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# Reciprocity and Related Topics in Elastodynamics

[+] Author and Article Information
Jan D. Achenbach

McCormick School of Engineering and Applied Sciences, Northwestern University, Evanston, IL 60208

Appl. Mech. Rev 59(1), 13-32 (Jan 01, 2006) (20 pages) doi:10.1115/1.2110262 History:

## Abstract

Reciprocity theorems in elasticity theory were discovered in the second half of the 19th century. For elastodynamics they provide interesting relations between two elastodynamic states, say states $A$ and $B$. This paper will primarily review applications of reciprocity relations for time-harmonic elastodynamic states. The paper starts with a brief introduction to provide some historical and general background, and then proceeds in Sec. 2 to a brief discussion of static reciprocity for an elastic body. General comments on waves in solids are offered in Sec. 3, while Sec. 4 provides a brief summary of linearized elastodynamics. Reciprocity theorems are stated in Sec. 5. For some simple examples the concept of virtual waves is introduced in Sec. 6. A virtual wave is a wave motion that satisfies appropriate conditions on the boundaries and is a solution of the elastodynamic equations. It is shown that combining the desired solution as state $A$ with a virtual wave as state $B$ provides explicit results for state $A$. Basic elastodynamic states are discussed in Sec. 7. These states play an important role in the formulation of integral representations and integral equations, as shown in Sec. 8. Reciprocity in 1-D and full-space elastodynamics are discussed in Secs. 9,10, respectively. Applications to a half-space and a layer are reviewed in Secs. 11,12. Section 13 is concerned with reciprocity of coupled acousto-elastic systems. The paper is completed with a brief discussion of reciprocity for piezoelectric systems. There are 61 references cited in this review article.

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## Figures

Figure 1

Body of volume V and boundary S subjected to concentrated loads P1 and P2

Figure 2

Inhomogeneous layer in between homogeneous domains

Figure 3

Directions of incident and scattered waves for the reciprocity relations

Figure 4

Half-space subjected to a time-harmonic line load

Figure 5

Configuration of the elastic layer

Figure 6

Compact inhomogeneity in an acoustic half-space

Figure 7

Rigid plate with a point source, A, and its image, A′

Figure 8

Configuration for application of electromechanical reciprocity relation

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