Techniques in oscillatory shear rheology - Indian Institute of ...

18 Abhijit P. Deshpandeics are observed. Therefore it is essential to establish the terminal periodic responsebefore analyzing the LAOS data [8].Due to small gaps of rotational rheometer geometries, it is assumed that flowis one dimensional with tangential velocities. However, at large amplitudes and atthe edges of the geometries, secondary flows can develop. These may also lead toseeming deviation from linear viscoelastic response or the appearance of higherharmonics. A homogeneous flow is also assumed (with constant velocity gradient,or strain rate) across the depth in the rheometer geometry. During LAOS, the flowmay become inhomogeneous (shear banding; regions of different strain rates) andtherefore, analysis of steady terminal response would be very difficult [17].Even though the presence of even harmonics is ruled out due to material symmetryconsiderations, they have been observed in several cases [4, 7]. These evenharmonicterms can be observed due to transient responses, secondary flows, viscousheating or dynamic wall slip. As an example, they have been shown to arise due tomisalignment of top and bottom geometry [4]. Edge fracture has also been shownto lead to higher harmonics including even ones [17].The intensity of (n+1) th harmonic is usually in the range of 1–10% of the n th harmonic.Therefore, intensities of successive harmonics are very low and it is difficultto ascertain there significance. Due to these experimental issues, LAOS data need tobe examined very closely before physical interpretation of material response.3 Constitutive models and LAOSThe linear viscoelastic models such as Maxwell, Jeffrey’s, standard linear solidand generalized Maxwell model (combination of Maxwell model as described inSect. 1.3 are commonly used to describe the oscillatory shear response [2, 16]. Someof the simplistic phenomenological models for large deformations or nonlinear viscoelasticitycan be considered as extensions of the linear viscoelastic models. Forexample, upper convected Maxwell model is,( )∂σσ+λ∂t + v · ∇σ −(∇v)T · σ − σ · ∇v = η ˙γ , (24)where the partial derivative from Eq. 4 has been replaced with upper convectedderivative (given in the parenthesis in the above equation).The quasi-linear model such as upper convected Maxwell model or Lodgerubber-like liquid and Oldroyd–B model lead to linear shear stress response forSAOS as well as LAOS [5, 21]. Therefore, they predict single harmonic stressresponse, at the same frequency as the strain. However, normal stress differencesare predicted to exhibit second harmonic [5] even at small strains. Additionally, allthese models predict the presence of higher harmonics during the initial stages ofLAOS (before the terminal periodicity is established). These results are not surprisingsince, the quasi-linear models exhibit no shear thinning behaviour. As is ap-