What is stress? Stress is internal force per unit area. Imagine cutting through a body along a plane with normal nˆ. The traction vector t = σ · nˆ gives the force per area on that cut face.
σxx, σyy: Normal stresses — force perpendicular to the face. Positive = tension (pulling), negative = compression (pushing).
τxy: Shear stress — force parallel to the face. The subscript means: force in x-direction on a face with y-normal (and vice versa, by symmetry).
Units: MPa = N/mm² = 10&sup6; Pa. Steel yields at ~250 MPa, concrete crushes at ~30 MPa.
Normal stresses (MPa):
σxx0—
σyy0—
Shear stress (MPa):
τxy0—
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Normal & Shear Stresses
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σ = Fn/A
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τ = Ft/A
Normal stress σ: force perpendicular to the cut — tension (+) or compression (−) Shear stress τ: force tangent to the cut — causes sliding/distortion Complementary shear: τxy = τyx (from moment equilibrium — NOT from symmetry of the tensor definition!)
Why is τxy = τyx? Consider a tiny square element. If τxy acts on the top and bottom faces, it creates a net moment. To prevent spin, complementary shear τyx = τxy must act on the left and right faces. This is why the stress tensor is symmetric.
Sign convention: Positive normal stress = tension (face pulls outward). Negative = compression (face pushes inward). Positive shear = stress on +x face pointing in +y direction.
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Principal Stresses & Mohr's Circle
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Transformation:T' = R(α) · T · RT(α)
σn(α) = ½(σxx+σyy) + ½(σxx−σyy)cos2α + τxysin2α
τ(α) = −½(σxx−σyy)sin2α + τxycos2α
Principal stresses: σI,II = C ± R = ½(σxx+σyy) ± √(¼(σxx−σyy)² + τxy²)Principal direction: tan(2α1) = 2τxy/(σxx−σyy) Max shear (Tresca): |τmax| = R at α1 + 45°
What does Mohr's circle show?
Imagine cutting through the material at angle α. On the cut face you measure:
• σn = normal stress perpendicular to cut
• τ = shear stress along the cut
Key points:
• Principal stresses = intersections with σn-axis (τ=0)
• Max shear = top/bottom of circle = at 45° to principal axes
• A=(σxx,τxy) and B=(σyy,−τxy) = starting points
• 2α on the circle = double the physical angle
• Tresca criterion: yielding when τmax = σY/2
Cut angle α (rotate to trace the circle)0°
Eigenvalues & Eigenvectors:
det(T − λI) = 0 gives σI ≥ σII. These are the stresses on faces where ALL shear vanishes. The eigenvectors define the principal directions.
Recipe: Drawing Mohr's circle
1. C = (σxx+σyy)/2
2. R = √((σxx−σyy)2/4 + τxy2)
3. Circle with center (C, 0), radius R
4. Plot A=(σxx, τxy)
5. Intersections with σn-axis = principal stresses
6. Top/bottom point = max shear stress = Tresca
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Stress Decomposition
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T = Thyd + s
Thyd = σm · Is = T − Thyd
In 2D: σm = ½(σxx + σyy), p = −σm (pressure)
Deviatoric s: shape-changing stress, tr(s) = 0 von Mises: σvM = √(σxx² + σyy² − σxxσyy + 3τxy²)
Why decompose? Hydrostatic stress (pressure) changes volume but NOT shape — materials rarely fail under pure pressure. The deviatoric part changes shape and drives yielding.
von Mises criterion: A material yields when σvM ≥ σY (yield strength). This is equivalent to: yielding depends ONLY on the deviatoric stress, not on pressure. That's why submarines don't crush under uniform pressure (if the shell is perfect), but fail at stress concentrations where σvM is high.
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