History effects in the creep of a disordered brittle material

We study the creep behavior of a disordered brittle material (concrete) under successive loading steps, using acoustic emission and ultrasonic sensing to track internal damage. The primary creep rate is observed to follow a (Omori-type) power-law decay in the strain rate, the number of acoustic emission events, as well as the amplitudes of the ultrasonic beams, supporting a brittle-creep mechanism. The distribution of acoustic emission event energies is observed to have a scale-free power-law distribution instead of a truncated one expected for a system approaching a critical point at failure. The main outcome is, however, the discovery of unexpected history effects that make the material less prone to creep when it has been previously deformed and damaged under primary creep. With the help of a progressive damage model implementing thermal activation, we interpret this as an aging-under-stress phenomenon: During an initial creep step at relatively low applied stress, the easy-to-damage sites are exhausted first, depleting the excitation spectrum at low stress gap values. Consequently on reloading, although previously damaged, the primary creep restarts but the material creeps (and damages) less than it would under the same stress but without precreeping. Besides shedding a new light on the fundamental physics of creep of disordered brittle materials, this has important practical consequences in the interpretation of some experimental procedures, such as stress-stepping experiments.

Figure 1

(a) Strain $\epsilon$ accumulated after the start of each creep step as a function of time $t$ after the start of the step at $t_r$. (b) Same as (a) for the number of acoustic events $n$. (c) Same as (a) for the damage parameter $D$, averaged over the sample. (d) Same as (a) for the simulations. The black lines represent fits according to Eq. (1).

Figure 2

The distribution of the Coulomb stress gaps $\mathrm{\Delta }\sigma$ at (a) $t-t_r=0.35$ s, (b) $t-t_r=10$ s, and (c) $t-t_r=700$ s in the simulations. Blue dots correspond to the initial step at $0.80\times {\sigma }_c$, red dots to the step at $0.85\times {\sigma }_c$, and yellow ones to $0.95\times {\sigma }_c$.

Figure 3

The cumulative distribution of AE event energies $E_{\mathrm{AE}}$ for each of the creep steps in one experiment. The inset shows the values of the power-law exponent $\beta$ obtained through a maximum likelihood estimation for all experiments.

Figure 4

A reference experiment with C-concrete where all the steps are performed at the same stress level (equal durations, unloading between steps). (a) The creep rates decrease quickly with increasing number of steps, and after around four steps the difference between steps becomes very hard to see. The black lines are fits corresponding to Eq. (1). (b) The number of acoustic events follows the behavior of strain in a linear fashion. The black lines are linear transformations of fits shown in panel (a). (c) The cumulative distribution of AE event energies shows a power-law distribution spanning several orders of magnitude for each creep step. The black line corresponds to $\beta =3/2$. (d) The exponent $p$ of the strain fits starts at a value $p\sim 1$ for the first steps (corresponding to logarithmic creep) and quickly decreases to a lower roughly constant value $p<1$. (e) The energy distribution power-law exponent $\beta$ is roughly constant, around the mean field value $\beta =3/2$. The increased variance in the latter steps is due to the reduced number of events in these steps. (f) The difference of the BICs for the power-law and truncated power-law models in the energy distribution fits shows that the truncated power law is preferred only for the first and third steps.

Figure 5

A schematic picture of the loading protocol used. After an initial ramp of constant stress rate (ending at time $t_r^1$) the stress is kept constant at ${\sigma }_1$ for a duration $T_1$. The stress is then increased with the same stress rate to ${\sigma }_2$ for a duration of $T_2$ and so forth. The test ends when the sample fails either during a creep step (stress step) or during one of the ramps (at the stress ${\sigma }_c$).

Figure 6

(a) The geometry of the sample cross section showing the initially cylindrical samples (radius $r$) and the three flattened sides. (b) Schematic illustration of the operating principle of the ultrasonic array, showing the sample between the source and receiver arrays, where the white dots represent individual transducers (for clarity an array with 20 transducers is drawn). The black lines represent the direct path from one source transducer to each receiver transducers. (c) pp1n image of the sample in the compression setup, showing the attachment of the ultrasonic arrays, as well as the heterogeneity of the sample microstructure. The height of the sample is denoted in the image as $h$.

Figure 7

An illustration of the Coulomb failure criterion used, represented in the Mohr plane. The stress gap $\mathrm{\Delta }\sigma$ is defined as the distance from the current stress state (circle defined by the applied normal stress ${\sigma }_1$) to the failure envelope (defined by the line with a slope $tan\phi$ and $y$ intercept $C$).

Figure 8

Figure 1 with double logarithmic axes.

Figure 9

Residuals of the fits shown in Fig. 1, and an additional one where Eq. (1) is fitted to the last creep step with fixed $p=1$.

Figure 10

The evolution of the $p$ exponent in Eq. (1) as a function of the scaled applied stress. The triangle symbols correspond to the simulations and the circles to the experiments.

Figure 11

The evolution of the difference of the BICs for the power-law and truncated power-law models [Eq. (E6)] in the experiments as a function of the scaled applied stress, showing that the truncated power law is significantly favored in only one of the stress steps.

Figure 12

An additional experiment (number 2) with C-concrete showing that the behavior with this microstructure is very similar to the M-concrete. (a) After an initial stress step the creep is slower and as the second step is fairly long, the creep is even slower in the third step. (b) The number of acoustic events shows behavior which is very similar to the time evolution of the strain.

Figure 13

An additional experiment (number 3) with M-concrete, showing the effect of an extremely long initial stress step (real duration $64036$ s, plot truncated for clarity) on subsequent, short stress steps. (a) After the extremely long initial step the next (fairly short) step shows very slow creep and in the third step the creep rate is only roughly equal to the first one. (b) Except for one sudden burst of events in the first stress step, the number of acoustic events roughly follows the behavior of the strain.

Figure 14

A reference experiment (number 4) with M-concrete where all the stress steps are kept short (equal durations). (a) The creep rates for the first three steps are roughly equal and after that start to increase. This behavior is not consistent with the power-law model $\dot{\epsilon }\propto {\sigma }^n$. (b) Also here the behavior of the number of acoustic events roughly matches the behavior of strain. There is again a sudden burst of events close to the end of one stress step.