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Research Papers

# Unique and Explicit Formulas for Green's Function in Three-Dimensional Anisotropic Linear Elasticity

[+] Author and Article Information
Federico C. Buroni

e-mail: fburoni@us.es

Andrés Sáez

School of Engineering,
University of Seville,
Camino de los Descubrimientos s/n,
Seville E-41092, Spain

1Corresponding author.

Manuscript received July 31, 2012; final manuscript received November 27, 2012; accepted manuscript posted February 11, 2013; published online July 12, 2013. Assoc. Editor: Marc Geers.

J. Appl. Mech 80(5), 051018 (Jul 12, 2013) (14 pages) Paper No: JAM-12-1362; doi: 10.1115/1.4023627 History: Received July 31, 2012; Revised November 27, 2012; Accepted February 11, 2013

## Abstract

Unique, explicit, and exact expressions for the first- and second-order derivatives of the three-dimensional Green's function for general anisotropic materials are presented in this paper. The derived expressions are based on a mixed complex-variable method and are obtained from the solution proposed by Ting and Lee (Ting and Lee, 1997,“The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids,” Q. J. Mech. Appl. Math. 50, pp. 407–426) which has three valuable features. First, it is explicit in terms of Stroh's eigenvalues $pα$ ($α=1,2,3$) on the oblique plane with normal coincident with the position vector; second, it remains well-defined when some Stroh's eigenvalues are equal (mathematical degeneracy) or nearly equal (quasi-mathematical degeneracy); and third, they are exact. Therefore, both new proposed solutions inherit these appealing features, being explicit in terms of Stroh's eigenvalues, simpler, unique, exact and valid independently of the kind of degeneracy involved, as opposed to previous approaches. A study of all possible degenerate cases validate the proposed scheme.

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## Figures

Fig. 1

Imaginary part of Stroh's eigenvalues when φ varies between 0 and 2π

Fig. 2

Comparison with the analytic solutions of ∂q0/∂φ, ∂q2/∂φ, and ∂q4/∂φ for materials A, B, and C when φ varies between 0 and π/2

Fig. 3

Comparison with the analytic solutions of ∂2q0/∂φ2 for materials A, B, and C when φ varies between 0 and π/2

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