What are Yield Criteria?
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What exactly is a "yield criterion"? I hear about materials yielding, but what are we comparing here?
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Basically, it's a mathematical rule that predicts when a material will start to deform permanently under stress. In this simulator, we compare three famous rules: von Mises, Tresca, and Drucker-Prager. For instance, if you set the principal stresses σ₁, σ₂, σ₃ on the left panel, each criterion draws a different "failure surface" on the 3D plot. Try changing σ₁ and watch the red dot (your stress state) move relative to these surfaces.
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Wait, really? So they give different answers? Which one is "correct"?
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Great question! They are models, so it depends on the material. Von Mises works well for most metals, like steel in a car frame. Tresca is more conservative and simpler. Drucker-Prager accounts for pressure sensitivity, crucial for soils or concrete. A common case is a pressurized pipe: von Mises and Tresca might disagree on the safety factor. You can see this instantly by adjusting the Yield Stress (σ_y) slider and watching the Safety Factor card update for each criterion.
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That makes sense. What's the "Friction Angle φ" for then? I only see it for Drucker-Prager.
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That's the key parameter that makes Drucker-Prager unique! It models how internal friction affects yielding. For granular materials like sand, strength increases under compression. Set φ to 0°, and D-P resembles von Mises. Crank it up to 30°, and you'll see the yield surface tilt and expand in the 3D plot—this is critical for geotechnical engineering. Try it while keeping the other stresses constant to see the dramatic change.
Physical Model & Key Equations
The von Mises criterion is based on distortional (shape-changing) energy. It states that yielding begins when the equivalent stress reaches the material's yield strength.
$$\sigma_{eq}^{VM}= \sqrt{\frac{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2}{2}}\leq \sigma_y$$
Here, $\sigma_{eq}^{VM}$ is the von Mises equivalent stress, $\sigma_1, \sigma_2, \sigma_3$ are the principal stresses, and $\sigma_y$ is the uniaxial yield stress. The criterion is a cylinder in principal stress space.
The Tresca (maximum shear stress) criterion is simpler: yielding occurs when the maximum shear stress reaches a critical value. The Drucker-Prager criterion extends this concept to pressure-sensitive materials by incorporating the hydrostatic stress and a friction angle.
$$\sigma_{eq}^{DP}= \alpha I_1 + \sqrt{J_2} \leq k$$
Where $I_1 = \sigma_1 + \sigma_2 + \sigma_3$ is the first stress invariant (hydrostatic pressure), $J_2$ is the second deviatoric stress invariant (related to von Mises stress), and $\alpha$, $k$ are material constants derived from the cohesion and the Friction Angle φ you set in the simulator. This creates a cone in principal stress space.
Real-World Applications
Metal Forming & Crash Simulation (CAE): In Abaqus or LS-DYNA, selecting von Mises plasticity is standard for simulating car body panel stamping or crashworthiness. Engineers use this simulator to quickly check how different stress states (like biaxial tension in a panel) compare to the yield surface before running a full, costly simulation.
Pressure Vessel & Pipeline Design: For safety analysis of boilers or gas pipelines, engineers must calculate safety factors against yielding under internal pressure and external loads. Using this tool, they can input the principal stresses from their FEA model and instantly compare the conservative Tresca result with the more common von Mises result to meet design codes.
Geotechnical & Concrete Analysis: The Drucker-Prager criterion is essential in LS-DYNA for modeling soil mechanics or concrete crushing (*MAT_SOIL_AND_FOAM, *MAT_CONCRETE_DAMAGE). The Friction Angle φ directly influences the predicted failure load of a soil slope or a concrete column, which you can explore here by varying φ.
Ductile Fracture Prediction: Advanced damage models (GTN, Lemaitre) in CAE use stress triaxiality—a ratio of hydrostatic to von Mises stress—to predict when a material will fracture after yielding. This simulator helps visualize how different stress paths (changing σ₁, σ₂, σ₃) affect triaxiality and thus the predicted damage.
Common Misunderstandings and Points to Note
When starting to use this tool, there are several pitfalls that users, especially CAE beginners, often fall into. A major misunderstanding is the belief that a larger safety factor absolutely guarantees safety. For instance, even if you calculate a safety factor of 1.5 using the Tresca criterion and 2.0 using von Mises, it is dangerous to use these values directly in your design. These yield criteria are merely benchmarks for "initial yield". Real-world components are subject to complex failure modes not considered by this tool, such as stress concentrations, fatigue, and creep. Treat the tool's results as a "guideline" for material selection and initial design; detailed simulation and experimental validation are ultimately necessary.
Next, regarding the automatic sorting of principal stress magnitudes. While you can freely input $\sigma_1, \sigma_2, \sigma_3$ into the tool, the Tresca criterion formula $\tau_{max}= (\sigma_1 - \sigma_3)/2$ implicitly assumes $\sigma_1 \geq \sigma_2 \geq \sigma_3$. The tool should handle this sorting internally, but if you perform manual calculations, forgetting this ordering leads to significant errors. For example, for (100, 0, 50) MPa, the maximum shear stress is not (100-0)/2=50 MPa, but is correctly based on the sorted values (100, 50, 0), resulting in (100-0)/2=50 MPa (coincidentally matching in this specific example).
Finally, the realistic range for the "friction angle" in the Drucker-Prager criterion. While the tool allows you to set values from 0° upwards, real materials have typical ranges: around 30°–40° for sand, about 30° for normal concrete, etc. Setting an unrealistic value like 60° for the friction angle results in an extremely open yield surface, meaning you're effectively modeling a fictional material that is "unrealistically strong in compression". The key is to first look up typical values for your material.
What is Yield criterionComparisonTool?
Yield criterionComparisonTool is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.