The central aspect of the sensitivity of the sensors that is under investigation is known as the Strain Sensitivity Coefficient, commonly known as the Gauge Factor (k_ε). It is given by:

$k_{\epsilon }=\frac{\left(\frac{\mathrm{\Delta }R}{R_0}\right)}{\left(\frac{\mathrm{\Delta }l}{l_0}\right)}$(1)

where R₀ and l₀ are the unstretched resistance and unstretched length, respectively, while ΔR is the value of resistance change for a change in length (a stretch) of Δl.

The ratio of the increase in length to the original length ∆l/l₀ is commonly referred to as the Strain, given by ε.

Therefore Eq. (1) above can be written as:

$k_{\epsilon }=\frac{\left(\frac{\mathrm{\Delta }R}{R_0}\right)}{\epsilon }.$(2)

The gauge factor, a dimensionless quantity, can be found by calculating the gradient of the characteristic curve on a graph of resistance change against strain. The ideal sensor would have the most significant gauge factor, meaning it has a high sensitivity to strain. This would make it better at detecting slight changes in strain as these would show significant changes in resistance, increasing the precision of the strain-based sensor [46].

Another sensitivity dimension seldom discussed when analysing the suitability of strain-based sensors is the Stress Sensitivity Coefficient (k_σ). It measures the force per unit area of the sensor required to cause a resulting resistance change [47]. It is given by:

$k_{\sigma }=\frac{\left(\frac{\mathrm{\Delta }R}{R_0}\right)}{\left(\frac{F}{A}\right)}$(3)

where R₀ is the initial resistance before applying any load, ΔR is the resistance change caused by a certain amount of load applied per unit cross-sectional area. F and A are the loads applied to the sensor sample and the sensor sample's cross-sectional area, respectively. The force per unit area, F/A is called the Stress (σ). Therefore, Eq. (3) above can be written as:

$k_{\sigma }=\frac{\left(\frac{\mathrm{\Delta }R}{R_0}\right)}{\sigma }.$(4)

If the force is in Newtons (N) and the Cross-Sectional Area is in square meters (m²), then the units for σ are given in Newtons per Square meter (N/m²) or Pascals (Pa), Where 1 N/m² = 1 Pa. Therefore, the units for k_σ are “per pascal” (Pa^-1). For this work, the sensor with the more significant value of k_σ was preferred because of its more diverse applications – the sensor gives a large resistance output for a small value of stress.

The experimental setup used for finding k_ε and k_σ is already shown in Fig. 3 above.

Linearity refers to the sensor’s ability to exhibit a consistent and proportional change in resistance with increased strain. Ideally, a perfect sensor would show a straight-line relationship on a graph of resistance against strain, indicating that the resistance changes linearly with strain. However, as authors such as [48, 49] have already reported, most resistance-based strain sensors suffer from non-linearity, if only for a specific strain range.

The nonlinearity of sensors makes their calibration process significantly more complicated, time-consuming, and costly. It also leads to reduced precision and resolution and makes overall system integration difficult, requiring additional circuitry, signal conditioning and sophisticated algorithms [29].

To find the Linear portion of the sensors under testing, the linear fit was found by computing the linear regression of the first order of the averaged values of the loading and unloading phases of the sensors. The “fitness” of this linear fit was confirmed in each case by visual inspection and calculating the Coefficient of Determination (R²). The relationship between the change in resistance and strain was only considered linear for the strain range of values when R² was significantly large (R² ≥ 0.9). This region was considered the Working Range of the sensor.

Resistive-type strain sensors with a wide working range are preferred because they can measure a broader spectrum of strain input values, allowing them to be used in diverse applications. They also tend to be more cost-effective, as using a single sensor with a wide range is better than procuring, installing, and maintaining multiple sensors to cover different measurement ranges.

In some cases, a more prominent gauge factor may limit the range of strain that can be accurately measured [50]. In such instances, the relationship between the change in resistance and the strain tends to become non-linear after applying a certain amount of strain. This behaviour is typical of strain-based resistive sensors, as demonstrated by the works of authors such as [51–53]. For the context of this work, the working range was the portion of the graph in which the resistance change and the change in strain were proportional.

While evaluating these three sensors, the preferred sensor would have a wider working range.

The hysteresis of a resistive-type strain sensor is the phenomenon where the output electrical resistance of the sensor for a given strain on the loading cycle is not equal to the electrical resistance for that same strain on the unloading cycle [54]. Ideally, the sensor would have the same resistance at that strain point regardless of whether it is loading or unloading, so one strain value can always be measured with one resistance value without knowing which cycle the sensor is in.

The hysteresis, represented by γ_H, can be given as a percentage given by:

${\gamma }_H=\frac{A_{\mathrm{L}}-A_{\mathrm{U}}}{A_{\mathrm{L}}}\times 100$(5)

where A_L and A_U are the areas of the Loading and Unloading characteristic curves, respectively.

When this equation cannot be used to find the value of hysteresis, one could approximate the sensor with the most prominent hysteresis by visual inspection. This is especially the case when comparing the hystereses of a limited number of sensors. The sensor whose resistance-strain graph has the most significant deviation from the characteristic curve has the most considerable hysteresis and vice-versa.

The hysteresis of a resistance-type strain sensor can be attributed to various factors, including material properties, mechanical design, and manufacturing processes. Works such as those by [55–57] go into a detailed investigation concerning this, whereas [58, 59] discuss some of the possible means, at least through fabrication considerations, to mitigate this.

A large hysteresis is undesirable because it can introduce errors in strain measurements by reducing the sensors’ accuracy, reliability, and stability, especially when the strain is applied cyclically or in dynamic applications [2]. Therefore, in this work, a sensor with a small hysteresis was preferred to one with a large hysteresis.

All these tests mentioned above were performed at room temperature.

Usually, strain-based sensors are tested and calibrated at room temperature (approx. 20 °C to 25 °C), or authors are completely mute about that aspect in their works. However, the intended end use for sensors was for wearable devices on a human body with a core temperature of around 36.5 °C or much higher than that in infants or when an adult patient is febrile [60]. Furthermore, some resistance-based strain sensors have been reported to increase in temperature over time as current passes through them in a process called self-heating caused by the excitation voltage [61, 62]. The work by [63, 64] concludes that temperature variations may introduce errors in resistance-strain readings if not appropriately compensated for.

This work investigated the relationship between temperature and electrical resistance output for each of the sensors at their unstretched length and Load of 0.0N.

In this investigation, the temperature was varied from approx. 0 °C to about 65 °C. This temperature range was of interest as the operation temperature range (for wearable devices) is well within this range, even after accounting for the temperature increase that has been associated with some resistance-based strain sensors mentioned earlier.

The varying temperatures were measured by a K-type thermocouple (nickel–chromium–Nisil), with a range from –20 °C to 1000 °C, a resolution of 1 °C and an accuracy and count of ±(1.0%+5).

The thermic ascent to the sensors was caused and controlled by altering the height of an infrared lamp above the sensors, as shown in Fig. 4 below. It was inspired by the setup in the work by [64].

Figure 4.

The experimental setup for thermal ascent

The relationship between temperature and resistance is called the Temperature Coefficient of Resistance (TCR), represented by α_t. The TCR indicates how much the resistance of a sensor changes with respect to temperature [65]. For a temperature range that is not too great, this variation can be represented approximately as a linear relation:

R_T = R_T0[1 + α_T (T – T₀)], (6)

where R_T and R_T0 are the resistances at temperatures of T and T₀, respectively. T₀ is often taken as room temperature or 0 °C. Rearranging Eq. (6) above gives:

$\frac{R_T^{\mathrm{\infty }}-R_{T_0}}{R_0}={\alpha }_T(T-T_0)$(7)

Which can also be presented as:

$\frac{\mathrm{\Delta }R}{R_{T_0}}={\alpha }_T\mathrm{\Delta }T,$(8)

where ∆R is the resistance change of the sensor due to a temperature change of ∆T. This means that from a graph plotted of ∆R/R_T0 against ∆T, the gradient will give the Thermal Coefficient of Resistance.

To confirm that the resistance change correlated to temperature change, the authors recorded, plotted, and compared the values for resistance when the temperature was being increased from minimum to maximum with those when the temperature was being decreased in the reverse order. The cooling of the sensors was done by placing each of the sensors’ samples in turn atop a generic 12706 thermoelectric cooling module that makes use of the Peltier principal to actively cool the temperature of one side of its two plates to as low as –30 °C [66, 67].

The sensor with the most negligible absolute Temperature Coefficient of Resistance is a more desirable one as it indicates that the sensor has greater thermal stability, meaning it exhibits minimal electrical resistance changes with temperature variations. This stability allows for more accurate and reliable temperature measurements over a wide range of temperatures. It also means less design and calibration considerations must be put towards temperature compensation when integrating the sensor for most real-world applications.

Although the effects of moisture on resistive-type strain sensors have been thoroughly investigated in works such as [68–73], in this body of work the authors will compare the differences in the Resistance change vs. Strain graphs for the wet, irrigated sensor, versus one that is dry (at a relative humidity of between 20 to 50%) to identify the sensor who’s resistance output might be affected the most by the various moisture for reasons already highlighted in works such as [74], i.e. how water molecules may sometimes act as an electrolyte and introduce conductive pathways.

This was an important aspect to investigate as the end use of the sensor will be as a wearable device and things such as human sweat tend to occur, not to mention the relative humidity that spontaneously may rise in the atmosphere from time to time. The sensor with the Change in Resistance vs. Strain that varies the least (ideally identical to) when the sensor is subject to moisture vs. when under dry conditions may be considered the most desirable.