Axisymmetric displacement boundary value problem for a penny-shaped crack

1995 ◽  
Vol 66 (1) ◽  
pp. 1 ◽  
Author(s):  
J. V. S. Krishna Rao ◽  
N. Hasebe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Displacement Boundary ◽  
Penny Shaped Crack ◽  
Shaped Crack

Diffraction of elastic waves by a penny-shaped crack

1981 ◽  
Vol 378 (1773) ◽  
pp. 263-285 ◽  
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Unique Solution ◽  
Double Layer ◽  
Elastic Waves ◽  
Boundary Value ◽  
Elastic Solid ◽  
Time Harmonic ◽  
Penny Shaped Crack ◽  
Shaped Crack

Consider an infinite elastic solid containing a penny-shaped crack. We suppose that time-harmonic elastic waves are incident on the crack and are required to determine the scattered displacement field u i . In this paper, we describe a new method for solving the corresponding linear boundary-value problem for u i , which we denote by S. We begin by defining an ‘elastic double layer’; we prove that any solution of S can be represented by an elastic double layer whose ‘density’ satisfies certain conditions. We then introduce various Green functions and define a new crack Green function, G ij , that is discontinuous across the crack. Next, we use G ij to derive a Fredholm integral equation of the second kind for the discontinuity in u i across the crack. We prove that this equation always has a unique solution. Hence, we are able to prove that the original boundary-value problem S always possesses a unique solution, and that this solution has an integral representation as an elastic double layer whose density solves an integral equation of the second kind.

scholarly journals A mixed boundary value problem of a cracked elastic medium under torsion

2021 ◽  
pp. 10-10
Author(s):  
Belkacem Kebli ◽  
Fateh Madani
Keyword(s):  
Boundary Value Problem ◽  
Elastic Medium ◽  
Integral Equations ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Crack Problem ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problem ◽  
Penny Shaped Crack

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

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