Unraveling Brittle-Fracture Statistics from Intermittent Patterns Formed During Femtosecond Laser Exposure

Femtosecond-laser-written patterns at the surface of brittle materials may show at random times spontaneous alternations from regular to disordered structures and vice versa. Here, we show that these random transitions carry relevant statistical information, such as the Weibull parameters characterizing the fracture of brittle materials. The regular-erratic cycles of random lengths of the observed patterns suggests a phenomenological analogy with the idle and busy periods arising in queuing systems. This analogy enables us to establish experimentally that the random durations of the successive cycles are statistically independent. Based on these observations, we propose an experimental method bypassing the need for many specimens to build up statistically relevant ensembles of fracture tests. Our method is potentially generic, as it may apply to a broad number of brittle materials.

Figure 1

(a) SEM image of the laser-processed line indicating the scanning direction ($v$), the laser-light polarization ($E$), the laser propagation direction vector ($k$), and the pattern that is written on the surface of a fused silica substrate. (b) and (c) show a magnified view of some line pattern transitions between regular and chaotic structures and vice versa. The black double arrows indicate regular patterns, while the red ones emphasize chaotic regimes. The pulse energy is $196\pm 1\mathrm{nJ}$. There are about 40 pulses overlapping each unit volume producing a net fluence of $3\mathrm{J}/{\mathrm{mm}}^2$. In all these experiments, the focusing objective numerical aperture is 0.4.

Figure 2

Overview of the transitions for different energy depositions and pulse energies for three different materials, illustrating the potential universality of this intermittent phenomenon for brittle materials. Black regions represent the self-organized patterns and the red ones the chaotic patterns.

Figure 3

(a) Schematic illustration of the femtosecond laser ellipsoid which corresponds to the laser-affected volume. The waist of the ellipsoidal is focused on the material-air interface, producing patterns on the surface of the material. (b) The average stress between the nanoplanes in fused silica for the case of 196 nJ per pulse at 800 kHz. The deposited energy is “tuned” by changing the translation velocity of the laser.

Figure 4

(a) Each element formation can be seen as an individual tensile-stress loading experiment. A testing cycle ($C$) is terminated a fracture occurs. (b) Illustration of the loading conditions for a single nanoplane formalized as an Inglis fracture problem 13].

Figure 5

The graph represents the probability of fused silica to fail under a certain stress level. The two curves correspond to two different surface qualities. For both cases, the 95% confidence band is given by the equation $x\pm Z\times \sigma /\sqrt{(n)}$, where $Z$ is the confidence level, $\sigma$ is the standard deviation, and $n$ is the sample size. For the two cases, the relative error along the $x$ axis is 3% of the values and finds its origin in the evaluation of the average stress between the nanoplanes (see Fig. 4). The relative error along the $y$ axis is estimated to be 5% of the values and is related to the uncertainty of the start position of the transition towards an erratic regime. For the sake of clarity, the error bars are not presented in the graphs.