Stress Triaxiality Calculator

Calculator Inputs

Input mode Principal stresses (σ1, σ2, σ3) Stress tensor components (σx, σy, σz, τxy, τyz, τzx)

Choose direct principal inputs or compute them from a tensor.

Stress unit PakPaMPaGPapsi

All inputs and outputs use the selected unit.

Decimal precision 2 decimals3 decimals4 decimals5 decimals6 decimals7 decimals8 decimals9 decimals10 decimals

Affects display and downloads.

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Principal stresses

σ1

Maximum principal stress (tension positive).

σ2

σ3

Minimum principal stress.

Stress tensor components

σx

σy

σz

τxy

τyz

τzx

Tensor is assumed symmetric: τxy=τyx, τyz=τzy, τzx=τxz.

Calculate Reset

Tip: If your state is nearly hydrostatic, σ_eq → 0 and η becomes undefined.

Stress triaxiality is defined as the ratio of mean (hydrostatic) stress to von Mises equivalent stress:

η = σm / σeq

Mean stress (tension positive):

σm = (σ1 + σ2 + σ3) / 3

Von Mises stress from principal stresses:

σeq = sqrt( ((σ1−σ2)² + (σ2−σ3)² + (σ3−σ1)²) / 2 )

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In tensor mode, the calculator first finds the principal stresses from the symmetric 3×3 stress tensor and then applies the same equations.

1. Select an input mode: principal stresses or stress tensor components.
2. Choose the unit you want to work in (e.g., MPa).
3. Enter stresses (tension positive, compression negative if applicable).
4. Click Calculate to view results above the form.
5. Use Download CSV or Download PDF after a successful calculation.

1) Why stress triaxiality matters

Stress triaxiality (η) tells you how hydrostatic stress compares with shear intensity. In ductile metals, higher positive η usually promotes void growth and earlier fracture, while low η is more shear dominated. Because η is dimensionless, it is easy to compare across tests and simulations.

2) Mean stress versus deviatoric stress

The mean stress σ_m is the “pressure-like” part of the state, while von Mises stress σ_eq measures distortional loading. Pure shear gives σ_m ≈ 0 and η near zero. A purely hydrostatic state makes σ_eq = 0, so η is undefined.

3) Typical ranges seen in lab data

Uniaxial tension commonly produces η ≈ 1/3. Plane strain tension often increases η toward about 2/3, and sharp notches can push η above 1. Compression can yield negative η when σ_m becomes compressive. These benchmarks help validate inputs quickly.

4) Role in ductile damage and fracture modeling

Many ductile damage models use η to scale strain-to-failure or to evolve porosity. Higher η tends to accelerate void growth, while low η supports shear localization. Reporting η together with σ_eq improves calibration because it separates hydrostatic effects from overall stress magnitude.

5) Using tensor inputs in simulations

Finite element outputs often provide σx, σy, σz and shear stresses. In tensor mode, this tool computes principal stresses from the symmetric 3×3 tensor and then evaluates η from those principals. Check that your shear terms match the symmetry assumption and your sign convention is consistent.

6) Why the Lode outputs matter

Two states can share the same η but have different deviatoric stress shapes, which can change fracture behavior. The Lode angle and normalized Lode parameter incorporate the third invariant (via J2 and J3). They help distinguish axisymmetric tension from shear-like states at similar η.

7) Units, scaling, and numerical sensitivity

Units only scale σ_m, σ_eq, and the invariants; η stays unitless. Near-hydrostatic conditions can make σ_eq very small, so η may fluctuate. Treat η as qualitative when σ_eq approaches zero, and verify principal stress ordering when results look odd.

8) Reporting and quality checks

Use CSV or PDF exports to capture σ_m, σ_eq, η, J2, J3, and Lode metrics in one place. σ_eq should be nonnegative. If you add tensile hydrostatic stress while keeping shear similar, η should increase. Re-check tensor symmetry if checks fail.

1) What does stress triaxiality represent physically?

It compares hydrostatic stress to shear-driven deformation. Higher positive values indicate more tensile “pressure,” which often increases void growth and fracture risk in ductile materials.

2) Why is η sometimes shown as undefined?

If von Mises stress is zero, the state is purely hydrostatic. Dividing by zero makes η undefined, so the tool flags that condition instead of returning a misleading number.

3) Can η be negative?

Yes. If the mean stress is compressive while shear exists, σ_m becomes negative and η follows. Negative η often indicates suppression of void growth under compression.

4) Which stresses should I enter as σ1, σ2, and σ3?

Use principal stresses from your analysis. If you only have tensor components, switch to tensor mode and the calculator will compute principal stresses internally.

5) How do I interpret η near zero?

η near zero suggests shear-dominated loading. This is common in torsion, simple shear, and some mixed-mode cases where hydrostatic stress is small compared with deviatoric stress.

6) What do the Lode outputs add beyond η?

Lode metrics capture third-invariant effects. They help distinguish different deviatoric stress shapes that can produce different ductile fracture behavior even when η is similar.

7) What is a quick sanity check for my inputs?

Try a simple uniaxial tension case: σ1 = S, σ2 = 0, σ3 = 0. You should get η close to 1/3 and σ_eq close to S.