A series of numerical simulations is carried on in order to understand the accuracy of dispersive wave models for microstructured solids. The computations are performed by means of the finite-volume numerical scheme, which belongs to the class of wave-propagation algorithms. The dispersion effects are analyzed in materials with different internal structures: microstructure described by micromorphic theory, regular laminates, laminates with substructures, etc., for a large range of material parameters and wavelengths

Berezovski, Arkadi

Berezovski, Mihhail

Engelbrecht, Juri

English

Embry-Riddle Aeronautical University

Publications 2012 Wave Propagation and Dispersion in Microstructured Solids Arkadi Berezovski Tallinn University of Technology, arkadi.berezovski@cs.ioc.ee Juri Engelbrecht Tallinn University of Technology, je@ioc.ee Mihhail Berezovski Worcester Polytechnic Institute, berezovm@erau.edu Follow this and additional works at: https://commons.erau.edu/publication  Part of the Mechanics of Materials Commons, Numerical Analysis and Computation Commons, and the Partial Differential Equations Commons Scholarly Commons Citation Berezovski, A., Engelbrecht, J., & Berezovski, M. (2012). Wave Propagation and Dispersion in Microstructured Solids. , (). Retrieved from https://commons.erau.edu/publication/943 This Conference Proceeding is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Publications by an authorized administrator of Scholarly Commons. For more information, please contact commons@erau.edu. Proceedings of the 25th Nordic Seminar on Computational MechanicsK. Persson, J. Revstedt, G. Sandberg, M. Wallin (Eds.)cLund University, 2012Wave propagation and dispersion in microstructured solidsArkadi Berezovski1, Juri Engelbrecht1, and Mihhail Berezovski1;2(1)Centre for Nonlinear Studies, Institute of Cybernetics at Tallinn University of Technology,Akadeemia tee 21, 12618 Tallinn, Estonia, Arkadi.Berezovski@cs.ioc.ee(2)Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Rd,Worcester, MA, 01609, USA, mberezovski@wpi.eduSummary. A series of numerical simulations is carried on in order to understand the accuracy ofdispersive wave models for microstructured solids. The computations are performed by means of thenite-volume numerical scheme, which belongs to the class of wave-propagation algorithms. The dis-persion eects are analyzed in materials with dierent internal structures: microstructure described bymicromorphic theory, regular laminates, laminates with substructures, etc., for a large range of materialparameters and wavelengths.Key words: microstructured solids, wave dispersion, numerical simulationIntroductionMicrostructural eects are observed in wave propagation in solids when the wavelength of atravelling signal becomes comparable with the scale of material heterogeneities. A vivid exampleof the inuence of microstructure on wave propagation is the wave dispersion that profoundlyalters both the shape and the velocity of propagating waves. The most recognizable signatureof the wave dispersion is that the phase and group velocities of propagating waves dier fromeach other. The dispersive eects of wave propagation in microstructured solids become non-negligible for suciently high frequencies.Wave propagation in heterogeneous solids has been a subject of considerable research formany years. However, micro-structural details are rarely taken into account in large-scale struc-tural dynamics or dynamic impact simulations. The reason is the enormous complexity of wavephenomena in highly heterogeneous media. There exist two alternative approaches to the de-scription of microstructural eects on wave propagation in solids. The rst one is focused onthe determining so-called eective properties of a material. It is expected that these averagedor smoothened properties reect in some global sense the response of specimens of the materialto external loads. Another approach to involve microstructural eects into the description ofwave propagation is provided by higher order or generalized theories of elastic continua. Thesetheories have been proposed in 1960s [1, 2], and later claried, classied, and extended [3]. Thewell-established framework for higher grade and higher order theories is, however, accompaniedby too many usually undetermined phenomenological coecients. Nevertheless, dispersive waveequations in solid mechanics are based either on a homogenization procedure or on a generalizedcontinuum theory.Dispersive wave modelsWave propagation in a homogeneous medium is a well known phenomenon in mechanics. Thecorresponding one-dimensional wave equation is a classical example of hyperbolic partial dier-ential equations in textbooksutt = c2uxx; (1)where u is the displacement, c is the elastic wave speed and subscripts denote derivatives. Thewave equation (1) possesses no dispersion. Considering a harmonic waveu(x; t) = u^ exp [i(kx  !t)] (2)with wave number k and frequency !, we obtain the dispersion relation!2 = c2k2: (3)It is easy to see that here the group velocity @!=@k is equal to the phase velocity c, which meansthat no dispersion is present.To describe wave propagation in heterogeneous materials reecting dispersion eects, sev-eral modications of the wave equation are proposed. The simplest generalization of the waveequation is the linear version of the Boussinesq equation for elastic crystals (cf. [4])utt = c2uxx + c2l2A11uxxxx; (4)where l is an internal length parameter and A11 is a dimensionless coecient. The dispersionrelation is obtained by using again the harmonic wave solution (2)!2 = c2k2   c2l2A11k4: (5)This dispersion relation is nonlinear, which means that phase and group velocities are dierent.Another generalization of the wave equation is the Love-Rayleigh equation for rods accountingfor lateral inertia (cf. [5], p.428)utt = c2uxx + l2A12uxxtt; (6)where A12 is again a dimensionless constant. The corresponding nonlinear dispersion equationhas the form!2 = c2k2   l2A12!2k2: (7)A more general equation combining the two dispersion models givesutt = c2uxx + c2l2A11uxxxx + l2A12uxxtt: (8)Similar model proposed by Engelbrecht and Pastrone [6] introduces additionally a contributionof microstructure on slowing down of the propagation velocity c2Autt = c2   c2Auxx + c2l2A11uxxxx + l2A12uxxtt: (9)Accordingly, it has the dispersion relation in the form!2 = (c2   c2A)k2   c2l2A11k4   l2A12!2k2: (10)Due to three additional terms combined, the last model has larger dispersion properties.In its turn, the Maxwell-Rayleigh model of anomalous dispersion [4] introduces in consider-ation the four-order time derivativeutt = c2uxx +l2A22c2 utt   c2uxxtt: (11)Four-order time derivatives are included also in the "causal" model for the dispersive wavepropagation proposed by Metrikine [7]utt = c2uxx   c2l2A11uxxxx + l2A12uxxtt   l2c2A22utttt; (12)and in the model based on the Mindlin theory of microstructure [8], which can be representedin the formutt = c2   c2Auxx + l2P utt   c2 uxxxx+l2c2Q utt   c2 uxxtt: (13)Here P and Q are dimensionless constants. It is clear hat corresponding dispersive relations arenonlinear.As it is shown recently [9], the last two models for dispersive wave propagation can be uniedas followsutt = c2   c2Auxx + l2P utt   c2 uxxxx+l2c2Q utt   c2 uxxtt+ c2l2Ruxxxx: (14)It is clear that the unied model (14) generalizes both approaches (12) and (13).Numerical simulationsIn order to understand the accuracy of the dispersive models, a series of numerical simulationsis carried on. The computations are performed by means of the nite-volume numerical scheme,which belongs to the class of wave-propagation algorithms [10]. Details of the numerical schemecan be found in [11]. The dispersion eects of 1D waves are demonstrated in materials with dif-ferent internal structures: microstructure described by micromorphic theory, regular laminates,laminates with substructures, etc.-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 3600  3800  4000  4200  4400  4600  4800Di me ns io nl es s st r es sSpace stepsMicrostuctured mediumPeriodic mediumFigure 1. Deformed shape of an initially Gaussian stress pulse in periodic and microstructured solids.One of the important problems is to compare the results obtained by means of various models.As an example, the comparison of a direct numerical simulation of a Gaussian stress pulsepropagation along the elastic bar containing an inhomogeneous part constructed by periodicallyalternating layers and a computation based on the Mindlin-type microstructure model (14) isshown in Fig.1.As one can see, the eect of microstructure in the model manifests itself only locally, whereasthe dispersion in the periodic laminate is non-local due to consecutive reections. In principle,the localization of the microstructure inuence is expected, since the presence of the microstruc-ture is invisible in the absence of loading.In order to get matching results, one should critically revise the free energy function inthe micromorphic theory for adequate modelling of interaction forces between macro{ and mi-crostructures. In the considered case, the pulse length is 5 times longer than the inhomogeneitysize. This particular case was chosen because it clearly shows the synergy of the two microstruc-ture models unied in [9]. The matching results are obtained by modifying the coupling betweenmacro- and microstructures including also the dependence on gradients of the internal variables.The correlation between models is analyzed in detail for a large range of material parametersand wavelengths.The research was supported by the EU through the European Regional Development Fundand by the Estonian Science Foundation (grant No. 8702).References[1] R.D. Mindlin. Microstructure in linear elasticity. Archive for Rational Mechanics and Anal-ysis, 16:51{78,1964.[2] A.C. Eringen and E.S. Suhubi. Nonlinear theory of simple microelastic solids I & II. Inter-national Journal of Engineering Science, 2:189{203,389{404,1964.[3] S. Forest. Micromorphic approach for gradient elasticity, viscoplasticity and damage. Journalof Engineering Mechanics, 135:117{131,2009.[4] G.A. Maugin. On some generalizations of Boussinesq and KdV systems. Proceedings of theEstonian Academy of Sciences. Physics. Mathematics, 44:40{55,1995.[5] A.E.H. Love. Mathematical Theory of Elasticity. Dover, New York, 1944.[6] J. Engelbrecht and F. Pastrone. Waves in microstructured solids with nonlinearities in mi-croscale. Proceedings of the Estonian Academy of Sciences. Physics. Mathematics, 52:12{20,2003).[7] A.V. Metrikine. On causality of the gradient elasticity models. Journal of Sound and Vibra-tion, 297:727{742,2006.[8] J. Engelbrecht, A. Berezovsk, F. Pastrone, M. Braun. Waves in microstructured materialsand dispersion. Philosophical Magazine, 85:4127{4141,2005.[9] A. Berezovski, J. Engelbrecht, and M. Berezovski. Waves in microstructured solids: a uniedviewpoint of modeling. Acta Mechanica, 220:349{363,2011.[10] A. Berezovski. Thermodynamic interpretation of nite volume algorithms. Journal of Struc-tural Mechanics (Rakenteiden Mekaniikka), 44/3,156{171,2011.[11] M. Berezovski, A.Berezovski and J.Engelbrecht. Waves in materials with microstructure:numerical simulation. Proceedings of the Estonian Academy of Sciences, 59/2, 99{107,2010.View publication stats

Wave Propagation and Dispersion in Microstructured Solids

https://commons.erau.edu/cgi/viewcontent.cgi?article=2040&amp;context=publication

Wave Propagation and Dispersion in Microstructured Solids

Abstract

Similar works

Full text

Available Versions

Embry-Riddle Aeronautical University