The crack and wedging problem for an orthotropic strip

In this paper the plane elasticity problem for an orthotropic strip containing a crack parallel to its boundaries is first considered. The problem is formulated under general mixed mode loading conditions. It is shown that the stress intensity factors depend on two dimensionless orthotropic constants only. For the crack problem the results are given for a single crack and two collinear cracks. The calculated results show that of the two orthotropic constants the influence of the stiffness ratio on the stress intensity factors is much more significant than that of the shear parameter. The problem of loading the strip by a rigid rectangular wedge is then considered. It is found that for relatively small wedge lengths continuous contact is maintained along the wedge-strip interface, at a certain critical wedge length the separation starts at the midsection of the wedge, and the length of the separation zone increases rapidly with increasing wedge length.

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Résumé Dans ce mémoire, le problème d'élasticité plane relatif à une bande orthotropique contenant une fissure parallèle à ses bords est considéré en premier lieu. Le problème est formulé pour des conditions générales de charge de mode mixte. On montre que les facteurs d'intensité de contrainte dépendent seulement de deux constantes orthotropiques sans dimension. Pour le problème de fissuration, les résultats sont fournis dans le cas d'une fissure simple et de deux fissures colinéaires. Les résultats calculés montrent que des deux constantes orthotropiques, c'est l'influence du rapport de rigidité sur les facteurs d'intensité de contrainte qui est beaucoup plus significative que celle du paramètre de cisaillement. Le problème de mise en charge de la bande par un coin rectangulaire rigide est ensuite considéré. On montre que pour des longueurs de coin relativement petites, un contact continu est maintenu le long de l'interface entre le coin et la bande. Pour une certaine longueur critique du coin, la séparation se produit à mi-section du coin et la longueur de la zone de séparation s'accroît rapidement lorsque augmente la longueur du coin.

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This work was supported by NASA-Langley under the Grant NGR 39-007-011.     

An interface crack between an orthotropic thin film and substrate

This work is concerned with a semi-infinite interface crack between a thin film and a substrate. The two materials are assumed to be linearly elastic and orthotropic. A solution is presented for the stress field due to an edge dislocation on the interface which is valid for any combinations of material parameters. It is found that the behavior of such a bi-material system is governed by 6 independent material parameters. The stress intensity factor is computed for general edge loadings by solving integral equations numerically, and the size of the K-dominant zone is also studied for a residually stressed thin film. The situations in which the K-field zone of dominance is very small are identified and discussed.     

Elastic equilibrium of an interface crack between rigid stamp and orthotropic strip

Rupture of the interface between an absolutely rigid stamp and an orthotropic infinite strip is investigated. A plane elasticity problem for an interface crack formally leads to oscillatory singularities at the crack tip. In order to overcome this nonphysical solution, a model of an interface crack with frictionless contact zones near the crack tips and the corners of the stamp is developed. By using the method of integral Fourier transforms the problem is reduced to a system of three singular integral equations. The system is solved by the method of collocations with the points of collocation chosen at zeros of the Chebyshev polynomials. The stress intensity factors at the crack tips and the stamp corner points are evaluated.     

General solution of stresses due to an adhesion crack for T-shaped junction of two plates

In this study, the general solution of stresses is derived for a T-shaped junction consisting of two thin plates with an adhesion crack. A shear force is applied to the crack surface. The analysis is based on the supposition that the stresses in each plate can be approximated by the plane stress condition. The results obtained are verified by a numerical calculation based on the finite element method. Moreover, a singular stress field is obtained from the solution for the vicinity of the crack.     

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.     

On the inclined crack problem in an orthotropic medium under biaxial loading

The problem of an inclined crack in an orthotropic medium under biaxial loading is analyzed. A suitable coordinate transformation is introduced and two decoupled systems of the Cauchy–Riemann type are obtained in terms of complex potentials. The crack problem is solved by using the method of analytic continuation and closed form expressions of the near tip stress and displacement fields are derived. The influence of load biaxiality on the stress intensity factors, as well as on the local stress components is studied and graphically represented. Moreover, the action of material orthotropy on various quantities describing the crack characteristic is pointed out.     

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.     

T-shaped crack problem for bonded orthotropic layers

The plane elasticity problem of two perfectly bonded orthotropic layers containing cracks perpendicular to and along the interface is considered. Cracks are extended to intersect the boundaries and each other in such a way that a crack configuration suitable to study the T-shaped crack problem is obtained. The problem is reduced to the solution of a system of singular integral equations with Cauchy type singularities. Numerical results for stress intensity factors and energy release rates are presented for various loading conditions and for isotropic as well as orthotropic material pairs. These results indicate that elementary strength of material type calculations for energy release rates provide a good approximation to the actual elasticity solution even for relatively short cracks, as long as the layer thicknesses are not very different.     

Nonlinear vibration and postbuckling of orthotropic thin annular plates

This paper presents an approximate analytical solution of the geometrically nonlinear elastic axisymmetric response of polar orthotropic thin annular plates. Plates with outer edges elastically restrained against rotation and inplane displacement and with unsupported inner edges are considered. Von Kármán type equations are employed. The deflection is approximated by a one term mode shape satisfying the boundary conditions. Galerkin's method is used to obtain Duffing's equation for the deflection at the inner edge. Nonlinear frequencies, postbuckling response, static response and maximum deflection response under a step load are obtained. It is shown that good engineering accuracy is achieved by the approximate method.     

Thermoelasticity problem of an orthotropic plate with two collinear cracks

In this article a rigorous formulation of, and an exact solution to the plane thermoelasticity problem of an orthotropic plate having two collinear cracks are presented. Explicit expressions for the temperature, thermal displacements, thermal stresses and thermal stress intensity factors are obtained assuming that uniform or linear heat flow has been applied on the crack surfaces. Numerical values of thermal stress intensity factors and thermal crack sliding displacements and other quantities are worked out and presented in graphic form. A number of conclusions of practical significance are derived from the above results, including the assertion that an extremely large magnitude of stress singularity may occur in the geometry of two closely neighbouring cracks under the action of linear heat flow.

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Résumé On présente une formulation rigoureuse et une solution exacte d'un problème de thermo-élasticité plane dans une tôle orthotrope comportant deux fissures colinéaires. On obtient les expressions explicites de la température, des dilatations thermiques, des contraintes thermiques et des facteurs d'intensité des contraintes thermiques en supposant appliquer un flux de chaleur uniforme ou linéaire sur les faces de la fissure. On élabore les valeurs numériques des facteurs d'intensité des contraintes thermiques, des déformations thermiques de la fissure par glissements, et d'autres grandeurs, et on les présente sous une forme graphique. On tire de ces résultats un certain nombre de conclusions pratiques, y compris la constatation que des singularités de contraintes d'extrêmmeent grande amplitude peuvent se rencontrer dans une géométrie de deux fissures très voisines sujettes à un flux de chaleur linéaire.

     

Cruciform crack in an orthotropic elastic plane

The problems of determining the stress and displacement fields in an infinite orthotropic plane containing a cruciform crack 387-1, y=0 and 387-2, x=0 when (I) the shape of the crack is prescribed and (II) the cracks are opened by given normal pressures, are reduced to mixed boundary value problems for the quarter plane. Using integral transform techniques, a closed form solution is obtained for problem I, whereas the solution of problem II has been reduced to solving a Fredholm integral equation of second kind with non-singular kernel. Numerical calculation of the stress intensity factor and crack energy in the case of a linear loading function for various crack lengths are presented for problem II, using the values of material constants for a Boron-Epoxy composite.     

Moving Griffith crack in an orthotropic material

In this paper, an integral transform technique is employed to solve the plane elastodynamic problem of a crack of fixed length propagating at a constant speed in a uniformly stressed medium. It is assumed that the crack is located in a plane of elastic symmetry of the material. The stresses and strains ahead of the crack tip are determined explicitly and the conditions governing the initial growth of the crack are investigated using two current theories of fracture (Maximum normal stress and Minimum strain-energy density). Based on these theories, it appears that, depending upon the particular orthotropy of the material, the crack may extend in a straight line for all velocities or may immediately branch out at low velocities (compared to the shear wave velocity of the material) or may start propagating along its initial position for small velocities and then, as the velocity increases, may curve and branch out.     

Kinking of an interface crack in an orthotropic bimaterial

We investigated the asymptotic problem of a kinked interface crack in an orthotropic bimaterial under in‐plane loading conditions. The stress intensity factors at the tip of the kinked interface crack are described in terms of the stress intensity factors of the interface crack prior to the kink combined with a dimensionless matrix function. Using a modified Stroh formalism and an orthotropy rescaling technique, the matrix function was obtained from the solutions of the corresponding problem in transformed bimaterial. The effects of orthotropic and bimaterial parameters on the matrix function were examined. A reduction in the number of dependent material parameters on the matrix function was made using the modified Stroh formalism. Moreover, the explicit dependence of one orthotropic parameter on the matrix function was determined using an orthotropic rescaling technique. The effects of the other material parameters on the matrix function were numerically examined. The energy release rate was obtained for a kinked interface crack in an orthotropic bimaterial.     

Elastic transverse impact on an orthotropic plate

The impact of a solid projectile via an elastic buffer on an orthotropic elastic plate possessing a cylindrical anisotropy has been numerically simulated. The dynamical behavior of the plate is described in terms of the Uflyand-Mindlin wave equations taking into account the rotary inertia and the transverse shear deformations. The wave equations are solved using the ray method and the matching of asymptotic expansions obtained for short times inside and outside the contact zone. The influence of the anisotropy of the plate material on the dynamic contact force and the plate inflection at the impact site has been studied.     

New exact solutions for free vibrations of thin orthotropic rectangular plates

In this paper, a novel separation of variables is presented for solving the exact solutions for the free vibrations of thin orthotropic rectangular plates with all combinations of simply supported (S) and clamped (C) boundary conditions, and the correctness of the exact solutions are proved mathematically. The exact solutions for the three cases SSCC, SCCC, and CCCC are successfully obtained for the first time, although it was believed that they are unable to be obtained. The new exact solutions are further validated by extensive numerical comparisons with the solutions of FEM and those available in the literature.     

Nonlinear transient response of orthotropic thin rectangular plates on elastic foundations

This paper presents geometrically nonlinear transient analysis of rectilinearly orthotropic thin rectangular plates resting on Winkler and Pasternak foundations for uniformly distributed step function and sinusoidal pulse loadings. The orthogonal point collocation method in the space domain and Newmark-β scheme in the time domain are employed. Immovable clamped and simply-supported plates are analysed. An approximate method is used to predict the maximum dynamic response to step loads from the results for static loads and is found to yield sufficiently accurate results.     

Thermoelastic analysis for an opening crack in an orthotropic material

The thermoelastic analysis of an opening crack embedded in an orthotropic material is made under applied uniform heat flux and mechanical loadings. To simulate the case of an opening crack filled with a medium, a thermal-medium crack model is proposed. The thermally permeable and impermeable cracks are the limiting ones of the proposed thermal-medium one. The crack-tip thermoelastic fields induced by a crack in an orthotropic material are determined in closed forms. The elastic T-stress can be also obtained explicitly. The effects of applied mechanical loadings and the thermal conductivity of crack interior on the heat flux at the crack surfaces and the mode-II stress intensity factor are investigated through numerical computations. The obtained results reveal that an increase of the thermal conductivity of crack interior decreases the mode-II stress intensity factor. And when an applied mechanical loading is increasing, the mode-II stress intensity factor is rising.     

The crack problem in a specially orthotropic shell with double curvature

In this paper the crack problem of a shallow shell with two nonzero curvatures is considered. It is assumed that the crack lies in one of the principal planes of curvature and the shell is under Mode I loading condition. The material is assumed to be specially orthotropic. After giving the general formulation of the problem the asymptotic behavior of the stress state around the crack tip is examined. The analysis is based on Reissner's transverse shear theory. Thus, as in the bending of cracked plates, the asymptotic results are shown to be consistent with that obtained from the plane elasticity solution of crack problems. Rather extensive numerical results are obtained which show the effect of material orthotropy on the stress intensity factors in cylindrical and spherical shells and in shells with double curvature. Other results include the stress intensity factors in isotropic toroidal shells with positive or negative curvature ratio, the distribution of the membrane stress resultant outside the crack and the influence of the material orthotropy on the angular distribution of the stresses around the crack tip.