Von Mises Criterion

The problem the criterion solves

A uniaxial tensile test is the simplest experiment in solid mechanics: pull a dog-bone specimen along its axis and read off the stress σ at which it stops returning to its original length. That number — the yield strength σ_y — sits in every materials handbook. For mild steel it is around 250 MPa; for 6061-T6 aluminium 276 MPa; for Ti-6Al-4V about 880 MPa.

But a real component rarely sees pure uniaxial tension. A pressure-vessel wall is in biaxial tension. A shaft sees combined torsion and bending. A spot-weld nugget is in something close to triaxial compression. The stress state at every point in a real part is a six-component symmetric tensor:

σ = ⎡ σ_xx  τ_xy  τ_xz ⎤
    ⎢ τ_xy  σ_yy  τ_yz ⎥
    ⎣ τ_xz  τ_yz  σ_zz ⎦

The yield strength σ_y is one number. The stress tensor is six. The yield criterion is whatever rule converts the six numbers into a one-number comparison against σ_y. Von Mises proposed in 1913 to use what we now call the second deviatoric invariant J₂, scaled to recover σ_y at uniaxial tension. That scaled scalar — the von Mises stress σ_VM — is the answer the criterion produces.

The formula, three ways

The same criterion can be written in three equivalent forms — pick whichever matches the data you have.

In principal stresses (σ₁, σ₂, σ₃):
  σ_VM = √( ½[ (σ₁−σ₂)² + (σ₂−σ₃)² + (σ₁−σ₃)² ] )

In Cartesian components:
  σ_VM = √( ½[ (σ_xx−σ_yy)² + (σ_yy−σ_zz)² + (σ_zz−σ_xx)²
              + 6(τ_xy² + τ_yz² + τ_zx²) ] )

In stress invariants:
  σ_VM = √(3 J₂)     where J₂ = ½ s_ij s_ij
                     and s_ij = σ_ij − (σ_kk/3) δ_ij     (deviatoric stress)

Yield condition:
  σ_VM ≥ σ_y   ⇒   plastic flow begins

All three forms give the same number. The invariant form is the elegant one: J₂ is invariant under rotation, so σ_VM does not care which axes you use. The Cartesian form is the one your FEA solver reports per element. The principal form is the one used to draw the yield surface in (σ₁, σ₂, σ₃) space.

The distortion-energy interpretation

Why does this particular combination of differences predict yielding? Because, as Heinrich Hencky pointed out in 1924, it is proportional to the strain energy associated with shape change as opposed to volume change.

Any stress tensor can be uniquely split into

σ_ij = p δ_ij + s_ij     where p = σ_kk / 3   (mean / hydrostatic stress)
                          and s_ij is the deviatoric stress

The hydrostatic part p δ_ij changes the volume of an isotropic linear-elastic element without changing its shape; the deviatoric part s_ij changes its shape without changing its volume. In a linear-elastic material the strain energy density splits cleanly too:

U_total = U_volume + U_distortion
       = p² / (2K)       +    s_ij s_ij / (4G)
       = p² / (2K)       +    J₂ / (2G)

where K is the bulk modulus and G the shear modulus. The von Mises criterion is exactly the statement that yielding begins when U_distortion reaches the value it has at uniaxial tensile yield (J₂ = σ_y²/3, hence σ_VM = σ_y). The hydrostatic energy U_volume is irrelevant: a metal under pure hydrostatic pressure — even tens of kilobars — does not yield. Bridgman's high-pressure metallurgy experiments in the 1940s confirmed this directly for every ductile metal tested, which is the empirical reason the criterion works.

Geometry of the yield surface

Plot the principal stresses (σ₁, σ₂, σ₃) as Cartesian axes. The set of stress states for which σ_VM = σ_y forms a yield surface — anything inside is elastic, anything on the surface is at incipient yielding, anything outside is plastically flowing. Because von Mises depends only on differences (σ_i − σ_j), the surface is unchanged when all three principal stresses shift by the same amount. That direction — the line σ₁ = σ₂ = σ₃ — is the hydrostatic axis, and the yield surface is invariant along it.

The surface is therefore a prism. Its cross-section, taken perpendicular to the hydrostatic axis in the so-called π-plane, is a circle of radius √(⅔) σ_y. Extruded, the surface is an infinite circular cylinder of that radius, tilted so its axis lies along the (1,1,1) direction of principal-stress space. Move any state along the hydrostatic axis and you do not change σ_VM; move perpendicular to that axis and σ_VM grows linearly with distance. The cylinder is the geometric statement of "pressure does not cause yielding".

Real engineering problems are often essentially two-dimensional — pressure-vessel walls, thin plates, sheet-metal stamping. There σ₃ = 0 is a good approximation. Slice the cylinder with the plane σ₃ = 0 and the cross-section is the famous plane-stress ellipse:

σ₁² − σ₁ σ₂ + σ₂² = σ_y²

The major axis runs along σ₁ = σ₂ (where the criterion is most lenient, with σ_VM = |σ₁| at uniaxial-equivalent loading), and the minor axis along σ₁ = −σ₂ (pure shear, where the criterion is most restrictive — σ_y/√3 ≈ 0.577 σ_y is enough to start yielding). The ellipse is the diagram every mechanical-design textbook shows.

Versus Tresca, side by side

Henri Tresca proposed a competing criterion in 1864, two generations earlier than von Mises. Tresca's idea was that yielding starts when the maximum shear stress on any plane reaches a critical value — empirically, σ_y/2 (since a uniaxial tensile test puts maximum shear at 45° equal to σ/2).

Tresca:    τ_max = (σ_max − σ_min) / 2 = σ_y / 2
       ⇔   σ_max − σ_min = σ_y     (in principal-stress terms)

In (σ₁, σ₂, σ₃) space this surface is a hexagonal prism along the hydrostatic axis. The von Mises cylinder circumscribes that hexagon — the two surfaces touch at six points corresponding to uniaxial tension, uniaxial compression and pure shear along each principal direction; between those points, Tresca lies inside, predicting yielding sooner.

State	σ_VM = σ_y at	Tresca yield at	Tresca more conservative by
Uniaxial tension	σ₁ = σ_y	σ₁ = σ_y	0 %
Equibiaxial tension	σ₁ = σ₂ = σ_y	σ₁ = σ₂ = σ_y	0 %
Plane strain	σ_y	2σ_y / √3 ≈ 1.155 σ_y	15.5 %
Pure shear	τ_y = σ_y / √3 ≈ 0.577 σ_y	τ_y = σ_y / 2 = 0.500 σ_y	15.5 %

Tabulated tensile-yield experiments on annealed copper, mild steel and aluminium (Taylor & Quinney 1931; Lode 1925) sit consistently closer to the von Mises locus than to Tresca, with deviations ≲ 5%. Tresca is still preferred in some pressure-vessel codes (ASME Section VIII Div. 1) on conservative-design grounds, but FEA solvers, ABAQUS, ANSYS, Nastran, COMSOL, and Optistruct all default to von Mises when you select "ductile-metal plasticity".

Worked example: a pressure-vessel wall

A thin-walled cylindrical pressure vessel of internal radius r = 250 mm and wall thickness t = 5 mm holds compressed gas at p = 8 MPa gauge. The principal stresses in the wall, away from the end caps, are the hoop, axial, and radial stresses:

σ_hoop  = p r / t        = 8 × 250 / 5      = 400 MPa
σ_axial = p r / (2t)     = 8 × 250 / (2×5)  = 200 MPa
σ_radial ≈ −p/2          ≈ −4 MPa            (thin-wall approximation, often dropped)

Take σ_radial ≈ 0 (the thin-wall convention). Then σ₁ = 400, σ₂ = 200, σ₃ = 0 MPa.

σ_VM = √( ½[ (400 − 200)² + (200 − 0)² + (400 − 0)² ] )
     = √( ½[ 40 000 + 40 000 + 160 000 ] )
     = √( 120 000 )
     = 346.4 MPa

For a Grade-50 carbon-steel vessel (σ_y ≈ 345 MPa), the von Mises stress is right at yield — the design has essentially no margin and would not meet code. Applying Tresca:

σ_max − σ_min = 400 − 0 = 400 MPa ≥ σ_y = 345 MPa  →  Tresca says yielded.

Tresca flags the same design as overstressed but at a stricter threshold, which is why ASME's code-allowable stress intensities use a Tresca-equivalent quantity multiplied by a safety factor.

Why FEA solvers default to von Mises

Open the post-processor of any commercial FEA package and the default field plotted on the deformed mesh is "von Mises stress", commonly with a rainbow color map: blue where σ_VM is well below σ_y, red where it approaches or exceeds it. The choice is overdetermined.

• Goodness of fit. The von Mises locus matches measured yield surfaces of ductile metals within roughly 5–10%, better than Tresca for most cases.
• Single scalar field. σ_VM is a strictly non-negative scalar, so it maps cleanly to a colour bar. The full stress tensor has six components and cannot be visualised as one image.
• Smoothness. The circular yield surface has continuous gradients everywhere. The Tresca hexagon has corners; numerical return-mapping algorithms must special-case them, which historically caused robustness problems.
• Associated flow rule, in closed form. The plastic strain increment perpendicular to the yield surface (the associated flow rule) is, for von Mises, simply proportional to the deviatoric stress — no case-splitting.
• Direct connection to J₂. Most of the apparatus of J₂-plasticity (Prandtl-Reuss equations, isotropic and kinematic hardening, return-mapping integration) is built around the von Mises surface as a starting point.

Where the criterion does not apply

The von Mises criterion bakes in four assumptions that are not always true: isotropy, pressure independence, equal yield in tension and compression, and no Bauschinger effect. Each failed assumption maps to a different alternative criterion.

Material	Why von Mises fails	Alternative
Soils, rocks, granular media	Strongly pressure-sensitive: higher confining pressure raises yield strength	Mohr-Coulomb, Drucker-Prager
Concrete	Pressure sensitive, asymmetric (compression-dominant)	Drucker-Prager cap, Willam-Warnke
Cast iron	Asymmetric: weaker in tension than compression by ~3×	Coulomb-Mohr, modified Mohr
Fiber-reinforced composites	Anisotropic: different yield in different fiber directions	Tsai-Wu, Tsai-Hill, Hashin, Puck
Rolled / textured sheet metal	Anisotropic plastic flow; r-values differ	Hill 1948, Barlat 2000 (Yld2000-2d)
Cyclically loaded metals	Bauschinger softening — reverse-load yield is lower	Kinematic hardening overlay (Armstrong-Frederick, Chaboche)
Brittle ceramics, glass	Fracture occurs before yielding	Maximum-normal-stress, Weibull
Polymers under multiaxial state	Pressure sensitivity at high triaxiality	Modified von Mises (Raghava), Drucker-Prager

For everyday static analysis of ductile steel, aluminium, copper, nickel-alloy, titanium, mild-steel weldments and most engineering thermoplastics under monotonic load, the plain von Mises criterion is the right default. Outside that envelope, the question switches from "does it yield" to "which yield surface is appropriate" — and choosing wrong can flip a design's predicted margin by tens of percent.

Beyond first yield — hardening

The criterion as stated tells you when first yielding begins. Past first yield, real metals do not simply flow indefinitely at σ_y; they harden — the yield strength grows with accumulated plastic strain. The von Mises framework extends naturally:

• Isotropic hardening. The yield-surface radius √(⅔)σ_y(ε_p) grows uniformly with plastic strain. Good for monotonic loading; predicts that reverse-load yield is at the new higher σ_y — wrong for real metals.
• Kinematic hardening. The yield surface translates rigidly in stress space along a "back-stress" α_ij. Captures the Bauschinger effect — reverse-load yielding at a lower stress than the most recent forward yield.
• Combined. Both grow and translate. The standard Chaboche model for cyclic plasticity stacks several kinematic terms with one isotropic term.

In all three cases the underlying yield surface is still a von Mises cylinder — only its size and centre evolve. The criterion is therefore the kernel of a much larger framework called J₂-flow plasticity, and that framework is built into every general-purpose FEA solver.

Common pitfalls

• Comparing σ_VM to ultimate strength. σ_VM is meant to be compared to yield, not ultimate. The criterion predicts onset of plastic flow, not fracture. Using σ_VM < σ_UTS as a margin claim hides plastic deformation that will accumulate fatigue damage.
• Reading σ_VM at a stress singularity. Re-entrant corners and point loads in an FEA model produce stress concentrations that diverge with mesh refinement. The reported σ_VM there is mesh-dependent garbage; use the elastic-plastic limit-load, a Neuber rule, or sub-modelling with realistic fillets.
• Using von Mises for concrete or soil. Pressure-insensitive surface predicts that hydrostatic compression cannot cause failure — but those materials fail readily under high mean stress.
• Forgetting that σ_VM is positive-only. The criterion cannot distinguish "yielding in tension" from "yielding in compression". For cast iron and other tension/compression-asymmetric materials, this is a serious limitation.
• Plane stress vs plane strain confusion. The plane-stress ellipse (σ₃ = 0) and the plane-strain locus (ε₃ = 0) are different curves. Plane strain pushes the third principal stress up to ν(σ₁ + σ₂), which inflates triaxiality and reduces the apparent margin — relevant for thick plates and welded joints.
• Trusting "von Mises stress" for fatigue without further care. Fatigue is driven by stress amplitude and mean stress, plus crack-tip details. A high-cycle fatigue check needs a Sines, Crossland, or Dang Van criterion built on top of, not in place of, the static-yield σ_VM check.