Weight Functions for Three-Dimensional Symmetrical Crack Problems

2009 ◽  
pp. 448-448-23 ◽  
Author(s):  
RC Labbens ◽  
J Heliot ◽  
A Pellissier-Tanon
Keyword(s):  
Three Dimensional ◽  
Weight Functions ◽  
Crack Problems

scholarly journals Approximation to Hadamard Derivative via the Finite Part Integral

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 983 ◽  
Author(s):  
Chuntao Yin ◽  
Changpin Li ◽  
Qinsheng Bi
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Numerical Methods ◽  
Three Dimensional ◽  
Cylindrical Wave ◽  
Physical Models ◽  
Finite Part ◽  
Numerical Examples ◽  
Crack Problems ◽  
Hadamard Derivative

In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique.

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