What is strain? When a body deforms, each point x moves to x' = x + u(x). The strain tensor E extracts the deformation part of ∇u, ignoring rigid body rotation.
In 2D: Only 3 independent components: εxx (stretch in x), εyy (stretch in y), εxy (shear). The canvas shows the deformed body (solid) and undeformed reference (dashed outline).
Normal strains:
εxx0.00—
εyy0.00—
Shear strain:
εxy0.00—
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Normal & Shear Strains
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εxx = ΔLx/Lx
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γxy = 2εxy
Normal strain: relative length change Shear strain: γxy = angle change between x,y edges Area change (2D): ΔA/A = εxx + εyy
Normal vs shear: Normal strains change lengths, shear strains change angles. A square under pure normal strain becomes a rectangle. Under pure shear it becomes a parallelogram. Under both, it becomes a general parallelogram.
Engineering vs tensorial: The engineering shear strain γ = 2εxy equals the total angle change. The tensorial εxy = γ/2 is half that, because each edge rotates by γ/2.
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Principal Strains & Mohr's Circle
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Transformation:E' = R(θ) · E · RT(θ)
εn(θ) = ½(εxx+εyy) + ½(εxx−εyy)cos2θ + εxysin2θ
εs(θ) = −½(εxx−εyy)sin2θ + εxycos2θ
Principal strains: εI,II = C ± R = ½(εxx+εyy) ± √(¼(εxx−εyy)² + εxy²)Principal direction: tan(2α1) = 2εxy/(εxx−εyy) Max shear: |εxy,max| = R at α1 + 45°
What does Mohr's circle show?
Imagine cutting through the material at angle θ. On the cut face you measure:
• εn = normal strain perpendicular to cut
• εs = shear strain along the cut
Key points:
• Principal strains = intersections with εn-axis (εs=0)
• Max shear = top/bottom of circle = at 45° to principal axes
• A=(εxx,εxy) and B=(εyy,−εxy) = starting points (diametrically opposite!)
• 2θ on the circle = double the physical angle
Cut angle θ (rotate to trace the circle)0°
Eigenvalues & Eigenvectors:
det(E − λI) = 0 gives ε1 ≥ ε2. These are the strains on faces where ALL shear vanishes. The eigenvectors define the principal directions.
Recipe: Drawing Mohr's circle
1. C = (εxx+εyy)/2
2. R = √((εxx−εyy)2/4 + εxy2)
3. Circle with center (C, 0), radius R
4. Plot A=(εxx, εxy)
5. Intersections with εn-axis = principal strains
6. Top/bottom point = max shear strain
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Strain Decomposition
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E = Evol + E'
Evol = ½ tr(E) · IE' = E − Evol
In 2D: Evol = ½(εxx + εyy) · I
Deviatoric: shape change only, tr(E') = 0
Why decompose? Volume change (hydrostatic) and shape change (deviatoric) are governed by different material properties. Yielding depends only on the deviatoric part. Toggle between views to see how the total deformation splits.
In 2D: The volumetric part is ½(εxx+εyy) times the identity. Note: in 2D we use ½ (not ⅓ like 3D) because there are 2 dimensions.
Display
Actions
Controls
Left-drag Pan view Scroll Zoom R Reset strain to zero