Problem of a crack on the boundary of a rigid inclusion in an elastic plate

The problem of equilibrium of a thin elastic plate containing a rigid inclusion is considered. On part of the interface between the elastic plate and the rigid inclusion, there is a vertical crack. It is assumed that, on both crack edges, the boundary conditions are given as inequalities describing the mutual impenetrability of the edges. The solvability of the problem is proven and the character of satisfaction of the boundary conditions is described. It is also shown that the problem is the limit problem for a family of other problems posed for a wider region and describing equilibrium of elastic plates with a vertical crack as the rigidity parameter tends to infinity.     

Antiplane shear crack in an infinite plate with a circular inclusion

Summary The effect of an elastic circular inclusion of a different material on the stress state of a single-cracked infinite sheet subjected to anti-plane shear is investigated. The proposed method makes use of complex variables in conjunction with Muskhelishvili's technique and has given interesting results for the stress field in the cracked plate. Numerical results, derived from this technique, investigate the influence of the regional inhomogeneity, produced by the inclusion in the structure, on the variation of stress-intensity factors at the crack tips.

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Nichtebener Schubspannungsriß in einer unendlichen Platte mit einem kreisförmigen Einschluß

Übersicht Es wird der Einfluß untersucht, den ein elastischer, kreisförmiger Einschluß aus verschiedenem Material in einer unendlichen Platte mit einem Riß, welche einer longitudinalen Schubspannung unterworfen ist, auf den Spannungszustand der Platte ausübt. Die vorgeschlagene Methode benutzt komplexe Variablen in Zusammenhang mit der Muskhelishvilischen Technik und gibt für den Spannungszustand in der Platte interessante Resultate. Numerische Ergebnisse wurden gefunden, um den Einfluß der Inhomogenität auf die Spannungsintensitätsfaktoren des Risses zu bestimmen.

     

A misfitting elastic inclusion in an infinite plane containing a crack

The problem discussed in this paper is that of a misfitting circular inclusion in an infinite elastic medium which contains a straight crack. The crack is stress free. The stresses develop in the elastic medium because of the misfit. The point force method is used to solve the problem. The problem reduces to finding two sets of complex potential functions: {(z), (z)}: One for the infinite medium and the other for the misfitting inclusion. The solution has been obtained in closed form. Graphs are drawn for stress intensity at the crack tip and also for normal, shear and hoop stresses at the common interface of medium and misfitting inclusion.     

Interaction between curved crack and elastic inclusion in an infinite plate

Summary In this paper, the curved-crack problem for an infinite plate containing an elastic inclusion is considered. A fundamental solution is proposed, which corresponds to the stress field caused by a point dislocation in an infinite plate containing an elastic inclusion. By placing the distributed dislocation along the prospective site of the crack, and by using the resultant force function as the right-hand term in the equation, a weaker singular integral equation is obtainable. The equation is solved numerically, and the stress intensity factors at the crack tips are evaluated. Interaction between the curved crack and the elastic inclusion is analyzed. Received 8 October 1996; accepted for publication 27 March 1997     

Deformation of an infinite plate with a crack

International Applied Mechanics -     

Magneto elastic stress subjected to uniform magnetic field over a thin infinite plate containing an elliptical hole with an edge crack

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack subjected to uniform magnetic field. Using a rational mapping function, each solution is obtained as a closed form. The linear constitutive equation is used for these analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. In the present paper, it raises a plane stress state for a thin plate, the deformation of the plate thickness and the shear deflection. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that those plane stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solutions of the magneto elastic stress are nonlinear for the direction of uniform magnetic field. Stresses in the direction of the plate thickness and shear deflection are caused and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length.     

含椭圆孔弹性平面基本解

运用复变函数保角变换与解析延拓方法,获得含椭圆孔无限弹性平面任意位置作用集中力的基本解,并由此获得含有限长裂纹弹性平面基本解,可作为弹性力学的典型问题.该方法较以往文献更为简捷.     

The problem of interaction between a misfitting inclusion and a crack in an infinite elastic medium

The problem of interaction between a curvilinear crack (or a system of such cracks) and a misfitting inclusion of arbitrary shape (or a system of such inclusions) inside an infinite isotropic elastic medium of the same material as the inclusion was solved by using the complex potential technique and reducing the problem to a complex Cauchy type singular integral equation along the crack only (or the system of cracks).

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Résumé Le problème de l'influence mutuelle d'une fissure curviligne (ou d'un système de telles fissures) et une inclusion malajustée de forme arbitraire (ou un système de telles inclusions) dans un milieu infini élastique isotrope du même matériau que l'inclusion a été résolu en utilisant la technique des potentiels complexes et en réduisant ainsi le probléme à une équation intégrale singulière complexe du type Cauchy seulement le long de la fissure (ou du système des fissures).

     

Magnetic field and magneto elastic stress in an infinite plate containing an elliptical hole with an edge crack under uniform electric current

Two-dimensional solutions of the electric current, magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack under uniform electric current. Using a rational mapping function, the each solution is obtained as a closed form. The linear constitutive equation is used for the magnetic field and the stress analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate which raises a plane stress state for a thin plate and the deformation of the plate thickness. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, electric current, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that the stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solving the present magneto elastic stress problem, dislocation and rotation terms appear, which makes the present problem complicate. Solutions of the magneto elastic stress are nonlinear for the direction of electric current. Stresses in the direction of the plate thickness are caused and the solution is also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length and the electric current direction.     

The effect of an elastic triangular inclusion on a crack

IntroductionWiththedevelopmentofparticleandfiberreinforcedcomposites,theinclusion_crackinteractionproblemisbecominganimportantfieldbeingstudied .Andasamodel,itisalsousedtostudytheeffectsofmaterialdefectsonthestrengthandfractureofengineeringstructure.TheinterationbetweencircularinclusionandcrackwasstudiedinRefs.[1 -6 ] ;InRefs.[7-1 2 ] ,theinterationbetweenlineinclusionandcrackswasdiscussed ;TheinterationbetweenellipticalinclusionandcrackwasstudiedinRefs.[1 3,1 4] .However,withthedevelopmento…     

Perturbed magnetic field of an infinite plate with a centered crack

Deforming a cracked magnetoelastic body in a magnetic field induces a perturbed magnetic field around the crack. The quantitative relationship between this perturbed field and the stress around the crack is crucial in developing a new generation of magnetism-based nondestructive testing technologies. In this paper, an analytical expression of the perturbed magnetic field induced by structural deforma- tion of an infinite ferromagnetic elastic plate containing a centered crack in a weak external magnetic field is obtained by using the linearized magnetoelastic theory and Fourier transform methods. The main finding is that the perturbed magnetic field intensity is proportional to the applied tensile stress, and is dominated by the displacement gradient on the boundary of the magnetoelastic solid. The tangential component of the perturbed magnetic-field intensity near the crack exhibits an antisymmetric distribution along the crack that reverses its direction sharply across its two faces, while the normal component shows a symmetric distribution along the crack with singular points at the crack tips.     

Analysis of a plate containing an elliptic inclusion with eigencurvatures

Summary An infinite plate containing an elliptic subregion in which a uniform eigencurvature is prescribed is analyzed. The problem is formulated by using the classical plate theory. Employing the Maysel's relation, an integral-type solution to the equilibrium equation is expressed in terms of the eigencurvature. Closed-form solutions of the displacement and corresponding resultant moment are obtained for interior points as well as for exterior points of the ellipse. An infinite plate containing an elliptic inhomogeneity in which a uniform eigencurvature is prescribed is also considered. The disturbance of the displacement and corresponding resultant moment due to the inhomogeneity is determined by the equivalent eigencurvature method. Solutions of a circular finite plate with uniform eigencurvature in a circular zone are also obtained analytically. Received 30 September 1997; accepted for publication 3 February 1998     

Scattering of elastic waves by an inclusion with an interface crack

A method of perturbation is used to derive an integral representation of the displacement field for the scattering of a plane wave from an inclusion with an interface crack. In the long-wave approximation it is shown that the solution of only an associated static problem is required and formal expressions are derived for the scattered far field amplitudes and scattering cross section. In the case of a cylindrical inclusion the solution of the associated static problem is then used to find in a closed form the corresponding expressions for plane incident P- and S-waves.     

Scattering of elastic waves in an elastic matrix containing an inclusion with interfaces

Based on the theory of elastic dynamics, the scattering of elastic waves and dynamic stress concentration in fiber-reinforced composite with interfaces are studied. Analytical expressions of elastic waves in different medium areas are presented and an analytic method of solving this problem is established. The mode coefficients are determined by means of the continuous conditions of displacement and stress on the boundary of the interfaces. The influence of material properties and structural size on the dynamic stress concentration factors near the interfaces is analyzed. It indicates that they have a great influence on the dynamic properties of fiber-reinforced composite. As examples, numerical results of dynamic stress concentration factors near the interfaces are presented and discussed. This paper provides reliable theoretical evidence for the study of dynamic properties in fiber-reinforced composite. Project supported by the National Natural Science Foundation of China (No. 19972018).