Bingham Plastic Yield Stress Calculator

The Bingham plastic model assumes a material needs a threshold stress before it flows. It can be written as:

τ = τy + μp γ̇

• τ is shear stress.
• τy is yield stress.
• μp is plastic viscosity.
• γ̇ is shear rate (s⁻¹).

One-point: τy = τ − μp γ̇. Two-point: μp = (τ2−τ1)/(γ̇2−γ̇1), then τy = τ1 − μp γ̇1.

1. Select a calculation mode that matches your data.
2. Enter shear stress values and choose the correct unit.
3. Enter shear rates in s⁻¹ from your measurement setup.
4. For one-point mode, enter plastic viscosity and its unit.
5. Press Calculate to show results above the form.
6. Use CSV or PDF buttons for exporting your results.

Sample values for a fluid measured at 25°C.

1. Why Bingham behavior matters

Many concentrated suspensions behave like a solid until a minimum stress is applied. The Bingham model captures that “no-flow then flow” transition with two parameters: yield stress and plastic viscosity. This calculator helps translate rheometer readings into engineering values you can compare, report, and reuse in design work.

2. Yield stress and start-up flow

Yield stress τ_y represents the threshold needed to initiate continuous shear. Below τ_y, the material may only deform elastically or creep very slowly. Once τ exceeds τ_y, the extra stress drives steady flow. In pipes, τ_y influences plug formation, restart pressure, and minimum pump requirements.

3. Plastic viscosity versus apparent viscosity

Plastic viscosity μ_p is the slope of τ versus shear rate in the linear region. Apparent viscosity, by contrast, is τ/γ̇ and changes with γ̇. For Bingham fluids, apparent viscosity drops as shear rate rises because τ_y becomes less dominant. Reporting both τ_y and μ_p gives a clearer material signature.

4. One-point estimation: when it is appropriate

One-point estimation is useful when μ_p is known from prior calibration, a datasheet, or a separate fit. You provide one measured shear stress and shear rate, and the calculator returns τ_y from τ_y = τ − μ_pγ̇. This method is fast for routine checks and quality-control sampling.

5. Two-point estimation: solving both parameters

With two measurements (τ₁, γ̇₁) and (τ₂, γ̇₂), the calculator computes μ_p as the slope (τ₂ − τ₁)/(γ̇₂ − γ̇₁) and then back-calculates τ_y. Choose points within the same linear regime. If your curve is strongly curved, consider fitting more points externally and using the one-point mode for τ_y validation.

6. Typical scales and unit handling

Yield stress can range from near 0 Pa for weakly structured fluids to hundreds of Pa for dense slurries. Plastic viscosity commonly spans from a few mPa·s (thin dispersions) to several Pa·s (heavy pastes). This tool converts between Pa, kPa, MPa, and psi, plus Pa·s, mPa·s, and cP, keeping your calculations consistent.

7. Practical data-quality checks

Stable temperature, steady shear, and repeatable geometry are essential. If τ_y comes out negative, it usually means the chosen point(s) sit below the linear Bingham region or μ_p is overestimated. Many labs clamp negative τ_y to zero, but you should also review measurement selection and instrument settings.

8. Engineering applications and reporting

Engineers use τ_y and μ_p to approximate start-up pressure, plug size, and energy demand in transport and mixing. Typical sectors include drilling fluids, wastewater sludge, mineral slurries, food pastes, and cementitious materials. Exporting CSV and PDF outputs supports lab notebooks, audits, and consistent specification sheets.

1) What is a Bingham plastic fluid?

A Bingham plastic behaves like a solid below a yield stress, then flows with an approximately linear stress–shear-rate relationship above that threshold.

2) What inputs do I need for one-point mode?

You need one measured shear stress, the corresponding shear rate, and a known plastic viscosity. The calculator then returns yield stress using the Bingham relation.

3) When should I use two-point mode?

Use it when you have two stress–shear-rate measurements in the same linear regime and want both yield stress and plastic viscosity from those points.

4) Why can the computed yield stress be negative?

Negative values typically indicate non-linear data selection, noise, or an overestimated plastic viscosity. Consider choosing higher shear-rate points or fitting more data.

5) What does plastic viscosity represent physically?

Plastic viscosity is the slope of the stress versus shear-rate line above yield. It reflects how much additional stress is needed for each increase in shear rate.

6) Do unit conversions affect the result?

Only if units are inconsistent. This tool converts stress and viscosity to internal base units before calculating, then converts results to your chosen output units.

7) Is this model valid for all non-Newtonian fluids?

No. Many materials are shear-thinning or time-dependent. Use the Bingham model when your stress–shear-rate data is approximately linear above a threshold.