Mohr's Circle for Plane Stress mechanism

Model scope

The numerical formulas for principal stresses and the in-plane Mohr-circle radius are correct. The specification needs explicit stress units, angle units, shear-sign convention, principal-angle branch handling, zero-stress handling, and an honest distinction between in-plane maximum shear and absolute 3D maximum shear. The blank honest_label is a concern because these conventions materially affect interpretation.

• The formulas are standard for a 2D plane-stress tensor when positive τxy is defined consistently with the stress-transformation convention σx'=σavg+((σx-σy)/2)cos(2θ)+τxy sin(2θ). The specification does not state that sign convention.
• θp=½atan2(2τxy,σx-σy) gives one principal-plane orientation; the complete family is θp+n·90° for integer n, and the other principal direction is θp+90°. The displayed formula alone does not specify this branch/family.
• τmax=R is the maximum in-plane shear stress. If the instrument claims absolute 3D maximum shear for plane stress, σz=0 must also be considered, so the result may instead be max(|σ1-σ2|,|σ1|,|σ2|)/2.
• The isotropic case σx=σy and τxy=0 makes atan2(0,0) undefined and the principal orientation non-unique; the instrument needs explicit handling.

Units and conventions

• σx, σy, and τxy must use the same stress unit; the outputs σ1, σ2, and τmax use that same unit.
• The specification does not state whether θp is returned in degrees or radians. Test vectors below use degrees.
• The plotted Mohr-circle axes use stress units: horizontal coordinate σ and vertical coordinate τ.
• The angle convention assumed here is counterclockwise rotation from the original x-plane to the principal x'-plane, with positive τxy following the stated stress-transformation equation.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "sigma_x": 80,
    "sigma_y": 20,
    "tau_xy": 30,
    "stress_unit": "MPa",
    "angle_unit": "degrees"
  },
  "expected_outputs": {
    "sigma_avg": "50 MPa",
    "R": "42.4264069 MPa",
    "sigma_1": "92.4264069 MPa",
    "sigma_2": "7.57359310 MPa",
    "tau_max_in_plane": "42.4264069 MPa",
    "theta_p": "22.5 degrees modulo 90 degrees",
    "circle_center": "(50, 0) MPa",
    "sampled_circle_points": [
      {
        "phi": "0 degrees",
        "sigma": "92.4264069 MPa",
        "tau": "0 MPa"
      },
      {
        "phi": "90 degrees",
        "sigma": "50 MPa",
        "tau": "42.4264069 MPa"
      },
      {
        "phi": "180 degrees",
        "sigma": "7.57359310 MPa",
        "tau": "0 MPa"
      },
      {
        "phi": "270 degrees",
        "sigma": "50 MPa",
        "tau": "-42.4264069 MPa"
      }
    ],
    "curve_features": {
      "sigma_axis_intersections": [
        "(92.4264069, 0) MPa",
        "(7.57359310, 0) MPa"
      ],
      "tau_extrema": [
        "(50, 42.4264069) MPa",
        "(50, -42.4264069) MPa"
      ],
      "center": "(50,0) MPa",
      "radius": "42.4264069 MPa"
    }
  },
  "working": "σavg=(80+20)/2=100/2=50 MPa. Half-difference=(80-20)/2=60/2=30 MPa. R=√(30²+30²)=√(900+900)=√1800=42.4264069 MPa. σ1=50+42.4264069=92.4264069 MPa. σ2=50-42.4264069=7.57359310 MPa. τmax=R=42.4264069 MPa. θp=½ atan2(2·30,80-20)=½ atan2(60,60)=½·45°=22.5°, with the second principal direction at 112.5°. The plotted circle is (σ-50)²+τ²=42.4264069²; evaluating φ=0°,90°,180°,270° gives the four listed points."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The all-zero stress state produces valid zero principal stresses and zero maximum shear, but theta_p requires atan2(0,0), which is undefined. The instrument must not silently return a numeric orientation for this input.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json