to the three orthogonal planes of material symmetry by introducing new complex variables. The complex theory was employed to find stress functions to satisfy the equilibrium equation, compatibility equation and boundary condition at infinite and crack surfaces. An analytical solution was examined in the case of isotropic materials. It is demonstrated that the analytical solution obtained is correct for the orthotropic composite plates.     

An orthotropic sandwich plate containing a part-through crack under mixed mode deformation

An orthotropic sandwich plate containing a part-through crack subjected to in-plane mixed mode tractions is analyzed, and the basic equations for estimating the stress intensity factors $\text{k}{}_1$, $\text{k}{}_2$ and the debonding naion around the crack are presented.     

Analytical solution for the singular stress distribution due to V-notch in an orthotropic plate material

On the basis of general solutions of two-dimensional linear elasticity, displacement and singular stress fields near the singular point in orthotropic materials are derived in closed form expressions. According to the presented expressions, analysis formulas of displacement and singular stress fields near the tip of a V-notch under the symmetric and the anti-symmetric modes are obtained subsequently. The open literatures devoted to developing stress singularity near the tip of the V-notch in anisotropic or orthotropic materials. In this study, however, not only direct eigenequations were derived, but also the explicit solutions of displacement and singular stress fields were obtained. At the end, the correctness of the formulas of the singular stress field near the tip of the V-notch has been verified by FEM analysis.     

A mode II crack problem with sliding contact in an orthotropic strip

The problem of a centrally cracked, linear elastic orthotropic strip loaded in bending by three point forces is analyzed and discussed. Coulomb friction is assumed between the crack faces to study the influence of the friction coefficient on the strain energy release rate. Under certain simplifying assumptions the problem is reduced to the solution of a singular integral equation which is evaluated numerically. The results are compared with the solution of the same problem obtained using the beam theory; limits of the application of beam theory for the reduction of experimental data are discussed.     

Closing of wedged crack in orthotropic viscoelastic composite

Quasi-static crack closing in a linear viscoelastic orthotropic composite with a crack which is opened by the wedge of constant width is analyzed. The use of superposition allows us to develop exact equations for predicting free crack length as a function of time. These equations are solved numerically using operator continued fractions method. As an example of the proposed approach some results for the composite of silicate glass fibers and epoxy resin binder are presented.     

Dynamic stress intensity factors around two parallel cracks in an infinite elastic plate

Summary Dynamic stresses around two parallel cracks in an infinite elastic plate are obtained. An incoming shock stress wave impinges on the cracks at right angles to their faces. The Fourier-Laplace transform technique is utilized to reduce the problem to dual integral equations. To solve these equations, the differences in the crack surface displacements are expanded in a series of functions which are zero outside the cracks. The unknown coefficients occurring in those series are solved using the Schmidt method. The stress intensity factors defined in the Laplace transform domain are inverted numerically, in the physical space.     

Crack tip rotation and shear strain on crack edge of mode II crack in anisotropic plate

Rigid body rotation is obtained at the points near the tip of a mode II crack in an infinite anisotropic plate. From Lekhnitskii's complex analysis the rotation is derived in terms of complex potentials and the material parameters. A relation of crack tip rotation is proposed by incorporating the mode II stress intensity factor and material constants of an anisotropic plate. The shear strain on the crack edge is calculated on the basis of rotation at the points lying on the crack edge.     

Steady-state propagation of a mode II crack in couple stress elasticity

The present work deals with the problem of a semi-infinite crack steadily propagating in an elastic body subject to plane-strain shear loading. It is assumed that the mechanical response of the body is governed by the theory of couple-stress elasticity including also micro-rotational inertial effects. This theory introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying microstructure and has proved to be very effective for modeling complex microstructured materials. It is assumed that the crack propagates at a constant sub-Rayleigh speed. An exact full field solution is then obtained based on integral transforms and the Wiener–Hopf technique. Numerical results are presented illustrating the dependence of the stress intensity factor and the energy release rate upon the propagation velocity and the characteristic material lengths in couple-stress elasticity. The present analysis confirms and extends previous results within the context of couple-stress elasticity concerning stationary cracks by including inertial and micro-inertial effects.     

The interacting stress field around a composite crack with a misfitting inclusion

The problem of a composite crack interacting with a circular misfitting inclusion in an infinite elastic medium is investigated. The close-formed solutions of the stress fields in the inclusion and the matrix are obtained by using the dislocation model of cracks and the point force method as well as the complex function technique. The stress intensity factors at two tips of the crack are calculated from the stress components.     

On closed form solution for the elastic stress field around holes in orthotropic composite plates under in-plane stress conditions

The present paper presents analytical work performed care of the Aerospace Department of the University of Rome ‘La Sapienza’. Using the classical Airy's function solution method the expression of the stress-strain field, in a composite material, has been determined for different simple hole geometries in a closed form. The results have been compared with several extensive finite element calculations that confirm the accuracy of the theoretical solutions found.     

Dynamic stress intensity factors around a rectangular crack in an infinite plate under impact load

A three-dimensional solution is presented for the transient response of an infinite plate which contains a rectangular crack. The Laplace and Fourier transforms are used to reduce the problem to a pair of dual integral equations. These equations are solved with the series expansion method. The stress intensity factors are defined in the Laplace transform domain, and they are inverted numerically in the physical space.     

Stress fields around a crack lying parallel to a free surface

A method of stress analysis for a two dimensional crack, which is subjected to internal gas pressure, and situated parallel to a free surface of a material, is presented. It is based on the concept of continuously distributed edge dislocations of two kinds, i.e. one with Burgers vector normal to the free surface and the other with Burgers vector parallel to it.Stress fields of individual dislocations are chosen so as to satisfy stress free boundary conditions at the free surface, by taking account of image dislocations. Distributions of both kinds of dislocations in the crack are derived so as to give the internal gas pressure and, at the same time, to satisfy shear stress free boundary conditions on the crack surface. Stress fields _xx, _yy and _xy in the sub-surface layer are then determined from them. They have square root singularities at the crack-up.

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Résumé On présente un méthode d'analyse de contrainte dans le cas d'une fissure bidimensionnelle soumise à la pression d'un gaz interne et située parallèlement à une surface libre d'un matériau. La méthode est basée sur un concept de dislocation-coin distribuée de manière continue et de deux types, c'est-à-dire l'une présentant un vecteur de Burgers normal à la surface libre et l'autre comportant un vecteur de Burgers parallèle à cette surface libre.On choisit le champ de contrainte de dislocations individuelles de telle manière qu'il satisfasse à des conditions de frontières libres de contrainte à la surface libre et en tenant compte des dislocations images. On en tire les distributions des deux types de dislocation dans la fissure de manière à fournir la pression interne des gaz et, dans le même temps, à satisfaire les conditions d'absence de contraintes de cisaillement aux frontières, sur la surface de la fissure. On en détermine les champs de contrainte et dans la couche subsuperficielle. Ces champs présentent des singularités d'ordre 1/2 à la face supérieure de la fissure.

     

Analytical solution for mode I crack orthogonal to free surface

In this paper, an analytical solution for the singular stress field near a flat crack which is orthogonal to the free surface is proposed. The singular stress field of a three-dimensional surface crack is obtained by the principle of superposition. The stresses of mode-I surface crack are derived by combining the solution of a three-dimensional crack and a second solution due to a singular traction on the free surface of a semi-infinite body. The closed form solution for the second solution is evaluated and the stress distribution at the free surface of the crack is compared with some existing results that appeared in the literature.     

Mixed mode crack in a periodically-layered composite

The problem on a crack in a bimaterial periodically-layered composite is considered. The single finite length crack parallel to the interfaces is loaded by normal opening tractions but the fracture mode is the mixed one as a result of non–symmetric crack location within the layer. The crack is presented as distributed dislocations with unknown density and the problem is reduced to a system of singular integral equations of the first kind. The coefficients of the system are derived from the application of the Green function for a single dislocation which is obtained in a closed form with the help of the representative cell approach. The dependence of the stress intensity factors K _I and K _II upon the geometric and elastic mismatch parameters is examined. The numerical study allowed to point out the cases in which the simplified sandwich model can be employed for the analysis. On the other hand, for the case of very thin and stiff non–cracked layers essentially dissimilar behavior of the stress intensity factors was revealed. In particular, we discovered that K _II may vanish not only for the symmetric crack position in the midplane of the layer but also in several additional ones. For some limiting cases the solution is seen to coincide with known results.     

Interface crack problems in graded orthotropic media: Analytical and computational approaches

Interface crack problems in graded orthotropic media are considered using analytical and computational techniques. In the analytical formulation an interface crack between a graded orthotropic coating and a homogeneous orthotropic substrate is considered. The principal axes of orthotropy are assumed to be parallel and perpendicular to the crack plane. Mechanical properties of the medium are assumed to be continuous with discontinuous derivatives at the interface. The problem is formulated in terms of the averaged constants of plane orthotropic elasticity and reduced to a pair of singular integral equations which are solved numerically to compute the mixed mode stress intensity factors and the energy release rate. In the second part of the study, enriched finite elements are formulated and implemented for graded orthotropic materials. Comparisons of the finite element and analytical results show that enriched finite element technique is capable of producing highly accurate results for crack problems in graded orthotropic media. Finally, periodic interface cracking and the four point bending test for graded orthotropic solids are modeled using enriched finite elements and the results are briefly discussed.     

Dynamic crack propagation and arrest in orthotropic DCB fiber composite specimens

An orthotropic double cantilever beam (DCB) model is used to study dynamic crack propagation and arrest in 90° unidirectional Hercules AS/3501-6 graphite fiber epoxy composites. The dynamic fracture toughness of the composite is determined from tests performed on the long-strip specimen and DCB crack arrest experiments are conducted. By using the dynamic fracture toughness in a finite-difference solution of the DCB governing partial differential equations, a numerical solution of the crack propagation and arrest events is computed. Excellent agreement between the experimental and numerical crack arrest results are obtained.     

The plastic stress ahead of a mixed mode I and II plane stress crack

The small scale yielding for mixed mode I and II plane stress crack problems in elastic perfectly-plastic solids is analysed by considering the stress field near the crack line. By expanding the stresses near the crack line and matching the stress field in the plastic zone with the elastic dominant field for a blunt crack near the crack line at the elastic-plastic boundary, the problem is reduced to solving a system of nonlinear algebraic equations. The relationship between the near-field mixity parameter M^p and the far-field mixity parameter M^e is detennined by solving the system of equations numerically. Analogous to Shih's calculation by the finite element method for the small scale yielding of mixed mode plane strain crack problems, the numerical results indicate that the shift from a mixed mode to a pure mode may not be a smooth one.     

Anti-plane shear problem for an edge crack in a finite orthotropic plate

The problem of an edge crack in a finite orthotropic plate under anti-plane shear is considered. The boundary collocation method is used to calculate the mode III stress intensity factor (SIF). For the case in which the material is isotropic, the present results agree very well with those obtained by using the integral equation method. Furthermore, the method can be extended readily for general cases with arbitrary geometrical and boundary loading conditions and material properties.     

Plastic zones in an orthotropic plate of finite width containing a Griffith crack

The problem of an orthotropic infinite plate of finite width containing a centrally located stressed Griffith crack is considered. The crack is located perpendicular to the edges of the orthotropic plate. The crack tips are fully yielded and in the inelastic zones the material carries only constant normal stresses equal to the yield stress. Dugdale's model is employed to find the effects of the material anisotropy on the size of the plastic zones around the crack tips. Graphical results showing the effects of anisotropy on the length of the plastic zone are also presented.