The large amplitude oscillatory strain response of aqueous...

DOI 10.1140/epje/i2008-10382-7

Eur. Phys. J. E 27, 309–321 (2008) THE EUROPEANPHYSICAL JOURNAL E

The large amplitude oscillatory strain response of aqueous foam:Strain localization and full stress Fourier spectrum

F. Rouyer1,a, S. Cohen-Addad1, R. Höhler1, P. Sollich2, and S.M. Fielding3

1 Université Paris-Est, Laboratoire de Physique des Matériaux Divisés et des Interfaces, UMR 8108 du CNRS, 5 BoulevardDescartes, 77 454 Marne-la-Vallée cedex 2, France

2 King’s College London, Department of Mathematics, Strand, London WC2R 2LS, UK3 Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Manchester M13 9EP, UK

Received 27 March 2008 and Received in final form 30 July 2008Published online: 10 November 2008 – c© EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2008

Abstract. We study the low frequency stress response of aqueous foams, subjected to oscillatory strain. Asthe strain amplitude is progressively increased starting from zero, the initially linear viscoelastic responsebecomes nonlinear as yielding sets in. To characterize this crossover from solid-like to liquid-like behaviourquantitatively, the full harmonic spectrum of the stress is measured. These results are compared to thesoft glassy rheology model as well as to elastoplastic models. Moreover, to check for strain localization, wemonitor the displacement profile of the bubbles at the free surface of the foam sample in a Couette cellusing video microscopy. These observations indicate that strain localisation occurs close to the middle ofthe gap, but only at strain amplitudes well above the yield strain.

PACS. 83.80.Iz Rheology: Emulsions and foams – 47.57.Bc Fluids dynamics: Foams and emulsions –83.60.La Viscoplasticity; yield stress

1 Introduction

Aqueous foam is a metastable disordered complex fluid,made up of gas bubbles closely packed in a surfactantsolution. It behaves like a viscoelastic solid or like a non-Newtonian liquid, depending on gas volume fraction andapplied stress [1–3]. In the work reported here, we focus onthe nonlinear viscous, elastic and plastic response of 3Dfoams at the shear-induced transition between solid-likeand liquid-like behaviour. Non-Newtonian steady flow hasbeen probed experimentally and interpreted in terms of in-teractions on the bubble scale [4–7]. Insight into the phys-ical processes explaining the transient dynamics that ac-company the onset of yielding requires additional experi-ments. Such information can be gained by applying oscilla-tory strain so that the sample switches back and forth be-tween solid-like and liquid-like behaviour on well-definedscales of time and strain amplitude. Previous studies of theoscillatory foam response as a function of strain amplitudewere focussed on the fundamental harmonic componentsof stress and strain [8,9] and discussed in terms of a strainamplitude dependent complex shear modulus. However,much additional information is contained in the spectrumof harmonics, or equivalently in the Lissajous representa-tion showing the evolution of stress versus strain with thetime as a parameter [10]. Indeed, for a fixed strain am-plitude and frequency, the complex shear modulus alone

a e-mail: [email protected]

does not allow to distinguish whether the dissipation of apredominantly elastic sample is of viscous or plastic ori-gin. In contrast, a Lissajous plot does directly allow sucha distinction since it exhibits a characteristic signature ofthe nonlinear response. It has the shape of an ellipse forviscous dissipation or of a parallelogram for plastic dissi-pation. Therefore, Lissajous data provide the opportunityto test physical foam rheology models predicting the on-set of plastic response with increasing strain amplitude,corresponding on the local scale to the onset of strain-induced irreversible bubble rearrangements. The evolutionof the complex shear modulus with strain amplitude alsoprovides information about the mechanisms involved inthe sample deformation, but the following example showsthat such evidence alone can be misleading. In a recentexperimental study of a biological yield stress fluid [11]the complex shear modulus was found to be almost inde-pendent of strain amplitude up to the yield strain so thatone might have expected that the rheological response islinear in this entire range. Remarkably, a Lissajous plotrevealed that this is not true: strong strain hardening isobserved within each strain cycle. Motivated by this con-text, we study in the present paper the oscillatory responseof 3D foam at the transition from solid-like to liquid-likebehaviour, detecting for the first time the full stress har-monics spectrum.

Rheological studies of flowing foam, at a macroscopicscale, require great care, because the engineering strain,

310 The European Physical Journal E

deduced from the relative motion of the sample bound-aries, can be different from the local strain relevant forphysical models, and it is possible that a solid-like regionand a liquid-like region coexist in the same sample. Thisphenomenon is known as flow localization [7,12]. Since theconditions under which the flow of foam is localized are notyet clear [4,5,7,8,12–14], a reliable interpretation of engi-neering strain in terms of local strain requires in situ ob-servation of the sample deformation during the rheologicalexperiment. We follow this approach and identify the lim-its of the range of strains where the flow is homogeneous.

The paper is organized as follows: In Section 2.1, wefirst provide a brief overview of nonlinear rheological datafor foams and concentrated emulsions. In Section 2.2, wepresent theoretical models and we focus on those where,in addition to conventional rheological characteristics, thefull harmonics spectrum of the oscillatory stress responsehas been predicted. So far, the soft glassy rheology (SGR)model and a simple elastoplastic model fulfil this criterion.Section 3 describes the sample preparation and the exper-imental procedures are given in Sections 4 and 5. In Sec-tion 6, experimental and theoretical results are comparedand discussed in the framework of other recent experi-ments. On this basis, directions for further developmentof foam rheology models are pointed out.

2 Previous nonlinear rheological experiments

and models

2.1 Experiments probing nonlinear rheology of 3Dfoams and concentrated emulsions

Experiments probing the solid-liquid transition

Experimentally, the nonlinear rheological response of 3Dfoam at the transition from solid-like to liquid-like be-haviour has been studied by measurements of the stressresponse to a steady applied shear rate [4–6,15], andto strain steps superimposed to such a constant strainrate [13]. These data alone do not fully characterize thenonlinear response: As the sample passes from solid-like to liquid-like behaviour, there is transient behaviour,evidenced by viscosity bifurcation dynamics [16] andthe stress overshoot reported in shear start-up experi-ments [15]. Transient dynamics have also been probed byapplying oscillatory strain [8,12,17]. At amplitudes far be-low the yield strain, the stress is sinusoidal and its am-plitude scales linearly with strain amplitude. When thestrain amplitude is increased nonlinear response such asshear-induced normal stresses sets in [18]. Beyond theyield strain, there is a crossover of the real and imagi-nary parts of the complex shear modulus as a function ofstrain amplitude. This feature is often used as a criterionto determine the yield strain [8]. Moreover, oscillatory ex-periments probing how soft disordered materials yield asa function of strain rate amplitude have recently shownthat frequency and strain amplitude dependence can beexplained in terms of a structural relaxation time whichdepends on strain amplitude [9]. However none of these

oscillatory experiments make use of the information con-tained in the harmonics spectrum of the stress response,which provides the signature of the physical processes in-volved in dissipation, as explained in the introduction.

Flow localization

Flow localization of a yield stress fluid can occur as a di-rect consequence of stress heterogeneity. For example, ifa steady torque is applied in a cylindrical Couette geom-etry, the stress increases with decreasing distance fromthe inner cylinder. If the stress remains larger than theyield stress only in the vicinity of this cylinder, localizedflow sets in. Previous authors have ruled out other formsof localized foam flow because they obtained compatibleresults in different rheological geometries [4,8]. Multiplelight scattering studies of the flow profile in Couette ex-periments [4] and observation of the bubble motion at thefree sample surface for samples obtained with a variety offoaming solutions in the plate-plate geometry [5] providedadditional evidence in favour of this conclusion. However,other experiments with 3D foams have evidenced flow lo-calization that cannot be interpreted as a direct conse-quence of stress heterogeneity. Coexistence of solid-likeand flowing regions has been observed inside dry 3D foamundergoing steady planar shear start-up flow where thestress is homogeneous [14]. Moreover, MRI observations ofthe flow in the cylindrical Couette geometry have shownthat steady flow is impossible below a critical strain rateγ̇c [7]. This means that for an engineering strain ratesmaller than γ̇c, part of the sample must remain solid-likewhereas another part flows at a shear rate γ̇c. At strainrates γ̇ larger than γ̇c, the stress was found to scale withγ̇ following a power law. In oscillatory shear experiments,shear localization in the middle of the gap has been ob-served [12]. This is an unexpected form of flow localizationbecause the stress is highest in the vicinity of the innercylinder. Let us note that the flow of 2D foams has recentlybeen studied by many authors, but we will not discuss thiswork in the present paper focussed on 3D foams.

The flow profile at the free surface of concentratedemulsions has been studied in steady-shear experimentsusing a Couette cell [19]. For emulsions with dispersedvolume fractions well above 0.7, localized flow is reportedwhereas in more dilute emulsions, the flow was foundto be homogeneous [19]. In other recent steady-flow ex-periments with emulsions, strain localization was foundonly in the presence of attractive interactions between thedroplets [20].

2.2 Models for the rheology of soft disorderedmaterials

Fluidity models are based on a scalar measure of the “de-gree of jamming” or structural relaxation time, coupledto the flow history of the material: Flow breaks up thejammed structure and reduces the viscosity. In contrast,aging re-establishes the jammed structure and enhancesthe viscosity [21–23]. The most detailed approach of this

F. Rouyer et al.: Large amplitude oscillatory strain response of aqueous foam 311

kind is based on a Maxwell equation ∂tσ = −aσ + Gγ̇,where the parameter a is the fluidity [22]. Aging andshear rejuvenation are expressed by a nonlinear differ-ential equation that describes the spatial and temporalevolution of a, as well as its coupling to stress and strainrate. This model predicts flow heterogeneities reminiscentof shear banding, but which are not intrinsic to the samplein the sense that they are dependent on the boundary con-ditions for the fluidity at the walls confining the sample.

The SGR Model

The soft glassy rheology (SGR) model is motivated bythe strikingly similar rheology of foams, emulsions, pastesand other close packings of small soft units, suggestingthat a generic mechanism may be involved. The modelexplains the rheological behaviour in terms of the dynam-ics of mesoscopic elements [24,25]. They are chosen suffi-ciently large compared to the bubbles, grains or dropletsso that the local packing structure is not resolved in detailand so that the local strain, stress and yield stress of anelement are well-defined quantities. At the same time, themesoscopic length scale is chosen small enough to capturespatial variations of local strain, stress and yield stress.The SGR model uses a simple mean-field coupling be-tween mesoscopic and macroscopic stress and it describesinteractions between mesoscopic regions in terms of aneffective noise temperature, denoted x.

In more detail, the SGR model describes an ensembleof mesoscopic elements of the material, each of which ischaracterized by its local yield energy density E and strainl, the latter being measured relative to a local equilibriumconfiguration. Elements are assumed to deform elastically,with l incrementing in step with the global strain γ, un-til their local elastic energy reaches values of order E;they then yield stochastically. The elastic energy is takento be of the simple quadratic form kl2/2, with k a lo-cal elastic constant that is assumed to be uniform acrossthe material for simplicity. The yield rate is the productof a microscopic inverse attempt time 1/τ0, and an acti-vation factor determined by the difference of the actualelastic energy to the yield value. After a yield, elementsare taken to have zero local strain (l = 0) relative to a newlocal equilibrium configuration, as well as a new yield en-ergy drawn randomly from some fixed distribution ρ(E).This distribution one presumes to be of exponential form,ρ(E) = (1/E0) exp(−E/E0), for E > 0. The mean E0of this is used to make the effective temperature parame-ter x dimensionless, i.e. the energy (density) available foractivation is written as xE0.

Mathematically, the above assumptions are summa-rized in the master equation for the probability P (E, l, t)of a given element having yield energy E and strain l attime t:

∂P

∂t=−γ̇(t)

∂P

∂l−

1

τ0exp

(−

E−kl2/2

xE0

)P +Y (t)δ(l)ρ(E).

(1)

The first term on the right corresponds to the uniformincrement in the local strains of all elements that do not

yield, according to l̇ = γ̇. The second gives the decrease inprobability due to yielding, and the third the correspond-ing increase in probability when elements are “reborn”after a yield; the factor Y (t) can be found from the re-quirement of conservation of probability.

Once P (E, l, t) has been determined, the global stressΣ is taken as the average of all local stresses kl across thisdistribution. It is possible, in fact, to solve for P (E, l, t)analytically for a given strain history γ(t), and this leadsto two coupled constitutive integral equations for Σ(t) andY (t) [24].

One can show that a glass (or jamming) transitiontakes place in the SGR model at x = 1 [25,26]. For smallerx and in the absence of steady shear, the system continuesto evolve without ever reaching equilibrium: it ages. Wewill see, however, that the experimental data presentedbelow can be reasonably fitted using values of x greaterthan unity. Consequently we will focus on the steady-statepredictions of the model. For oscillatory strain, in particu-lar, the amplitude and frequency dependent complex shearmodulus G∗(Γ0, ω) and the residual q (see Sect. 4 for def-initions) can then be found by solving the constitutiveequation numerically as outlined in reference [26].

We briefly discuss which parameters appear when fit-ting predictions of the SGR model to experimental data.The elastic (G′) and viscous (G′′) components of the com-plex modulus G∗ = G′ + iG′′ both scale with the localelastic constant k; this can be eliminated by normalizingby the linear response limit value (denoted G below) ofG′ at each given angular frequency ω. The dependenceon frequency of the SGR predictions is through the di-mensionless combination Ω = ωτ0. Similarly, the depen-dence on the strain amplitude Γ0 is through a scaled strainΓ0/ΓSGR; the relevant scale is set by ΓSGR =

√E0/k. The

fitting procedure therefore involves finding optimal valuesfor x, Ω and ΓSGR. The result cannot be cast in any sim-ple explicit form but examples of the variation of G∗ withamplitude can be found in Section 6.4 below.

Viscoelastoplastic models

Several constitutive laws [1,26–29], based on the associa-tion of springs, sliders and viscous elements, have beenproposed to describe phenomenologically the homoge-neous flow of complex yield strain fluids similar to foams.If the deformation is quasistatic, viscous effects vanish andelastoplastic behaviour is expected to be dominant. In thiscase, foam may be mimicked by a spring connected in se-ries with a slider. The spring represents the static elasticresponse, characterized by a shear modulus G, whereas theslider schematically describes the yielding (cf. Fig. 1a). Itbehaves as a rigid link when subjected to a force whosemagnitude is below a threshold and it slides freely forforces of larger magnitude (cf. Fig. 1b). In a material, theforce and its threshold value, respectively, correspond tothe stress Σ and the yield stress Σy where plastic flow setsin. To our knowledge, the first analytical prediction of thestress response to an oscillatory strain of an elastoplasticmaterial is presented in [28]. The complex modulus andthe stress harmonic spectrum are given in Appendix A.


Fig. 1. (a) Schematic model of elastoplastic behaviour basedon a spring and a slider, representing, respectively, an elasticmodulus G and a yield stress Σy. (b) Stress-strain relationshipof such an elastoplastic element.

To introduce schematically viscoelastic behaviour atlow strain amplitudes and to fit the yielding transitionmore accurately, an additional viscous element and a phe-nomenological yield function have been introduced [26,27]. A similar model has also recently been proposed in thecontext of complex polymeric fluids [30]. To summarize,these viscoelastoplastic models involving springs, slidersand viscous elements do not explain the physical mecha-nisms involved in foam rheology on the bubble or meso-scopic scale, but they have the merit of providing simpleconstitutive laws where the information contained in ex-perimental data is reduced to a relatively small number ofparameters. However, they are restricted to flow regimeswhere strain localization does not occur.

3 Samples

We use stable foams for which drainage, coarsening andbubble coalescence are negligible over the duration ofa rheological measurement. We study Gillette shavingcream which has been used in many rheological exper-iments published in the literature [2]. Its measured gasvolume fraction is 92.5 ± 0.5%. We also investigate a sec-ond kind of foam that will be called AOK. It is generatedby injecting a gas and a polymer surfactant based aqueoussolution into a column filled with glass beads, as describedelsewhere [14]. This solution contains 1.5% g/g sodiumα-olefin sulfonate (AOK, Witco Chemicals), 0.2% g/gPolyethylene-oxide (Mw = 3 × 10

5 g mol−1, Aldrich) and0.4% g/g dodecanol (Aldrich). Its surface tension and vis-cosity are similar to those of the Gillette foaming solution.The gas is nitrogen containing perfluorohexane vapour.The measured gas volume fraction of the AOK foam sam-ples is 97.0 ± 0.3%. All experiments are performed at atemperature of 21 ± 1 ◦C. To mimimize the rheologicalmemory of the flow history due to sample injection intothe rheometer, Gillette samples are allowed to coarsen foreither 30 or 60 minutes before the experiment is started.This leads to average bubble diameters equal to 28μm and36μm, respectively. For AOK foam, the experiments aredone 20 minutes after sample injection when the averagebubble diameter is 50μm. Note that AOK foams coarsenmuch more slowly than Gillette foams.

4 Rheological experiments and data analysis

Immediately after its production, the foam is injected intothe cylindrical Couette cell of a rheometer (Bohlin CVOR-150) with an inner radius ri = 21 mm, an outer radiusro = 25mm and a rotor height H = 48 mm. To preventwall slip, closely spaced 0.2mm deep grooves parallel tothe axis of the cylinders are carved into the surfaces in con-tact with the foam. The air in contact with the samplesis saturated with humidity to avoid evaporation. Whenthe sample has aged as described in the previous section,we apply a sinusoidal shear strain by rotating the innercylinder. The frequency ν = ω/2π, is fixed either to 1Hz or0.3 Hz and the strain amplitude Γ0 is increased from 10

−3

to 3. This sweep is divided into steps where the amplitudeis constant, lasting 5 seconds for 1Hz and 10 seconds for0.3Hz. The rheometer is operating in a controlled strainmode. During the experiment it records as a functionof time the engineering strain Γ (t) = Γ0Re[exp(−iωt)],which is deduced from the angle of rotation of the innercylinder θ(t) as

Γ (t) =8r2i r

2oθ(t)

(ri + ro)3(ri − ro). (2)

Note that for linear elastic or Newtonian materials, thismacroscopic quantity is equal to the strain in the middleof the gap. The rheometer also records the shear stressin the middle of the gap Σ(t) which is deduced from thetorque M(t) applied to the inner cylinder

Σ(t) =2M(t)

πH(ro + ri)2. (3)

To first order in (ro−ri)/ri, this stress is equal to the spa-tially averaged stress in the sample, and in the following,we will neglect the distinction between these two stresses.To analyze the stress data, we decompose them as follows:

Σ(t) = Γ0Re[G∗ (Γ0, ω) e

iωt]+ ΔΣ(t). (4)

The first term is the fundamental harmonic component. Itis related to the applied strain via G∗(Γ0, ω), defined as theratio of the fundamental harmonic components of shearstress and strain. Note that in the limit of small strainamplitude, G∗(Γ0, ω) does not depend on Γ0 and convergesto the usual linear complex shear modulus. The secondterm in equation (4), ΔΣ(t), contains the contributionsof all the higher stress harmonics

ΔΣ(t) =

∞∑n=2

hn cos(nωt + ϕn), (5)

where

hn =π

ω

∣∣∣∣∣∫ 2π/ω

0

e−inωtΣ(t)dt

∣∣∣∣∣ . (6)As shown in Figure 2 the third harmonic (n = 3) providesthe dominant part of the nonlinear response, and contri-butions from harmonics where n is even or greater than 9


Fig. 2. Stress residual q defined in equation (7) and odd har-monics from n = 3 to 9 versus normalised strain amplitude,for AOK foam at 1 Hz. The harmonics are normalized in thesame way as the stress residual.

are found to be insignificant. To quantify the nonlinearityof the rheological response in a global way, we define thestress residual q as the dimensionless root-mean-squarevariation of ΔΣ(t)

q =

√∫ΔΣ2(t)dt∫Σ2(t)dt

=

√√√√ ∑4i=1 h22i+1h21 +

∑4i=1 h

22i+1

. (7)

In our experiments, q is almost equal to the amplitude ofthe third harmonic. We nevertheless discuss our data interms of q to provide a connection with the SGR model.We also analyse the Fourier spectrum of the recordedstrain signal Γ (t) which should ideally be perfectly sinu-soidal. Its anharmonic residual, defined in analogy with q,is found to be smaller than 0.002. Therefore, we concludethat nonlinear stress response with q well above 0.002 isdue only to the nonlinear rheological response of the foam.

5 Measurement of the local strain and stress

Before the shear experiment we draw a black radial lineon the free foam surface between the two cylinders, usinga suspension made of carbon black particles dispersed inthe foaming solution. When oscillatory shear is applied byrotating the inner cylinder, the bubbles move to a goodapproximation on trajectories whose radial coordinate ris constant (cf. Fig. 3). The black line is used as a tracerand we record its deformed shape as a function of time us-ing a CCD camera. The curvilinear displacement of tracerpoints along their circular trajectories is observed as afunction of r and time t. For r = ri, the bubbles movewithin experimental accuracy at the same velocity as theinner cylinder and at r = ro they are immobile. This ob-servation excludes any wall slip.

Fig. 3. Top view of the Couette cell illustrating the coordi-nates used in the analysis. The dashed line represents the tracerposition before shear, and the continuous thick line shows itsposition after rotation of the inner cylinder. The thick arrowrepresents the amplitude of the curvilinear tracer displacements0(r), measured at a time t when Γ (t) reaches its maximumvalue Γ0.

We focus next on the amplitude of the bubble displace-ments during one oscillation period. The amplitude of thecurvilinear bubble displacement at a radius r is denoteds0(r). The local shear strain amplitude γ0(r) is extractedfrom this function by the following expression, applied fora closely spaced set of radii, covering the range ri < r < ro:

γ0(r) =∂s0(r)

∂r−

s0(r)

r. (8)

The derivative in this expression is evaluated approxi-mately by constructing a tangent to the experimentals0(r) curve. Due to the imperfection of the tracer, thisconstruction is not accurate for r very close to either ro orri and the corresponding points are omitted. The secondterm that is subtracted on the right-hand side representsthe change of s0 with r expected for a rigid rotation with-out any strain.

Similarly, we can look at stress amplitudes, denotingΣ0 and σ0(r) the amplitudes of the fundamental harmoniccomponents of the shear stress Σ(t) and of the local stressσ at a radius r. Since inertial forces acting on the foamare negligible in the investigated range of frequencies andstrain amplitudes, and in view of equation (3), they arerelated as follows [31]:

σ0(r) = Σ0(ro + ri)

2

4r2. (9)

6 Results and discussion

6.1 Onset of strain localization

To monitor flow homogeneity, we observe the bubble mo-tion at the free surface of Gillette samples as described in


Fig. 4. a) Normalized amplitude of the curvilinear displace-ment versus radial distance for increasing strain amplitudes Γ0pplied to the same sample (Gillette foam, bubble diameter =28 μm) at 1 Hz. The dashed line corresponds to the displace-ment expected for linear elastic deformation or Newtonian flow(Eq. (10) with A = −0.11 mm−1 and B = 71 mm). Typical er-ror bars are indicated. b) Local strain amplitudes derived from(a) using equation (8) plotted versus the radial coordinate. Thedotted lines correspond to the local amplitude expected in themiddle of the gap for linear elastic deformation or Newtonianflow (γ0((ri + ro)/2) = Γ0).

Section 5. As a reference, we calculate the displacements0(r) for a Newtonian fluid or a linearly elastic solid. Inboth cases the displacement amplitude s0(r), normalizedby the displacement amplitude of the rotor, s0(ri) is pre-dicted by

s0(r)

s0(ri)= Ar +

B

r, (10)

where the constants A and B are chosen to fit the bound-ary conditions at ri and ro. Figure 4a compares equa-tion (10) to the normalized displacement amplitudes mea-sured for Gillette foam during a strain amplitude sweep. Inview of the experimental uncertainty, the data for Γ0 = 0.3and 0.5 are in good agreement with equation (10). How-

Fig. 5. Image of the free top surface of Gillette foam in theCouette cell. The applied strain amplitude Γ0 is 0.9 and thepicture is taken close to an instant where Γ (t) ∼= 0. The rotorand the stator appear in black. A dotted line is drawn at theradius where the local strain is maximum.

ever, for the largest amplitude Γ0 = 0.7 there are devia-tions larger than the error bar. These results are furtheranalyzed in Figure 4b where the local strain amplitudeγ0(r) is derived from the measured values of s0(r) usingequation (8). For a linearly elastic material or a Newto-nian fluid, γ0(r) is expected to scale with 1/r

2, as thestress. In our geometry, this implies γ0(ri) ∼= 1.20Γ0 andγ0(ro) ∼= 0.85Γ0, in rough agreement with our data forΓ0 = 0.3 and 0.5. However, for Γ0 = 0.7 the deformationhas a maximum near the middle of the gap where the localstrain reaches 2Γ0. This is the signature of strain local-ization, setting in above Γ0 = 0.6± 0.1 as reported previ-ously [12]. Moreover, above a strain amplitude of 0.9±0.1,we systematically observe a sharp discontinuity in the dis-placement profile, indicating a strongly localized strain asshown in Figure 5. The picture is blurred close to the rotorbecause in this region the bubbles motion is faster thanclose to the stator. The quantitative analysis presented inFigure 4 was carried out only for one type of foam sam-ple but, qualitatively, strain localization only close to themiddle of the gap for strain amplitudes larger than about0.6 is observed for all of the investigated foams and fre-quencies. Since in cylindrical Couette geometry the stressincreases with decreasing distance from the axis of thecylinders, localization would a priori be expected close tothe inner cylinder as mentioned in Section 2.1. Thus, wecan exclude stress heterogeneity as the direct origin of theobserved localization. Moreover, at least in simple shearand steady flow, the fluidity model outlined in Section 2.2predicts flow that is localized in the vicinity of boundaries,contrary to our observations.

These features demonstrate that for the foams inves-tigated here, there exists a regime of localized oscillatoryflow which is explained neither by stress heterogeneity,nor by fluidity models. Our results also show that this lo-calized flow only sets in at strain amplitudes well abovethe yield strain. As a consequence, there is a well-definedregime where it is legitimate to compare our data to mod-els of yielding that do not predict flow localization.


Fig. 6. Comparison of local and macroscopic normalised stressamplitudes versus strain amplitude. The lines correspond tothe macroscopic stress amplitude Σ0/Σy versus engineeringstrain amplitude Γ0. The symbols represent the local stressamplitude normalized by the yield stress, σ0(r)/Σy versus lo-cal strain amplitude γ0(r). These data are deduced from Fig-ure 4. Typical error bars for γ0(r) and σ0(r)/Σy are drawn.The sample is Gillette foam (average bubble size = 28 μm)and the frequency is 1 Hz.

6.2 Local constitutive relation

We now use equation (8) and equation (9) to deduce fromour observations the constitutive law relating local stressamplitude σ0(r) and strain amplitude γ0(r). In Figure 6,this relation is compared to the one between Σ0 and Γ0.Here stresses are normalized by the yield stress Σy whichis determined as explained in Section 6.3. For Γ0 < 0.6,all the data are consistent, showing that in our case thecontinuum description of foam flow is indeed valid. Thisverification is necessary, since it has been suggested onthe basis of previous MRI measurements of Gillette foamssteadily flowing in a Couette geometry that in regions ofthe order of 25 bubble diameters, the continuum descrip-tion can break down [7]. In our experiments this wouldcorrespond to a significant fraction of the gap.

At strain amplitudes Γ0 larger than 0.6 there clearlyare deviations between local and macroscopic rheologicalbehaviours. Indeed, a narrow range of σ0(r) correspondsto a wide range of γ0(r). This local rheological behaviourconfirms that the observed strain localization is not in-

Fig. 7. a) Parametric plots of stress Σ(t) versus strain Γ (t) forapplied strain amplitudes Γ0 = 0.055, 0.15, 0.25, 0.43, 0.72 and1.2 corresponding to contours of increasing thicknesses. Thefrequency is equal to 1 Hz. The sample is Gillette foam (bub-ble size 28 μm). Panels b) and c) schematically illustrate theresponse expected for viscoelastic and elastoplastic response,respectively.

duced by the stress variation in the gap. Moreover, it isconsistent with the coexistence of liquid-like and solid-likeregions reported in [32]. This finding is robust with respectto changes of bubble size in the investigated range.

6.3 Nonlinear rheological response

A typical Lissajous plot showing Γ (t) versus Σ(t) is shownin Figure 7a for a Gillette sample. The same qualitativebehaviour is found for all investigated samples. At smallamplitudes, the plot has an elliptic shape (Fig. 7b), as ex-pected for linear viscoelastic behaviour. With increasingstrain amplitude, a crossover to a parallelogram-shapedLissajous plot is observed, as expected for elastoplasticmaterials (Fig. 7b) (for further details see App. A), andin qualitative agreement with Lissajous plots predictedin the framework of the SGR model [24]. A variety ofcomplex fluids showing similar behaviour have been clas-sified in a phenomenological survey of large amplitude os-cillatory strain responses [10]. These materials typicallypresent a yield stress, and the flat and steep parts ofthe Lissajous diagram for large strain amplitudes corre-spond to liquid-like and solid-like response which alter-nately predominate during the oscillations. To analyze thisbehaviour quantitatively, we first focus on the fundamen-tal frequency components of stress and strain from whichwe deduce the amplitude dependent complex shear mod-ulus G∗(Γ0, ω) = G

′(Γ0, ω) + iG′′(Γ0, ω) defined in equa-

tion (4). The insert of Figure 8a shows the typical evolu-tion of G′(Γ0, ω) with stress amplitude Σ0. There is a lin-ear regime at small Σ0, characterized by a constant mod-ulus that will be denoted G, and followed by the nonlinearregime where G′(Γ0, ω) decreases following a power law.The intersection between these two asymptotic regimes isused to estimate the yield stress Σy as illustrated by thegeometrical construction. Similar criteria have been usedin many previous studies to detect yielding [8,10,19]. Thecriterion that we use here has been shown to be consistent


Fig. 8. Normalised elastic and loss shear moduli versus nor-malised strain amplitude. The symbols correspond to Gilletteand AOK foams at two frequencies, as shown in the legendof panel (b). The full line corresponds to the elastoplastic os-cillatory response (Eq. (A.2)). The other lines show the pre-diction of the SGR model (see text) (dashed line: x = 1.07,Ω = 0.3 and ΓSGR = Γy, dotted line: x = 1.05, Ω = 0.3 andΓSGR = 0.85Γy). In the hatched region, the onset of strain lo-calization is observed. The insert in a) shows the evolution ofthe elastic modulus with stress amplitude for AOK foam.

with other methods for detecting the yield stress in 3Dfoams [12]. From this estimation, we derive the yield strainamplitude using the relation Σy = |G

∗(Γy, ω)|Γy. Withinthe studied range of frequency and bubble size, we obtainΓy = 0.15 ± 0.01 for Gillette foam, and Γy = 0.22 ± 0.02for the AOK foam. Figure 8a and b show the evolutionof the elastic and loss moduli normalized by G as a func-tion of the normalized strain amplitude Γ0/Γy. We no-tice the evolution of G∗ with strain amplitude usually ob-served for foams: At small Γ0 a solid-like regime whereG′′(Γ0, ω) � G, at large Γ0 liquid-like behaviour withG′(Γ0, ω) � G

′′(Γ0, ω) and a crossover between G′(Γ0, ω)

and G′′(Γ0, ω) close to the maximum of G′′(Γ0, ω). Our

results are in qualitative agreement with previously re-ported data for moderately dry foams [8,12] and con-centrated emulsions [33]. Let us recall that G scales assurface tension divided by the mean bubble radius andthat the complex shear modulus at low frequency scales

Fig. 9. Stress residual q defined in equation (7) versus nor-malised strain amplitude, for Gillette and AOK foams as shownin the legend. The continuous line corresponds to the elasto-plastic oscillatory response (Eq. (A.2)). The other lines cor-respond to the SGR model (dashed line: x = 1.07, Ω = 0.3and ΓSGR = Γy, dotted line: x = 1.05, Ω = 0.3 and ΓSGR =0.85Γy). In the hatched region, the onset of strain localizationis observed.

as G. Therefore, the normalization by G allows variationsin average bubble size between different samples to betaken into account. For Γ0 < Γy, only the normalizedG′(Γ0, ω) data are similar for all kinds of foams, whereasthe normalized G′′(Γ0, ω) is smaller for AOK foam thanfor Gillette foam. This can be explained by the differencesin gas volume fraction [8] and rate of coarsening inducedbubble rearrangements [34,35]. For Γ0 > Γy the G

∗ dataobtained at 1Hz and 0.3Hz with AOK foam and withGillette foam for both investigated bubble sizes superim-pose to a good approximation. Remarkably, at the strainamplitude where the onset of localized flow is observed,there are no distinctive features on the curves representingthe G∗ data. Moreover, we notice that yielding and strainlocalization induced by oscillatory deformation are clearlydistinct phenomena, since they set in at well-separatedstrain amplitudes. This result is qualitatively consistentwith previous observations of yielding and localization insteady shear start-up experiments with dry foam [14].

Finally, we plot in Figure 9 the anharmonic resid-ual stress q defined in equation (7). For strain amplitudeΓ0 � Γy, the observed small level of anharmonicity is tooclose to the limit of detection to be interpreted quantita-tively. However, for Γ0 ≥ Γy, q increases by more thanone order of magnitude as the strain increases by a fac-tor of 3. At amplitudes where strain localization sets in,the residual stress saturates. In contrast to the modulusG∗(Γ0, ω) which is very similar for Gillette and AOK foamfor Γ0 > Γy, the respective stress residuals are clearly dis-tinct: q rises much faster with strain amplitude for AOKfoam compared to Gillette foam. We finally note that thestress residuals measured for an applied strain oscillationfrequency of 1Hz and 0.3Hz are very similar. These lat-


ter findings strongly argue against artifacts related to thelimited frequency bandwidth of the rheometer.

6.4 Comparison of our data with models

To introduce the discussion of specific models, we first re-call recent measurements showing that the complex shearmodulus of foams scales with strain amplitude and fre-quency as expected if a single structural relaxation timedependent on strain rate amplitude governed the stressrelaxation and the rearrangement processes on the bub-ble scale. This relaxation time characterizes the dynamicsglobally, but it does not indicate how bubble rearrange-ment events are distributed in time within a strain cycle.This is a new feature that our harmonics (or Lissajous)data allow us to probe. An analogy with the Green To-bolsky model [36], well known in polymer rheology, sug-gests that rearrangements that relax stress and that oc-cur randomly in time would lead to Maxwell liquid be-haviour. (Here, we make the simplifying assumption thatfoam elasticity in the absence of any rearrangements islinear). For a Maxwell liquid the Lissajous plot has anelliptical shape. This is approximately true for strain am-plitudes up to the yield strain, but at higher amplitudesthe plots clearly have a different shape, resembling a par-allelogram. We find this behaviour up to the highest in-vestigated strain amplitudes where we observe flow local-ization. The parallelogram-shaped plot arises because re-arrangements typically occur only if enough elastic energyhas been built up locally. Our data contain the signatureof this process, and therefore allow us to refine the physi-cal description of the process of yielding. On this basis, wediscuss in the following paragraphs the elastoplastic andSGR models. It would also be of great interest to com-pare our data to fluidity models, but this would requirecomplex calculations which are beyond the scope of thepresent paper. A similar remark applies to models basedon a minimal strain rate [23] which successfully describesteady foam flow. In their present form, they do not yetcapture transient viscoelastic effects. Further theoreticalwork allowing additional models to be compared to ourdata would be of great interest.

Comparison with viscoelastoplastic models

For elementary elastoplastic behaviour, an analytical cal-culation first presented in [28] and recalled in Appendix Apredicts the stress response to oscillatory shear, given byequation (A.2). At strain amplitudes Γ0 � Γy, this resultreduces to the asymptotic power law decays

G′(Γ0, ω) =16G

3π

(Γ0Γy

)−3/2

(11)

and

G′′(Γ0, ω) =4G

π

(Γ0Γy

)−1

. (12)

Note that the full analytic solution presented in Ap-pendix A shows that equation (11) is already reachedat strain amplitudes close to Γy, whereas the asymptoticpower law for G′′(Γ0, ω) is reached only at strain ampli-tudes more than a decade larger than Γy. Figure 8a showsgood agreement between equation (11) and the experi-mental data in the whole range of Γ0/Γy. The predictionfor G′′(Γ0, ω) given by equation (A.2) is in good agreementwith the loss moduli data only for Γ0/Γy > 1 (cf. Fig. 8b).This is consistent with the fact that, for Γ0/Γy � 1,the dissipation is mainly due to coarsening induced bub-ble rearrangements which are not taken into account bythis elastoplastic model. Moreover, it is remarkable thatthe predictions of the elastoplastic model still hold in theregime where we observe a cross-over to localized flow.Here, one might identify the slider of the model with theregion where the shear is localized and the spring with thepart of the foam that may remain elastic.

The merit of the elastoplastic model is to predict thenonlinearity of G∗(Γ0, ω) with strain amplitude using onlytwo free parameters G and Γy, whose physical origin onthe scale of the microstructure is known [1]. The scalingof G with surface tension T and bubble radius R arisesfrom the change of interfacial energy induced when thefoam is sheared. Yielding in foams and emulsions is due tobubble or droplet rearrangements that arise when, underthe effect of an applied strain, unstable configurations arecreated. Dimensionally, the yield stress also scales as T/R.The quantitative evolution of yield strain and elastic shearmodulus with gas volume fraction has been described byphenomenological expressions [8].

Despite its success in describing the behaviour at largestrain amplitudes, basic elastoplasticity remains an incom-plete phenomenological model. It predicts a sharp transi-tion from linear elastic behavior to plastic flow as a func-tion of Γ0. As a consequence, the predicted anharmonic-ity for strain amplitudes just above Γy is much largerthan the measured data, as shown in Figure 9. A relatedweak discrepancy appears in the decay of G′ with Γ0 closeto Γy. All these features suggest that a distribution ofyield strains is necessary to describe the data more accu-rately. Marmottant et al. have recently introduced a phe-nomenological yield function that describes such a distri-bution and which allows experimental complex shear mod-ulus data to be fitted accurately [27]. Explicit predictionsfor the anharmonicity derived in this framework have notyet been published. Only if the same distribution of yieldstrains can be used to predict strain amplitude dependen-cies of both the stress Fourier spectrum and G∗ close toΓy, is elastoplasticity a good concept for modelling foamsaccurately. A second shortcoming of the basic elastoplas-tic model is that it does not account for the dissipationexperimentally observed for Γ0 � Γy. To describe this fea-ture, a viscous element connected in parallel to the plasticand elastic elements of the model has been proposed [27].It allows the foam data to be fitted at a given frequency,but this model is not able to predict the experimentallyobserved G∗(Γ0) over an extended range of frequencies.Indeed, it reduces to a Voigt model in the linear viscoelas-tic regime, which is incompatible with the Maxwell liquid


behaviour experimentally observed [13,34,37,38] at lowfrequencies and the power law increase G∗ ∼ ω1/2 foundat high frequency [13,37,39].

Comparison with the SGR model

For noise temperatures x close to 1, the SGR model pre-dicts a strain amplitude dependence of the shear modulussimilar to the experimental results shown in Figure 8. Theratio G′′(Γ0, ω)/G

′(Γ0, ω) at low strain amplitude and lowfrequency is predicted to depend on x as follows:

G′′(Γ0, ω)

G′(Γ0, ω)= tan

(π2

(x − 1))

. (13)

We have G′′(Γ0, ω)/G′(Γ0, ω) ∼= 0.10 for the Gillette

foam for both bubble sizes and frequencies, andG′′(Γ0, ω)/G

′(Γ0, ω) ∼= 0.055 for AOK foam. Therefore,relation (13) suggests x ∼= 1.07 and x ∼= 1.05 for theGillette and AOK foams, respectively. These values thenhave to be tuned slightly to allow for the fact that thefitted (dimensionless) frequency Ω will not necessarily besmall enough for equation (13) to apply exactly. In prac-tice, we fit x and Ω to the data for G′′/G, as this hasthe most structure in its dependence on strain amplitudeΓ0. This is done by minimizing the error in the predictedvalues for the maximum of G′′/G and the height of theplateau in G′′/G in the linear response regime of smallΓ0. The strain scale ΓSGR is then fitted afterwards to re-produce the experimental value of the strain where themaximum of G′′/G occurs.

The best fits obtained in this way are for x = 1.07(Gillette) and x = 1.05 (AOK), close to the values ex-pected from relation (13). Fits of similar quality can be ob-tained by varying x within the range ±0.01. The optimalassociated values of the fitted frequency vary over a ratherbroad range, Ω = 0.3 ± 0.2 for both Gillette and AOK.Translating to the attempt frequency via Ω = 2πντ0with ν = 0.3Hz or 1Hz, one gets values of τ0 in therange from 0.01 s to 0.3 s. As regards the strain scale, fi-nally, ΓSGR = Γy is satisfactory for the Gillette foam,while for AOK the best fit value is slightly smaller atΓSGR ≈ 0.85Γy. The fitted strain scale is therefore closeto Γy as expected.

The SGR predictions using the parameter values givenabove are shown in Figures 7 and 8. For all foams and fre-quencies investigated, the predicted elastic modulus G′/Gagrees with the experimental data for strain amplitudesup to Γ0/Γy around 2, with deviations setting in at largerstrain amplitudes. The predictions for the loss modulusG′′/G agree quite well across the whole range of Γ0 calcu-lated. For the AOK foam, also the anharmonic residual qis predicted rather well by the model, while for the Gillettefoam the onset of anharmonicity is predicted to be muchsteeper than observed experimentally. The lower qualityof the predictions for G′/G and q may in part be due tothe fact that the SGR model cannot capture the observedonset of strain localization at large values of Γ0/Γy, giventhat it treats elements from all areas of the sample as sta-tistically indistinguishable. In addition, it is likely that the

exponential form of the yield rate assumed in the model,cf. equation (1), becomes unrealistic for large strain am-plitudes: Physically one expects that even elements whichare strained beyond their yield barrier cannot yield ar-bitrarily quickly, so that the exponential increase of theyield rate with l2 must be cut off eventually. One mightexpect that this would restore a somewhat more gradualdecrease of G′/G, in line with experimental data, thanpredicted by the SGR model. Independently of these twoshortcomings, why the model prediction for the anhar-monicity q is so much more accurate for the AOK thanfor the Gillette foam remains unclear.

To explore the above suggestion further, we have inves-tigated a modified SGR model where the exponential inequation (1) is replaced with unity when the exponent be-comes positive, i.e. once an element has been strained pastits notional yield point. Because the l dependence then nolonger appears as a simple factor, this model is not easy tostudy analytically, but can be simulated efficiently usingwaiting time Monte Carlo techniques [40]. At large strainamplitudes most SGR elements are pushed quickly intothe regime where the cutoff on the yield rate kicks in, andone can show analytically that the predictions for G∗(ω)then become amplitude independent and reduce to thoseof a Maxwell model with relaxation time τ0.

To rectify this clearly unphysical behaviour, one can inaddition postulate that the basic yield rate 1/τ0 in equa-tion (1) should increase with strain rate and be replacedwith 1/τ0 + bγ̇ for some constant b, see, e.g. [9,41]. Thepredictions of this version of the SGR model can still beworked out analytically in the limit of large strain ampli-tudes. Intriguingly, we find —and have confirmed by com-paring with simulation results— that G′ and G′′ decaywith strain amplitude according to the same power lawsas in the elastoplastic model, see equations (11, 12). Thissupports the view expressed above that such a modifiedSGR model should be capable of producing a good overallfit to the experimental data. Because results at interme-diate strain amplitudes currently have to be obtained bysimulation, we have not, however, attempted detailed fitsof the relevant model parameters.

6.5 Connection with the foam microstructure and itsdynamics

From a physicist’s point of view, a complete model offoam rheology should explain how the macroscopic re-sponse arises due to processes on the scale of the bub-bles and the liquid films. At this level of understandingit should be possible to predict all the free parameters ofphenomenological models. In his pioneering work, Princenhas provided insight about the dependence of the shearmodulus and the yield stress on bubble size [1,2]. This isenough to predict the free parameters from a basic elasto-plastic model from foam liquid fraction, surface tension ofthe foaming liquid and bubble size, or to fix the elasticconstant of the mesoscopic regions and the scale of thedistribution of trap depths in the SGR model. Providinga microscopic interpretation of the effective noise temper-


ature x in the SGR model is more challenging. To discussthis point, we recall that quiescent coarsening foams haveintrinsic, non-thermal dynamics due to the Laplace pres-sure driven diffusive gas exchange between neighbouringbubbles [1]. It induces intermittent structural rearrange-ments in small bubble clusters. One might therefore cor-relate the rate of coarsening to the effective noise temper-ature. Indeed, the Gillette foam that coarsens faster alsohas the higher noise temperature, according to the fits re-ported above. However, the rate of bubble rearrangementsis known to be enhanced by steady flow [41], so that thenoise temperature would have to depend on shear rate, apossibility which is not considered in the standard SGRmodel. Moreover, previous experiments and simulationshave shown how coarsening induced rearrangements relaxstress and lead on a macroscopic scale to Maxwell liquidbehaviour in the low frequency limit [34,35,38]. A Maxwellliquid, mimicked by a spring in series with a viscous ele-ment, is characterized by a single characteristic time τ . Asa consequence, G′ progressively goes to zero for frequen-cies such that ωτ < 1, in agreement with experimentaldata reported in [13]. The results of creep experimentsare consistent with these findings [34]. In the SGR modela large spectrum of relaxation times is predicted and theexperimentally observed very low frequency behaviour ofthe complex shear modulus is not found for x = 1.07. Anadditional feature not captured by the SGR model is thetransient slowing down of bubble dynamics observed fol-lowing a transient flow [17], which is the opposite of theshear rejuvenation effect observed in glasses.

7 Conclusion and outlook

The recent literature on liquid foam rheology raises thequestion whether the stress in a liquid foam is a well-defined function of strain and strain rate, and, if so, howthis constitutive law is related to physico-chemical pro-cesses and structures on a mesoscopic, bubble, or filmscale. To answer the first question, we probe in situ thebubble displacement profile at the free sample surface ofa Couette cell and show that strain localization clearlydistinct from wall slip and not directly induced by het-erogeneous stress is a robust feature of the investigatedtypes of foams and oscillatory flows. Such behaviour is in-compatible with conventional constitutive laws. However,localization only sets in at strain amplitudes far abovethe yield strain, demonstrating that there is indeed an ex-tended regime of strains where it makes sense to comparethe data to models.

This conclusion allows us to investigate the secondquestion raised above, and we focus on a previously unex-plored feature of the nonlinear rheological response. A va-riety of models of complex yield strain fluid rheology pre-dict how the fundamental harmonic component of stressinduced by an oscillatory strain depends on strain ampli-tude at low frequency. This information is generally repre-sented in terms of the amplitude dependent complex shearmodulus G∗(Γ0, ω). To assess to what extent such modelsactually capture the relevant physics for foams, we mea-

sure and report for the first time in addition to G∗(Γ0, ω)the full harmonic spectrum of the stress response of foamwhich can equivalently be represented in the form of a Lis-sajous plot. These data resolve quantitatively how stressrelaxation is distributed in time within a cycle of defor-mation, so that elastoplastic and viscoelastic behaviourwith a strain rate dependent relaxation time can be dis-tinguished. We find that this feature depends significantlyon the physicochemical constitution of the sample, in con-trast to the G∗(Γ0, ω) data which do not show such a de-pendency in the vicinity of the yield strain.

For the driest investigated foam and at strain ampli-tudes well below the onset of flow localization, the SGRmodel accurately predicts the evolution with strain ampli-tude of three a priori independent functions, describingthe nonlinear rheology: G′(Γ0, ω), G

′′(Γ0, ω) and the mea-sure of the anharmonic response q. The fit is obtained withonly two adjustable parameters (x,Ω). Phenomenologicalelastoplastic models capture the power law decay of thereal and imaginary parts of the complex shear modulus inthe liquid-like regime. The two respective exponents arepredicted correctly, without any free parameters. However,neither of the two mentioned approaches predicts the pre-viously reported frequency dependence of the viscoelasticbehaviour over an extended range [13,37,39] and accountsfor the flow localization observed at high strain ampli-tudes. As a perspective for future work, it would be ofgreat interest to construct a model of foam viscoelasticityand plasticity by combining the known facts about foamphysics on the bubble scale with features of successful phe-nomenological models. We expect that the data reportedin this paper will be useful in this context.

We acknowledge stimulating discussions with C. Gay, P.Marmottant and S. Cox and during the informal workshopon Foam mechanics, Grenoble, 21st-27th January 2008. Wethank D. Hautemayou for his technical help and we ac-knowledge financial support from E.S.A. (MAP AO99-108:C14914/02/NL/SH).

Appendix A. Oscillatory response of an

elastoplastic material

Figure 10 illustrates the basic elastoplastic response thatcan be mimicked by a combination in series of a springand a slider (cf. Fig. 1). Plastic strain and elastic strain,respectively, correspond to elongations of the slider and ofthe spring. The total strain Γ in the material that can beapplied experimentally is given by the sum of its elasticand plastic components. Figure 10a illustrates the relationbetween shear stress Σ and strain Γ predicted for an idealelastoplastic material subjected to the oscillatory strainshown in Figure 10b.

We now calculate the stationary response of suchan elastoplastic element to an applied strain Γ (t) =Γ0 cos(ωt). If Γ0 is smaller than or equal to Γy, the re-sponse is purely elastic and G∗(Γ0, ω) = G. In this case,the stress oscillates sinusoidally with the same phase as


Fig. 10. a) Elastoplastic stress-strain relation, in response toa sinusoidal applied strain of period T and amplitude Γ0 (cf.Fig. 1a). The instant t1 is defined by equation (A.1). b) Timeevolution of the stress (thick line) if a sinusoidal strain (thinline) is imposed with an amplitude exceeding the yield strain.

the strain. If the strain amplitude is larger than Γy, thestress-strain relation becomes hysteretic, as shown in Fig-ure 10b. As long as the slider is stuck, the plastic strainis constant so that any variation of the applied strain ΔΓinduces an equal change of the elastic strain and a stressvariation ΔΣ = GΔΓ . If the stress magnitude reachesthe yield stress, the slider becomes mobile and any fur-ther increase of the applied strain magnitude induces onlyplastic strain while the elastic strain remains constant inthis case.

As the applied strain decreases after having reached itsmaximum value, as for instance at t = 0 in Figure 10b, theslider becomes stuck. From here on, the evolution of theapplied strain induces a negative variation of the elasticstrain equal to Γ (t) − Γ0 and therefore a stress Σ(t) =GΓy + GΓ0(cos(ωt) − 1). Indeed, this expression satisfiesthe initial condition at t = 0 as well as the relation ΔΣ =GΔΓ that must be valid as long as the slider is stuck.As time goes on, the stress decreases to −Σy at a timedenoted as t1 (cf. Fig. 10b). An elementary calculationyields

t1 =1

ωarccos(−2Γy/Γ0 + 1). (A.1)

The stress then remains saturated until the minimum ofthe applied strain is reached at a time which is half theoscillation period T . As the strain starts increasing again,the elastic strain increases by Γ (t) + Γ0. Therefore thestress increases as Σ(t) = −GΓy + GΓ0(cos(ωt) + 1) untilit reaches the value Σy, at the time T/2 + t1. The evo-lution then remains saturated until the time t = T , andfrom then on the whole process continues periodically. Todetermine G∗(Γ0, ω), we calculate the Fourier componentof the stress oscillation at the frequency ω and divide it bythe strain amplitude [28]. These expressions are simplifiedwithout any loss of generality by scaling stress and strainso that G = 1 and Γy = 1

G∗(Γ0, ω) =2

Γ0T

{ ∫ t10

(1 + Γ0(cos(ωt) − 1)) e−iωtdt

−

∫ T/2t1

e−iωtdt+

∫ T/2+t1T/2

(−1+Γ0(cos(ωt)+1))e−iωtdt

+

∫ TT/2+t1

e−iωtdt

}=

1

π

(arccos

(−2 + Γ0

Γ0

)

−2(−2 + Γ0)

Γ0

√−1 + Γ0

Γ 20+ i

4(Γ0 − 1)

Γ 20

). (A.2)

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