Strain gauges are widely used sensors for converting mechanical strain into an electrical resistance change. When bonded to the surface of a structural or machine element, a strain gauge experiences the same local deformation as the underlying material. This deformation changes the gauge's length, cross-sectional area, and resistivity, which in turn changes its resistance. By incorporating the gauge into a suitable electrical circuit (typically a Wheatstone bridge), we obtain a voltage signal that is proportional to the strain in the structure.
In this section we summarize the basic principles, construction, and practical characteristics of strain gauges, with emphasis on bonded metal-foil gauges as commonly used in mechatronics and measurement laboratories.
The resistance of a conductor is \[ R = \rho \,\frac{L}{A}, \] where \(L\) is the length, \(A\) the cross-sectional area, and \(\rho\) the resistivity of the material. When a strain gauge is bonded to a surface that is stretched or compressed, all three quantities may change:
For metallic strain gauges, most of the resistance change arises from the geometric changes in \(L\) and \(A\); the piezoresistive contribution is relatively small. For semiconductor gauges, the piezoresistive effect dominates.
To make strain measurement practical, the conductor is arranged in a long, thin grid pattern on an insulating backing. The grid maximizes total conductor length within a compact footprint, increasing the sensitivity to strain while keeping resistance within a convenient range (typically 120–1000 \(\Omega\)).
In most laboratory applications, we use bonded metal-foil strain gauges. These consist of:
The metal grid is on the order of \(0.0002\) in thick and is carried on a flexible backing film about \(0.001\) in thick. When the gauge is bonded to a specimen, the backing and adhesive ensure that the grid follows the local surface strain. The gauge therefore reports an average strain over its active length, not the strain at a mathematical point. For regions with high strain gradients (e.g., near stress concentrations), small-gage lengths are desirable to reduce averaging errors.
Bonded metal-foil strain gauges are popular because they offer:
Common nominal resistances are 120 \(\Omega\), 350 \(\Omega\), and 1000 \(\Omega\). Allowable excitation currents are limited by self-heating and typically fall in the range of 5–40 mA, depending on the gauge and mounting conditions.
The sensitivity of a strain gauge is characterized by its gauge factor, defined as \[ GF = \frac{\Delta R / R}{\varepsilon}, \] where \(R\) is the nominal gauge resistance, \(\Delta R\) is the change in resistance due to strain, and \(\varepsilon\) is the axial strain along the gauge grid.
Equivalently, \[ \frac{\Delta R}{R} = GF \,\varepsilon. \]
For typical bonded metal-foil gauges, \(GF\) is in the range of 2–4, with a common value around 2.0. Semiconductor strain gauges can have gauge factors as high as 50–150, but at the cost of increased temperature sensitivity, nonlinearity, and mounting difficulty.
In practice, the manufacturer determines the gauge factor by bonding sample gauges to a calibration specimen with a well-known strain field and measuring \(\Delta R\) versus \(\varepsilon\). The gauge factor printed on each gauge package is then an average value for that production batch. A gauge-factor tolerance on the order of \(\pm 1\%\) is typical and represents a fundamental accuracy limit for stress analysis based on that gauge alone.
When a strain gauge is used in a Wheatstone bridge, the relationship \(\Delta R / R = GF\,\varepsilon\) is what links the mechanical strain to the bridge output voltage. For example, in a quarter-bridge configuration with one active gauge, the bridge output is approximately proportional to \(V_{\text{ex}} \, GF \, \varepsilon\) (see the Wheatstone-bridge section).
Although bonded metal-foil gauges are the most common in teaching labs, several other types are used in practice:
Thin-film metal gauges
These are deposited directly onto the transducer's elastic element using vacuum deposition or sputtering. They eliminate organic adhesives, improving long-term stability and high-temperature performance.
Semiconductor (piezoresistive) gauges
Small slices of doped silicon with very high gauge factors. They are often used inside pressure and acceleration transducers where small strains must be detected with high sensitivity. Their drawbacks include strong temperature sensitivity and greater nonlinearity.
Diffused semiconductor gauges
Strain-sensitive regions formed directly in a silicon diaphragm using integrated-circuit diffusion processes. Many commercial pressure sensors use this approach for low-cost, batch-fabricated transducers.
Packaged/high-environment gauges
For harsh environments, gauges may be built into welded metal housings and attached by spot welding. Internal arrangements provide both active and dummy filaments for temperature compensation and mechanical robustness in pipelines, power plants, and similar applications.
In this lab we will primarily use conventional bonded metal-foil gauges, but it is useful to be aware of these alternatives.
Strain gauges are sensitive not only to mechanical strain but also to temperature:
A common way to mitigate temperature effects is to use a dummy gauge in an adjacent leg of the Wheatstone bridge. The dummy gauge is bonded to an unloaded piece of the same material and kept at the same temperature as the active gauge. Temperature-induced resistance changes then occur in both gauges and cancel in the bridge output, while true strain affects only the active gauge.
Commercial gauges may also be manufactured with inherent temperature compensation for specific materials, so that thermal expansion effects nearly cancel without a separate dummy gauge.
In many applications, the direction of the principal strains is not known in advance. To recover the strain state at a point, manufacturers provide strain-gauge rosettes: pre-arranged combinations of two or three gauges on a single backing, with accurately known relative orientations.
Common rosette patterns include:
By measuring the strains in each element of a three-gauge rosette, we can calculate the principal strains and directions at the measurement point. Stacked rosettes place the gauges on top of each other for “point” measurement, while planar rosettes spread them out to reduce self-heating and simplify installation.
For metallic bonded strain gauges, the dynamic response is excellent. Over a wide frequency range (tens of kilohertz in typical applications), the gauge and adhesive layer follow the specimen strain with negligible dynamic lag. The gauge can therefore be modeled as a quasi-static sensor of strain for most mechatronics and vibration experiments.
The ultimate limit on the smallest measurable strain is set by electrical noise, particularly Johnson (thermal) noise in resistors and noise in the amplification electronics. For a single gauge in a bridge, the resulting output is often in the microvolt to millivolt range, so high-quality instrumentation amplifiers and careful shielding are needed. In practice, amplifier noise, bridge resistor noise, and environmental pickups (e.g., from fluctuating magnetic fields) usually limit system resolution well before the theoretical thermal-noise limit of the gauge itself.
Strain gauges provide a practical and versatile means of converting mechanical strain into an electrical signal. Bonded metal-foil gauges are the workhorses of experimental stress analysis and many homemade transducers, offering reasonable gauge factors, good linearity, and manageable temperature effects. The key relationships are:
In subsequent sections, we will see how to integrate strain gauges into Wheatstone-bridge circuits and how to interpret the resulting voltage signals in terms of mechanical strain and stress.