Summary
The validity of a one-dimensional Cauchy type singular integral equation of the first kind at points where the known right-hand side function presents jump discontinuities and, therefore, the unknown function of the integral equation presents logarithmic singularities, is proved under two assumptions: (i) the Cauchy type integral in the singular integral equation is given the interpretation of a finite-part integral, and (ii) the values of the right-hand side function at the points of the jump discontinuities are the mean values of the corresponding limiting values. These results apply directly to elasticity problems, generally reducible to Cauchy type singular integral equations with right-hand sides the loading distributions and unknown functions appropriate dislocation distributions. A quadrature rule for the numerical evaluation of the finite-part integrals at the points of loading discontinuities is also proposed.
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Ioakimidis, N.I. On the validity of the singular integral equations of elasticity problems at points of loading discontinuities. Acta Mechanica 50, 127–134 (1983). https://doi.org/10.1007/BF01170446
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DOI: https://doi.org/10.1007/BF01170446