Calculator
Formula used
A flow curve relates shear stress (τ) to shear rate (γ̇). This calculator estimates yield stress (τy) by fitting a constitutive model to your measured data.
1) Bingham Plastic
τ = τy + μp γ̇
- τy is the yield stress (intercept).
- μp is the plastic viscosity (slope).
2) Herschel–Bulkley
τ = τy + K γ̇n
- K is the consistency index.
- n is the flow behavior index (shear-thinning if n<1).
Fit metrics shown: R² and RMSE. Residual = measured − predicted.
How to use this calculator
- Collect a flow curve: shear rate (γ̇) and shear stress (τ) pairs.
- Select a model: Bingham for near-linear curves, Herschel–Bulkley for curved behavior.
- Paste your data with one pair per line (spaces, tabs, commas, or semicolons).
- Press Calculate. Results appear above the form.
- Use residuals and R² to judge fit quality, then download CSV or PDF.
Example data table
| Shear rate (1/s) | Shear stress (Pa) |
|---|---|
| 10 | 120 |
| 20 | 160 |
| 50 | 260 |
| 100 | 380 |
| 200 | 560 |
Tip: click “Load example data” to paste these pairs automatically.
Professional guide to yield stress from flow curves
1) Why a flow curve can reveal yield stress
A flow curve maps shear stress (τ) versus shear rate (γ̇). For many suspensions, pastes, and gels, the curve does not extrapolate to zero stress at low shear. The nonzero stress level is often treated as an apparent yield stress, meaning the stress required to initiate steady flow under the chosen test conditions.
2) Selecting the best model for your material
Use the Bingham option when your data are close to linear over the measured shear-rate window. Choose Herschel–Bulkley when the slope changes with shear rate (curvature on a τ vs γ̇ plot). Comparing R², RMSE, and residual patterns helps you select the most defensible model.
3) Recommended data range and sampling density
Yield stress estimates improve when you sample a broad shear-rate span rather than a tight cluster. Many rotational rheometers cover roughly 0.1 to 1000 s−1, but your setup may differ. Aim for at least 8–15 well-spaced points, repeat measurements when possible, and exclude obvious outliers caused by slip or bubbles.
4) Units, scaling, and consistent labeling
The fit uses your numeric pairs exactly as entered, so keep the stress unit consistent across all rows (Pa, kPa, MPa, or psi) and keep the shear-rate unit consistent (commonly s−1). If you convert units, convert every data point. Unit mismatches can inflate RMSE and shift the estimated yield stress.
5) Understanding fit quality metrics
R² summarizes how much of the stress variation the model explains, while RMSE reflects the typical prediction error on the stress axis. Always judge RMSE relative to your stress scale. For example, an RMSE of 2 Pa may be minor for stresses near 200 Pa, but significant if your stresses are only about 10 Pa.
6) Residuals tell you what the metrics hide
Residuals (measured minus predicted) should scatter around zero without a clear trend. Systematic patterns—such as positive residuals at low shear and negative at high shear—often indicate the wrong model form, a structural transition, or a limited shear-rate window. Residual tables make your conclusions easier to audit.
7) Reporting yield stress with traceable context
Yield stress depends on temperature, time history, preshear, and geometry, so report test conditions alongside τy. Record the shear-rate range used for fitting and whether you used an up-curve, down-curve, or averaged curve. Keeping the parameter set, R², RMSE, and residuals supports repeatability and comparison across batches.
8) Common pitfalls and how to avoid them
Wall slip and poor trimming can bias low-shear points, artificially increasing the fitted yield stress. If your first points look inconsistent, repeat the low-rate region with a roughened geometry or a smaller gap. Avoid mixing ramp and steady-state points in one dataset, and prefer the model that stays stable when you remove one or two points.
FAQs
1) What input format does the calculator accept?
Enter one pair per line as shear rate then shear stress. Separate values with spaces, tabs, commas, or semicolons. Lines starting with # or // are ignored.
2) When should I use the Bingham model?
Use Bingham when your flow curve is approximately linear in τ versus γ̇ over the measured range. It provides a yield stress (intercept) and plastic viscosity (slope).
3) When is Herschel–Bulkley a better choice?
Choose Herschel–Bulkley when the curve is nonlinear and the apparent viscosity changes with shear rate. It captures shear-thinning or shear-thickening behavior through the flow index n.
4) Why does the tool need positive stresses for Herschel–Bulkley?
The fitting step uses logarithms of (τ − τy) and γ̇. Logarithms require positive values, so each point must satisfy τ > τy and γ̇ > 0.
5) What does a negative yield stress estimate mean?
A negative estimate usually indicates the data pass near the origin or low-stress values are noisy. In many reports it is interpreted as no measurable yield within the tested range.
6) How many data points should I use?
Two points are enough for a linear fit, but more points are strongly recommended. For stable yield stress, target at least 8–15 points spanning low to high shear rates.
7) How do I decide if the fit is acceptable?
Use R² and RMSE together, then inspect residuals for trends. A good fit shows small RMSE relative to your stress scale and residuals scattered around zero.