# Detecting Criticality by Exploring the Acoustic Activity in Terms of the \"Natural-Time\" Concept
## 3 Analysis of the Acoustic Activity by Means of the \(F\)-Function in the Natural Time Domain
### Uniaxial Compression of Intact Prismatic Marble Specim  
Intact prismatic Dionysos marble specimens (dimensions 45 mm \(\times\) 45 mm \(\times\) 100 mm) were subjected to uniaxial compression. A strain gauge was attached at the center of one of the specimens' lateral surfaces in order to measure the axial strain developed, while an acoustic sensor was attached at the center of the adjacent lateral surface (Figure 6a). The compressive force was applied at a constant rate equal to 475 N/s. During the experiments, the force and the strain developed were recorded at a sampling rate of 75 values per minute. In addition, the acoustic signals of amplitude \(A\geq\) 45 dB were recorded. The time evolution of their amplitudes is shown in Figure 6b together with the respective evolution of the force applied. The axial stress-axial strain (\(\sigma\)-\(\epsilon\)) curve for a typical experiment of the protocol is shown in Figure 7a. The linear relationship, expected for brittle rocks, characterizes the major portion of the \(\sigma\)-\(\epsilon\) curve, corresponding to Young's modulus equal to about 70 GPa (the initial non-linearity for \(t<7\) s is attributed to unavoidable bedding errors). The time evolution of the axial strain is shown in Figure 7b together with the respective linear fitting, which governs the response of the material up to the \(t=\) 181 s time instant. At this time instant, the stress level attained is equal to about 84 MPa, which corresponds to about 88% of the specimen's compressive strength. For the specific experiment more than 1000 acoustic hits were recorded until the fracture of the specimen, most of which (more than 50%) were recorded after the specimen abandoned its linear response. Taking into account the overall number of hits recorded the sliding window used for the calculation of the \(F\)-function included \(n=50\) consecutive hits.  
The evolution of the \(F\)-function in Natural Time \(\chi\) is shown in Figure 8 in juxtaposition to the respective evolution of the axial strain recorded. In the same figure, the linear fit of the axial strain in Natural Time terms is plotted. It is clearly seen from this figure that, as soon as the material ceases responding linearly, the evolution of the \(F\)-function in the Natural Time domain is, again, governed by the power law of Equation (2).  
Proper fitting of the data gives the exponent \(m\) a value equal to \(m=5.26\). The range of applicability of the power law covers the interval from \(\chi=0.58\) to \(\chi=0.93\). The corresponding conventional-time interval extends from \(t=189\) s to \(t=198\) s, while the corresponding values of the axial stress are 89.9 MPa and 94.2 MPa, respectively, or, equivalently, 94% and 99% of the material's compressive strength.  
Figure 5: The evolution in the Natural Time domain of the \(F\)-function and the load imposed for the notched specimens under three-point bending.  
Figure 6: **(a)** Sketch of the intact prismatic specimen. The red arrow indicates the direction of the imposed load and the green rectangle the strain gauge. **(b)** The temporal evolution of the amplitudes of the acoustic hits in juxtaposition to that of the compressive force imposed.  
Figure 7: **(a)** The axial stress versus the axial strain for the intact prismatic marble specimen. **(b)** The temporal evolution of the strain for the intact prismatic marble specimen.