Tresca vs Mises Yield Criteria Simulator — Principal Stress Space Comparison
Compare the von Mises (ellipse) and Tresca (regular hexagon) yield surfaces in the principal stress plane while you tune the three principal stresses and the yield stress sigma_y. The tool reports the two equivalent stresses and safety factors in real time so you can see how Tresca always lies on the conservative side of Mises and grasp the onset of plasticity for ductile metals at a glance.
Parameters
Principal stress sigma1
MPa
Principal stress sigma2
MPa
Principal stress sigma3
MPa
Yield stress sigma_y
MPa
sigma_VM = sqrt(0.5*[(s1-s2)^2 + (s2-s3)^2 + (s3-s1)^2])
sigma_TR = max(|s1-s2|, |s2-s3|, |s3-s1|)
SF = sigma_y / sigma_eq (>1 is safe)
Blue ellipse = Mises, red hexagon = Tresca. The plane is normalised by sigma_y with sigma3 set to zero. The yellow marker is the current (sigma1, sigma2) state. Inside the surface = elastic, outside = plastic. The Tresca hexagon is inscribed inside the Mises ellipse.
Principal stress bars and yield limit
Three bars show sigma1, sigma2, sigma3. The yellow dashed line marks +sigma_y and the red dashed line marks -sigma_y. A bar tip crossing a dashed line yields under uniaxial tension or compression (multi-axial states still need an equivalent-stress check).
Yield occurs at $\sigma_{VM} = \sigma_y$ (Mises) or $\sigma_{TR} = \sigma_y$ (Tresca). The inequality $\sigma_{TR} \geq \sigma_{VM}$ always holds, with the maximum ratio $2/\sqrt{3} \approx 1.155$ in pure shear ($\sigma_1=-\sigma_3$, $\sigma_2=0$). Tresca is conservative; Mises matches experiments on ductile metals more closely.
About the Tresca vs Mises Yield Criteria Simulator
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With the defaults the Mises stress is 173.2 MPa but Tresca is 200 MPa. Why is Tresca always larger?
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Good catch. Tresca only looks at the maximum shear, so it takes the largest pairwise difference of the principal stresses (200-0=200) as the equivalent stress. Mises is built on the second deviatoric invariant J2 and adds the three differences in a quadratic way, so it usually gives a smaller number. For any state we have sigma_TR >= sigma_VM. Here (200, 100, 0) gives Mises = sqrt((100^2+100^2+200^2)/2) = sqrt(30000) = 173.2 and Tresca = max(100, 200, 100) = 200. The gap is about 15.5%, which is the worst case and shows up in pure shear (sigma1=-sigma3, sigma2=0).
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So the smaller Tresca safety factor of 1.25 means it is the more conservative theory?
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Exactly. SF = sigma_y / sigma_eq, so the larger Tresca equivalent gives the smaller SF and predicts yielding sooner. In practice the ASME Boiler and Pressure Vessel Code Section VIII Division 1 uses Tresca because it is on the safe side, while Abaqus and ANSYS pick Mises as the default J2 plasticity model thanks to its smooth ellipse and good agreement with classic Lode, Taylor and Quinney experiments on ductile metals.
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When I press the sigma1 sweep button the yellow marker keeps entering and leaving the ellipse and hexagon. Is that the elastic-plastic transition?
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Yes. Inside the blue ellipse (Mises) and red hexagon (Tresca) is elastic, outside is plastic. Green marker = safe under both, orange = plastic under Mises but still elastic under Tresca, red = plastic under both. For example sigma1=400 (with sigma2=100, sigma3=0, sigma_y=250) gives Mises = sqrt((300^2+100^2+400^2)/2) = 360.6 MPa and Tresca = 400 MPa, so both criteria predict yielding. Checking the position in principal stress space before you launch a plasticity run is a good habit.
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Sliding sigma3 changes the equivalent stresses but the ellipse and hexagon stay still. Why?
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Sharp eye. The yield surface plot is a 2D slice at sigma3 = 0 (the textbook view). The actual surface lives in 3D principal stress space: Mises is an infinite cylinder around the hydrostatic axis sigma1=sigma2=sigma3, Tresca is a regular hexagonal prism on the same axis. Moving sigma3 pushes the marker out of the page, so the 2D slice cannot show it, but the equivalent stress numbers above carry the full 3D information. Read the figure and the numbers together — that is exactly how a CAE post-processor turns a 3D stress field into a Mises contour plot.
Physical model and key equations
A yield criterion decides when a material under a multiaxial stress state begins to yield. Richard von Mises (1913) and Henri Tresca (1864) proposed two criteria built on different physical pictures.
The Mises equivalent stress is built from the second invariant $J_2 = \tfrac{1}{2}s_{ij}s_{ij}$ of the deviatoric stress tensor $s_{ij} = \sigma_{ij} - \tfrac{1}{3}\sigma_{kk}\delta_{ij}$, so it can also be written $\sigma_{VM} = \sqrt{3 J_2}$. Physically it is the distortion energy per unit volume, which removes the hydrostatic part and only retains the pure shear contribution.
The Tresca equivalent stress is twice the maximum shear stress $\tau_{\max} = (\sigma_{\max} - \sigma_{\min})/2$, so the yield condition is equivalent to $\tau_{\max} = \sigma_y / 2$. The physical idea is the simplest possible: yielding starts when the shear stress on a slip plane exceeds a critical value.
In principal stress space the yield surfaces are an infinite cylinder for Mises and a regular hexagonal prism for Tresca, both centred on the hydrostatic axis $\sigma_1=\sigma_2=\sigma_3$. The pi-plane projection (perpendicular to that axis) is a circle of radius $\sqrt{2/3}\,\sigma_y$ for Mises and a regular hexagon inscribed inside it for Tresca, with maximum ratio $2/\sqrt{3} \approx 1.155$.
Real-world applications
Default plasticity model in commercial CAE: Abaqus, ANSYS, LS-DYNA, Marc and COMSOL all default to von Mises for ductile metals because the criterion is differentiable (numerically stable), matches Lode, Taylor and Quinney experiments very well and admits a simple associated flow rule. For steel, aluminium, copper or nickel alloys, Mises with isotropic hardening (bilinear or multilinear) is usually accurate enough.
Pressure vessel and piping codes (ASME): The American Society of Mechanical Engineers Boiler and Pressure Vessel Code Section VIII Division 1 traditionally uses Tresca as a conservative way to set allowable stresses. Division 2 also offers a Mises-based linearised stress route for finer design, and Japanese JIS pressure vessel codes mix both. Tresca is also easier to evaluate by hand because it is simply a max of differences.
Sheet metal forming simulations: Stamping, deep drawing, forging and rolling involve large plastic deformation and rely on Mises plasticity combined with anisotropic hardening (Hill or Barlat yield functions). Pure isotropic Mises is not accurate when the rolling and transverse directions of a sheet have different yield stresses, so Hill 1948 or Barlat YLD2000 are layered on top. The Mises and Tresca pair in this tool is the foundation that those richer models extend.
Non-associated flow for soils and concrete: Unlike metals, soils and concrete depend strongly on the hydrostatic pressure, so neither Mises nor Tresca can describe them. Drucker-Prager, Mohr-Coulomb and cap models are needed instead. In geotechnical CAE the Mohr-Coulomb model is standard, with Tresca recovered as the special case where the friction angle phi equals zero.
Common misconceptions and pitfalls
The most frequent misunderstanding is, "Mises and Tresca are nearly identical, so it does not matter which one I use." The two criteria do agree under uniaxial tension or compression, but in pure shear (sigma1=-sigma3, sigma2=0) the ratio sigma_TR / sigma_VM grows to 2/sqrt(3) approx 1.155, a 15.5% gap. For pressure vessels and shear-dominated structures this gap directly affects the safety factor, so the right choice must follow the relevant code.
The next pitfall is, "An equivalent-stress check replaces a full plasticity analysis." Comparing the Mises stress from a linear-elastic run to sigma_y only flags the onset of yielding. Once yielding is predicted, plastic redistribution, residual stresses and accumulated plastic strain can only be captured by a true elastic-plastic analysis that integrates the incremental constitutive law along the load path.
Finally, "Tresca is an old theory with no modern value." Tresca is still widely used for hand calculations, conservative quick checks, pressure vessel codes and nuclear plant design, where being on the safe side is a hard requirement. Mises is correct on average but can underestimate yielding in shear-dominated local regions, so a sound design report often quotes both.
FAQ
The von Mises criterion uses the second invariant J2 of the deviatoric stress tensor and predicts plastic yielding when the equivalent stress sigma_VM = sqrt(0.5[(s1-s2)^2 + (s2-s3)^2 + (s3-s1)^2]) reaches the yield stress sigma_y. Because it agrees well with experiments on ductile metals (steel, aluminium, copper) it is the default plasticity model in CAE codes. With the default values s1=200, s2=100, s3=0, s_y=250 MPa the tool reports sigma_VM=173.2 MPa and SF_VM=1.44.
The Tresca criterion is the maximum shear stress theory: yielding occurs when sigma_TR = max(|s1-s2|, |s2-s3|, |s3-s1|) reaches sigma_y. In the principal stress plane the Tresca surface is a regular hexagon inscribed inside the Mises ellipse, so sigma_TR is always greater than or equal to sigma_VM, making Tresca the more conservative (safer) estimate. With the default state the tool returns sigma_TR=200 MPa and SF_TR=1.25, lower than the Mises factor.
The safety factor SF = sigma_y / sigma_eq tells you how many times the present stress state could be scaled before yielding occurs. SF > 1 means elastic (safe), SF = 1 is incipient yield and SF < 1 means the state is already plastic. Design codes typically demand SF >= 1.5 for static loads and SF up to 3 for fatigue or impact. The tool shows both the Mises and Tresca SF, and you can confirm that Tresca is always the smaller (more conservative) of the two.
Modern CAE codes (Abaqus, ANSYS, LS-DYNA) default to von Mises for ductile metals because the smooth ellipse is differentiable and matches experimental data. Traditional hand calculations and pressure vessel codes such as ASME Section VIII Division 1 often pick Tresca for its conservative estimate. Sweep sigma1 in the tool and watch the maximum gap between the two equivalents grow to about 15.5% in pure shear (sigma1=-sigma3, sigma2=0).