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

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

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

Guided Waves

Lord Rayleigh ( ) 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

Surface Waves

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

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

State A: State B: virtual surface wave

Lamb’s Problem P z x

Comparison of LAMB’s and Rec. Thm. Results Result of FIT: Result of Rec. Thm.:

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.