(a) Strain-controlled and (b) stress-controlled time sweeps measured under oscillatory conditions that are in the linear regime [circles (amplitudes equal to 1.6% and 3.2 Pa)], yielding regime [squares (amplitudes equal to 6.3% and 12 Pa)], and thinning region [triangles (amplitudes equal to 50% and 30 Pa)] at an applied angular frequency of 1 rad/s. Symbols: (filled) and (open).
Combined stress- and strain-controlled OS response for the (a) soft and (b) solid states to amplitude sweeps. Symbols: (squares), (circles), filled (strain controlled), and unfilled (stress controlled). Lines: —dotted (strain controlled) and dashed (stress controlled).
Comparison between amplitude and time sweeps at three different times. The circles (strain controlled) and squares (stress controlled) are reproductions of the data presented in Fig. 2 for the specific waiting times, while the triangles represent the concatenated data from time sweeps. Upward-pointing triangles come from strain-controlled time sweeps and downward-pointing triangles were collected in controlled-stress tests. Symbols/lines: (closed), (open), and (curve).
Cage modulus (upward-pointing triangles) and storage modulus (downward-pointing triangles) calculated from the strain-controlled time sweep data displayed as a time series. Horizontal lines—storage modulus of the linear regime of the soft (lower) and solid (upper) states. Vertical line—yielding amplitude. Inset: Close-up showing the cage modulus of the solid state close to the critical stress amplitude.
(a)–(c) Strain acquired at the point of maximum stress and (d)–(f) stress at zero instantaneous rate calculated from strain sweeps and strain-controlled time sweeps. Filled circles—soft state strain sweep response. Triangles—solid state strain sweep response. Unfilled circles—static yield strains. Unfilled squares—evolution of strain acquired at maximum stress and stress at zero rate. Vertical line—yielding amplitude. Solid lines are power-law fits (discussed in the text).
Examples of the apparent cage modulus (dashed line) and yield modulus (dotted line) applied to the full stress response to oscillatory strain of amplitude (curve). (b) is a close-up of the box in (a). The amount of strain acquired at the static yield point, B is equal to twice the strain implied by the static yield stress and the apparent cage modulus, A. A large arrow points to the part of the curve where linear elastic behavior ends.
Stress response to step-rate tests taken from Erwin et al. (2010). Rates are indicated at right. Dotted angled lines overlaid make clear the similar power-law stress increase at early times for all but the lowest rates (marked by stars) which are below the transition noted in this Appendix.
Coefficient as a function of shear rate showing remarkably well-defined power-law behavior.
State diagram with the previously defined soft–shear-thinned transition marked by a star. Line: The stress achieved at yielding, giving the same power-law increase in yield strain as seen in oscillatory shearing.
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