Micromechanical approach for the analysis of wave propagation in particulate composites

Modelamiento micromecánico de la propagación de ondas en materiales compuestos con refuerzos esféricos

Cristhian Fernando Rojas-Cristancho^1a*, Florence Dinzart^1b, Octavio Andrés González-Estrada²

¹ Estrada21Laboratoire d’Etude des Microstructures et de Mécanique desMatériaux(LEM3), Université de Lorraine, 1 route d’Ars Laquenexy, 57078 Metz, France. Email:
^acristhian-fernando.rojas-cristancho1@etu.univ-lorraine.fr

^bflorence.dinzart@univ-lorraine.fr

² Grupo de Investigación en Energía y Medio Ambiente (GIEMA), School of Mechanical Engineering, Universidad Industrial de Santander, Colombia. Email:
agonzale@saber.uis.edu.co

Forma de citar: How to cite:C. Rojas, F. Dinzart, O.A. González-Estrada, “Micromechanical approach for the analysis of wave propagation in particulate composites,” Rev. UIS Ing., vol. 18, no. 2, pp. 41-50, 2019. doi: 10.18273/revuin.v18n2-2019004

Received:6June2018. Accepted: 11 October, 2018. Finalversion:12January2019

Abstract

Laser ultrasonic non-destructive testing is widely used for the inspection of mechanical structures. This method uses the propagation of ultrasonic guided waves (UGW) in the media. It has been demonstrated that the addition of a thin composite layer between the laser source and the structure for inspection is necessary. Consequently, this composite is an optoacoustic transducer composed of an absorption material as carbon for inclusions and an expanding material as an elastomer for the matrix. Thus, optimal fabrication of this composite should enable the amplification of the signal for inspection. Indeed, experimental research has demonstrated that variation in the volumefraction of carbon inclusions, their shape and the nature of the matrix affect the amplification of the signal directly. The aim of this study is to analyse the wave propagation in particulate viscoelastic composites by a dynamic self-consistent approach

Keywords: particulate composites; self-consistent; viscoelastic composites; wave propagation.

Resumen

La inspección de componentes mecánicos por ultrasonido láser es uno de los controles no destructivos (CND) más utilizados en la industria, ya que permite inspeccionar rápidamente piezas de gran tamaño y de formas complejas por medio de la propagación de ondas guiadas. Ha sido demostrado que, para obtener la mejor calidad posible de la señal acústica, es necesario integrar una fina capa de material compuesto entre la placa y la fuente láser. Dicha capa de material compuesto permitiría la amplificación de la señal acústica;esta capa está formada por refuerzos de carbono que dan una característica de absorción térmica y de una matriz elastómera que otorga una característica de expansión volumétrica. Por tanto, la fabricación óptima de dicho compuesto permitiría la amplificación de la señal de inspección. De hecho, experimentalmente ha sido demostrado que la variación de la fracción volumétrica del refuerzo, de su forma (esférica o elipsoidal) y del tipo de matriz (silicona o resina), afecta directamente la amplificación de la señal. El objetivo de este trabajo es realizar un estudio micromecánicode tipo autocoherente de la propagación de ondas elásticas en un medio heterogéneo compuesto por una matriz viscoelástica y refuerzos esféricos elásticos.

Palabras clave: micromecánica; autocoherente dinámico; viscoelasticidad; compuestos particulados; propagación de ondas

INTRODUCCIÓN

Reliability in mechanical engineering structures after fabrication depends on how well these are controlled in a lifetimeperiod[1], [2]. For that, non-destructivetesting(NDT)isa useful tool. Ultrasonic guided waves (UGW) for inspection in mechanical structures havebeenwidely investigated and developed[3]. This is because this method allows for evaluating large and complex form structures with only one measurement, unlike the conventional ultrasonic method using a transmitter-receiver system for local measurement.

A variation forUGWgeneration consists ofusing the power density of an incident laser beam[4].For low power densities, ultrasonic waves are caused by the rapid thermal expansion of the material being irradiated. For this purpose, an optoacoustic transductor must transform in an easy and efficient way laser energy into acoustic (elastic) waves. This method to generate mechanical energy has been the focus of research in[5]–[9], where many different configurations ofcomposite materialssuch as gold nanoparticles, carbon nanofiber and carbon nanotubeshave been studied experimentally to obtain high-performancelaser ultrasound transducers. In the same way, Ref.[10], [11]have focused on viscoelastic particulate composites

According to [10], carbon nanoparticles produced by candle soot (CSPs) is an efficient light absorbing materialand viscoelastic Polydimethylsiloxane (PDMS) has a high thermal coefficient of volume expansion. Therefore, CSPs –PDMS composite is found to be a high-performanceoptoacoustic transducer since it could generate a high frequency, broadband and highamplitude ultrasound wave.

Inspired by the aforementioned results,the authorsare interested in carrying out an analysisof elastic wave propagation in heterogeneous media. A first theoretical thermo-acoustic validation is proposed in[10], where the generated pressure gradient is directly associated withthe temperature gradient generated by the laser impact. Under meaningful assumptions, the pressure gradientmay be related solely to the setting parameters of the lasersource.

Another analysis consists ofconsidering the propagationprocess being in steadystate. Then,it is possible to consider elasticbehaviourand temperaturegradient separately, as proposedin[12].In this case, the displacement field ofthe propagating waveis supposedlycoincident with the displacement field of solid media. Therefore, the solution searched is the eigenvalues and eigenvectors velocities of the propagatingwave in the solid media.Solvingthe wavepropagation problemisstandardfor homogeneous media[12]. The micromechanicalapproach is a traditional way to estimate the effectivebehaviourof heterogeneous materials, especiallyfor staticproblems[13]. Withinthe framework of dynamicproblems, the acceleration term in the Navierequation has to be retainedto evaluate the dynamic response of the compositethrough amicromechanical self-consistent approach[14].

The dynamicresponse in a solid homogeneous media for the wave propagation problem must be evaluatedby the wave number 푘훾, where 훾=훼for longitudinal waves and 훾=훽for shear waves. This wavenumber belongs to the complex domain:

where, 푉(휔)and 훼(휔)are the phase propagation velocity and the propagation attenuation coefficient, respectively. Dissipative propertiesof viscoelastic composites directly affect the attenuation coefficientas demonstrated experimentally in[15], [16].

This work presents in Section 2 a general micromechanical self-consistent formulation based on[14]for analysing the elastic wave propagation in a heterogeneous solid media.Multiple scattering caused by the random distribution and interaction between particlescomplicates considerably the wave propagation analysis. Therefore, we considered a single scattering problem described in Section 2. Afterward, the additionof viscoelastic properties to the micromechanical formulation is presented. In Section 3, the results obtained are shown and discussed.Finally,in Section 4,some conclusions and limitations of the study are presented.

2.Methodology

2.1.Dynamic self-consistent formulation

Micromechanical methods arewidespread to estimate composite behaviourwithin the static framework. For the dynamic casewhere acceleration terms areconsidered, the perturbation in the infinitemedia is supposed to be the mean propagating wave.Moreover, volume inclusion concentration in composites is available information in practiceand notthe distribution and interaction between particles. Accordingly, a first approach consists of considering the single inclusion problemembedded in the matrixpossessing effective media properties

A composite is considered comprising a matrix with elastic moduli tensor 퐿푛+1and density 휌푛+1, in which there are embedded 푛different types of inclusions;an inclusion of family phase 푟has an elastic modulustensor 퐿푟and density 휌푟. Volume concentration in the composite is such that:

Solving the dynamic problem consist ofdetermining a mean response to the displacement field on the solid<푢>. For this purpose, the starting point is the Navier equation, that in the absence of body forcereads:푑푖푣휎

where 휎is the stress field,and 푝is the momentum density. Stress and momentum density are related to elasticity Hooke’s law and displacement wave field,respectively:

At the same time, deformation푒is related to displacement field through Cauchy’s law in elasticity for small displacement assumption

Analysis of the compositebehaviourby a classical micromechanical approach comprising 푛phases supposesthat an effective response of the composite behaviouris the result of the micromechanical behaviourof each constituent;thus:

Correspondingly, other quantities describing compositebehaviourare averagedso thatwe obtain:

Effective deformation field:

Effective displacement field:

Effective stress field:

Effective momentum density field:

By using perturbation theory shown in[17]for the static problem, the elastic and momentum density field in (5) can be separatedby splitting intoa homogenous part (comparison media) and a fluctuating part (perturbation).Consequently, (8) and (9) are written as follows:

The final solution is then obtained for 푒푟and푢̇푟expressed as a function of the mean fields <푒>and<푢̇>. Forthis aim,the problem is simplified by introducing the single scattering problem for spherical inclusions, as presented below

2.2.Single scattering problem

This problem considersa single inclusion of volume Ω푟embedded in an elastic or viscoelastic effective matrix. The displacement field of the compositeis coincident with the displacement field of the wave that is represented through the wave propagation equation in solids:

where m is the wave amplitude, w is the wavenumber,and 휔is the frequency. Consequently, the dynamic response of an effective media of subscript ‘0’,Equation (13),must be equivalent to Equations (10) and (11)

The mathematical treatment that follows requiresof perturbation theory, micromechanical Green’s function, ‘polarization’ theory and operation of convolution. This development is well presented in previous works[14], [18]. Final equations represent the dynamic response of effective properties (퐿0,휌0)to the elastic wave propagation:

where ℎ푟(푘),ℎ푟(−푘)and 푆̅푥(푟),푀̅푥(푟)are localization functions and dynamic tensors,respectively. They are dependent onthe shape and size of the inclusion, the wave frequency as well as effective properties. As show n (14) and (15), the equation system is implicit,and it must be solved by iteration.

In addition, the static response of a composite canbe estimated from (14) and (15), consideringthe zero-frequency limit of equations, 휔≈0, 푘=0, ℎ푟(푘)=1and 푀̅푡(푟)=0. Thus, equations are reduced to the classical micromechanical approach for the static problem:

where 퐴푟are localization tensorsin the classical micromechanical theory.

2.3.Spherical inclusions case

As proposed in Section 2.2, Equations (14) and (15) are dependent on the shape of the inclusion. In this section,the system of equationsthat describesthe dynamic effective response to the elastic wave propagating in asphericalparticle compositeis presented.Since a heterogeneous mediahas become homogenised by a self-consistent approach, the effective media can bedescribed with the aid of elastic moduli. Thus, the behaviourof the phase 푟is represented by two elastic constants[12],퐿푟=(3휅푟,2휇푟).The dynamic response for a phase ofrandom spherical inclusions, each of radius 푎, is as follows:

Subscripts ‘1’ and‘2’ represent the inclusion and matrix response. The Equations (18-20) are functionof volume inclusion concentration 푐1, the functions ℎ1(푘)and휀훾, which are presented for the spherical case below:

Subscript 훾denotestwo possible polarizations for the wave propagation problem in homogeneous solid media,훾=훼and 훾=훽for thelongitudinal and transversal wave propagation, respectively. Polarization for the wave propagation problem is displayedin detail in[12]with the aid of the wavenumberin(1); therefore:

Finally, (18–24) is a system of implicit equations, which is solved by iteration. The firstiteration is doneby assigning to the effective properties the matrix properties(퐿0=퐿2,휌0=휌2). Because this study is interested in seeking the dynamic response of the composite adding viscoelastic properties,thenext sectionpresentsthedynamic viscoelastic response (퐿2,휌2) that will be integrated to the formulation in (18–20).

2.4.Viscoelasticproperties

Viscoelastic homogeneous behaviouris represented in the complex domain, acknowledgingthe elastic-viscoelastic duality of material. Several rheological models have been proposed in the literature[19]. The chosen model to apply in the present study, due to its simplicity, istheMaxwellrheologicalmodel. In this model, mechanical behaviourof the material is representedwith the aid of a spring for elasticity behaviourand a damper for viscosity behaviour, both connected in series.

For homogeneousviscoelasticmaterials, thedynamic behaviourisdescribed by two mechanical constants just as the homogeneous elastic material. This time, these properties are described in the complex domain. For this purpose, the fundamental equation stress-deformation for Maxwell materials

is introduced in (3). Since the solicitation remains in a propagating wave as (12), the equations obtained[12]for the dynamic response in viscoelastic materials are:

where, 푐푇and 푐퐿denote the phase velocity, and훼푇and 훼푇are the attenuation coefficients of the longitudinal and transverse waves, respectively

2.4.1.Dynamiccharacterization of viscoelasticmatrix

To calculate(26) and (27) to be introduced in the formulationmodel (18–20), viscoelastic properties of the matrix must be experimentally characterised through the dynamic values (푐푇, 푐퐿, 훼푇and 훼퐿)and beintroduced as frequency dependent parameters defined by (26) and (27) in the propagation wave model (18-20)

Several previous experimental studies for the measurement of longitudinal wave propagation in viscoelastic materials have been carried out. It has been found that for a large frequency interval, phasevelocity is invariant. Furthermore, the attenuationcoefficient increases linearly with frequency (훼=푚휔+훼0) [15].This behaviouris presentedin Table 1 and Figure 1. As EPOXY is often used as a matrix in the literature, we have also analysed the wave propagation problem with this material in Section 3. EPOXY and PMMA experimental characterization are taken from[16]. In addition, an experimental characterization for the PDMS has been studied in[20].

3.Results

As mentioned before, the dynamic response to the wave propagationin a solid can be estimated with the wavenumberin(1). The phase velocity as well as attenuation coefficientare evaluated using (23,24).

Three special studies have been carried out in this work. First, it was the validation of micromechanical self-consistent, adding the viscoelastic properties to the formulation (18–20) developed in Section 2and presented in[14], where the formulation is solved for an elastic particulate composite. Second, once the model is validated, itis subjected to the change of different parameters such as the volumeconcentration of inclusions and their size. Finally, these results are compared with experimental results, with [21]for phase velocity and[16]for the attenuationcoefficient. The labels corresponding to the various configurations illustratedin the following figures are presented in Table2

The next figures have been normalised for phase velocity, attenuation coefficient and frequency. The phase velocity measurement is taken as 휔/Re{푘훾}, where 훾=훼and 훾=훽for longitudinaland transversal wave propagation, respectivelyand where 푘훾is defined by (23, 24). Phase velocities are normalised to the phasevelocity of a longitudinal wave propagation in the matrix material(C2L).The measure of attenuation is Im{푎푘훾}.Normalisedfrequency is 푘2푎, where 푘2=Re{푘훼}and isevaluated for 휅0=휅2,휇0=휇2,휌0=휌2

3.1.Phase velocity

At first, the study focused on acomposite with a large contrast in density between its constituents, e.g.an Epoxy-leadcomposite. It has been demonstrated that this large difference directly affects the dynamic response, mainly the effective density, which becomesthe complex domain[22].

3.1.1.Validation of the self-consistentmodel adding the dynamicviscoelastic response

Lead-epoxy composite is first considered, with the sizeof inclusionsset at 660[휇푚]. Figures 2 and 3 show the study carried out for an epoxy matrix containing 5% and 15% of volume concentration spherical lead inclusions. The resultsare obtained for elastic and viscoelastic matrices.However, it was noticed that accounting for dynamic viscoelastic responseleads to a global increase of the phase velocity. The experimental study carried out by Kinra[21](yellow point in Figures 2 and 3) shows some peaks due to the resonancephenomenon that are also clearly representedby the micromechanical approach. For this particular composite, resonancephenomenon is close to 푘훼푎=0.5.Noticethat the micromechanical approach gives good approximations to experimentation, especially for upper-frequencyvalues.

Finally, the micromechanicalself-consistent approach does not take into account the information on spatial correlations between inclusions[14]. Thus, this fact is a source of error and can be seen in the large difference in the resonancezone. In the literature, theself-consistent micromechanical approach gives a goodapproximation to lowvolume concentration of inclusions, typically 30% as in static problem.

3.1.2.Variation ofvolume concentration inclusion

The experimentalstudy in [21]for Epoxy –Lead particulate composite for longitudinal phase velocity shows a displacement of resonancephenomenon when the volume concentration of inclusions increases(Figure4).

3.2.Attenuation coefficient

In Section 2.3, it wasshown thefirstcharacterization of attenuation coefficient in the viscoelasticmatrix(26) and (27)that wasnecessary for the evaluation ofdynamic effective properties in (18-20). In this section, the imaginarypartsof (23,24) areanalysed.

3.2.1.Validation of self-consistent micromechanical model with viscoelastic matrix

Accounting for the viscoelastic behaviourof the matrix leads to change in the attenuation coefficient,in particular,at the high-frequencyregime. Attenuation coefficient becomes zero athigh frequency for the elasticmatrix. This attenuation coefficient has a defined value different from zerowhen viscoelasticity in the matrixis considered. Dynamic response of the effective media at high frequency corresponds to viscoelastic dynamic linear behaviour, as introducedin 2.4.1.Thevalidation has been carried out for an Epoxy –Lead particulate composite, with the size of particles supposed at 푎=660[휇푚]. Volume concentrations of5 and 15%have been analysedin Figure 5.

The theoreticalmodel developed in[16]evaluates attenuation coefficient by absorption due to viscoelastic behaviourand by scattering due to inclusions, separately. One of the particulate composites used for that study has been the Epoxy –Glass.In order to validatethis approach with the present work, Epoxy and Glass properties are taken from Table 3, and the results are demonstrated in Figure 6.ComparingFigures 5 and 6,it can be observed that the tendency of the curves isdifferent for the same frequency range. Therefore, it is demonstrated that dynamic composite response depends on the inclusion nature

3.2.2.Variation of volume concentration of inclusions

Figure 7shows epoxy –lead composite with different values of volume concentration.Atlow volume concentration of inclusions, the attenuation coefficient seems to belineardue to viscoelastic behaviour. That means dissipation in wave propagation energy due to the absorption in the matrix. On the other hand, when the volume concentration increases, attenuation is due to the scattering insingle inclusion

3.2.3.Variation of thesizeof inclusions

As mentioned before,two factors cause attenuation of a wave propagating in a viscoelastic matrix composite. First, it is absorption due to the viscoelastic behaviourin he matrix,and second, it is scattering due to the inclusions. In this section, the micromechanicaldynamic self-consistent approach is compared to the theoreticalmodel proposed by Biwa [16], who has studied the influence of size variation ofinclusionson the attenuation coefficient. Figure 8 depicts the results obtained by Biwa for Epoxy –Glass composite;volumeconcentration of inclusions is 20%.It is worth to note that theattenuationcoefficient in particulate composites is always higherthan attenuation in pure viscoelastic material, contrary to the unidirectional fibrecomposites case[16]. Figure 9 shows the results for the micromechanicalself-consistent approach.

The increase of the attenuation value is greater for a largersize of inclusions and at low frequencies(Figure8). However, for the dynamic –self-consistent approach(Figure9),this behaviouris rather observed at higher frequencies

3.2.4.Experimental validation

The self-consistent model is comparedto experimental results carried out by Kinra[21]. Attenuation coefficient for Epoxy -Glass composite has been analysed. The size of particles is150[휇푚],and the volume concentration of inclusions issupposed to be 8,6%.

Both theoretical approaches presented in Figure10showa good approximation with experimentation work for a low volume concentration in inclusions (8,6%) and at low frequency. When thesevaluesincrease,both approaches give an incorrectapproximation;in the case of dynamic self-consistent,this is due to the no experimental information relating to the correlationbetween particles. Thus, the singlescattering problem does not give a good approximation at high volume concentration values

4.Conclusions

In this work, elastic wave propagation in a heterogeneousviscoelastic particulate media has been studied.This approach is based ona dynamic micromechanical self-consistent approach. The dynamicterm has been included in the Navierequation;it represents the displacement field of the wavewhich is coincidentwith the displacementfield of effective media. In the literature, this approach gives good approximations at low concentration volume of inclusion, 30%maximum

The dynamicmicromechanical self-consistent approach does not take into account correlationsbetween nclusions.Therefore, this is a source of error compared with experimental works.Thedynamic composite response also depends on the inclusion nature.The integration of dynamic viscoelastic response of matrix gives an increase in bothphase velocity and attenuation coefficient values in comparison with the elasticmodel

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