Modeling the propagation of ultrasonic waves in the interface region between two bonded elements P.P. Delsanto a, * , Sigrun Hirsekorn b,1 , V. Agostini a , R. Loparco a , A. Koka b a Dipartimento di Fisica, INFM – Dip. Fisica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy b Fraunnhofer Institut fuer zerstoerungsfreie Pruefverfahren (IZFP), Universitaet, Geb. 37, D-66123 Saarbruecken, Germany Abstract An important task in nondestructive materials evaluation is the development of techniques to characterize the bond quality of adherent joints. Binding forces are nonlinear and cause a nonlinear modulation of transmitted and reflected ultrasonic waves. As a consequence,thehigherharmonicsgeneratedbyaninsonifiedmonochromaticwavegiveinformationabouttheadhesivebonds.The localbindingforcesinthinbondedinterfacescanbeobtainedbytheamplitudesoftheultrasonicwavesoftheinsonifiedfrequency and its higher harmonics as transmitted through the interface. Additional phase measurements may enable one to obtain the evaluationofthefullhystereticcycleoftheinteractionforce.Inordertogainadeeperunderstandingoftheinterfaceregionandto improvethetechnique,numericalsimulationsoftheultrasonicwavepropagationthroughspecimensoftwobondedelementscanbe used.Asimplemodelbasedonthelocalinteractionsimulationapproach(LISA)isdescribedinthiscontribution,andacomparison between the results of the simulations and the experimental data is presented. Besides its intrinsic relevance for NDE, the problem considered in this paper may be very useful to analyze and test models for the simulation of ultrasonic wave propagation in nonclassical nonlinear mesoscopic elastic materials. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Adhesion; Binding forces; Higher harmonics; Nonlinear ultrasound; Numerical simulations 1. Introduction For traditional nonlinear materials the one-dimen- sional wave equation governing the propagation of acoustic (or ultrasonic) pulses may be written by simply expressing the driving force for the local displacement u as a power series in the strain e ¼ ou=ox. For these materials,calledatomicelasticmaterials,elasticityarises fromwellknownforcesattheatomicormolecularlevel. These materials are well described by the classical (Landau) theory of elasticity [1]. There are, however, several materials, which we shall call nonlinear meso- scopic elastic (NME), for which such a description is inadequate. For them it is necessary to add a nonana- lytical term Aðe; _ eÞ in the wave equation: o 2 u ot 2 ¼ S q o ox ou ox " þ b ou ox   2 þ d ou ox   2 þ # þ Aðe; _ eÞ; ð1Þ where q is the density and S the elastic modulus. These materialsincludemanyofinterestinseismology,suchas rock, sand, and soil, and others of relevance in industry andcivilengineering,suchasconcreteandatomicelastic materials with microscale damages [2]. All these mate- rials share several important properties, such as a hys- teretic behavior and discrete memory in quasi-static experiments, and the appearance of effects in resonant wave experiments, such as a shift of the resonance fre- quency when increasing the driving amplitude, the generation of higher harmonics at a prescribed rate in the response, and the phenomenon known as slow dy- namics.Sincetheseeffectsmaybeverylargetheycanbe advantageously exploited for NDE purposes [3,4]. Nonclassicalnonlineareffectsarebelievedtobedueto the presence of softer regions in hard materials. Typi- cal examples are microcracks, flat pores, and soft bonding regions between grains in granular matter. In Ultrasonics 40 (2002) 605–610 www.elsevier.com/locate/ultras * Corresponding author. Tel.: +39-011-5647320; fax: +39-011- 5647399. E-mail addresses: delsanto@polito.it (P.P. Delsanto), hirsek- orn@izfp.fhg.de (S. Hirsekorn). 1 Tel.: +49-681-9302-3836; fax: +49-681-9302-5903. 0041-624X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0041-624X(02)00183-X