Time-harmonic body force loading of a mode-I penny-shaped crack

1996 ◽  
Vol 33 (2) ◽  
pp. A98
Keyword(s):  
Body Force ◽  
Mode I ◽  
Force Loading ◽  
Time Harmonic ◽  
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.

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