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The method of lines in analyzing solids containing cracksA semi-numerical method is reviewed for solving a set of coupled partial differential equations subject to mixed and possibly coupled boundary conditions. The line method of analysis is applied to the Navier-Cauchy equations of elastic and elastoplastic equilibrium to calculate the displacement distributions in various, simple geometry bodies containing cracks. The application of this method to the appropriate field equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. When decoupling of the equations and their boundary conditions is not possible, the use of a successive approximation procedure permits the analytical solution of the resulting ordinary differential equations. The use of this method is illustrated by reviewing and presenting selected solutions of mixed boundary value problems in three dimensional fracture mechanics. These solutions are of great importance in fracture toughness testing, where accurate stress and displacement distributions are required for the calculation of certain fracture parameters. Computations obtained for typical flawed specimens include that for elastic as well as elastoplastic response. Problems in both Cartesian and cylindrical coordinate systems are included. Results are summarized for a finite geometry rectangular bar with a central through-the-thickness or rectangular surface crack under remote uniaxial tension. In addition, stress and displacement distributions are reviewed for finite circular bars with embedded penny-shaped cracks, and rods with external annular or ring cracks under opening mode tension. The results obtained show that the method of lines presents a systematic approach to the solution of some three-dimensional mechanics problems with arbitrary boundary conditions. The advantage of this method over other numerical solutions is that good results are obtained even from the use of a relatively coarse grid.
Document ID
19910010164
Acquisition Source
Legacy CDMS
Document Type
Conference Paper
Authors
Gyekenyesi, John P. (NASA Lewis Research Center Cleveland, OH, United States)
Date Acquired
September 6, 2013
Publication Date
November 1, 1990
Subject Category
Structural Mechanics
Report/Patent Number
E-5794
NASA-TM-103626
NAS 1.15:103626
Meeting Information
Meeting: International Symposium on Methods of Lines, Surfaces and Dimensional Reduction in Computational Mathematics and Mechanics
Location: Athens
Country: Greece
Start Date: April 25, 1991
End Date: April 30, 1991
Sponsors: Polytechnic of Central London
Accession Number
91N19477
Funding Number(s)
PROJECT: RTOP 505-63-5B
Distribution Limits
Public
Copyright
Work of the US Gov. Public Use Permitted.

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