Enter the 6 components of a 3D stress tensor to instantly calculate principal stresses, Von Mises stress, and Tresca stress. Visualize 3D Mohr's circles, stress element diagram, and yield margin across tabs.
Stress Tensor [MPa]
Blue: normal stress / orange: shear stress. Because the tensor is symmetric, only the upper-triangular shear terms are editable.
Results
Principal stress σ₁
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MPa
Principal stress σ₂
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MPa
Principal stress σ₃
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MPa
Von Mises σvm
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MPa
τmax (Tresca)
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MPa
Hydrostatic stress σm
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MPa
Cvmohr
Elem
Theory & Key Formulas
$$\sigma^3 - I_1\sigma^2 + I_2\sigma - I_3 = 0$$
$I_1=\sigma_{xx}+\sigma_{yy}+\sigma_{zz}$ (first invariant)
Von Mises: $\sigma_{vm}=\sqrt{\tfrac{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2}{2}}$
What Does the Stress Tensor Represent?
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The term 'stress tensor' sounds really complicated just from the name... σxx, τxy — do all these have different meanings?
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Roughly speaking, it represents the forces acting on each of the six faces when you cut out a tiny cube (like a die) at a point inside an object. σxx is the force pushing (or pulling) in the x-direction on a face oriented in the x-direction; τxy is the force trying to shear the x-face in the y-direction. In 3D, there are 6 faces (actually 3 due to symmetry), each with a normal and two shear forces, giving a total of 6 independent components.
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I see. But then, what is 'principal stress'? When I select the 'Pure Tension' preset in the tool, only σ1 has a value, and σ2 and σ3 are zero...
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Sharp question! Principal stress is the normal stress when you rotate the cube to a special orientation where shear stresses become exactly zero. For pure tension, τ is already zero, so it comes out directly. The interesting part is the 'Pure Shear' preset. Try entering only τxy. You'll see values appear for σ1 and σ3, and they are symmetric positive and negative, right? When you look at shear from a rotated perspective, it becomes tension in one diagonal direction and compression in the other.
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Wow, it's true! Even with just shear, principal stresses appear. Also, Von Mises stress is used to compare with the material's yield strength, right? When I moved the 'σy' slider in the Yield Margin tab, the bar chart changed.
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Exactly, that σy is the 'limit beyond which the material does not yield'. If the Von Mises stress exceeds it, yielding occurs. In practice, CAE analysis shows the Von Mises stress distribution across a part, and you check 'what percentage of the yield strength is being used?'. If the margin is small, the design is revised. In the 3D Mohr's Circle tab, the radius of the outermost circle is the maximum shear stress τmax, which is the criterion for the Tresca criterion. The Tresca criterion is slightly more conservative (safe-side) than Von Mises and is used in design.
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When I tried the 'Combined Loading' preset, Von Mises came out a bit lower than Tresca. Why is that?
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The Von Mises criterion uses an elliptical yield surface based on constant shear strain energy, while the Tresca criterion uses a hexagonal yield surface. Since the ellipse fits inside the hexagon, the Von Mises criterion sometimes judges there is a bit more 'margin'. Experimentally, for ductile metals, Von Mises often better reproduces actual yielding. However, Tresca is sometimes used deliberately for conservative design.
Theory of Principal Stress Calculation
To obtain principal stresses from an arbitrary stress tensor, solve the tensor eigenvalue problem. Expanding the characteristic equation gives a cubic equation:
$$\sigma^3 - I_1\sigma^2 + I_2\sigma - I_3 = 0$$
The stress invariants are $I_1 = \sigma_{xx}+\sigma_{yy}+\sigma_{zz}$, $I_2 = \sigma_{xx}\sigma_{yy}+\sigma_{yy}\sigma_{zz}+\sigma_{zz}\sigma_{xx} - \tau_{xy}^2 - \tau_{yz}^2 - \tau_{zx}^2$, and $I_3 = \det(\boldsymbol{\sigma})$.
From the principal stresses $\sigma_1 \ge \sigma_2 \ge \sigma_3$, compute equivalent stresses for yield checks:
Shaft strength design: A rotating shaft can experience normal stress from bending and shear stress from torque at the same time. Enter bending stress as σxx and torsional shear stress as τxy to evaluate the combined stress state and choose a shaft diameter that keeps Von Mises stress below the yield strength at critical sections.
Pressure vessels: Cylindrical vessels under internal pressure experience hoop stress and axial stress. Use the biaxial preset as a starting point, then compare Von Mises stress with the allowable stress for the selected material to size wall thickness.
Geotechnical and civil structures: Principal stress directions help estimate potential failure-plane orientation in soil or structural members. The outer radius of the 3D Mohr circle gives maximum shear stress and can be compared with Mohr-Coulomb failure criteria.
Frequently Asked Questions
The stress tensor has 9 components, but due to the symmetry τij = τji (derived from the balance of angular momentum), there are only 6 independent components (σxx, σyy, σzz, τxy, τyz, τzx). In other words, τxy (shear on the x-face in the y-direction) and τyx (shear on the y-face in the x-direction) are always equal. This simulator also displays only the reference values for symmetric components, and you input 6 components.
For ductile metals, when the Von Mises stress exceeds the yield strength (0.2% proof stress), plastic deformation (permanent deformation) begins. It's not 'breaking (fracture)' but 'deformation that remains'. Fracture occurs when the tensile strength is exceeded. However, for brittle materials (cast iron, ceramics), yielding and fracture happen almost simultaneously, so the yield strength and tensile strength are nearly the same. The appropriate failure criterion must be selected based on the material type.
Because a 3D stress state has three principal stresses, and Mohr's circles can be drawn for each pair of principal stresses (σ1-σ2, σ2-σ3, σ1-σ3). The radius of the outermost circle (σ1-σ3) is the maximum shear stress τmax = (σ1-σ3)/2. The stress state on any arbitrary plane always lies within the region formed by these three circles. In 2D (plane stress), only one circle is needed, but in 3D, three must always be drawn.
$I_1$ (first invariant) is the trace of the stress tensor, the sum of the three normal stresses. It relates to hydrostatic volume change (expansion/contraction). $I_2$ (second invariant) is related to shear strain energy and is used to calculate Von Mises stress. $I_3$ (third invariant) is the determinant of the tensor, used to compute the Lode angle, which indicates the 'skewness' of the stress state (tension-dominated or compression-dominated). These invariants are 'essence of physical quantities' that do not change regardless of the coordinate system.
Major CAE software like Abaqus, Ansys, and Nastran can output stress components (labeled S11, S22, S33, S12, S23, S13, etc.) numerically for each element or node. By identifying the coordinates of a point of interest (e.g., stress concentration area) and reading its 6 components, you can input them into this simulator for more detailed hand-calculation verification. Also, while CAE software directly outputs Von Mises stress, using this tool to check individually deepens understanding.
It comes from the difference in the shape of the yield surface in stress space. The Von Mises criterion has a cylindrical (in 3D) or elliptical (in π-plane projection) yield surface. The Tresca criterion has a regular hexagonal (prismatic) yield surface that lies inside the Von Mises ellipse. This means the Tresca criterion judges yielding to occur about 15% earlier than Von Mises in the edge regions of the yield ellipse (biaxial stress states). For pure shear, the yield strength is $\sigma_y/\sqrt{3}$ for Von Mises and $\sigma_y/2$ for Tresca.
What is Stress Tensor?
Stress Tensor 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.
Physical Model & Key Equations
The simulator is based on the governing equations behind 3D Stress Tensor Analyzer. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: The concepts behind 3D Stress Tensor Analyzer are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.
Enter the six independent stress tensor components in MPa: σxx, σyy, σzz (normal stresses on diagonal), τxy, τyz, τzx (shear stresses off-diagonal).
Click Analyze to compute principal stresses (σ1, σ2, σ3) via eigenvalue decomposition and Von Mises equivalent stress using √[0.5((σ1−σ2)² + (σ2−σ3)² + (σ3−σ1)²)].
Review the 3D Mohr circle diagram showing stress state graphically, plus Tresca maximum shear stress τmax = (σmax − σmin)/2 for material yield comparison.
Worked Example
For a titanium alloy (Ti-6Al-4V) landing gear subject to combined landing loads: σxx=450 MPa, σyy=−280 MPa, σzz=120 MPa, τxy=85 MPa, τyz=60 MPa, τzx=75 MPa. The analyzer computes principal stresses σ1≈520 MPa, σ2≈180 MPa, σ3≈−410 MPa. Von Mises stress = 862 MPa exceeds Ti-6Al-4V yield (~880 MPa), indicating near-critical conditions. Tresca stress = 465 MPa provides conservative alternative failure criterion.
Practical Notes
For FEA validation: export nodal stress tensors from Abaqus or NASTRAN output decks directly into this tool to verify principal stress orientations and material failure margins in complex loading scenarios.
Anisotropic composites: use local material coordinate frame stresses; misaligned tensor input may mask fiber-dominated failure modes like matrix shear cracking at τyz > 80 MPa.
Pressure vessels: hoop stress typically dominates σyy; ensure radial gradients and bending components are separated before inputting averaged tensor.