Gauge factor of resistance strain gauge

The gauge factor (G) of a strain gauge is the ratio of the fractional change in resistance to the fractional change in length (strain):

G = (ΔR/R) / (ΔL/L) = (ΔR/R) / ε

It is the sensitivity of the gauge; a higher G gives a larger output.

Derivation

The resistance of a conductor of resistivity ρ, length L and cross-section area A is:

R = ρL/A

Take logarithms and differentiate:

dR/R = dρ/ρ + dL/L − dA/A

For a circular wire A = πD²/4, so dA/A = 2 dD/D.

Poisson's ratio ν = −(lateral strain)/(axial strain) = −(dD/D)/(dL/L), thus dD/D = −ν dL/L, and

dA/A = −2ν dL/L

Substituting:

dR/R = dL/L + 2ν dL/L + dρ/ρ = (1 + 2ν) dL/L + dρ/ρ

Dividing by dL/L = ε:

G = 1 + 2ν + (dρ/ρ)/ε

Meaning of the terms

• 1 + 2ν: change in resistance due to the geometry (dimensions) change.
• (dρ/ρ)/ε: change due to the piezoresistive effect (change in resistivity with strain).

For metallic gauges, ν ≈ 0.3, so 1 + 2ν ≈ 1.6; with the resistivity term G ≈ 2 (constantan 2.0 to 2.1). For semiconductor gauges, the piezoresistive term dominates and G ≈ 100 to 200.

If the resistivity change is negligible, G ≈ 1 + 2ν. Note ν = 0.5 would imply no volume change, G = 2.