New integral equation for plane elasticity crack problems

In the usual formulation of singular equation approach for crack problems in plane elasticity [1,2], if one changes the right-hand term of the integral equation from tractions to resultant forces, a new integral equation can be obtained and is presented in this paper. The newly obtained integral equation has a log singular kernel. Interpolation equation for the dislocation functions (the undetermined functions in the integral equations) is proposed. Numerical examination is used to demonstrate the efficiency of the present technique, and a number of numerical examples are given.     

A new integral equation formulation of two-dimensional inclusion–crack problems

A new integral equation formulation of two-dimensional infinite isotropic medium (matrix) with various inclusions and cracks is presented in this paper. The proposed integral formulation only contains the unknown displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks. In order to solve the inclusion–crack problems, the displacement integral equation is used when the source points are acting on the inclusion–matrix interfaces, whilst the stress integral equation is adopted when the source points are being on the crack surfaces. Thus, the resulting system of equations can be formulated so that the displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks can be obtained. Based on one point formulation, the stress intensity factors at the crack tips can be achieved. Numerical results from the present method are in excellent agreement with those from the conventional boundary element method.     

Solution of a dual integral equation for crack and indentation problems

Using the Sonine integrals, some dual integral equations with the Bessel kernels were reduced to a single integral equation. Then the closed form solutions of these dual integral equations were obtained. Based on the method, the anti-plane shear of an elastic layer was solved exactly.     

An approach to solving mechanics problems for materials with multiscale self-similar microstructure

This article proposes an efficient method for solving mechanics boundary value problems formulated for domains with multiscale self-similar microstructure. In particular, composite materials for which one of the phases has a fractal-like structure with scale cut-offs are considered. The boundary value problems are solved using a finite element procedure with enriched shape functions that incorporate information about the geometric complexity. The use of these shape functions makes possible the definition of a unique, parametrically defined model from which the solution for configurations with an arbitrary number of scales can be derived. The proposed method is primarily useful for structures with a large number of self-similar scales for which using the usual finite element method would be too expensive. In order to exemplify the method, a 2D composite with fractal microstructure is considered and several boundary value problems are solved.     

An integral equation approach to finite deflection of elastic plates

The present paper concerns an approximate integral equation approach to finite deflection of elastic plates with aribitrary plan form. With the combined help of the Berger equation governing non-linear bending and a weighted residual technique for boundary-value problem, a boundary integral equation is formulated for immovable edge conditions. We here clarify the formulation and show that the calculation can be performed, with a slight modification, through a procedure similar to that conventionally employed in the linear bending analysis. Availability of the derived integral equation and the solution scheme is shown by way of simple numerical examples.     

Elliptical crack and punch problems solved by integral equation method for piezoelectric media

The problem of an elliptical crack embedded in an unbounded transversely isotropic piezoelectric media with the crack-plane parallel to the plane of isotropy of the media and subjected to remote normal mechanical as well as electric loading is considered first. The problem has been successfully reduced to a pair of coupled integral equations that are suitable for the application of an integral equation method developed earlier for three-dimensional problems of LEFM. Solution to the mechanical displacement and electric potentials are obtained for prescribed uniform loadings and expressions for corresponding intensity factors and crack opening displacement are deduced. The above method has further been applied to solve the problem of a rigid flat-ended elliptical punch indenting a transversely isotropic piezoelectric half-space surface with the plane of isotropy parallel to the surface. Solutions to mechanical stress and electric displacement are obtained for prescribed constant normal displacement and constant electric potential interior to the elliptical region and expression for the total force required to maintain a prescribed indentation is deduced.     

Two dimensional self-similar expanding crack problems in elastic half-space

The application of the property of dynamic similarity is useful to the solution which admits a self-similarity or homogeneous form. One independent variable has been dropped in the present equivalent set of the governing equations. The displacement discontinuity on the crack face and also the displacement field on the surface due to an in-plane shear model over an expanding zone of slippage of arbitrary dip have been obtained. The moving slip edge extends towards the surface with a constant velocity. Cagniard De-Hoop technique has been used here to obtain the two dimensional exact transient response due to the slip in the vertical mode via body force equivalent. The results of the present paper are valid at least up to the time when the diffracted waves from the crack edge have not reached the receiving station. The spectral behavior of the source time function has also been discussed.     

A boundary integral equation method for the solution of a class of crack problems

A boundary integral equation method for the solution of an important class of crack problems in elasticity is outlined. The method is applicable for deformations in which the crack faces remain in contact. Specific numerical examples are considered to illustrate the application of the method.     

Singular integral equations in Hilbert space applied to crack problems

Crack problems are solved by application of a system of singular integral equations in Hilbert space. The method consists of replacing the singular integral equation by a system of linear algebraic equations for the values of the unknown function at specially chosen points within the range of integration. Obtained is a solution for a nonsymmetric cross-shaped crack in an infinite and isotropic solid subjected to a constant pressure, the symmetric configuration being a special case.     

An integral equation for dynamic elastic response of an isolated 3-D crack

By use of a steady state (e^−iωt) dynamic elastic representation theorem for fields created by relative motions ΔU_k on the faces of a crack, we reduce the problem of steady state response of an isolated three-dimensional planar crack, loaded by tractions on its surfaces, to an integral equation for ΔU_k.     

An integral equation method for stress concentration problems in cylindrical shells

A method is described for determining the stresses in perforated cylindrical shells. The method is applicable to cases where several holes of arbitrary form are present. The problem is formulated as a system of four coupled integral equations together with a number of compatibility relations in integral form. A numerical procedure for solving the equations is also described and some simple applications of the method including the case of one elliptical hole with arbitrary orientation relative to the generators are presented.

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Zusammenfassung Eine Methode zur Berechnung der Spannungen in zylindrischen Schalen mit Löchern willkürlicher Form wird beschrieben. Das Problem ist als ein System von vier gekuppelten Integralgleichungen mit einer Anzahl von Kompatibilitätsbedingungen auf Integralform formuliert. Ein numerisches Verfahren zur Lösung der Gleichungen ist beschrieben und auf einfache Rechnungsbeispiele angewandt, darunter der Fall, dass ein elliptisches Loch vorhanden ist, dessen Orientierung bezüglich der Erzeuger willkürlich ist.

     

Viscoelastic crack analysis by the boundary integral equation method

Summary The linear viscoelastic three-dimensional crack problem is analyzed by combining the correspondence principle and the boundary integral equation method. In a general crack analysis the usual boundary integral equations lead to a nonunique formulation of the problem, because they do not involve information about the loading on the crack surface. Here, the boundary integro-differential equations are applied to the numerical calculation of the crack opening displacement of a penny-shaped crack in an infinite linear viscoelastic body. Moreover, the influence of several parameters of the three-parameter viscoelastic model on the crack opening displacement and the incubation time is shown.

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Viskoelastische Rißanalyse durch Randintegralgleichungen

Übersicht Das linear viskoelastisch räumliche Rißproblem wird mit Hilfe einer Kombination von Korrespondenzprinzip und Randintegralgleichungsverfahren gelöst. In einer allgemeinen Rißanalyse führen die üblichen Randintegrale zu einer nicht eindeutigen Formulierung dieses Problems, weil die Angaben über Belastung und Rißoberfläche fehlen. Das Randintegralgleichungsverfahren wird für die numerische Berechnung der Rißerweiterung eines münzförmigen Risses in einem unendlich linear viskoelastischen Körper angewendet. Weiterhin wird der Einfluß von verschiedenen Parametern des räumlich viskoelastischen Modells auf die Rißerweiterung und die Inkubationszeit gezeigt.

     

An integral equation formulation of three dimensional anisotropic elastostatic boundary value problems

An approximate solution capability is developed to handle three dimensional anisotropic elastostatic boundary value problems. The method depends crucially on the existence and explicit definition of a fundamental solution to the governing partial differential equations. The construction of this solution for the anisotropic elastostatic problem is presented as is the derivation of the expression for the surface tractions necessary to maintain the fundamental solution in a bounded region. After the fundamental solution and its associated surface tractions are determined, a real variable boundary integral formula is generated which can be solved numerically for the unknown surface tractions and displacements in a well-posed boundary value problem. Once all boundary quantities are known, the field solution is given by a Somigliana type integral formula. Techniques for numerically solving the integral equations are discussed.

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Zusammenfassung Es wird eine Näherungslösung entwickelt die es dreidimensionale anisotrope elastostatische Randwertprobleme zu lösen. Die Methode hängt entscheidend vom Vorhandensein und expliziten Bestimmung einer Grundlösung der zugeehörigen partiellen Differentialgleichungen. Die Êntwicklung dieser Lösung fur den Fall eines anisotropen elastostatischen Problems wird gegeben wie auch die Ableitung einer Formel fur Oberflächenspannung die erforderlich ist, um die Gultigkeit der Grundlösung im Randgebiet fortbestehen zu lassen. Nachdem die Grundlösung und die mit ihr verbundenen Oberflächenspannungen gefunden worden sind, wird eine Randwertformel in Integralform fur reelle Variablen entwickelt, die numerisch fur unbekannte Oberflächenspannungen und Verschiebungen jedoch mit gut definierten Randwerten gelöst werden kann. Nachdem alle Randwerte bekannt sind ist die Feldlösung durch eine Integralformel vom Typ Somigliana gegeben. Verfahren zur numerischen Lösung der Integralgleichungen werden erörtert.

     

Finite-part integral and boundary element method to solve flat crack problems

Using the Somigliana formula and the concepts of finite-part integral, a set of hypersingular integral equations to solve the arbitrary fiat crack in three-dimensional elasticity is derived and its numerical method is then proposed by combining the finitepart integral method with boundary element method. In order to verify the method, several numerical examples are carried out. The results of the displacement discontinuities of the crack surface and the stress intensity factors at the crack front are in good agrernent with the' theoretical solutions.     

Application of hypersingular integral equation method to three-dimensional crack in electromagnetothermoelastic multiphase composites

A three-dimensional crack problem in electromagnetothermoelastic multiphase composites (EMTE-MCs) under extended loads is investigated in this paper. Using Green’s functions, the extended general displacement solutions are obtained by the boundary element method. This crack problem is reduced to solving a set of hypersingular integral equations coupled with boundary integral equations, in which the unknown functions are the extended displacement discontinuities. Then, the behavior of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of hypersingular integral equations. Analytical solutions for the extended singular stresses, the extended stress intensity factors (SIFs) and the extended energy release rate near the crack front in EMTE-MCs are provided. Also, a numerical method of the hypersingular integral equations for a rectangular crack subjected to extended loads is put forward with the extended displacement discontinuities approximated by the product of basic density functions and polynomials. In addition, distributions of extended SIFs varying with the shape of the crack are presented. The results show that the present method accurately yields smooth variations of extended SIFs along the crack front.     

Singular integral equation for antiplane-wave scattering by a semi-infinite crack

The problem of antiplane-wave scattering by a semi-infinite crack is reduced to a singular integral equation of the Cauchy type. This equation is obtained by treating the problem as the limiting case of a sequence of problems for which the crack-opening displacements decay exponentially at infinity, and by using real-variable (as opposed to complex-variable) Fourier-transform methods. An integral identity is used to obtain the solution of the singular integral equation. The solution is shown to coincide with the classical Wiener-Hopf solution of the problem.