Calculator Inputs
Use one consistent stress unit for every tensor component.
Example Data Table
| σx | σy | σz | τxy | τyz | τzx | I1 | Von Mises | Principal stresses |
|---|---|---|---|---|---|---|---|---|
| 120 MPa | 80 MPa | 40 MPa | 25 MPa | 10 MPa | 15 MPa | 240 MPa | 87.464 MPa | 135.361, 68.175, 36.464 MPa |
| 90 MPa | 60 MPa | 30 MPa | 0 MPa | 0 MPa | 0 MPa | 180 MPa | 51.962 MPa | 90, 60, 30 MPa |
Formula Used
Stress tensor:
σ = [ [σx, τxy, τzx], [τxy, σy, τyz], [τzx, τyz, σz] ]
Stress invariants:
I1 = σx + σy + σz
I2 = σxσy + σyσz + σzσx − τxy² − τyz² − τzx²
I3 = det(σ)
Deviatoric measures:
Mean stress = I1 / 3
J2 = 1/6[(σx−σy)² + (σy−σz)² + (σz−σx)²] + τxy² + τyz² + τzx²
J3 = det(σ − mean stress × I)
Equivalent and shear stresses:
Von Mises stress = √(3J2)
Maximum shear stress = (σ1 − σ3) / 2
Tresca equivalent stress = σ1 − σ3
Octahedral shear stress = √(2J2 / 3)
How to Use This Calculator
- Enter the three normal stress components: σx, σy, and σz.
- Enter the three shear stress components: τxy, τyz, and τzx.
- Choose one unit and use it consistently for all inputs.
- Enter an allowable stress if safety factors are needed.
- Select the decimal precision for formatted results.
- Press the calculate button to view results above the form.
- Use the CSV or PDF button to save the calculation report.
Understanding Invariant Stress Calculations
What the Tensor Shows
A three dimensional stress state can look complex. It contains normal stresses and shear stresses. These values depend on the chosen coordinate axes. A rotated part may show different component values. Stress invariants solve this problem. They remain unchanged when the axes rotate.
Why Invariants Matter
Invariants help engineers compare stress states with confidence. The first invariant gives the trace of the tensor. It also relates to mean stress. The second and third invariants describe deeper tensor behavior. They support principal stress calculations and failure checks.
Principal Stress View
Principal stresses are normal stresses on planes where shear stress becomes zero. They are useful in brittle material checks. They also help identify maximum tension and compression. This calculator solves the full symmetric tensor. It then sorts the principal values from largest to smallest.
Deviatoric Stress Measures
Mean stress changes volume. Deviatoric stress changes shape. Many ductile material theories focus on the deviatoric part. The J2 invariant gives the basis for von Mises stress. This value is often compared with yield strength. J3 and the Lode angle add extra detail about the stress path.
Design Use
The calculator also estimates maximum shear stress, Tresca equivalent stress, and octahedral shear. These outputs support early design reviews. They do not replace full finite element validation. Boundary conditions, fatigue, temperature, welds, defects, and standards can change the final decision. Use the exported results as a clear calculation record.
FAQs
1. What is a stress invariant?
A stress invariant is a tensor value that stays the same after coordinate rotation. It helps compare stress states without depending on axis direction.
2. What sign convention does this calculator use?
Positive normal stress means tension. Negative normal stress means compression. Shear stress signs follow the entered symmetric tensor convention.
3. Can I use psi instead of MPa?
Yes. Choose psi, ksi, Pa, kPa, MPa, or GPa. Keep every entered component in the same unit for correct results.
4. What is von Mises stress?
Von Mises stress is an equivalent stress based on J2. It is commonly used for ductile material yield checks.
5. What is the difference between I2 and J2?
I2 belongs to the full stress tensor. J2 belongs to the deviatoric tensor, which removes the hydrostatic part.
6. Why are principal stresses useful?
Principal stresses show the normal stresses on planes with zero shear. They simplify failure checks and stress interpretation.
7. Does this calculator handle plane stress?
Yes. For plane stress, set σz, τyz, and τzx to zero. Then enter σx, σy, and τxy normally.
8. Is this enough for final design approval?
No. It supports stress review, but final approval should consider standards, fatigue, geometry, load cases, and professional engineering judgment.