Two-dimensional elastic Herglotz functions and their applications in inverse scattering

• Published in: Journal of elasticity 2002-01, Vol.68 (1-3), p.123-144
• Main Authors: SEVROGLOU, Vassilios, PELEKANOS, George
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Numerical approximation and analysis; Ordinary and partial differential equations, boundary value problems; Physics; Solid mechanics; Static elasticity; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Studies
• ISSN: 0374-3535; 1573-2681
• DOI: 10.1023/A:1026059224433

In this work solutions of the spectral Navier equation that satisfy the Herglotz boundedness condition in two-dimensional linear elasticity are presented. Navier eigenvectors in polar coordinates are introduced and it is established that they form a linearly independent and complete set in the L^sup 2^-sense on every smooth curve. It is also proved that the classical solutions of the spectral Navier equation are expressed via Navier eigenvectors, and this expansion converges uniformly over compact subsets of R^sup 2^. Two far-field patterns, the longitudinal and the transverse one corresponding to the displacement field are introduced, and the Herglotz norm is expressed as the sum of the L^sup 2^-norms of these patterns over the unit circle. It is also established that the space of elastic Herglotz functions is dense in the space of the classical solutions of the spectral Navier equation. Finally, connection to inverse elasticity scattering is established and reconstructions of rigid bodies are presented.[PUBLICATION ABSTRACT]