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A New Use of the Elastodynamic Reciprocity Theorem Jan D. Achenbach McCormick School of Engineering and Applied Sciences Northwestern University Evanston, IL, 60208 Symposium in Honor of Ted Belytschko April 18-20, 2013

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Elastodynamic Reciprocity Betti (1872), Rayleigh (1873), Graffi (1947) time-harmonic fields:,, Consider two elastodynamic states Then for a region V with boundary S Linear stress-strain relation: Solids may be: anisotropic inhomogeneous linearly viscoelastic

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Lamb’s Problem (1904) Time Harmonic line load radiates: Circular longitudinal wave Circular transverse wave Wedge wave Surface (Rayleigh wave)

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Guided Waves

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Lord Rayleigh (1842-1919) J.W.S. Raleigh, “On waves propagated along the plane surface of an elastic solid,” Proc. London Math. Soc., Vol. 17 (1885) pp. 4-11

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Surface Waves

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Equation for Phase Velocity The condition that yields the well known equation

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STEADY-STATE TIME-HARMONIC CASE State A: State B: Applied body or surface force

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State A: State B: virtual surface wave

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Lamb’s Problem P z x

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Comparison of LAMB’s and Rec. Thm. Results Result of FIT: Result of Rec. Thm.: 0.10001.50000.8931 -0.9651 -0.9651 0.16001.57180.9090 -0.8682 -0.8682 0.20001.63300.9110 -0.8068 -0.8068 0.26001.75590.9210 -0.7199 -0.7199 0.30001.87080.9274 -0.6655 -0.6655 0.40002.44950.9422 -0.5420 -0.5420

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Conclusion For every configuration that supports guided waves: e.g., surface waves (Rayleigh) waves in a layer (Lamb) waves in a film/substrate configuration (Sezawa) and if the free time-harmonic form of such waves is known, then the reciprocity theorem of elastodynamics provides a simple approach to determine the amplitude of guided waves that radiate from time-harmonic external excitation. For pulsed excitations such as laser generated heating, Fourier superposition gives the signal strength of the radiated guided waves.

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