Vol. 6, No. 1-4, 2011

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ISSN: 1559-3959
Singular harmonic problems at a wedge vertex: mathematical analogies between elasticity, diffusion, electromagnetism, and fluid dynamics

Alberto Carpinteri and Marco Paggi

Vol. 6 (2011), No. 1-4, 113–125
Abstract

Multimaterial wedges are frequently observed in composite materials. They consist of two or more sectors of dissimilar materials joined together, whose interfaces converge at the same vertex. Due to the mismatch in material properties such as Young’s modulus, thermal conductivity, dielectric permittivity, or magnetic permeability, these geometrical configurations can lead to singular fields at the junction vertex. This paper discusses mathematical analogies, focused on singular harmonic problems, between antiplane shear problem in elasticity due to mode III loading or torsion, the steady-state heat transfer problem, and the diffraction of waves in electromagnetism. In the case of a single material wedge, a mathematical analogy between elasticity and fluid dynamics is also outlined. The proposed unified mathematical formulation is particularly convenient for the identification of common types of singularities (power-law or logarithmic type), the definition of a standardized method to solve nonlinear eigenvalue problems, and the determination of common geometrical and material configurations allowing the relief or removal of different singularities.

Dedicated to the memory of Marie-Louise Steele.

Keywords
singularities, multimaterial wedges, elasticity, diffusion, electromagnetism, fluid dynamics
Milestones
Received: 19 June 2010
Revised: 9 September 2010
Accepted: 10 September 2010
Published: 28 June 2011
Authors
Alberto Carpinteri
Department of Structural and Geotechnical Engineering
Politecnico di Torino
Corso Duca degli Abruzzi, 24
I-10129 Torino
Italy
http://staff.polito.it/alberto.carpinteri
Marco Paggi
Department of Structural and Geotechnical Engineering
Politecnico di Torino
Corso Duca degli Abruzzi, 24
I-10129 Torino
Italy
http://staff.polito.it/marco.paggi