This article was downloaded by: [University of Leeds]On: 19 October 2014, At: 05:31Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Philosophical MagazinePublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tphm20

Shear stress at the interface ofcoaxial composites determined bythe resonance of low-amplitudevibrationsÉlida B. Hermida a & Diego G. Melo ba Consejo Nacional de Investigaciones Científicas y Técnicas ,Avenida Rivadavia, 1917, Buenos Aires, C1033AAJ, Argentinab Facultad de Ciencias Exactas y Naturales, DepartamentoFísica , Universidad de Buenos Aires , Ciudad Universitaria,Pab. I, Buenos Aires, C1428, ArgentinaPublished online: 15 Nov 2010.

To cite this article: Élida B. Hermida & Diego G. Melo (2003) Shear stress at the interface ofcoaxial composites determined by the resonance of low-amplitude vibrations, PhilosophicalMagazine, 83:15, 1761-1773, DOI: 10.1080/0141861031000104154

To link to this article: http://dx.doi.org/10.1080/0141861031000104154

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoeveras to the accuracy, completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions and views of theauthors, and are not the views of or endorsed by Taylor & Francis. The accuracyof the Content should not be relied upon and should be independently verifiedwith primary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connectionwith, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.

Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

PHILOSOPHICAL MAGAZINE, 2003, VOL. 83, NO. 15, 1761–1773

Shear stress at the interface of coaxial compositesdetermined by the resonance of low-amplitude vibrations

E¤ lida B. Hermidayzk and Diego G. Meloz}yConsejo Nacional de Investigaciones Cientıficas y Tecnicas, Avenida Rivadavia

1917, Buenos Aires C1033AAJ, Argentina}Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales,Departamento Fısica, Ciudad Universitaria, Pab. I, Buenos Aires C1428,

Argentina

[Received 5 April 2002 and accepted in revised form 10 February 2003]

Abstract

The aim of this work is to determine the viscoelastic behaviour of the interfacein a coaxial composite material made of a tough shield and a ductile core. Theelastic modulus and the amplitude-independent internal friction are measuredusing a longitudinal oscillating resonant system at 50 kHz. The contribution ofthe interface is modelled as a shear stress that modifies the elastic behaviour of theconstituents. The value of this shear stress is determined for different interfaces(epoxy resin–brass, epoxy resin–Pyrex and paraffin–Pyrex). The model isautovalidated by the excellent agreement between the calculated andexperimental values of the internal friction (damping) of the composites.

} 1. IntroductionOne of the desirable characteristics of composite materials is that they can be

tailored according to a particular purpose. When the goal is to optimize the visco-elastic behaviour of the composite, tailoring is based on the elasticity and damping ofthe constituents of the composite and the interface. Furthermore, the properties atthe interface play a major role in determining the mechanical behaviour of compositematerials (Tsai and Hahn 1980, Hashin 1983, Hull and Clyne 1996, Meyers andChawla 1999). Usually an optimum interfacial shear resistance is expected inorder to transfer stresses between the fibre and the matrix and to achieve goodmechanical properties. Marshall (1985) and Marshall and Oliver (1987) developedan extensively adopted experimental technique to obtain a qualitative estimation ofthe interfacial shear stress. In this technique, indentation is used to push anembedded fibre to slide and the analysis assumes a constant interfacial shear stressalong the sliding zone length. Fibre pull-out tests are also frequently applied toobtain the frictional resistance at the interface. More accurate experimentaltechniques, such as a vibrating reed, led to the measurement of the load transferthrough the interface in thin films and multilayer membranes. This technique,

Philosophical Magazine ISSN 1478–6435 print/ISSN 1478–6443 online # 2003 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

DOI: 10.1080/0141861031000104154

{Mailing address: Comision Nacional de Energıa Atomica, Centro AtomicoConstituyentes, Unidad de Actividad Materiales, Avenida Gral Paz 1499, B1650 KNA SanMartın, Argentina.

kAuthor for correspondence. Email: ehermida@cnea.gov.ar.

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

introduced and pursued by Berry and Pritchet (1983) and Berry (1988), ischaracterized by a resonance frequency related to the elastic modulus of thecomposite and a ratio of the dissipated to the stored mechanical energies, which iscalled internal friction. Recently, Rokhlin et al. (1995), in a detailed study on theultrasonic phase velocity behaviour in fibre reinforced composites, found that thisvelocity is significantly affected by the degradation in the elastic properties of theinterface.

The elastic modulus and internal friction are the two main characteristic para-meters of a linear viscoelastic composite; various expressions have been reported inthe literature to relate them to the viscoelastic properties of the constituents of acomposite material. Some of these are based on the law or rule of mixtures (RM),known to be accurate for long-fibre composites in the direction of reinforcement(Gibson et al. 1982). In fact, this law, which was developed to calculate the elasticmodulus ERM

c of a composite material as a function of the moduli of the matrix andthe fibre and the geometry of both constituents, is based on the following twoassumptions.

(i) The reinforcement is parallel to the tensile direction.(ii) There is no load transfer between reinforcement and matrix at the interface.

It establishes that (Hashin 1983)

ERMc Ac ¼ E1A1 þ E2A2; ð1Þ

where Ai is the cross-section of the respective constituent (indicated by subscript 1 or2) or of the composite (indicated by subscript c). Since the materials are not idealelastic but dissipate energy when submitted to different mechanical excitations, theRM was extended also to describe the internal friction (dissipated energy) of aparallel composite in terms of the energies dissipated by its constituents. The damp-ing of a composite system consisting of a fully adherent matrix on parallel fibresresults (Nishino and Asano 1993, Lesieutre 1994)

�c ¼E1A1

EcAc

�1 þE2A2

EcAc

�2; ð2Þ

where �1 and �2 are the internal frictions of the constituents evaluated individuallyunder the same conditions as the composite. In addition to the two assumptionsstated earlier, equations (1) and (2) are strictly valid for both quasistatic anddynamic tests if Poisson’s ratio of both phases are equal (Crema et al. 1989,Meyers and Chawla 1999). However, when this condition is not fulfilled, differencesare no higher than 2% of this calculation (Meyers and Chawla 1999). This differenceis not very important when compared with differences between the predictions of theRM and the experimental results (Hull and Clyne 1996). In fact, different kinds ofexperiment indicated that the RM gives only a rough approach to the mechanicalresponse of a parallel composite material. Flexion and torsion at low (Crema et al.1989, Brodt and Lakes 1995) and medium frequencies (Wuttig and Su 1992, Su andWuttig 1994) performed with parallel composites showed that the measured elasticmodulus is always lower and the measured internal friction higher than the corre-sponding values determined by the RM. These differences come from ignoring theload transfer through the interface, that is assuming that the fibre displacement iselastically coupled to the matrix or, equivalently, the matrix adheres perfectly to thefibre. Wuttig and Su (1992) and Su and Wuttig (1994) found that the internal friction

1762 E. B. Hermida and D. G. Melo

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

of Cu/Ni multilayer membranes and Ni50Ti50=SiO2=Si composites were lower thanthe values predicted by the RM. Then they introduced an adherence coefficient�ð0 < �4 1Þ that takes into account the imperfection in the adherence betweenlayers. According to this analysis the lack of adhesion should be correlated with adecrease in the internal friction. However, some composite materials, such as poly-propylene with glass fibres, dissipate more energy than the value predicted byequation (2) with Ec ¼ ERM

c and the damping of the respective constituentsmeasured under the same stress–strain conditions as the composite (Crema et al.1989). In these cases the adherence coefficient does not provide a good solution tothe misfit between the RM and the empirical results.

The aim of the present work is twofold. Firstly, it aims to show that theresonance of longitudinal wave vibrations of very low amplitude is a propertechnique to determine the elastic modulus and the internal friction of parallelcomposites. The samples chosen are practical in terms of the processing techniqueand the piezoelectric excitation device. Secondly, it aims to develop a model topredict the elastic modulus and the internal friction of the composite in terms ofthe respective properties and geometry of the constituents and the shear stress at theinterface due to an imperfect bonding. The simple geometry of the samplesdetermines that a simple analytical solution is attainable.

} 2. Experimental procedure

2.1. SamplesThe composite samples are made of two coaxial cylinders with an external

diameter less than 5 mm. The inner cylinder or nucleus is made of a soft and high-damping material and the external cylinder or shield is made of a low-dampingmaterial harder than the nucleus. Four different materials were used for the nuclei:the epoxy resin Araldite1 AY103 with the hardener HY956 (in the proportions1 : 18) cured in vacuum at room temperature for 24 h, and three paraffins A(C27H55), B (C28H57Þ and C (C55H111Þ with melting points at 568C, 588C and 908Crespectively. The external shields were tubes of Pyrex1 for the nuclei made ofparaffins, and brass or Pyrex1 for the Araldite1 nuclei. We employed two differenttubes of Pyrex1, denoted Pyrex1 t and Pyrex1 T, with slightly different mechanicalproperties; Pyrex1 T has an interface area 57% greater than Pyrex1 t. The innerradii a and outer radii b of the tubes are detailed in the first two columns of table 1.

Shear stress at interface of coaxial composites 1763

Table 1. Inner (2a) and outer (2b) diameters, elastic modulus (E), internal friction (�Ind andcritical strain amplitude ð"IndÞ.

Constituents2a

(mm)2b

(mm)E

(GPa)�Ind

ð�103Þ"Ind

ð�105Þ

Pyrex t 2.55 3.95 68 0.58 0.7Pyrex T 3.20 4.75 63 0.56 0.7Brass 2.65 3.90 116 0.11 1.1Annealed brass 2.65 3.90 116 0.09 1.1Paraffin A — 4.55 7.3 101 1.3Paraffin B — 4.55 9 82 0.8Paraffin C — 4.30 13 158 1.3Araldite — 4.40 4.6 70 2.0Annealed Araldite — 4.40 4.6 64.5 1.2

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

By melting in vacuum (102 Torr) we produce cylindrical samples of Araldite1

and the three different paraffins with diameters less than 5 mm (limit imposed by thecross-section of the piezoelectric quartz crystals employed). The lengths of thesesamples and the tubes of Pyrex1 and brass were adjusted to obtain resonance ofthe ultrasonic vibrations applied to the sample. It is noted that the parallelismbetween the opposite faces of the cylindrical samples guarantees proper resonancewith a high quality factor.

In order to modify the interface adhesion and to establish whether theexperimental technique enables those changes to be detected, the brass–Araldite1

composites were annealed for 4 h at 708C.

2.2. Experimental device: piezoelectric oscillatorThe three-component piezolectric oscillator depicted in figure 1 is a suitable

device for measuring the elastic modulus and the internal friction of materials athigh frequencies (Marx 1951, Povolo and Gibala 1969). It consists of the sample andtwo piezoelectric crystals, each with metallized opposite faces with a wire, formechanical support and electric connections, soldered in its centre. One of the crys-tals is used for excitation and the other for detection of the mechanical longitudinalvibrations. Each crystal is 5 cm long and with a square cross-section of5 mm � 5 mm, its mass is 3:86 0:01 g and its resonance frequency49 848:2 0:1 Hz. The three components are joined with cyanoacrylate, which doesnot affect the experimental determination of the mechanical properties of the sam-ples (Hermida et al. 2000). The composite oscillator was placed in a stainless steel

1764 E. B. Hermida and D. G. Melo

lc

lc

ls 2a

2b

Figure 1. Three-component piezoelectric oscillator; the schematic diagram on the rightshows the relevant dimensions of the composite sample.

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

test chamber under dynamic vacuum of 102 Torr. The temperature inside thechamber, measured with a K-type thermocouple, was held at 5:0 0:58C for allsamples.

When a sinusoidal voltage of amplitude Vd is applied to the piezolectric driver,longitudinal sonic waves propagate along each component of the oscillator: drivercrystal, gauge crystal and sample. The supports in the middle of the driver crystalproduce a boundary condition; the displacement of the crystal is null at the supportsand maximum at the ends of the crystal. Thus, the wavelength of the principal modeof the mechanical vibration propagating in the crystal is �c ¼ lc=2, where lc denotesthe length of the crystal (see appendix A). The source of the driver crystal was afrequency synthesizer, in series with a matching transformer and a power amplifierwhen an input voltage higher than 30 V is required.

The vibration produces a voltage Vg between the metallized faces of the gaugecrystal, proportional to the amplitude of the vibration. This voltage reaches amaximum when the frequency ft of the oscillator is approximately equal to theresonance frequency fc of the crystals. The voltages were measured with high-frequency voltmeters.

The internal friction �t of the composite oscillator at the resonance is (Povoloand Gibala 1969)

�t ¼c

mt ft

Vd

Vg

; ð3Þ

where c ¼ 38:3 0:2 g Hz is a calibration constant for the crystals, mt is the totalmass of the oscillator, and Vd and Vg are the voltages on the driver and gaugecrystals respectively at resonance.

The equivalent electric circuit of the composite oscillator leads us to relate theresonance frequency and the internal friction of the oscillator with those ones of thecomponents (Robinson and Edgar 1974):

mt f2t ¼ mc f

2c þms f

2s ; mt ft�t ¼ mc fc�c þms fs�s; ð4Þ

where mi is the mass, fi the resonance frequency and �i the internal friction of theelement i of the oscillator (the subscripts t, c and s denote the whole oscillator, thecrystals and the sample respectively). These equations are valid if fs � fc � ft. Thiscondition is fulfilled if the length of the sample is properly adjusted (Robinson andEdgar 1974). In fact, on considering that fs is the ratio of the sound velocity cs in thesample to the wavelength �s, of the mechanical waves propagating in the sample andthat �s ¼ ls=2, it follows that fs ¼ 2cs=�s.

Furthermore, if �2s � 1, the elastic modulus Es of the sample is simply related to

the resonance frequency as follows (Marx 1951):

Es ¼ 4s lsfsð Þ2; ð5Þ

where s and ls are the density and length respectively of the sample.The strain amplitude of the mechanical vibrations in the sample can be calcu-

lated as (Povolo and Gibala 1969)

" ¼ S1lcls ft

Vg; ð6Þ

Shear stress at interface of coaxial composites 1765

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

where S1 is a calibration factor; for our experimental device,S1lc ¼ ð2:0 0:1Þ � 102 Hz m V1. From the Vd versus Vg plots illustrated infigures 2–5, we obtain the maximum voltage for a linear response and therefore,the critical strain amplitude "Ind above which the viscoelastic behaviour of thesample becomes nonlinear.

1766 E. B. Hermida and D. G. Melo

Figure 2. Gauge voltage versus driver voltage for the materials used as the nucleus of thecomposites.

Figure 3. Gauge voltage versus driver voltage for the materials used as the shields of thecomposites.

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

At a constant temperature the internal friction of the sample includes twocontributions: one independent ð�Ind

s Þ of and other dependent ð�Deps Þ on the strain

amplitude. The independent component is (Povolo and Gibala 1969)

�Inds ¼ bc mc fc�c

ms fs; ð7Þ

Shear stress at interface of coaxial composites 1767

Figure 4. Driver voltage required to increase the amplitude of the mechanical vibrationsð" / VgÞ in the composites with a paraffin nucleus and Pyrex shield.

Figure 5. Driver voltage required to increase the amplitude of the mechanical vibrationsð" / VgÞ in the composites with an Araldite nucleus.

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

where b is the slope of the Vd versus Vg curve at low strain amplitudes.

} 3. Results

The diameters of the nuclei and tubes as well as the viscoelastic properties of theconstituents, measured at 5:0 0:58C, 102 Torr and 50 kHz are summarized intable 1. The elastic modulus and internal friction measured and calculated usingthe RM are indicated in table 2.

From table 2 it follows that for all the other composites except Pyrex1

t–Araldite1, Eexp � ERM. The difference between these values increases when theinterfacial surface increases. It is noted that the change in the elastic modulus of thebrass–Araldite1 composite after the heat treatment is only 3%, that is of the orderof the error of the measurement.

The larger differences between the measured internal friction and the valuecalculated using the RM correspond to the composites with greater interfacialarea (paraffin with thick Pyrex1 tubes).

The best fit of the properties calculated using the RM to those measured is forthe Pyrex1 t–paraffin C and brass–Araldite1 composites (without annealing); thisindicates that the interface has no influence on the elastic response of theconstituents. However, when the brass–Araldite1 composite is annealed, thedifference between the measured and calculated values increases significantly. Thisresult arises from the change in the adhesion at the interface after the thermaltreatment. Effectively, the nature of the interfacial contact is strongly influencedby the presence of residual stresses; one of the most important sources of thesestresses is the thermal contraction which occurs during post-fabrication cooling(Hull and Clyne 1996). In our case, the interface is modified during the post-anneal-ing cooling. This example emphasizes the dominant role of the interface in theanalysis of a model to describe the viscoelastic properties of parallel composites.

} 4. Discussion

In order to model the resonance of longitudinal sound waves in the compositesample we consider a cylindrical coordinate system where the origin of the axial axisz is at the centre of the sample of length ls. It is noted that longitudinal waves meansthat �l � 2a, that is extensions and compressions propagate along its length.

Previous work and our experimental results indicate that the RM does not fitproperly the linear viscoelastic behaviour of composites because it does not include a

1768 E. B. Hermida and D. G. Melo

Table 2. Elastic modulus and internal friction � measured (expÞ and calculated usingequations (1) and (2) (RMÞ.

CompositeEexp

(GPa)ERM

(GPa)�exp

ð�103Þ�RM

ð�103Þ

Pyrex t–Araldite 42 2 38 2 5:12 0:01 4:23 0:04Pyrex T–Araldite 28 1 36 2 69:9 0:1 4:29 0:06Pyrex t–Paraffin B 39 1 46 2 7:39 0:01 6:34 0:04Pyrex T–Paraffin B 30 1 38 2 19:1 0:1 9:18 0:08Pyrex t–Paraffin A 40 1 44 2 7:85 0:01 7:13 0:04Pyrex T–Paraffin A 29 1 38 2 67:0 0:1 9:36 0:06Pyrex t–Paraffin C 47 1 47 2 17:9 0:1 17:4 0:4Brass–Araldite 64 2 64 2 2:49 0:01 2:45 0:04Brass–Araldite (annealed) 62 1 64 2 11:4 0:1 2:43 0:04

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

mechanism associated with the interface. The interaction between the constituents ofa composite were considered as a shear stress at the interface (Hsueh 1990, Lu andHsueh 1990). Furthermore, the preparation of the composites (melting of the par-affins or curing of the epoxy resin inside the shields) should guarantee that theadherence of the interface does not depend on the azimuthal coordinate ’. Then,the propagation of longitudinal waves in our cylindrical samples will produce in theconstituent i a local displacement of mass ui that depends on the time t, the axialcoordinate z and the radial coordinate r. On considering the anelastic dissipativeeffects as a perturbation, the equation of motion for any particle of each constituentis (Landau and Lifshitz 1959)

Ei

o2u

oz2þ Ei

2r

o

orrouior

� � i

o2uiot2

¼ 0; i ¼ 1; 2: ð8Þ

The first term of these equations represents the elastic restoring force in the zdirection, the second one the shear elastic stress present in each phase due to theinterface and the third one the inertial contribution.

A sinusoidal voltage of angular frequency ! ¼ 2pf applied to the driver crystalproduces the mechanical excitation of the composite oscillator; then, the boundaryconditions at the bonding between the crystals and the composite sample areuiðr; ls=2; tÞ ¼ u0 sin ð!tÞ; ðoui=ozÞðr; ls=2; tÞ ¼ 0. Hence, the solution of equation(8) reduces to

uiðr; z; tÞ ¼ uuiðr; zÞ sin ð!tÞ; ð9Þ

where the spatial factor uuiðr; zÞ solves the differential equation

Ei

o2uuioz2

þ Ei

2r

o

orrouuior

� �þ i!

2uui ¼ 0; ð10Þ

with the boundary conditions

ouuioz

ðr; ls=2Þ ¼ 0;ouuioz

ðr;ls=2Þ ¼ 0

(see appendix A).The experimental device detects the mean longitudinal displacement along the z

axis for each constituent, that is

UiðzÞ ¼1

p R2i r2

ið Þ

ð2

0

ðRi

ri

uuiðr; zÞr dr d’; i ¼ 1; 2; ð11Þ

where r1 ¼ 0, R1 ¼ r2 ¼ a and R2 ¼ b. On applying this integration to equation (5)the equations for the mean longitudinal displacement of each constituent result:

E1

d2U1

dz2þ E1

a

ouu1

or

����r¼a

þ1!2U1 ¼ 0; ð12Þ

E2

d2U2

dz2 E2a

b2 a2

ouu2

or

����r¼a

þ2!2U2 ¼ 0 ð13Þ

and ðdUi=dzÞðls=2Þ ¼ 0: In order to solve these equations we assume that the shearstress is proportional to the mean longitudinal displacement of each phase, that is

Shear stress at interface of coaxial composites 1769

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

Eiaouuior

����r¼a

¼ �iUi; ð14Þ

where �i is a constant. Under this assumption (the interface introduces an elasticinteraction), equations (12) and (13) reduce to

d2Ui

dz2þ k2

i Ui ¼ 0; ð15Þ

where

k21 ¼ 1

E1

1!2 þ �1

a2

� �; k2

2 ¼ 1

E2

2!2 �2

b2 a2

� �:

ki ¼ 2p=ls is the wave number of the mechanical vibration in each phase. It is notedthat when ! ¼ 0, equation (15) reduces to the expression proposed by Hsueh (1990)and Lu and Hsueh (1990) in the model of the delayed stress developed for staticconditions.

The solution of equation (15) is Ui ¼ U0isin pz=lsð Þ; that is, both mean displace-

ments have the same spatial distribution. Furthermore, if we take into account thatthe shear stress must be equal at both sides of the interface, that is

G1

ouu1

or

����r¼a

¼ G2

ouu2

or

����r¼a

ð16Þ

with Gi ¼ Ei=2ð1 þ �iÞ, and �i Poisson’s ratio, and that the shear strain at both sidesof the interface is expressed in terms of equation (14), equation (16) leads to

U02¼ �1

�2

1 þ �2

1 þ �1

U01: ð17Þ

On the other hand, since in a homogeneous material k2 ¼ !2=E, we can define theeffective elastic moduli

Eeff1 ¼ E1

�1l2s

p2a2; Eeff

2 ¼ E2 þ�2l

2s

p2 b2 a2ð Þ: ð18Þ

If we consider these effective moduli as the moduli of the components given inequation (1), we obtain

Ec ¼ ERMc 2l2s

2b2�1 �2ð Þ: ð19Þ

This is the elastic modulus of the composite with a correction due to the shearstresses at the interface. According to our model this value coincides with thatmeasured with the three component oscillator, Eexp, and is equal to the valuepredicted by the RM only if the shear stress at the interface vanishes, as stated inone of the postulates of this RM. The difference between ERM

c and Ec ¼ Eexp leads usto calculate the net shear stress �1 �2 at the interface. The first column of table 3shows these values for our composite samples. In general, �1 �2 > 0, indicating thatthe shield reinforces the mechanical vibrations in the nucleus of the composite.However, the Araldite1 nucleus is sufficiently hard to modify the vibrations of thethin Pyrex1 shield, as shown in the negative value of �1 �2: �1 �2 ¼ 0 shows thelack of adherence between the constituents of the Pyrex1 t–paraffin C and brass–Araldite1 samples. Furthermore, we can include the contribution of the interface in

1770 E. B. Hermida and D. G. Melo

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

the calculation of the internal friction of the composite. In fact, by substitutingequation (19) in equation (2) we obtain

�c ¼Eeff

1 A1

EcAc

�1 þEeff

2 A2

EcAc

�2: ð20Þ

The values of the internal friction calculated with the corrected elastic modulus ofthe composite are in excellent agreement with those measured with the resonancetechnique as shown in the last two columns of table 3.

} 5. Conclusions

The viscoelastic behaviour of a coaxial cylindrical binary composite cannot bepredicted by the RM; neither the elastic modulus nor the internal friction (damping)estimated by this RM agrees with the experimental results. In fact, Eexp � ERM and�exp > �RM; these differences increase as the interface surface increases. Thismismatch indicates that the fields of displacements in both components at the inter-face are not identical and, in consequence, there should be a dependence on theradial coordinate. This dependence leads to a shear stress on each constituent thatalso depends on the radial coordinate; our model considers the mean shear stress at acertain cross-section of the sample. The stronger assumption of our model is that themean shear stress is proportional to the mean displacement of the cross-section(equation (14)). The model enables us to calculate the amplitude of the shear stressat the interface. Furthermore, the excellent agreement between the internal frictionof the composites calculated by the model and the experimental values are the key tovalidating the assumptions that we made.

ACKNOWLEDGEMENT

This research has been partially supported by Consejo Nacional deInvestigaciones Cientıficas y Tecnicas.

Shear stress at interface of coaxial composites 1771

Table 3. Shear stress at the interface ð�1N�2Þ, experimental internal friction (�exp) andcalculated using equation (14).

Composite�1N�2

(MPa)�exp

ð�103Þ�c

ð�103Þ

Pyrex t–Araldite 30 5.12 5.12Pyrex T–Araldite 97 69.9 69.8Pyrex t–paraffin A 30 7.85 7.8Pyrex T–paraffin A 96 67.0 66.9Pyrex t–paraffin B 56 7.39 7.39Pyrex T–paraffin B 84 19.1 19.1Pyrex t–paraffin C 0 17.9 17.4Brass–Araldite 0 2.49 2.45Brass–Araldite (annealed) 36 11.4 11.4

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

APPENDIX AThe equation of motion for longitudinal elastic waves that propagate along the z

axis of a piezoelectric single crystal, denoted as driver, such as that illustrated infigure 1 is

Eo2ud

oz2¼

o2ud

ot2ðA 1Þ

where ud is the displacement along the z axis, t is the time, and E and are the elasticmodulus and density respectively of the crystal. The present experimental methodrequires an expression relating the particle velocity at the sample to the frequency ofexcitation of the system. If the potential difference applied to the metallized faces ofthe crystal is V ¼ V0 exp ði!tÞ; the amplitude of vibration at any point of the crystalwill vary harmonically with time and

u ¼ u0 exp ½ið!tþ kczÞ þ u1exp ½ið!t kczÞ� ðA 2Þ

will be a solution of equation (A 1) provided that kc ¼ ð=EÞ1=2.The driver crystal is supported by two hard wires, one at the centre of each of

two opposite longitudinal faces, as shown in figure 1. Thus no displacement ispossible at the centre of the crystal, that is uðz ¼ 0Þ ¼ 0. Hence,

u ¼ u0 sin ð!tÞ sin ðkczÞ: ðA 3Þ

Furthermore, in our experiment the frequency of the sinusoidal electric excitationapplied to the piezoelectric sample is tuned to obtain resonance of stationary waves(Balamuth 1934). This means that the displacement at the ends of the crystal ismaximum, that is ðoud=ozÞjz¼ls=2 ¼ 0. Thus, k ¼ hð =lcÞ; with h a positive integer.From now on we shall consider h ¼ 1; that is, we shall analyse the mechanicalresponse of the oscillator for its fundamental mode.

If a second piezoelectric crystal, namely the gauge, identical with the driver isjoint to the top face of the driver, the elastic waves that it generates will propagateinto the gauge with the same frequency and produce its resonance, since bothcrystals have the same length. The wave equation for the displacement ug atthe gauge is the same as that for the driver with the boundary conditionðoug=ozÞ

��z¼lc=2

¼ ðoud=ozÞjz¼lc=2 ¼ 0. That is ug ¼ ug0sin ðkczÞ sin ð!tÞ:

Analogously, when a homogeneous sample is cemented to the bottom face of thedriver (as shown in figure 1), ðoud=ozÞjz¼lc=2 ¼ ðous=ozÞjz¼lc=2 ¼ 0. If the sample iscut to obtain resonance of the elastic waves, then the elastic displacement inside thesample will be

us ¼ us0sin ðkszÞ sin ð!tÞ;

where ks ¼ !ðs=EsÞ ¼ p=ls, and s and Es are the density and elastic modulusrespectively of the sample.

Moreover, it can be shown that

ðA

uddA ¼ðA

us dA;

where A is the cross-sectional area of the surfaces of driver and sample joined by thecement (Balamuth 1934).

1772 E. B. Hermida and D. G. Melo

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014

References

Balamuth, L., 1934, Phys. Rev., 45, 715.

Berry, B. S., 1988, Diffusion Phenomena in Thin Films and Microelectronic Materials, editedby D. Gupta and P. S. Ho (Park Ridge, New Jersey: Noyes), pp. 73–143.

Berry, B. S., and Pritchet, W. C., 1983, Rev. scient. Instrum., 54, 254.

Brodt, M., and Lakes, R. S., 1995, J. Composite Mater., 29, 1823.

Crema, L. B., Castellani, A. and Serra, A., 1989, J. Composite Mater., 23, 978.

Gibson, R. F., Chaturvedi, S. K., and Sun, S. T., 1982, J. Mater. Sci., 17, 3499.

Hashin, Z., 1983, J. appl. Mech., 50, 481.

Hermida, E¤ . B., Melo, D. G., Aguiar, J. C., and Lo¤ pez, D. E., 2000, J. Alloys Compounds,310, 91.

Hsueh, C.-H., 1990, Acta metall. mater., 38, 403.

Hull, D., and Clyne, T. W., 1996, An Introduction to Composite Materials, second edition(Cambridge University Press), chapter 7.

Landau, L. D., and Lifshitz, E. M., 1959, Theory of Elasticity (Oxford: Pergamon).

Lesieutre, G. A., 1994, J. Composite Mater., 28, 1770.

Lu, M. C., and Hsueh, C.-H., 1990, J. Composite Mater., 24, 572.

Marshall, D. B., 1985, J. Am. Ceram. Soc., 67, 259.

Marshall, D. B., and Oliver, W. C., 1987, J. Am. Ceram. Soc., 70, 542.

Marx, J., 1951, Rev. scient. Instrum., 22, 503.

Meyers, M. A., and Chawla, K. A., 1999, Mechanical Behaviour of Materials (EnglewoodCliffs, New Jersey: Prentice-Hall), appendix 15: composite materials.

Nishino, Y., and Asano, 1993, S., Phys. Stat. sol. (a), 139, K97.

Povolo, F., and Gibala, R., 1969, Rev. scient. Instrum., 40, 817.

Robinson, W. H., and Edgar, A., 1974, IEEE Trans. Ultrasonics, 21, 98.

Rokhlin, S., Huang, W., and Chu, Y. C., 1995, Ultrasonics, 33, 351.

Su, C. M., and Wuttig, M., 1994, J. Alloys Compounds, 211/212, 428.

Tsai, S. W., Hahn, H. T., 1980, Introduction to Composite Materials (Westport, Connecticut:Technomic).

Wuttig, M., and Su., C. M., 1992, Proceedings of the Damping of Multiphase InorganicMaterials Symposium (Chicago, Illinois: ASM), 1992, pp. 159–163.

Shear stress at interface of coaxial composites 1773

Dow

nloa

ded

by [

Uni

vers

ity o

f L

eeds

] at

05:

31 1

9 O

ctob

er 2

014