Enter stress components and plane direction
Use the grid below. Large screens show three columns, smaller screens show two, and mobile shows one.
Sample input cases
| Case | σxx | σyy | σzz | τxy | τyz | τzx | σ1 | σ2 | σ3 | Von Mises |
|---|---|---|---|---|---|---|---|---|---|---|
| Uniaxial tension | 100 | 0 | 0 | 0 | 0 | 0 | 100.000 | 0.000 | 0.000 | 100.000 |
| General 3D state | 120 | 80 | 60 | 35 | 25 | -20 | 140.642 | 91.276 | 28.082 | 97.724 |
| Pure shear in xy | 0 | 0 | 0 | 50 | 0 | 0 | 50.000 | 0.000 | -50.000 | 86.603 |
Core equations behind the calculator
σ = [ [σxx, τxy, τzx], [τxy, σyy, τyz], [τzx, τyz, σzz] ]
The principal stresses are the eigenvalues of the symmetric tensor σ.
I1 = tr(σ)
I2 = σxxσyy + σyyσzz + σzzσxx − τxy² − τyz² − τzx²
I3 = det(σ)
p = I1 / 3
σhyd = pI
s = σ − pI
J2 = 1/2 · sijsij
J3 = det(s)
σvM = √(3J2)
τmax = (σ1 − σ3) / 2
t = σn
σn = n · t
τ = t − σnn
|τ| = shear magnitude on the plane
Steps for accurate calculations
- Enter the three normal stress components.
- Enter the three independent shear components.
- Provide a plane normal direction using nx, ny, and nz.
- Select the preferred units and output precision.
- Press the calculate button.
- Review principal stresses, invariants, and matrix decomposition.
- Inspect the Plotly charts for principal values and Mohr circles.
- Use CSV or PDF buttons to export your result summary.
Frequently asked questions
1. What does this stress tensor calculator compute?
It computes the full symmetric stress tensor, principal stresses, tensor invariants, hydrostatic and deviatoric parts, von Mises stress, maximum shear stress, and stresses acting on a user-selected plane.
2. Why are only six stress inputs required?
A physical Cauchy stress tensor is symmetric for standard continuum mechanics problems. That means τxy = τyx, τyz = τzy, and τzx = τxz, so only six independent components are needed.
3. What are principal stresses?
Principal stresses are the eigenvalues of the stress tensor. They act on planes where shear stress becomes zero, making them essential for failure checks, material comparisons, and Mohr circle interpretation.
4. What is the difference between hydrostatic and deviatoric stress?
Hydrostatic stress represents the mean pressure part of the tensor. Deviatoric stress measures distortion-producing effects. Their separation helps explain yielding, shape change, and pressure-related behavior in solids and fluids.
5. When is von Mises stress useful?
Von Mises stress is widely used for ductile material assessment. It converts a multi-axial stress state into one equivalent scalar value that engineers often compare with yield strength.
6. What does the selected plane normal do?
The plane normal defines an orientation in space. The calculator uses it to find the traction vector on that plane, plus the normal stress and shear magnitude acting there.
7. Do units affect the mathematics?
No. Units only label the values. The numerical relationships stay consistent as long as all entered stress components use the same unit system.
8. What do the Mohr circles show?
The Mohr circles show normal stress on the horizontal axis and shear stress on the vertical axis. They reveal principal stresses, stress ranges, and maximum shear values visually.