How does a resistive strain gauge detect stretching or compression?

A resistive strain gauge detects stretching or compression by measuring changes in electrical resistance that occur when the device is deformed.1

The gauge consists of a fine wire or metallic foil grid bonded to the surface of a solid body. As the underlying material undergoes stress, the gauge follows its deformation, causing the conductor to stretch or contract. This physical change alters the conductor's resistance because its length, cross-sectional area, and resistivity are modified.12

The relationship between resistance ($R$), length ($l$), area ($A$), and resistivity ($\rho$) is defined by the equation:
$$R=\rho \frac{l}{A}$$
When the gauge is deformed, the resulting fractional change in resistance ($\delta R/R$) is directly proportional to the strain ($\delta l/l$). This change in resistance is then amplified and converted into a measurable electrical signal, allowing for the precise calculation of strain.12