Piezoresistive Strain Gauge Rosette

Applied Load

Strain ε_x 0 µε

Strain ε_y 0 µε

Shear γ_xy 0 µε

Gauge factor GF 2.1

Rosette Readings (ΔR/R)

Gauge A (0°)

0.00%

Gauge B (45°)

0.00%

Gauge C (90°)

0.00%

ε_a, ε_b, ε_c

0,0,0 µε

Recovered Principal Strain

ε₁ (max)

0 µε

ε₂ (min)

0 µε

Angle θ_p

0°

Max shear γ_max

0 µε

How it works

A piezoresistive gauge's resistance changes with strain along its sensing axis:

ΔR/R = GF · ε
GF ≈ 2.0–2.2  metal foil (mostly geometric)
GF ≈ 50–200   doped-silicon piezoresistive
              (dominated by the piezoresistive
               coefficient π, not shape change)

A single gauge only measures strain along its own axis. A 45° rosette bonds three gauges (0°, 45°, 90°) at one point so the full 2D strain state can be reconstructed. A gauge at angle θ from the x-axis reads:

ε(θ) = εx·cos²θ + εy·sin²θ + γxy·sinθ·cosθ

Inverting the three readings (ε_a=ε_0°, ε_b=ε_45°, ε_c=ε_90°) recovers the strain tensor and its principal values:

εx = εa,  εy = εc,  γxy = 2εb − εa − εc

ε1,2 = (εx+εy)/2 ± √[ ((εx−εy)/2)² + (γxy/2)² ]
θp   = ½·atan2(γxy, εx−εy)

ε_x, ε_y, γ_xy sliders — the true strain field applied to the plate (µε = microstrain, 10⁻⁶). The plate mesh deforms visually (exaggerated) to match.
Gauge factor GF — swap between a metal-foil gauge and a semiconductor piezoresistive gauge; the ΔR/R readout scales directly with GF at fixed strain, showing why piezoresistive gauges give a much larger electrical signal per unit strain.
Each foil grid on the plate visually stretches/compresses along its own measurement axis by its own ε(θ) — gauge B (45°) responds to both normal strains and shear, which is exactly why a third gauge is needed to separate them.

Real-world relevance: this is the standard rectangular (45°) rosette used on load cells, aircraft wing spars and pressure-vessel test rigs to find principal stress directions from surface strain measurements.

Gauge A · 0° Gauge B · 45° Gauge C · 90°

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