Crackling Dynamics in the Mechanical Response of Knitted Fabrics

Crackling noise, which occurs in a wide range of situations, is characterized by discrete events of various sizes, often correlated in the form of avalanches. We report experimental evidence that the mechanical response of a knitted fabric displays such broadly distributed events both in the force signal and in the deformation field, with statistics analogous to that of earthquakes or soft amorphous materials. A knit consists of a regular network of frictional contacts, linked by the elasticity of the yarn. When deformed, the fabric displays spatially extended avalanchelike yielding events resulting from collective interyarn contact slips. We measure the size distribution of these avalanches, at the stitch level from the analysis of nonelastic displacement fields and externally from force fluctuations. The two measurements yield consistent power law distributions reminiscent of those found in other avalanching systems. Our study shows that a knitted fabric is not only a thread-based metamaterial with highly sought after mechanical properties, but also an original, model system, with topologically protected structural order, where an intermittent, scale-invariant response emerges from minimal ingredients, and thus a significant landmark in the study of out-of-equilibrium universality.

Figure 1

Experimental system and typical force response. (a) A knitted fabric is stretched uniaxially while its mechanical response is recorded and the position of the stitches is tracked through digital image processing, with a precision of approximately $10\mu \mathrm{m}$. Typical picture of the knit; the stretching direction is materialized by the arrow associated to the force $F$, while $L$ denotes its elongation. The scale bar is 25 mm long. Inset: Enlargement over a few stitches tagged by a red dot indicating their position defined as their geometric center. Here the scale bar is 4 mm long. (b) Typical mechanical response of the fabric while stretched between $L_m=230\mathrm{mm}$ and $L_f=234\mathrm{mm}$ at a constant speed of $v=5\mu \mathrm{m}/\mathrm{s}$. Stick-slip events at the contact points generate an intermittent signal, typical of crackling dynamics. Inset: Enlargement over a small interval, the definition of $\mathrm{\Delta }f$ is emphasized, and the two vertical lines distant by $\mathrm{\Delta }L=0.2\mathrm{mm}$ point to elongations at which two successive pictures of the fabric are taken.

Figure 2

Local detection of slip events. (a) Displacement field corresponding to the inset in Fig. 1. Each stitch is tagged by its nonaffine displacement $\overrightarrow{u}$, portrayed by red arrows magnified by a factor of 35. Black scale bar (position), 25 mm; red scale bar (displacement), 0.6 mm. (b) Vorticity $\omega$ and (c) deviatoric strain ${ϵ}_d$ of the displacement field in the stitch network. Each arrow of the displacement field is associated to a single stitch.

Figure 3

Event size distribution measured from external and internal quantities. (a) Probability distribution of global event size measurement $\mathrm{\Delta }f$. The dotted line is the best linear fit with a slope of $-1.50\pm 0.03$. Inset: For each interval between two images, the sum of event sizes measured externally is compared to the sum of events measured internally. (b) Probability distribution of local event size measurements from vorticity $S_{\omega }$ and deviatoric strain $S_d$ of the displacement field. Dotted lines are the best linear fit with a slope of $-1.51\pm 0.05$ and $-1.61\pm 0.03$ for, respectively, $S_{\omega }$ and $S_d$. Inset: Comparison between the sum of $S_{\omega }$ and the sum of $S_d$ for each interval. The loading speed for the data shown in this figure is $v=5\mu \mathrm{m}/\mathrm{s}$. The uncertainty in the exponents is evaluated from the standard error and a 95% confidence interval.

Figure 4

Experimental spatial correlation and theoretical elastic deformation of positive events. (a) Amplitude of the correlation function $C_{\omega }^{+}$ of the vorticity, showing the events propagating along diagonal stitches, together with the relative displacement field during positive events ${\overrightarrow{\tilde{u}}}_{+}(\delta x,\delta y)=⟨\overrightarrow{u}(x+\delta x,y+\delta y)-\overrightarrow{u}(x,y){⟩}_{+}$. Data shown for $v=5\mu \mathrm{m}/\mathrm{s}$. (b) Analytical response of the elastic network to a local perturbation shown through the displacement field around the perturbation, along with the angular dependence of the vorticity field $r^2\omega (r,\theta )$. The direction of maximum vorticity redistribution matches the orientation of the slip lines, suggesting that the events are formed by triggered successive slips.