J. Ind. Eng. Chem., Vol. 12, No. 5, (2006) 653-662 REVIEW

Yielding Behaviour of Viscoplastic Materials

C. Tiu†, J. Guo, and P. H. T. Uhlherr

Department of Chemical Engineering, Monash University, Clayton 3800, Australia

Received September 4, 2006

Abstract: Complex fluids exhibiting a have extremely variable shear properties ranging from elasto-plastic behaviour below the yield stress to viscous behaviour beyond that. The progressive elastic to plastic transition of viscoplastic materials can be explored by a number of rheological techniques including stress sweep, stress ramp, and strain recovery. The non-linear behaviour of such materials under different loading conditions can be investigated using a large amplitude oscillatory shear flow (LAOS) combined with Fourier-Transform (FT) harmonic analysis. Results from these rheological measurements provide a char- acteristic “rheological fingerprint” for complex materials exhibiting yield stress.

Keywords: yield stress, viscoplasticity, creep, recovery strain, large amplitude oscillatory shear

Introduction responding to the transition between plastic 1) and purely viscous flow [4-6]. Although the yielding Many industrial materials, such as filled , response is conventionally interpreted in terms of a gels, thickeners, foodstuffs, cosmetics, concen- “critical stress”, measurements with highly-filled suspen- trated mineral suspensions, electro-rheological and mag- sions such as ER fluids have shown that it is equally neto-rheological fluids behave rheologically in a manner possible to consider the existence of a critical strain at that varies between elastic solid and viscous liquid any level of stress [7,8]. A fundamental interpretation of depending upon the processing conditions. The exis- this transition with respect to processing conditions is of tence of a “yield stress” is traditionally recognised to be great importance to industrial applications. responsible for the complicated transition between The techniques developed for measuring yield stress classical solid-like and liquid-like behaviour. Although can be categorized into two groups: indirect and direct the inadequacy of the rheological definition of this methods. The indirect methods involve the fitting of the property has generated worldwide debate on the exis- shear stress-shear rate data to rheological models such as tence of a true yield stress [1-3], the concept of the yield the Bingham, Casson and Herschel-Bulkley models, etc. stress remains an important parameter in rheological The yield stress is determined by the extrapolation of the characterisation of complex fluids and in understanding flow curve at low shear rates to zero shear rate [9]. The their processing characteristics in industrial applications. results obtained using indirect methods are very sensitive The yield stress is generally regarded as the transition to the models used for fitting the rheological data and the stress between elastic solid-like behaviour and viscous accuracy of the rheological measurement in the low shear liquid-like behaviour, and is related to the internal rate region. particulate network structure. However, this transition Over the last two decades, a variety of direct mea- typically occurs not at a single point, but instead over a surement techniques have been developed to obtain the range of stresses starting at a lower limit, corresponding yield stress based either on dynamic or static principles. to progressive transition between elastic and plastic de- Some examples of these techniques include the im- formation and ultimately ending at a higher limit, cor- mersed plate [10], vane torsion [11], falling and rolling ball [12], inclined plate [13,19], pendulum [14] and slotted plate [15], cylindrical penetrometer [17], and † To whom all correspondence should be addressed. falling needle [18] methods. Among these methods, the (e-mail: [email protected]) 654 C. Tiu, J. Guo, and P. H. T. Uhlherr vane torsion technique is probably the most recognized understand the creep behaviour of a complex material. In and accepted method for yield stress measurement [16]. order to elucidate the transition mechanisms, only the With this method, a well-designed four-bladed vane fully results obtained for a polymeric microgel dispersion are immersed in the test liquid or suspension is rotated at a discussed hereafter. The experimental techniques em- sufficiently low shear rate, and the torque generated is ployed are divided into two parts. Firstly, the transition recorded as a function of time. The vane torsion method behaviour of yield stress materials is fully explored using measures the dynamic yield stress as the sample is the experimental results obtained from various steady sheared from its rest period until the stress reaches its and transient shearing tests. The non-linearity of such peak. The yield stress is calculated from the peak torque materials is investigated in the second part under dif- measurement and the vane dimensions. Recently, the ferent loading conditions using the LAOS and Fourier- vane torsion device has been incorporated into a transform techniques. The large amount of information stress-controlled rheometer to carry out the stress ramp/ generated by LAOS measurements and subsequent FT sweep and creep tests for yield stress materials [6]. On harmonic analysis is represented by some less common the other hand, the static yield stress is the equilibrium formalisms including Pipkin diagrams, Lissajous figures, stress at which the viscoplastic material attains the static FT amplitude and phase angle spectra. This information equilibrium with the measuring devices. With the static provides a unique “rheological fingerprint” for charac- methods, the yield stress is usually approached from the terising the linear and nonlinear behaviour of any non-equilibrium stresses above it. complex fluid. Such rheological fingerprinting will shed Yield stress measurements are notoriously difficult to light on the structural changes from a linear elastic solid interpret. It is not surprising to see that vastly different to a non-Newtonian viscous liquid, occurring in complex values of yield stress of the same material are reported in materials exhibiting a yield stress under the action of an the literature. The variations may be attributed to the applied stress. differences in the measurement techniques, definition of the yield stress, sample preparation, time scale of measurement, and sample pre-sheared history, etc Steady and Transient Shear Flow [6,9,20]. A recent inter-laboratory study [21] was conducted to evaluate the reliability and reproducibility The progressive transition of rheological behaviour of of several common yield stress measuring devices. In viscoplastic materials can be revealed by the shear-stress general, yield stress obtained by the direct methods is sweep or ramp test using a stress-controlled rheometer. A more reliable than the indirect methods. A uniform stress plateau coincides with an unambiguous transition sample preparation, conditioning procedure and a well- from solid-like behaviour to liquid-like behaviour for controlled shear history are paramount in obtaining yield stress materials [6]. Figure 1 shows a typical plot of accurate and reproducible results by direct measurement shear stress versus shear strain and shear stress versus methods. shear rate obtained for a 0.5 % Carbopol gel. Carbopol is The yielding process of viscoplastic materials is closely a high molecular weight polyacrylic acid, cross-linking linked to the transition between linear and non-linear with polyalkenyl and polyester. In an aqueous solution, it rheological behaviour. In recent years, large amplitude takes the form of a microgel dispersion and is often used oscillatory shear (LAOS) flow has been used increas- as a model yield stress material because of its stability ingly to investigate the non-linear rheological responses and transparent nature. At lower stress values, the shear of complex materials such as polymer melts, ER fluids strain appears to be a unique function of the shear stress, and highly concentrated suspensions [8,22-24]. Sig- and is independent of the rate of stress increment, as seen nificant progress has been made in the application of a in Figure 1. A linear relation with a slope approximately methodology known as Fourier-Transform (FT) rheology equal to unity implies a Hookean solid-like behaviour. At [25,26]. The accurate acquisition and processing of large stress values, the shear stress is a unique function highly sensitive FT spectra enable one to systematically of strain rate, which represents a fluid behaviour. The quantify the non-linearity of materials by monitoring the transition from solid to liquid is not abrupt, but goes appearance and growth of higher harmonics. through a region of a stress plateau. The extent of the In this review paper, the yielding process of nonlinear stress plateau reflects the breaking process of the internal viscoplastic materials, covering the entire range of their structure of the material, the slope of which represents transition from elastic to plastic to viscous behaviour, is the uniformity of the molecular bonding strength, or explored using a variety of rheological techniques. A inter-particle forces for the case of a particulate system quantitative understanding of this progressive transition such as suspension. A large slope implies that the in rheological behaviour from linear elastic response to breaking process occurs in a wide range of stress; hence, non-linear elasto-plastic deformation is necessary to the structure of the material exhibits ductile-type failure. Yielding Behaviour of Viscoplastic Materials 655

Figure 2. Effect of concentration on yield stress and slope of stress plateau for Carbopol 934 gel. Figure 1. Transition behaviour of Carbopol gel. “cellular lattice”. The closer the particle packing, the Conversely, a small slope reflects a brittle-type failure of larger the stress required to permanently deform the the material structure. The bonding strength of a material structure. As a result, the neutralized Carbopol gels is controlled by a variety of physical and chemical exhibit a higher yield stress. properties, such as solids concentration, extent of particle The yielding behaviour of a yield stress material can flocculation for suspensions and polymer chain entan- also be explored using a creep test. Figure 4 shows some glements for polymers. For polymeric gels, the bonding typical creep profiles obtained at different applied strength of the structure is expected to be a function of stresses for a 0.5 % Carbopol gel. The deformation of a pH and the concentration of the polymer resin. material during creep can be classified into three regions. The effect of resin concentration on the yielding There is an initial elastic response when stress is applied behaviour is illustrated in Figure 2. It is seen that the to the sample, followed by a creep process. After a yield stress increases steadily with increasing concentra- certain time, the creep rate increases slowly at first foll- tion, while the slope of the stress plateau decreases with owed by a sharp upturn, indicating that some viscous increasing concentration. Since the Carbopol microgel component was involved. A steady strain rate is finally dispersion is consisted of highly cross-linked (interior), developed, corresponding to a viscous flow condition in enormously swollen and deformable particles [27-29], the material. The region of creep is shortened with in- increasing the resin concentration effectively increases creasing applied stress. Once the stress reaches a certain the coil density in the gel. Thus, the swellable volume of level, this region almost disappears. The results indicate each particle decreases, and the size of the particles in the that the yielding process of a material is time-dependent. gel is eventually reduced. The particles become harder A prolonged creep time can cause failure of the material and less deformable. This may contribute to the in- structure at lower stresses, even at well below the creased uniformity of structure and bonding strength. In conventional yield stress value. The strain generated by addition, the chain entanglements between particles, creep appears to be cumulative, i.e. the rate of structure caused by the dangling ends on the surface of the regeneration is slower than the rate of structure break- particles, increase significantly with increasing concen- down even at very low creeping rate. Thus, the structure tration. This also enhances the particle-particle inter- of a visco-plastic material seems to fail at a certain strain actions. Therefore, with increasing resin concentration, regardless of the level of applied stress. However, the the gel would become harder, tougher and more brittle. time leading to such structure failure and catastrophic Figure 3 shows the pH effect on the yield stress for two flow could be very long when the applied stress is concentrations of Carbopol gel. The yield stress of significantly lower than the conventional yield stress Carbopol gel appears to be very sensitive to pH. It value. The time-dependent behaviour of the yielding increases sharply to its maximum value when the gel is process is directly responsible for the diverse yield stress neutralised. When the carboxyl groups in Carbopol are results obtained using different techniques due to the progressively ionised, the intra-molecular repulsion of different time frames involved in the measurement [20]. the charges extends the coils and results in dramatic In addition, the results presented show that continuous swelling of the Carbopol particles. Hence, the particles flow could exist even at stresses well below the yield become closer packed as neutral pH is approached. In stress, but at an extremely slow shear rate. This shear rate terms of the cellular model [30,31], the yield stress can could be of the order 10-5 to 10-7, which implies a time be defined microscopically as the stress required to move frame of weeks or even months during the creeping the swollen and packed particles past one another in the process. At such a large time scale, it is appropriate to 656 C. Tiu, J. Guo, and P. H. T. Uhlherr

Figure 3. Effect of pH on yield stress for Carbopol 934 gels. Figure 5. Variation of apparent critical strain with applied stresses from creep tests for Carbopol gel.

Time (s) Figure 6. Creep and recovery course of Carbopol gel at a Figure 4. Creep profile of Carbopol gel. stress of 4 Pa. classify the flow as “creeping” rather than “viscous [4,32]. flow”. The difference in the classification of flow has Results obtained from strain recovery measurements caused the continuous debate over the existence of yield can be used to examine the solid-like behaviour of a stress [4,20]. yield stress material. The strain recovery test is carried The yield point of a material at different shear stresses out by applying a constant stress to a sample for a certain can be determined graphically from the creep profiles period of time and then the stress is removed. The entire shown in Figure 4 by extrapolating the two straight lines strain history is recorded. Figures 6～8 show the corresponding to the solid-like and the liquid-like progress of strain build-up (creep) under a constant stress behaviour. The intersection of the two lines represents and strain recovery after the removal of stress for a 0.5 % the yield point, or the critical strain read on the ordinate- Carbopol gel. This gel has a conventional yield stress of axis. Figure 5 plots the variation of the critical strain with 62 Pa, obtained from the vane torsion method. The creep shear stress for the same Carbopol gel. The critical strain generally consists of three regions: instantaneous elastic, appears to be independent of the stress and is of the order retarded elastic and viscous responses, depending on of unity. Different stress levels simply cause catastrophic both the applied stress and the creep time. The strain flow after different times, once the solid structure is recovery undergoes two stages: an instantaneous re- deformed to a certain amount. This seems to suggest that covery followed by a gradual recovery. At very low the structure failure only occurs once the accumulated stresses, the strain can be recovered fully, which is strain reaches a certain value. Therefore, using a critical characteristic of an ideal elastic solid behaviour. At a strain to characterise structure failure or yielding of stress near or beyond the conventional yield stress value, viscoplastic materials may be more appropriate than a there is no strain recovery at all. In between these two yield stress. Using a critical strain to characterise the stresses, there is only a partial recovery of strain that is yielding process has also been suggested elsewhere similar to the behaviour of viscoelastic polymer solu- Yielding Behaviour of Viscoplastic Materials 657

Figure 7. Creep and recovery course of Carbopol gel at a Figure 9. Variation of strain recovery ratio with different total stress of 45 Pa. stains for Carbopol gel.

Figure 8. Creep and recovery course of Carbopol gel at a Figure 10. Determination of yield stress from strain recovery stress of 62 Pa. ration. tions. There is clearly a fundamental difference in the for visco-plastic materials, only a partial recovery can stress-strain response between the visco-plastic material ever be achieved for viscoelastic liquids. and conventional viscoelastic fluid. Two critical values of strain can be identified if a The extent of strain recovery can be expressed by a straight line is used to represent the recovery curve. strain recovery ratio, defined as the ratio of recoverable Strain recovery is expected to be complete if the total strain to total strain. Figure 9 shows the variation of strain in creep is less than the lower critical value; strain recovery ratio with total strain, calculated from the whereas little or no strain recovery is observed if the total results of the creep and strain recovery tests given above. strain exceeds the higher critical value. These two critical It can be observed that the strain recovery ratio is only values can be determined by superimposing the strain dependent on the total strain attained at the end of the recovery curve (Figure 9) with the stress-strain curve creep period, and is independent of the magnitude of the (Figure 1), as shown in Figure 10. Projecting the 0 and applied stress and creep time. The decrease in strain 100 % recovery values onto the stress-strain curve yields recovery ratio from 100 to 0 % represents a change of the two critical stresses, which defines the lower and upper materials’ behaviour from elastic solid to viscous liquid. limits of the stress plateau region. The lower critical For comparison purposes, the recovery behaviour of a stress represents the transition from purely elastic to viscoelastic liquid, (a 0.9 wt% Separan AP-903 in 55.3 plastic behaviour; and the upper critical stress represents wt% glycerine) is also included in the figure. There is a the transition of plastic to viscous flow. The higher value fundamental difference in the strain recovery behaviour of stress is generally in good agreement with the between visco-plastic materials and a viscoelastic liquids. traditional value of yield stress as measured by the vane Whilst a full recovery of strain can always be attained torsion method. The two stresses obtained from the re- 658 C. Tiu, J. Guo, and P. H. T. Uhlherr

Figure 11. Pipkin diagram of dynamic rheological behaviour Figure 12. Comparison of complex and steady of of Carbopol gel (yield stress: σy = 55 Pa). Carbopol gel. covery ratio can be interpreted as the limits of solid-like until it exceeds the critical yield value. In the course of behaviour and liquid-like behaviour, representing the two the experiment, the large amplitude spectral response can yield stresses, static and dynamic [20,33,34]. be obtained as a function of shear strain (or stress). It is useful to construct a measurement map showing the dependence of rheological behaviour of yield stress ma- Large Amplitude Oscillatory Shear Flow terial on operating conditions. The Pipkin diagram re- (LAOS) presents the time-varying response of a complex material as a two-dimensional plot of strain on the ordinate axis Oscillatory shear flow is an appropriate and versatile against frequency on the abscissa [35]. Different charac- experimental tool to study the rheological behaviour of teristic rheological regimes can be identified from the nonlinear materials over a broad range of deformations. diagram; the abscissa corresponds to the linear visco- This is because both the amplitude and the rate of the elastic limit of oscillatory flow with vanishing small imposed sample deformation can be specified indepen- strains, and the ordinate axis corresponds to steady dently. This technique is commonly used under suffi- viscometric deformations of arbitrarily large amplitude. ciently small strain amplitudes to obtain a linear response The Newtonian fluid response corresponds to the single (small amplitude oscillatory shear flow, or SAOS). When point at the origin according to the ‘Ordered Fluid the sample is subjected to a sinusoidal oscillating shear Theories’ [35]. Oscillatory stress-sweeps with increasing strain or shear stress in the linear regime, the output imposed frequency correspond to the iso-stress trajec- signal containing the sample response is also sinusoidal tories through this operating space. Figure 11 shows the but phase-shifted by an amount that depends on the Pipkins diagram obtained from the LAOS test using the viscoelastic nature of the material being tested [35]. controlled-stress mode for the Carbopol gel. It is clearly Small-amplitude oscillatory shear is especially suitable seen that at stresses well below the conventional yield for polymer melts and solutions that generally have a stress (62 Pa), the measured material strain is independent large linear viscoelastic domain. However, complex flui- of frequency. This behaviour is typical for elastic solids. ds such as highly filled suspensions or other complex As the stress approaches the yield value, the strain fluids such as polymer gels that exhibit a critical yield response becomes progressively non-linear. At stresses stress have extremely variable viscoelastic properties significantly above the yield stress, two different regions ranging from elasto-plastic behaviour below the yield can be observed: a purely viscous region with a slope of stress to viscous behaviour beyond that. Thus, the linear -1 on log-log axes at low frequency (in which the viscoelastic domain of yield stress materials is typically response at constant stress corresponds to a constant limited to very low strain, and is not relevant to many shear rate γ ̇≡ γ 0ω) and a nonlinear viscoelastic region industrial applications in which the mechanical load in in the moderate to high frequency range. the material is usually above the linear region [9]. Complex viscosities obtained at different strain am- The large amplitude oscillatory shear (LAOS) tests can plitudes are compared with the steady for the be performed in either a stress-controlled mode or in a Carbopol gel in Figure 12. It can be seen that complex strain-controlled mode. The sample is subjected to a viscosity is dependent not only on frequency, as for harmonic oscillating stress (or strain) excitation of a linear materials, but also on the strain amplitude. constant angular frequency and a maximum amplitude. Obviously, the Cox-Merz Rule is not obeyed for this type The stress (or strain) amplitude is then increased in steps of material. Only when the strain amplitude approaches Yielding Behaviour of Viscoplastic Materials 659

Figure 13. A Lissajous Figure for 0.5 wt% Carbopol Gel at a Figure 14. A Lissajous Figure for 0.5 wt% Carbopol Gel at a Frequency of 0.01 rad/s. Frequency of 100 rad/s.

100 % does the dynamic viscosity curve coincide with of the sinusoidally varying stress wave. the steady viscosity curve. Doraiwamy and coworkers Considering an experiment with an imposed oscillatory [36] proposed a model for yield stress materials which strain, the time-varying input and output signals can be predicts the superimposition of steady viscosity and represented in the following Fourier decomposition: complex viscosity as functions of the shear rate and the effective shear rate respectively. However, the present Input: γ(t)=γ 0sin(ωt) (1) results show that the modified Cox-Merz Rule [36] is only valid at sufficiently high strains. The differences σ γ 0 * ω γ 0 ω δ Output: (t)= ∑|G i ( , )|sin( t+ i) (2) observed at intermediate strains can be explained by the existence of a large domain of plastic strain below the The output is composed of a set of harmonic responses yield stress, as suggested by Mas and Magnin [37]. with their magnitude depending on both the frequency The time-varying shear stress and shear strain data and amplitude of the imposed deformation. In the limit of obtained from the LAOS test can also be plotted directly small deformations, the response signal is sinusoidal and in the form of a Lissajous figure. For a linear viscoelastic all higher harmonics tend to zero. The amplitude of the material, the shape of the resulting trajectory {γ(t), σ(t)} first term is only a function of frequency, independent of is an ellipse with major and minor axes that depend on strain. This is the linear viscoelastic limit and the the in-phase and out-of-phase components of the response signal is linearly related to the input with a complex modulus G* (ω). Non-linear viscoelastic be- magnitude given by the complex modulus haviour is characterised by a deviation from a perfect elliptic form [22,38]. Figures 13 and 14 show the * ω ' ω '' ω 0.5 (3) Lissajous figures for the 0.5 wt% Carbopol gel obtained |G 1( )| = [G 1( )+G 1 ( )] at two frequencies. The rheological evolution of a yield stress material can be followed as it transforms under and the phase angle or loss tangent stress from a linear elastic solid (corresponding to a line δ ω ω ' ω on these axes, with stress perfectly in phase with strain) tan 1( )=G''( )/G ( ) (4) to a linear viscoelastic material (elliptic loop) to a non-linear viscoplastic liquid (distorted elliptic loop). At However, as the applied strain amplitude increases, the low stress amplitudes, the material response is linear sinusoidal response signal is progressively distorted as viscoeslastic, resulting in an elliptical Lissajous figure. shown in the stress versus time curves obtained at ω= 1 As the imposed stress increases in magnitude, the rad/s in Figure15. response becomes increasingly nonlinear. The area The primary signals of shear stress and shear strain enclosed by the Lissajous figure, which corresponds to from LAOS tests can also be transformed into amplitude the dissipated energy, increases significantly when the and phase spectra with respect to frequency using the fast imposed stress exceeds the yield stress. At very large Fourier transform (FFT) algorithm. If the material is stress amplitude, the Lissajous figure becomes almost a deformed in the linear regime, the response signal parallelogram reflecting very large changes in defor- contains no higher harmonics. Non-linear behaviour can mation rate (and total strain) accumulated in the portions be detected by the appearance of higher order odd 660 C. Tiu, J. Guo, and P. H. T. Uhlherr

Figure 15. Variation of stress response with increasing Figure 17. Increase in relative intensity of third harmonic as a applied strain amplitude. function of strain amplitude at different frequencies.

Figure 18. Variation of relative intensity of the strain Figure 16. A typical FT spectra measured for 0.5 wt% amplitude of third harmonics as a function of the imposed Carbopol gel at a frequency of 1 rad/s. stress amplitude. harmonics in the Fourier spectra of the response signals also seen that at small strain amplitudes, increasing the [39]. Figure 16 shows a typical FT amplitude spectrum frequency has little effect on the relative intensity of the of stress response for the 0.5 % Carbopol gel measured at 3rd harmonic, which implies the behaviour of an elastic ω=1 rad/s and γ0 = 8.2 in the controlled-strain mode. It solid. The increasingly elastic behaviour at higher fre- is seen that the main peak or ‘fundamental’ spectrum ap- quency causes the decrease in the relative intensity of pears exactly at the imposed frequency. The higher order high harmonic at large strain amplitude. FT rheology harmonics occur at only the odd spectral frequencies and makes it possible to quantify the degree of non-linearity can be detected up to the 9th order. This is a typical under different oscillatory conditions, and allows the characterisation of the non-linearity of the material. determination of the crossover between linear elastic and The magnitude of each harmonic can also be repre- non-linear visco-elastic regimes. sented by the relative intensity with respect to the As stated earlier, the LAOS test can also be conducted fundamental frequency. Figure 17 shows the increase in in a stress-controlled mode. The development in the the relative intensity of the stress amplitude of the 3rd relative intensity of the strain amplitude of the high harmonics as a function of the imposed strain amplitude harmonics as a function of the imposed stress amplitude at different frequencies. The third harmonic is first is shown in Figure 18. It is seen that the crossover detectable at a strain amplitude as low as γ0 = 0.04 and between linear and non-linear regimes, which marks the becomes increasingly important as the strain amplitude onset of non-linear plastic deformation, occurs at a stress approaches unity, which is the critical strain identified by almost an order of magnitude below the conventional steady-shear experiments. This value represents the tran- yield stress (62 Pa). This finding lends strong support sition in rheological behaviour from linear elastic solid to that the progressive transition of yield stress materials non-linear viscous fluid. Therefore, the yielding response from solid to liquid occurs over a range of stresses, of the material is directly related to the occurrence and which is consistent with the behaviour observed under growth of higher harmonics and should influence their steady shear conditions reported in the previous section. relative intensity in a characteristic and generic way. It is The measurement of storage modulus G' and loss Yielding Behaviour of Viscoplastic Materials 661

Conclusions

This paper is a comprehensive review of a variety of rheological techniques used to explore the yielding be- haviour of a viscoplastic material. The results indicate that the transition of a yield stress material from solid to liquid states occurs over a range of stresses, which is represented by a stress plateau. The slope of the stress plateau reflects the uniformity of the structure, and hence the distribution of bonding strength. Depending on the nature of structure, the yielding process can occur over a Figure 19. Variation of modulus with increase in strain wide or narrow range of stress, which corresponds to amplitude. either a ductile-type or a brittle-type failure. Yield stress is a time-dependent property. The time modulus G'' is commonly used to characterise complex scale for a certain application must be taken into account fluids. However, the traditional definition is limited to in choosing an appropriate technique for measuring yield the linear viscoelastic regime. For the non-linear stress. The yielding process appears to be determined by response to LAOS, the moduli need to be defined for a critical strain rather than a critical stress. The transition each harmonic [40]. The complex modulus of the ith regime can be well defined by the strain recovery ratio. harmonic in Equation 1 can be expressed as: Two critical strain values corresponding to the top and bottom limits in the stress plateau leads to the definition σ of two yield stresses - the static and dynamic yield * i (5) G i = γ 0 stresses. The Large amplitude oscillatory shear test, coupled with Thus, the in-phase and out-of-phase moduli of the ith and the Fourier-Transform analysis is a powerful tool to harmonic are, respectively, study the progressive rheological transition from linear elastic solid-like to non-linear liquid-like behaviour of viscoplastic materials. G ' = G * cos δ (6) i i i The stress and strain amplitudes, together with the phase spectra obtained by the LAOS test can be '' = * δ (7) G i G i sin i conveniently presented in the Pipkin diagram, Lissajous plot and Fourier-Transform spectra. These plots are more Figure 19 shows the moduli of the first harmonic for a functional and informative than those obtained by ' 0.5 wt% Carbopol gel. The in-phase (elastic) modulus G i conventional shear-flow methods in characterising the * is superimposed on the complex modulus G i at small nonlinear materials exhibiting yield stress. strain amplitude. This implies a linear elastic regime and ' therefore represents an elastic modulus. However, G i * Acknowledgment deviates markedly from G i with increasing strain am- plitude. On the other hand, the out-of-phase (viscous) The authors are grateful for financial supports from the modulus '' is relatively small at small strain G i Australian Research Council and the Faculty Grants amplitudes, but ultimately superimposes on * at high G i Scheme of Monash University. strain amplitudes. This regime corresponds to the viscous flow regime. It is interesting to note that the crossover of modulus occurs at a strain amplitude of approximately References unity which is the same as the critical stain identified from the steady creep tests reported above. Hence, the 1. H. A. Barnes and K. Walters, Rheol. Acta, 24, 323 behaviour of moduli obtained by the LAOS tests can also (1985). be used to characterise the yielding process of the 2. I. D. Evans, J. Rheol., 36, 1313 (1992). material. 3. H. A. Barnes, J. Non-Newtonian Fluid Mech., 81, 133 (1999). 4. R. Houwink and H. K. de Decher, , Plas- ticity and Structure of Matter, Cambridge University Press, London (1971). 662 C. Tiu, J. Guo, and P. H. T. Uhlherr

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