Department of Mathematical Sciences
Loughborough University
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Mathematical Modelling of Nonlinear Waves in Layered Elastic Waveguides with Inhomogeneities

Funded by the EPSRC (2006-2008, Grant EP/D035570/1, 121,689 pounds)

Principal Investigator: Dr. Karima Khusnutdinova (Loughborough University, UK)
Research Associate: Dr. Alexey Zakharov (Loughborough University, UK)
Visiting Fellow: Prof. Alexander Samsonov (Ioffe Institute of the Russian Academy of Sciences, St. Petersburg, Russia)
Sub-contracted experimental group: Dr. Galina Dreiden, Prof. Alexander Samsonov, Dr. Irina Semenova (Ioffe Institute of the Russian Academy of Sciences, St. Petersburg, Russia)

Aims and Objectives

This 2 year long project lies at the intersection of Mathematical Theory of Nonlinear Waves, Solid Mechanics and Non-destructive Testing of Materials and Structures. The aim of the project is to develop a theory of nonlinear waves propagating in layered elastic waveguides with extended inhomogeneities (e.g., representing damage / delamination). The objectives of this research are:

Main Results and Publications to Date

Main results:

(a) Derivation of some model equations for long nonlinear longitudinal waves in elastic waveguides with perfect and imperfect interfaces, using nonlinear elasticity approach and lattice considerations [1, 2, 3, 4]
(b) Asymptotic theory of nonlinear wave processes in layered elastic waveguides with extended inhomogeneities modelling poor adhesion / delamination with an emphasis on the case of an incident solitary wave, and related experimental studies [1, 5, 6, 7, 8]
(c) Description of the evolution of nonlinear longitudinal waves in inhomogeneous layered waveguides with imperfect bonding, and related experimental studies [2, 3, 9, 10, 11]

Publications:

1. K.R. Khusnutdinova, A.M. Samsonov, Fission of a longitudinal strain solitary wave in a delaminated bar, Phys. Rev. E 77 (2008) 066603.
2. K.R. Khusnutdinova, A.M. Samsonov, A.S. Zakharov, Nonlinear long wave models for imperfectly bonded layered waveguides, Theor. Math. Phys. 159 (2009) 475-489.
3. K.R. Khusnutdinova, A.M. Samsonov, A.S. Zakharov, Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures, Phys. Rev. E 79 (2009) 056606.
4. K.R. Khusnutdinova, Coupled Klein-Gordon equations and energy exchange in two-component systems, EPJ Special Topics, Volume 147 (2007) 45-72.
5. G.V. Dreiden, K.R. Khusnutdinova, A.M. Samsonov, I.V. Semenova, Longitudinal strain solitary wave in a two-layered polymeric bar, Strain 46 (2010) 589-598.
6. A. Samsonov, G. Dreiden, K. Khusnutdinova, and I. Semenova, Elastic Solitons in Delaminated Bars: Splitting Leads to Fission, Nonlinear Acoustics- Fundamentals and Applications, ISNA18, AIP Conference Proceedings, ed. B.O. Enflo, C.M. Hedberg, L. Kari, Stockholom, Sweden (2008) 307-310.
7. G. Dreiden, K. Khusnutdinova, A. Samsonov, I. Semenova, Nonlinear bulk strain waves in layered elastic waveguides with delamination: theory and experiments, XXII International Congress of Theoretical and Applied Mechanics, CD-ROM Proceedings, Adelaide, Australia, ISBN 978-0-9805142-1-6 (2008) 11385.
8. G.V. Dreiden, K.R. Khusnutdinova, A.M. Samsonov, I.V. Semenova, Splitting induced generation of soliton trains in layered waveguides, J. Appl. Phys. 107 (2010) 034909.
9. G.V. Dreiden, K.R. Khusnutdinova, A.M. Samsonov, I.V. Semenova, Comparison of the effect of cyanoacrylate- and polyurethane- based adhesives on a longitudinal strain solitary wave in layered polymethylmethacrylate waveguides, J. Appl. Phys. 104 (2008) 086106.
10. G.V. Dreiden, A.M. Samsonov, I.V. Semenova, K.R. Khusnutdinova, Observation of a radiating bulk strain soliton in a solid waveguide, Techn. Physics 56 (2011) 889-892.
11. K.R. Khusnutdinova, K.R. Moore, Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations, Wave Motion 48 (2011) 738-752.

More...

Most important physical effects revealed in this study are the following:

(a) Splitting of a layered bar can lead to the generation of more than one secondary solitary wave from a single incident soliton. The number and parameters of generated secondary solitons depend on the number of layers in a waveguide and potentially can be used to detect poor adhesion / delamination, complementing currently existing techniques. This effect has been predicted in theory (see [1, 6, 7]) and confirmed in experiments [6, 7, 8]. Our theory is based on a weakly nonlinear solution for a Boussinesq-type equation derived for a waveguide made of isotropic nonlinearly hyperelastic Murnaghan material.The developed approach uses matching of two asymptotic multiple-scales expansions and results from the theory of integrable systems. Experiments were performed using holographic interferometry and laser induced generation of an incident compression solitary wave in two- and three-layered polymethylmethacrylate (PMMA) bars, bonded using ethyl cyanoacrylate-based (CA) adhesive.

Below: observation of two secondary solitons in a split three-layered PMMA bar (from [6]).


(b) Difference in the behaviour of nonlinear waves in a nearly symmetric layered structure with an imperfect interface compared to the limiting case of the perfect (or ideal) interface: classical solitons, exponentially decaying to zero in their tail regions are generally replaced with generalized solitary waves, having a copropagating oscillatory tail. This effect, described theoretically and numerically in [3] and [11], is believed to be linked to the observed increase in the decay rate of a nonlinear longitudinal bulk strain solitary wave in some imperfectly bonded structures compared to the decay rate in a perfectly bonded structure [9]. Indeed, the amplitude of the core solitary wave gradually decreases due to the radiation of a copropagating wave. First observation of the radiating bulk strain solitary wave in a layered solid waveguide is reported in [10].

Below left: generation of a generalized solitary wave in a two-layered imperfectly bonded waveguide from localized initial data for two sets of parameters (from [3]).
Below right: Exponential decay curves for two- and three-layered PMMA waveguides with canoacrylate- (CA) and polyurethane (PU)-based adhesives (from [9]).
 

Below: visiting experimental group from the Ioffe Institute in June 2007 and June 2008. First photograph, left to right: Dr I.V. Semenova, Prof. A.M. Samsonov, Dr G.V. Dreiden next to their unique experimental set-up.

 

This webpage was created on 25 November 2008.

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