3D Principal Stress Simulator — Eigenvalues of the Stress Tensor

This tool solves the eigenvalue problem of the stress tensor sigma_ij to extract the three principal stresses. A principal stress is a normal stress acting along an axis (the principal axis) on which all shear components vanish. Because the stress tensor is symmetric, its three eigenvalues are guaranteed to be real and the corresponding eigenvectors are mutually orthogonal.

$$\sigma = \begin{bmatrix} \sigma_x & \tau_{xy} & 0 \\ \tau_{xy} & \sigma_y & 0 \\ 0 & 0 & \sigma_z \end{bmatrix}$$

This simplified tensor assumes tau_yz = tau_xz = 0, so the z axis is decoupled from in-plane shear. The eigenvalue decomposition splits exactly into the 2x2 in-plane block and the scalar sigma_z. The in-plane eigenvalues are $\lambda_{1,2} = c \pm r$ with centre $c = (\sigma_x+\sigma_y)/2$ and radius $r = \sqrt{((\sigma_x-\sigma_y)/2)^2 + \tau_{xy}^2}$, while sigma_z is always one principal stress by itself.

$$\tau_{\max} = \frac{\sigma_1 - \sigma_3}{2}$$

After sorting $\sigma_1 \geq \sigma_2 \geq \sigma_3$, the maximum shear stress is half the gap between the extreme principal stresses. Geometrically it is the radius of the largest of the three Mohr circles (the sigma1-sigma3 circle) and by Mohr's three-circle theorem no shear stress on any plane can exceed it. tau_max also defines the Tresca yield condition tau_max = sigma_y/2.

$$\sigma_m = \frac{\sigma_1+\sigma_2+\sigma_3}{3} = \frac{I_1}{3}$$

The mean stress (hydrostatic stress) sigma_m equals one third of the first invariant I1 of the stress tensor. Subtracting it gives the deviatoric tensor $s_{ij} = \sigma_{ij} - \sigma_m \delta_{ij}$. Mises plasticity in ductile metals depends only on the deviator, while sigma_m drives elastic volume change (proportional to the bulk modulus K). The stress triaxiality eta = sigma_m / sigma_VM is a critical parameter for ductile fracture and fatigue.

Standard CAE post-processing output: Abaqus, ANSYS, NASTRAN and Marc all expose sigma1, sigma2, sigma3 contours, the maximum shear stress tau_max contour and the Mises equivalent stress as default outputs of a linear-elastic or plastic run. Understanding the descending sort and the three-way relationship between principal stresses, as visualised by this tool, is the literacy needed to read those plots. The maximum principal stress sigma1 is used directly in brittle fracture checks (cast iron, concrete, ceramics), while sigma3 (minimum principal = maximum compression) is the headline number in geotechnics and rock mechanics.

Multiaxial fatigue: Fatigue assessment frequently uses principal-stress-based formulations (maximum principal stress method, Sines, Findley, Crossland). The maximum principal stress method compares sigma1_max with the fatigue limit sigma_w, while Findley searches across principal-axis pairs for the maximum shear stress amplitude combined with the normal stress. Multiaxial fatigue loading such as bending plus torsion or internal pressure plus axial force requires principal stress analysis at each instant of the load history.

Geotechnics and rock mechanics: Soil and rock stress states are expressed in principal stresses by convention. The ratio sigma1/sigma3 (or the combination of sigma1 and sigma3) drives failure via the Mohr-Coulomb or Hoek-Brown criterion. In-situ stress measurements (hydraulic fracturing, stress-relief overcoring) report the full principal stress tensor, and tunnelling or slope stability analyses routinely visualise the principal stress vector field in post-processing.

Stress triaxiality and ductile damage: Stamping and forging simulations use the stress triaxiality eta = sigma_m / sigma_VM as the central parameter for fracture prediction. The LS-DYNA GISSMO model, the Abaqus Ductile Damage model and almost every ductile fracture surface take eta as input. The mean stress sigma_m from this tool, together with the principal stresses, lets you build a quick eta estimate and feel the regimes: eta < 0 (compression-dominated, drawing operations gain ductility), eta ~ 0.33 (uniaxial tension), eta > 1 (brittle-like fracture).

The most common confusion is "principal stress equals Mises equivalent stress." Principal stresses sigma1, sigma2, sigma3 are three eigenvalues of the stress tensor, while the Mises equivalent is a single scalar built from squared differences of principal stresses. Comparing only sigma1 with the yield stress sigma_y is the maximum principal stress theory used for brittle materials. Comparing Mises with sigma_y is the J2 flow rule used for ductile metals. Mixing the two can flip a design from safe to unsafe.

The second pitfall is "plane stress has one Mohr circle, so 3D has one too." Plane stress assumes sigma_z=0, which is itself a hidden principal stress, so two additional unseen Mohr circles exist. The maximum shear stress must be computed as (sigma1-sigma3)/2 across all three principal stresses, not as half the in-plane principal difference (sigma1-sigma2)/2. Forgetting this leads to "the plate surface is elastic" conclusions when in fact the true tau_max criterion is being violated through the thickness.

Third, do not treat "descending sort as a trivial post-processing step." In real CAE the ranks of sigma1, sigma2, sigma3 can swap across load steps or integration points, especially near hydrostatic states where principal stresses are nearly equal. Numerical noise alone can flip the order. Apparent jitter in principal-stress contours often means the rankings are reshuffling, not that the simulation has broken. Sweeping tau_xy in this tool around the sigma_z swap point demonstrates the phenomenon in slow motion.