Thursday, 11 June 2020

Speaker: Prof. Emer. John R. Willis

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Cambridge

Transmission and reflection at the boundary of a random two-component composite

Abstract

Description: A half-space x₂ > 0 is occupied by a two-component statistically-uniform random composite with specified volume fractions and two-point correlation. It is bonded to a uniform half-space x₂ < 0 from which a plane wave is incident. The transmitted and reflected mean waves are calculated via a variational formulation that makes optimal use of the given statistical information. The problem requires the specification of the properties of three media: those of the two constituents of the composite and those of the homogeneous half-space. The complexity of the problem is minimized by considering a model acoustic-wave problem in which the three media have the same modulus but different densities. It is formulated as a problem of Wiener–Hopf type which is solved explicitly in the particular case of an exponentially decaying correlation. A striking feature in this case is that the composite supports exactly two mean acoustic plane waves in any given direction. Each decays exponentially. At low frequencies the rate of decay of one wave is much slower than that of the other; at higher frequencies the decay rates of the two waves are comparable. Thus, in general, there are two transmission coefficients and one reflection coefficient, and the conditions of continuity of traction and displacement of the mean waves do not suffice to determine them: the solution absolutely requires a more complete calculation, such as the one presented.

Biography

John Raymond Willis, Professor of Theoretical Solid Mechanics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge since 2001 (also, 1994–2000); Emeritus from 2008.

Previous Positions: Professeur de Mécanique, Ecole Polytechnique (part-time, 1998–2004); Professor of Applied Mathematics, University of Bath (1972–1994; also, 2000–2001), Senior Assistant in Research/Assistant Director of Research, University of Cambridge (1965–1972), Research Associate, Courant Institute of Mathematical Sciences, New York University (1964– 1965), Assistant Lecturer, Imperial College (1962–1964).

Degrees and Awards: BSc, First Class Honours (London, 1961), PhD (London, 1964), MA (Cambridge, 1966), Honorary DSc (Bath, 2007). Lubbock Prize, University of London, 1961. Governors’ Prize in Mathematics, Imperial College, London, 1961. Sherbrooke Research Studentship, University of London, 1961. Adams Prize, University of Cambridge, 1971. Timoshenko Medal, American Society of Mechanical Engineers, 1997. Prager Medal, Society of Engineering Science, 1998. Euromech Solid Mechanics Prize, 2012.

Societies: Fellow, Institute of Mathematics and its Applications (FIMA), 1968. Fellow, Royal Society of London (FRS), 1992. Foreign Associate, U.S. National Academy of Engineering, 2004. Foreign Associate, French Academy of Sciences, 2009.

Editorial Activities: Member of editorial boards of: J. Mech. Phys. Solids (editor, 1982–2006), Comptes Rendus Mécanique, Acta Mechanica Sinica.

Research Interests: Mathematical investigation of problems arising mostly in the mechanics of solids, including the statics and dynamics of composite materials, dislocation theory, nonlinear fracture mechanics, elastodynamics of crack propagation, strain-gradient plasticity, rate and state models of friction.

Notes

by Zahra Hooshmand Ahoor and Kübra Sekmen

The analysis of the equations is exact but the equations themselves are approximate, relying on a Hashin-Shtrikman type of approximation to which the QCA (Quasi-Crystalline approximation) is closely related.

In this study, except for the dominant wave, only one other wave has been found, however changing the correlation function will lead to more other waves, probably an infinite number.

At low frequencies, there’s one dominant wave, which does not decay much, and every other wave will decay rapidly. If the dominant wave has been found, an ordinary transmission-reflection calculation can be done. This calculation is fine at low frequencies, but at high frequencies not anymore. It means that we have to do a more detailed calculation to deal with high frequencies. This issue reminds us of the limitation of employing any local approximation for ‘effective properties’. It would be a better prediction if operators have been used for approximation of the effective constants.

The results presented in this work can be interpreted for conventional acoustics. To do so, displacement u becomes pressure, modulus µ becomes specific volume (1/ρ) and density becomes compressibility. Thus, for conventional acoustics, our results are for uniform density but varying compressibility.