## 2 Electrical Resistivity/Impedance Measurement: Is It Really that Simple?
### Stress and Strain
The same materials combinations was tested by [PERSON] and [PERSON] [49] in conductive asphalt concrete; they noticed that FCR slows down with more loading cycles, meaning that piezoresistivity tends to decrease because of viscoelasticity. Also **steel fibers** can be used. According to [PERSON] and [PERSON] [51], the relationship between the percent change in electrical resistance and tensile strain is almost linear before cracking (R\({}^{2}=0.92\) and R\({}^{2}>0.94\) for mixtures without and with fibers, respectively; the correlation coefficient converges at 0.99 approximately at percolation threshold). The gauge factor increases with fibers volume, converging approximately in correspondence of the percolation threshold (1 vol.%). [PERSON] et al. [53; 54] evaluated different types of steel fibers, namely smooth, twisted, and hooked, noticing that the most effective for enhancing mechanical resistance are those twisted (better meso than macro twisted), followed by smooth and hooked ones. Electrical resistivity decreases with tensile strain until post-cracking, due to the formation of multiple cracks due to strain-hardening. **Cementitious strain sensors** can be realized to be embedded in structures to be monitored, providing a high monitoring efficiency in continuous, low-cost, high durability, and simple construction technology to be embedded only in critical points of concrete structures, forming a distributed monitoring network. Strain sensing provides valuable information for the control of structural vibrations, traffic monitoring, and weighing [213]. [PERSON] and [PERSON] [121] highlighted a non-linear and rate-dependent relationship between change in resistivity and compressive stress, measured in electrically conductive cementitious composites with carbon fibers and CNTs (better when used in conjunction). Also, [PERSON] et al. [38] used CNTs to realize sensors to be embedded on a concrete beam (\(250\times 250\times 2200\) mm) at a distance of 250 mm from each other, forming a durable distributed SHM solution able to provide results in agreement with traditional strain gauges. [PERSON] and [PERSON] [67] employed carbon fibers and carbon black to realize piezoresistive sensors embedded in concrete structures. They obtained a linear relationship between FCR and compressive stress (y = -1.35x); another linear relationship was established between FCR and compressive strain (y = \(-\)0.0227x). The response of the contact resistance to elastic deformation is reversible [214], whereas becomes irreversible in the case of plastic deformation. [PERSON] et al. [40] developed cementitious sensors, manufactured with the addition of CNT/NCB (nano carbon black) composite filler, to incorporate in concrete columns. These sensors presented high stability and repeatability within the elastic regime, providing a polynomial relationship between stress/strain and FCR (y = 7.743\(\cdot 10^{-5}\)x\({}^{3}-0.005\)x\({}^{2}\) + 0.510x, with R\({}^{2}\) = 0.999 and y = 0.018x\({}^{3}-0.759\)x\({}^{2}\) + 28.00751 with R\({}^{2}\) = 0.997 for stress and strain, respectively). During loading, FCR decreases; no damages to the sensor are caused by the compressive stress. The change of electrical resistance is reversible upon loading and unloading phases, even if the amplitude of FCR is slightly different in subsequent load cycles; under monotonic loading, when the sensor ultimate stress has been reached, FCR remains quite stable. Similarly, [PERSON] et al. [55] developed highly sensitive piezoresistive sensors using nickel powder-filled cement-based composites; they found that electrical resistivity decreases monotonously as compressive load increases, showing a decrease by approximately 63% in the elastic regime (investigated range: 0-12.5 MPa). FCR is highly correlated to compressive force (Boltzmann sigmoidal regression curve, R\({}^{2}\) > 0.99). Nickel powder enables contacting and tunneling conduction effects, causing a decrease in the electrical resistivity, besides enhancing the sensitivity to stress/strain. [PERSON] et al.