| Mode | Inputs | Mean p | von Mises | ||s|| |
|---|---|---|---|---|
| Tensor | σx=120, σy=60, σz=30, τxy=25, τyz=10, τzx=15 | 70.000000 | 101.242284 | 82.649556 |
| Principal | σ1=150, σ2=80, σ3=20 | 83.333333 | 112.249722 | 91.612700 |
The Cauchy stress tensor is decomposed into hydrostatic and deviatoric parts: σ = pI + s, where p = (σx + σy + σz)/3 is the mean stress and s = σ − pI is the deviatoric stress tensor.
The second deviatoric invariant is computed as J2 = 1/2 · (s:s), with s:s = sxx² + syy² + szz² + 2(sxy² + syz² + szx²).
Common measures derived from J2 are ||s|| = √(s:s) and the von Mises equivalent stress σvm = √(3J2). For principal stresses, σvm = √( ( (σ1−σ2)² + (σ2−σ3)² + (σ3−σ1)² ) / 2 ).
- Select an input method: full stress tensor or principal stresses.
- Choose the unit that matches your data, then enter values.
- Press Calculate to display results above the form.
- Review mean stress, deviatoric components, invariants, and equivalent stress.
- Use Download CSV or Download PDF after computing.
1) Why deviatoric stress matters
Many failures are controlled by shape‑changing (distortional) stress rather than purely volumetric loading. The deviatoric stress tensor isolates that distortional part by removing the hydrostatic mean stress, p. This is why metal yielding, ductile fracture, and plastic flow models often depend on deviatoric measures.
2) Mean stress versus shear‑driven response
Mean stress, p = (σx+σy+σz)/3, represents uniform compression or tension. It strongly influences void growth, cavitation, and pore pressure effects, but it does not create distortion by itself. Deviatoric components capture the directional imbalance that drives slip and shear bands.
3) Using full tensor inputs in engineering
When you have normal and shear components from strain‑gauges, lab tests, or finite‑element output, the tensor route is the most general. The calculator forms s = σ − pI, then evaluates invariants and norms using s:s with the correct shear weighting.
4) Principal stresses for quick assessments
If you already computed principal stresses (σ1, σ2, σ3), deviatoric values are immediate: subtract the mean to get the deviatoric principals. The equivalent stress check based on pairwise differences helps confirm consistency when principal values come from an external tool.
5) J2, von Mises, and common reporting
The second deviatoric invariant, J2, is widely used because it is objective and smooth. The von Mises equivalent stress σvm = √(3J2) is typically compared to yield strength. For example, structural steels often report yield around 250–550 MPa, while many aluminum alloys are near 150–500 MPa.
6) Octahedral shear as an interpretation aid
In principal form, octahedral shear τoct = (√2/3)σvm provides an intuitive “average shear” level across planes. In geomechanics, it can complement mean stress when interpreting triaxial tests and shear failure envelopes.
7) Practical unit and magnitude checks
Keep units consistent across inputs and outputs. If you enter values in MPa, all results stay in MPa, while invariants appear as squared or cubed units. As a sanity check, pure hydrostatic loading gives near‑zero deviatoric values, while large shears or unequal normals increase σvm noticeably.
8) How to use results in workflows
Export CSV for reports or batch post‑processing, and export PDF for quick sharing. In simulation pipelines, store mean stress and σvm together because damage criteria may depend on both. When comparing cases, report the input mode, units, and the computed invariant set for traceability and auditability.
1) What is deviatoric stress?
It is the stress component that causes distortion without changing volume. It equals the total stress tensor minus the mean stress times the identity tensor.
2) What does the mean stress represent?
Mean stress p is the average of the three normal stresses. It represents the hydrostatic part of stress, linked to uniform compression or tension and volumetric response.
3) Why is von Mises related to yielding?
For many ductile metals, yielding correlates with distortional energy rather than hydrostatic pressure. The von Mises stress is a scalar derived from J2 that tracks that distortional intensity.
4) Do shear stresses affect J2 and von Mises?
Yes. Shear components contribute through the double‑contraction term s:s, with a factor of two for off‑diagonal terms, increasing J2 and therefore increasing the von Mises equivalent stress.
5) When should I use principal stresses instead of tensor inputs?
Use principal stresses when they are already available from Mohr’s circle, a solver, or a test report. It is faster and avoids needing shear components in a specific coordinate system.
6) What should deviatoric stress be under pure hydrostatic loading?
It should be approximately zero, aside from rounding. Equal normal stresses with zero shear produce no distortion, so deviatoric components vanish and von Mises approaches zero.
7) Why do invariants show squared or cubed units?
J2 is built from products of stress components, so it carries stress squared. J3 is a determinant‑type measure of the deviatoric tensor, so it carries stress cubed.