Large amplitude oscillatory shear of pseudoplastic and elastoviscoplastic materials

Abstract

We explore the utility of strain-controlled large amplitude oscillatory shear (LAOS) deformation for identifying and characterizing apparent yield stress responses in elastoviscoplastic materials. Our approach emphasizes the visual representation of the LAOS stress response within the framework of Lissajous curves with strain, strain rate, and stress as the coordinate axes, in conjunction with quantitative analysis of the corresponding limit cycle behavior. This approach enables us to explore how the material properties characterizing the yielding response depend on both strain amplitude and frequency of deformation. Canonical constitutive models (including the purely viscous Carreau model and the elastic Bingham model) are used to illustrate the characteristic features of pseudoplastic and elastoplastic material responses under large amplitude oscillatory shear. A new parameter, the perfect plastic dissipation ratio, is introduced for uniquely identifying plastic behavior. Experimental results are presented for two complex fluids, a pseudoplastic shear-thinning xanthan gum solution and an elastoviscoplastic invert-emulsion drilling fluid. The LAOS test protocols and the associated material measures provide a rheological fingerprint of the yielding behavior of a complex fluid that can be compactly represented within the domain of a Pipkin diagram defined by the amplitude and timescale of deformation.

Appendix

Raw data from LAOS experiments

The full LAOS response of the xanthan gum, including transients, is shown in Fig. 13. The Lissajous curves of the raw measured stress vs. strain indicate an initial transient response which quickly settles into steady oscillatory orbits.

Fig. 13

Raw LAOS data for the xanthan gum solution (0.2 wt.% aqueous). Individual orbits are Lissajous curves of normalized stress vs. strain. The initial condition at the beginning of each test is γ = 0, σ = 0. Each individual plot is positioned within a Pipkin space according to the associated LAOS input parameters {ω, γ ₀}. The peak stress within each cycle (including the initial transient) is indicated above the individual curves

The drilling fluid LAOS response is shown in Fig. 14. Lissajous curves of apparent stress vs. rim shear rate are arranged within a Pipkin space according to the LAOS input conditions {ω, γ ₀}. The startup transients settle into steady oscillations, which are used for the quantitative LAOS analysis presented here.

Fig. 14

Experimental LAOS data for the drilling fluid, including transients, shown as normalized Lissajous curves of stress vs. strain. The maximum absolute value of stress is shown above each curve. The test sequence consists of strain-amplitude sweeps (γ ₀ = 0.0056–10) at constant frequency, in the order ω = (15, 4.75, 1.5, 0.475) rad s^− 1. The total number of cycles for each frequency is N = (19, 12, 20, 12) cycles for ω = (15, 4.75, 1.5, 0.475) rad s^− 1, respectively. The approach to the final periodic orbit can be identified visually. The data shown in Fig. 1 correspond to the test shown here at ω = 15 rad s^− 1, γ ₀ = 3.16. The final six steady oscillatory cycles are used for quantitative analysis of the limit cycle behavior

Both materials settle into steady oscillations which have the expected symmetry for shear-symmetric responses, i.e., any curve can be rotated by 180° about an axis out of the page and result in the same steady oscillatory curve. This corresponds to the existence of only odd harmonics in the Chebyshev/Fourier representation, Eq. 6.

Interpolation method used for Pipkin space fingerprints

To improve visualization of trends and contour lines in the rheological fingerprints, the data are interpolated to produce smooth gradients, as in Fig. 15b. The figures presented here have been interpolated to include 100 points per decade in both frequency ω and strain amplitude γ ₀. The collection of experimental data occurs at discrete values within the Pipkin space of {ω, γ ₀}. For example, the drilling fluid data is spaced at two points per decade in frequency ω and four points per decade in strain amplitude γ ₀. The viscoelastic parameters corresponding to these discrete sampling points can be visualized by plots such as Fig. 15a. Color blocks are used for this plot, which are centered about the imposed values of {ω, γ ₀}. The width and height of the block area is determined by the spacing of the data. The finite width and height of each block increases the limits of the plot beyond where data was actually collected, since the blocks are centered over the corresponding {ω, γ ₀} location. Only four strain sweeps (at different fixed frequency) were used to create this fingerprint.

Fig. 15

Rheological fingerprint for the first-harmonic elastic modulus of the drilling fluid. Discrete experimental data points shown in (a) are interpolated to allow for smoothed gradients and contour lines within the Pipkin space (b)