Data collected from strain sweeps at 1 rad/s. The dynamic moduli are displayed as functions of strain amplitude. Circles—response of the soft state; squares—solid state response. Filled symbols show ; open symbols show . The arrow indicates the strain above which the elastic moduli of the soft and solid states converge (see text).
Waveforms of the response to oscillatory shear at 1 rad/s of the soft state (left panels) and solid state (right panels). (a) Stress response to small amplitudes (1.3%–7.5%); (b) medium amplitudes (10%–56%); and (c) large amplitudes (75%–420%), regardless of initial conditions.
(a) Selected waveforms of the stress response to large-amplitude applied strains highlighting the constant initial elasticity. These waveforms are taken from experiments on the initially solid state, but in this large-amplitude regime, the response of the initially soft state is identical. (b) The apparent cage modulus is plotted on the same set of axes of the data presented in Fig. 1, indicating that elastic processes are present at all strain amplitudes and have the same magnitude, a response that is not accounted for by Fourier-based analyses. Upward-pointing triangles: soft state/downward-pointing triangles: initially solid state. The driving force for solidification is time (on the order of 4000 s) and rejuvenation occurs through LAOS.
A perfectly elastic material (a) acquires strain at the point of maximal stress while a perfectly viscous material (b) acquires only .
(a) Waveforms of the stress response of the soft state to selected strain amplitudes (percentage indicated by numbers) with solid circles indicating the point of maximum stress and unfilled circles indicating the point of maximal elastic stress where the viscous stress exceeds the elastic. (b) Strain acquired at the point of maximum stress (filled symbols) and the strain acquired at the static yield point (unfilled symbols) since reversal of the shear rate. Filled circles: soft state/triangles: solid state.
[(a) and (b)] Stress in the system at the maximum elastic point (open symbols) and moment of zero instantaneous rate (filled symbols), as indicated in (a) for four representative responses of the soft state and in (b) for all amplitude responses of the soft and solid states. Circles: soft state/squares: solid state. Solid lines: stress amplitude reported by the software. Dashed and dotted lines: power-law fits of identical index to the large-amplitude limit of the static and dynamic yield stresses, respectively.
The flowing portion of the oscillatory waveform is highlighted in (a) for three different amplitudes [from inside to out (bowties), 177% (hourglasses), and 420% (triangles)] and plotted as a function of the instantaneous shear rate in (b) on the same axes as the flow curve.
A representative power-law flow response [solid line in (a), corresponding to a strain amplitude of 100% and angular frequency of , raw waveform as dotted line] has a power-law index of 0.34, the Fourier-transform of which (b) contains infinitely many higher harmonics.
Assuming the higher harmonics come only from the power-law fluid response of the material allows a modeling of their relative magnitudes. Points (circles, squares, and triangles) represent measured magnitudes of the third, fifth, and seventh harmonics and the lines come from the theory discussed in the text.
Waveform (a) and state diagram (b) indicating the different responses of the material. At (1), the cages begin deforming. By (2), the cages have weakened and the modulus has dropped and yielding takes place. The material then behaves as indicated by the steady-state flow curve (3) and as the instantaneous rate falls to zero at (4) cages reform and the process begins again.
Typical viscoelastic response constructed with . Phase angles can be calculated from Lissajous–Bowditch curves by taking the inverse sine of the ratio between the strains at which the stress is zero, , marked by circles, and the strain amplitude, , marked by squares. In this illustrative figure, that ratio is the ratio of distances between circles and squares, the inverse sine of which is 45°.
The phase angle between the stress and strain vectors is calculated from the Lissajous–Bowditch curves by assuming linear elastic behavior following the dynamic yield point [indicated by the lower dashed line, in (a)]. The form of the expression resulting from this assumption [shown in (b) as dashed line] matches well the measured values from amplitude sweep experiments on the soft (circles) and solid (squares) states.
The relations found previously for the stress amplitude and dynamic yield stress are used to calculate the dynamic moduli for large strain amplitudes. Symbols are recreations of Fig. 1 and dotted and dashed lines are the expected forms of and , respectively.
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