Yield-Stress Measurements

There is no such thing as "the" yield stress

This app is based on an excellent paper¹ by Maureen Dinkgreve and colleagues at U Amsterdam. They took some well-known yield-stress solutions and carefully measured "the" yield-stress σ_y by some of the standard techniques. The values are different. This is no great surprise, but it at least gives us a formal set of results to see what is going on. I leave you to read their analysis. This app simply shows the broad trends of what they found. The six graphs represent four measurement techniques with two ways of showing the data with two different ways (top and bottom) of looking at the data from the top-left and -middle techniques. Where appropriate I have added the sort of levels of noise that one might expect to find in some regions of the plots.

Flow curve

This simply runs a series of measurements of stress σ at different shear rates γ̇ and plots the stress versus strain rate, fitted to the standard Herschel-Bulkley equation:

σ = σ_y+Kγ̇ ⁿ

For convenience the app uses K=1, n=0.5 and illustrates what you would see in a real-world experiment.

The plot below it is a popular way of using data from the same experiment - though is not included in the paper. The "viscosity" is calculated during the shear rate ramp and reaches a peak at the yield stress. My attempts to understand how they calculate this "viscosity" from the raw data have so far failed and the current plot is only a place-holder using some plausible Gaussian curves till I understand better.

Oscillatory

A series of oscillatory experiments at increasing stress gives a plot of G' and G'' versus stress. There are at least three ways of getting σ_y

1. Use the G'/G'' cross-over point. This will give a higher value for σ_y and is sometimes called the Flow-stress rather than Yield-stress.
2. Find the point where the G' value starts to reduce from its elastic value. With real-world data this is somewhat hard to pin down, but the average human eye can do a reasonable job given the uncertainty of what σ_y actually means.
3. Plot (graph underneath the G'/G'' graph) the data as log(σ) versus log(γ) calculated directly from the data used to generate G'/G'' [γ = σ/G* where G*² = G'² + G''²], and define σ_y as the point where the two straight-ish lines intercept.

Stress Growth

At a constant, small, shear rate, the stress versus strain is measured. At the yield stress, the stress remains constant for increased strain.

Compliance Curves

For an ideal elastic solid before yield stress is exceeded, the Compliance, strain/stress, is a constant. So you'd apply an initial stress, σ and see that the compliance, J=γ/σ, is constant over time. You'd then apply a higher stress until at σ_y J would show some steady increase with time. For real-world materials it's not so simple; there is some increase in J over time and it also increases with σ rather than stay constant. However, at σ_y there should be a significant and obvious change in the curve. The curves in the app are highly idealised versions of the best ones shown in the paper. For some materials there is no clear-cut and obvious change (so there's viscoelastic behaviour before and after the yield point).