Complete Williams Asymptotic Expansion near the Crack Tips of Collinear Cracks of Equal Lengths in an Infinite Plane

2016 ◽  
Vol 258 ◽  
pp. 209-212 ◽  
Author(s):  
Larisa Stepanova ◽  
Pavel Roslyakov ◽  
Tatjana Gerasimova
Keyword(s):  
Plane Elasticity ◽  
Analytical Determination ◽  
Geometrical Parameters ◽  
Infinite Plane ◽  
Variable Theory ◽  
Complex Variable Theory ◽  
Plane Medium ◽  
Crack Tips ◽  
Plane Elasticity Theory

The present study is aimed at analytical determination of coefficients in crack tip expansion for two collinear finite cracks of equal lengths in an infinite plane medium. The study is based on the solutions of the complex variable theory in plane elasticity theory. The analytical dependence of the coefficients on the geometrical parameters and the applied loads for two finite cracks in an infinite plane medium is given. It is shown that the effect of the higher order terms of the Williams series expansion becomes more considerable at large distances from the crack tips. The knowledge of more terms of the stress asymptotic expansions allows us to approximate the stress field near the crack tips with high accuracy.

scholarly journals Modern interpretation of Saint-Venant’s principle and semi-inverse method

2020 ◽  
Vol 16 (5) ◽  
pp. 390-413
Author(s):  
Evgeny M. Zveryaev
Keyword(s):  
Boundary Conditions ◽  
Small Area ◽  
Inverse Method ◽  
Deformable Body ◽  
Plane Elasticity ◽  
Variable Theory ◽  
Progressive Development ◽  
Complex Variable Theory ◽  
Convergence Results ◽  
Classic Theory

Relevance. The progressive development of views on the Saint-Venant formulated principles and methods underlying the deformable body mechanics, the growth of the mathematical analysis branch, which is used for calculation and accumulation of rules of thumb obtained by the mathematical results interpretation, lead to the fact that the existing principles are being replaced with new, more general ones, their number is decreasing, and this field is brought into an increasingly closer relationship with other branches of science and technology. Most differential equations of mechanics have solutions where there are gaps, quick transitions, inhomogeneities or other irregularities arising out of an approximate description. On the other hand, it is necessary to construct equation solutions with preservation of the order of the differential equation in conjunction with satisfying all the boundary conditions. Thus, the following aims of the work were determined: 1) to complete the familiar Saint-Venants principle for the case of displacements specified on a small area; 2) to generalize the semi-inverse Saint-Venants method by finding the complement to the classical local rapidly decaying solutions; 3) to construct on the basis of the semi-inverse method a modernized method, which completes the solutions obtained by the classical semi-inverse method by rapidly varying decaying solutions, and to rationalize asymptotic convergence of the solutions and clarify the classical theory for a better understanding of the classic theory itself. To achieve these goals, we used such methods , as: 1) strict mathematical separation of decaying and non-decaying components of the solution out of the plane elasticity equations by the methods of complex variable theory function; 2) construction of the asymptotic solution without any hypotheses and satisfaction of all boundary conditions; 3) evaluation of convergence. Results. A generalized formulation of the Saint-Venants principle is proposed for the displacements specified on a small area of a body. A method of constructing asymptotic analytical solutions of the elasticity theory equations is found, which allows to satisfy all boundary conditions.

Magnetic Pulse Welding by Electromagnetic Compression: Determination of the Impact Velocity

2014 ◽  
Vol 966-967 ◽  
pp. 489-499 ◽  
Author(s):  
Joern Lueg-Althoff ◽  
Amanda Lorenz ◽  
Soeren Gies ◽  
Christian Weddeling ◽  
Gunther Goebel ◽  
...  
Keyword(s):  
Impact Velocity ◽  
Measurement Data ◽  
Analytical Determination ◽  
Geometrical Parameters ◽  
Magnetic Pulse ◽  
Engineering Structures ◽  
Magnetic Pulse Welding ◽  
Pulse Welding ◽  
The Impact

The implementation of multi-material concepts and the manufacturing of modern lightweight structures, for example in automotive engineering, require appropriate joining technologies. The ability to join dissimilar materials without additional mechanical elements, chemical binders, or adverse influences of heat on the joining partners is key in reaching the desired weight reduction in engineering structures. The Magnetic Pulse Welding (MPW) process meets these demands, making it a viable alternative to conventional thermal welding and mechanical joining processes. The present paper focuses on the analytical determination of the impact velocity as one of the key parameters of MPW processes. On the basis of experimentally recorded data concerning the course of the discharge current and geometrical parameters of the welding setup, the respective velocity is determined. A comparison with measurement data gained by Photon Doppler Velocimetry is performed.

scholarly journals Determination of shrinkage of weld

2018 ◽  
Vol 224 ◽  
pp. 01108 ◽  
Author(s):  
Nikolay Velikanov ◽  
Sergey Koryagin ◽  
Oleg Sharkov
Keyword(s):  
Residual Stresses ◽  
Elasticity Theory ◽  
Experimental Technique ◽  
Heat Input ◽  
Experimental Studies ◽  
Plane Elasticity ◽  
Theoretical Solution ◽  
Strain Gauges ◽  
Plane Elasticity Theory

A calculation-experimental technique of determination of weld shrinkage is described in that work. Experimentally, with the help of strain gauges, residual stresses are determined at two points, which are at known distances from the weld. Shrinkages are found out from the theoretical solution of the plane elasticity theory about the insertion of bodies with interference. Calculations have been carried out for experimental studies of a weld 400 mm long for different thicknesses of metal. It is found that when the thickness of the metal increases, the shrinkage values also increase. This is due to the fact that as the thickness of the metal increases, the heat input required to penetrate it also increases.

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