Papers by Keyword: Complex Function

Abstract: By the theory of complex functions, symmetrical dynamic propagation problems of mode Ⅲ interface crack were investigated. The problems considered can be very easily translated into Riemann-Hilbert problem by the methods of self-similar functions, and the universal expressions of analytical solutions for the surfaces of symmetrical mode Ⅲ interface crack subjected to moving alterable loadings Pt3/x3 and Px4/t3 were obtained, respectively. After those solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be acquired.

Abstract: Coal usually contains many cracks which affect the stability of mining zone. In order to research the fracture mechanism of cracked coal, a special case of three collinear cracks filling with gas is considered. A set of complex stress functions which satisfy all the required conditions are first derived, based on Muskelishvili’s results. The least square boundary collocation method is used to enforce the external boundaries approximately, and the stress intensity factors (SIF) can then be computed according to the related formulae. The value of SIFs for different crack orientations, different crack distance under different loading are calculated. The results shows that the most unfavorable crack orientation is in the 45° angle and the confining stress and the distance between cracks have a considerable effect on the SIF value.

Abstract: The problem of dynamic response of multiple circular cavities near multiple semi-cylindrical alluvial valleys under incident plane SH-waves is investigated by the methods of complex function and multi-polar coordinates in this paper. Firstly, the solution domain is divided into two parts, Domain I is multiple semi-cylindrical alluvial valleys, and Domain Ⅱ is an elastic half space with several subsurface circular cavities near multiple semi-cylindrical alluvial valleys. A series of infinite algebraic equations is then obtained based on the displacement and stress continuity condition on “common boundary” of two parts after constructing the associated displacement and stresses expressions of each part. Finally some numerical expamples are prensented and dynamic response of subsurface circular cavities near semi-cylindrical alluvial valleys with respect to different parameters is discussed.

Abstract: Conformal mapping with complex function based on plane elasticity mechanics is an analytical method for resolving stress and displacement at any point of a half-plane domain. Using complex function conformal mapping method in this article we investigated the relationship between load on tooth surface and maximum stress at tooth root for calculating the maximum compressive stress on the opposite side of working flank and maximum tensile stress on working flank side when loads are applied to tooth top and root of working flank side, respectively. The maximum tensile and compressive stress at the tooth root are the main forces that cause fatigue cracking of the tooth root, which may extend along the elastomer compound-cord interface resulting in shear cracking of the belt tooth. The results of our calculation reveal the mechanisms whereby tooth shear cracking causes fatigue failure of synchronic belt, which are consistent with the experimental research results of Lizuka.

Abstract: Firstly several seismic simplified methods commonly used for deep circular tunnel are evaluated and the difficulties in response displacement method are pointed out. Then the analytical solution of soil spring coefficient and soil response of deep circular tunnel is derived from using complex variable theory of planar elastic theory based on pseudo-static hypothesis. The analytical solution has been verified by comparing its predictions with results from an analysis in finite element method. It is concluded that the analytical solution can be regarded as one feasible reference for the simplification of response displacement method.

Abstract: By means of the complex variable functions, dynamic expension problems on symmetrical mode Ⅲ interface crack were researched. The problems considered can be very facilely transformed into Riemann-Hilbert problem by the measures of self-similar functions, and the general expressions of analytical solutions for the edges of mode Ⅲ symmetrical interface crack subjected to motive variable loadings Px²/t³ and Pt⁴/x³ were obtained by means of self-similar functions, respectively. After those solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be readily attained.

Abstract: By the approaches of complex variable functions, two dynamic propagation problems of mode Ⅲ interface crack were researched. The problems considered can be very facilely changed into Riemann-Hilbert problem by means of self-similar functions, and analytical solutions of the stresses, displacements, dynamic stress intensity factors for the edges of mode Ⅲ symmetrical dynamix interface crack subjected to moving increasing loads Pt2/x2 and Px3/t2, respectively, were obtained by the methods of self-similar functions. After those solutions were utilized by superposition theorem, the solutions of arbitrary complex problems can be readily attained.

Abstract: By application of the theory of complex functions, the problem for mode Ⅲ dynamic crack under concentrated force were researched. The Riemann-Hilbert problem is formulated according to relationships of self-similar functions. Analytical solutions of stresses, displacements and dynamic stress intensity factors were obtained by the measures of self-similar functions and corresponding differential and integral operation.

Abstract: Dislocation distribution functions of mode I dynamic crack subjected to two loads were studied by the methods of the theory of complex variable functions. By this way, the problems researched can be translated into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements and dynamic stress intensity factors were obtained by the measures of self-similar functions and corresponding differential and integral operation. The analytical solutions attained relate to the crack propagation velocity and time, but the solutions have nothing to the other parameters. In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were gained, and variation rules of dislocation distribution functions were depicted.

Abstract: By the theory of complex functions, dynamic propagation problems on symmetrical mode Ⅲ interface crack were researched. The problems considered can be very facilely transformed into Riemann-Hilbert problem by the methods of self-similar functions, and the general expressions of analytical solutions for the surfaces of mode Ⅲ symmetrical interface crack subjected to motive variable loadings Px²/t² and Pt³/x² were obtained, respectively. After those solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be attained.