Understanding 3D Stress Results
A real machine part can carry load in many directions. A 3D stress tensor describes that state with three normal stresses and three shear stresses. This calculator treats the tensor as symmetric. That matches most solid mechanics problems when angular momentum is balanced.
Principal stresses are the normal stresses found on planes where shear stress becomes zero. They are useful because they show the strongest tensile and compressive directions. Engineers often compare the largest principal stress with brittle material limits. They also compare stress differences with ductile yield rules.
Why Invariants Matter
Stress invariants do not change when the coordinate axes rotate. That makes them dependable checks. The first invariant is the trace of the tensor. It gives three times the mean stress. The second and third invariants help define the characteristic equation. Their values lead to the principal stress roots.
Deviatoric invariants remove the hydrostatic part. They describe distortion rather than volume change. J2 is the basis of von Mises stress. J3 helps locate the Lode angle. Together, they explain how close the state is to tension, compression, or shear dominated loading.
Advanced Review
The calculator also reports maximum shear stress. It uses half the difference between the largest and smallest principal values. Octahedral shear is included for comparison. Stress triaxiality is reported when von Mises stress is not zero. It indicates how much mean stress exists relative to distortional stress.
Use consistent units for every input. Do not mix psi with MPa. Positive shear signs should follow your chosen tensor convention. For finite element checks, copy tensor components directly from the same coordinate system. Then compare the sorted principal stresses with your solver output.
Results are rounded only for display. The internal calculation keeps full floating precision. CSV export is helpful for spreadsheets. PDF export is useful for design notes. Always review boundary conditions and load cases before using results for final decisions.
Practical Notes
A repeated principal value can appear in uniform or nearly uniform stress states. In that case, the principal direction may not be unique. A zero shear tensor gives diagonal roots immediately. Large shear terms can rotate the critical planes strongly, even when normal stresses look moderate. Check signs carefully.