A New Use of the Elastodynamic Reciprocity Theorem Jan D. Achenbach

1. 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

2. 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

3. Lamb’s Problem (1904) Boundary Conditions At • Time Harmonic line load radiates: • Circular longitudinal wave • Circular transverse wave • Wedge wave • Surface (Rayleigh wave) Reciprocity Theorem yields simple solution for J.D. Achenbach (2003), Reciprocity in Elastodynamics. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, UK.

4. Guided Waves Along the surface of a half-space: Rayleigh Waves Along the interface between two solids: Stonely Waves Along the interface between a fluid and a solid: Scholte Waves Along an elastic layer: Lamb Waves Harmonic Waves: propagates with phase velocity c No characteristic length in the geometry: c= constant If there is a characteristic length:

5. 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

6. Surface Waves velocity of surface waves are simple expressions in terms of and

7. Equation for Phase Velocity The condition that and at yields the well known equation for the velocity of surface waves. Note that depends only on

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

9. State A: Radiated Waves: for surface waves dominate State B: virtual surface wave has been omitted

10. Only counterpropagating waves along contribute where can be evaluated analytically

11. P Lamb’s Problem x Boundary conditions: at z Apply the exponential Fourier transform over x Apply the FT to the system of pde’s, with respect to Solve resulting ode’s with respect to Apply BC’s at and radiation condition as Apply inverse transform w.r.t. Extend integral over to complex -plane 2 branch points and a pole Branch cut integrals give cylindrical waves Pole gives surface wave

12. Comparison of LAMB’s and Rec. Thm. Results is the solution of Result of FIT: Result of Rec. Thm.: 0.1000 1.5000 0.8931 -0.9651 -0.9651 0.1600 1.5718 0.9090 -0.8682 -0.8682 0.2000 1.6330 0.9110 -0.8068 -0.8068 0.2600 1.7559 0.9210 -0.7199 -0.7199 0.3000 1.8708 0.9274 -0.6655 -0.6655 0.4000 2.4495 0.9422 -0.5420 -0.5420

13. 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 guidedwaves 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. Note: body with dependent elastic moduli and mass density: O. Balogun and J.D. Achenbach: Wave Motion, in press