Fluctuations and Scaling in Creep Deformation

The spatial fluctuations of deformation are studied in the creep in Andrade’s power law and the logarithmic phases, using paper samples. Measurements by the digital image correlation technique show that the relative strength of the strain rate fluctuations increases with time, in both creep regimes. In the Andrade creep phase characterized by a power-law decay of the strain rate ${ϵ}_t\sim t^{-\theta }$, with $\theta \approx 0.7$, the fluctuations obey $\Delta {ϵ}_t\sim t^{-\gamma }$, with $\gamma \approx 0.5$. The local deformation follows a data collapse appropriate for a phase transition. Similar behavior is found in a crystal plasticity model, with a jamming or yielding transition.

Figure 1

The experimental scenario. From a pair of digital images at a time interval $\Delta t$, the local deformations are extracted at a grid of $N\times M$ points, with $N=49$ and $M=39$ on the lower half-plane of the sample. The strain fluctuations are measured via the time-dependent standard deviation, and compared to the mean creep rate, here in primary or Andrade creep. In the DDD simulations, the system is similarly divided into boxes of linear size $l$, with local strain rates ${ϵ}_t^{i,j}$. A digital image of a paper sample on a scale of $40{\mathrm{mm}}^2$. Superposed is a typical deformation grid for a time difference $\Delta t$ of 10 s. The color scale indicates the degree of local creep deformation (blue, small; red, large). In the background is the experimental setup. The visible speckle pattern is printed and designed to have a structure and contrast appropriate for the DIC method.

Figure 2

Average creep rate and the standard deviation of the local creep rates. (a) Typical experimental data. The sample average lifetime is 800–1600 s. The primary-to-secondary creep transition takes place continuously at 150–200 s. Sixteen experiments are used. The effective Andrade exponent is about 0.7 in the range 10–50 s, and becomes close to 1 (signaling secondary creep) above 150 s. (b) The Andrade law and the fluctuation scaling for the DDD model for three different box sizes to compute the local rates, at $\sigma \approx {\sigma }_c$. The early-time crossover in the fluctuations is due to computation time interval of $\Delta t=100$.

Figure 3

(a) Experimental results from logarithmic creep using a smaller imaging area, 5 s image intervals, and a smaller load, showing the average creep rate and the fluctuations. The former decays faster, and at the end these two become comparable, on the observation scale. The data are an average over 9 experiments. (b) Data collapse of the primary creep strain distribution $P(ϵ,t)$ from the experiments. The local deformation distributions can be collapsed using the creep strain as the scaling parameter.