Multiaxial Yield Criteria

Multiaxial Yield Criteria (structures/materials/multiaxial-yield-criteria)

Use when the task is a hand yield check of an isotropic metal part under a multiaxial stress state: von Mises and Tresca equivalent stresses, Mohr-circle principal stresses, yield margins, and the von Mises envelope verdict for the biaxial plane. It pairs with the materials pack siblings around it: ramberg-osgood builds the uniaxial stress-strain curve whose yield point feeds these criteria, and mmpsd-allowables supplies the statistically based yield strength design value that Sy comes from. Anisotropic composite ply strength work (composites pack) and finite-element equivalent stress post-processing (fem pack) are different tasks; this leaf evaluates an isotropic metal stress state by hand with the classical criteria.

Domain quick reference

• von Mises equivalent stress in plane stress (sx, sy, txy): vm = sqrt(sx2 - sx*sy + sy2 + 3*txy**2). Uniaxial tension s reduces it to |s|; pure shear tau reduces it to sqrt(3)*tau.
• von Mises equivalent stress in full 3D (sx, sy, sz, txy, tyz, tzx): vm = sqrt(0.5*((sx-sy)*2 + (sy-sz)*2 + (sz-sx)2) + 3(txy*2 + tyz*2 + tzx*2)).
• Mohr circle plane-stress principals: center (sx+sy)/2, radius sqrt(((sx-sy)/2)2 + txy2), so sigma_1 >= sigma_2 with sigma_1,2 = (sx+sy)/2 +/- sqrt(((sx-sy)/2)2 + txy2).
• Tresca equivalent stress: the maximum principal stress difference. In plane stress the out-of-plane principal is zero, so tresca = max(sigma_1 - sigma_2, sigma_1, -sigma_2): uniaxial tension gives sigma, pure shear gives 2*tau, and the Tresca to von Mises ratio for pure shear is 2/sqrt(3) = 1.1547.
• Yield margin: margin = Sy/equivalent - 1, positive below yield, zero at yield, negative past yield (Sy = tensile yield strength).
• Shaft under combined bending sigma_b and torsion tau: von Mises equivalent sqrt(sigma_b2 + 3*tau2), the classical hand form.
• von Mises yield envelope in the biaxial (sx, sy) plane: sx2 - sx*sy + sy2 <= Sy**2; the boundary counts as within. The pure shear edge of the envelope sits at tau = Sy/sqrt(3).
• For the same plane-stress state the Tresca equivalent never falls below the von Mises equivalent; the ratio reaches 2/sqrt(3) in pure shear.

Workflow

1. Collect the plane-stress state (sigma_x, sigma_y, tau_xy) and the tensile yield strength Sy in one consistent unit (MPa or Pa).
2. Compute the von Mises equivalent stress in plane stress with von_mises_plane_stress, or in full 3D with von_mises_3d when sigma_z and the out-of-plane shears are present.
3. Resolve the plane-stress principal stresses from the Mohr circle with plane_stress_principals (sigma_1 >= sigma_2).
4. Compute the Tresca equivalent stress with tresca_plane_stress, the maximum principal stress difference including the zero out-of-plane principal; use tresca_equivalent(sigma_1, sigma_3) when the ordered extremes are known directly.
5. Form the yield margins with yield_margin on the von Mises and on the Tresca equivalent stress; the governing (lower) margin decides the hand check.
6. Run the von Mises combined bending-plus-torsion margin for a shaft section with combined_bending_torsion_margin.
7. Issue the von Mises yield envelope verdict for the biaxial point with is_within_von_mises_envelope.
8. Confirm every computed number with the contract test scripts/test_multiaxial_yield_criteria.py.

Worked example

Plane stress sx = 200 MPa, sy = -50 MPa, txy = 60 MPa, Sy = 400 MPa:

• von_mises_plane_stress = 251.594913 MPa.
• plane_stress_principals = (213.654246, -63.654246) MPa, sigma_1 at +69.1 deg by the Mohr circle geometry.
• tresca_plane_stress = 277.308492 MPa.
• yield_margin on von Mises = 400/251.594913 - 1 = 0.589857.
• yield_margin on Tresca = 400/277.308492 - 1 = 0.442437.
• is_within_von_mises_envelope(200, -50, 400) = True.
• The Tresca margin governs: 0.442437 versus 0.589857 on von Mises.

Combined bending and torsion sigma_b = 180 MPa, tau = 100 MPa, Sy = 350 MPa:

• von Mises sqrt(1802 + 3*1002) = 249.799920 MPa.
• combined_bending_torsion_margin = 350/249.799920 - 1 = 0.401121.

Pure shear checks: von_mises_plane_stress(0, 0, 100) = 173.205081 MPa = sqrt(3)*100; the envelope edge for pure shear sits at tau = Sy/sqrt(3) = 230.940108 MPa.

Verification

• Confirm the worked anchors: von Mises 251.594913 MPa, principals (213.654246, -63.654246) MPa, Tresca 277.308492 MPa within 1e-4; margins 0.589857 and 0.442437 within 1e-5; envelope verdict True.
• Confirm the combined case 249.799920 MPa and margin 0.401121.
• Confirm the closed-form identities: uniaxial von Mises equals |s|, pure shear von Mises equals sqrt(3)*tau, von_mises_3d with sz = tyz = tzx = 0 equals the plane-stress form exactly, the Tresca equivalent never falls below the von Mises equivalent, the uniaxial point (Sy, 0) sits on the envelope boundary, and the biaxial corner (Sy, -Sy) lies outside.
• Confirm non-physical inputs raise ValueError: a zero or negative equivalent stress, a zero or negative yield strength, and unordered principals in tresca_equivalent.
• Run the contract test offline: python3 scripts/test_multiaxial_yield_criteria.py (35 tests, deterministic).

Related leaves

• structures/materials/ramberg-osgood: the uniaxial stress-strain curve whose 0.2 percent offset yield point feeds Sy here.
• structures/materials/mmpsd-allowables: statistically based design values that set the yield strength Sy used by the margins.
• structures/materials/fracture-toughness: crack instability after the part passes its yield check.
• structures/composites/failure-criteria: anisotropic lamina strength assessment for composite plies, the fence on this leaf.
• structures/fem/calculix-linear: element-basis equivalent stress post-processing of a solved model, not a hand yield check.

Pitfalls

• Mixing units: every stress and the yield strength must share one unit (MPa or Pa); the sqrt form is unit-homogeneous, so a Pa/MPa mix quietly shifts the margin.
• Using the principal difference sigma_1 - sigma_2 alone for Tresca in plane stress: the zero out-of-plane principal matters, and the maximum is over sigma_1 - sigma_2, sigma_1 and -sigma_2, so a uniaxial compression state governs on -sigma_2.
• Assuming the Tresca von Mises ratio is always 2/sqrt(3): that ratio belongs to pure shear; the ratio is not fixed for general biaxial states, so compute both equivalents and take the governing margin.
• Reading a yield margin of zero as a pass: zero sits exactly at yield, so any margin at or below zero fails the hand check.
• Forgetting the -sx*sy cross term in the plane-stress von Mises form: dropping it overstates the equivalent stress for biaxial tension and understates it for tension-compression states.
• Confusing the envelope verdict with the margin: the envelope takes a biaxial (sx, sy) point with no shear and returns a bool, while the margin is a number on an equivalent stress.
• Applying isotropic criteria to a composite lamina: anisotropic strength work belongs to structures/composites/failure-criteria.

Behavior contract (gate 3)

The von Mises, Tresca, principal stress, margin, shaft, and envelope logic is exercised by the gate 3 contract test: scripts/test_multiaxial_yield_criteria.py against scripts/multiaxial_yield_criteria_logic.py (stdlib unittest, offline). Run: python3 scripts/test_multiaxial_yield_criteria.py

Compliance

• Standards referenced, not reproduced: MMPDS (SAE) is proprietary and is named only as the source of statistically based metallic yield strength design values; no tables or values are copied. The von Mises and Tresca yield criteria and the Mohr circle are classical strength-of-materials methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.