Reversible viscoelasticity and irreversible elastoplasticity in the power law creep and yielding of gels and fibre network materials under stress

Abstract

We study computationally the creep and yielding of athermal gels and fibre network materials under a constant imposed shear stress, within a minimal model of interconnected filaments with central forces in $d=2$ spatial dimensions. Each filament is assumed Hookean initially, then breaks irreversibly above a threshold strain. At early times after the imposition of a small stress, we find purely viscoelastic creep response associated with non-affine deformations within the material, with solid terminal behaviour for a network coordination $Z>2d=4$ and initially floppy response for $Z<4$. For a marginally connected network, $Z=4$, we find sustained power law creep with a strain rate $\dotγ\sim t^{-1/2}$ and strain $γ\sim t^{1/2}$ as a function of time $t$ after the imposition of the stress. This viscoelastic regime gives way at later times to irreversible elastoplastic creep arising from filament breakage, broadening the range of values of $Z$ and time over which power law creep occurs, compared to a network with filament breakage disallowed. This accumulating damage can weaken the network to such an extent that catastrophic material failure then occurs after a long delay, which we characterise. Finally, we consider the implications of viscoelastic versus elastoplastic deformation for the extent to which a material will recover its original shape if the load is removed after some interval of creep.

Figures (10)

• Figure 1: Validation of our numerical method for simulating a random network at constant imposed stress. a) Strain versus time using algorithm I for viscosity $\eta=0.00, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64, 1.28$ in curves left to right at an imposed stress $\Sigma = 0.18095$. b) Strain versus time computed using algorithm II (solid lines) and algorithm I (dashed lines) for viscosity $\eta = 0.16$ and stress $\Sigma = 0.154, 0.177, 0.200, 0.223$ in curves upward. Filament breakage threshold $b = 1$ and initial network connectivity $Z=3.5$ in both panels.
• Figure 2: Steady state strain attained at long times $t\to\infty$ as a function of imposed stress $\Sigma$, without filament breakage, $b\to\infty$.
• Figure 3: Creep without filament breakage, $b\to\infty$. Strain (left panels a+c) and strain rate (right panels b+d) against time for several imposed stress values $\Sigma = {0.00005, 0.0001, 0.0002, 0.0004, 0.0008, 0.0016, 0.0032}$ in curves from black to violet. Network connectivity $Z=4.50$ (top panels a+b, curves for different stresses indistinguishable) and $Z=3.75$ (bottom panels c+d). Axes have been scaled in each panel to achieve data collapse to a master scaling curve in the limit $\Sigma\to 0$, as discussed in the main text.
• Figure 4: Creep without filament breakage, $b\to\infty$. a) Strain rate versus time for an imposed stress $\Sigma = 3.162$x$10^{-5}$ and several values of network connectivity $Z$. Blue and black dashed lines show power law fits to early and late power law regimes respectively. Green dots show crossover points between these two regimes. b) Exponent of early and late power law regimes versus $Z$ c) Crossover time between early and late regimes versus $Z$.
• Figure 5: Creep and yielding with filament breakage. a) Strain versus time for imposed stress values $\Sigma=0.0300,0.0305,0.0310,\cdots 0.0350$ in curves right to left. Dashed line indicates the yield strain $\gamma_{\rm y}$, defined as the strain attained at long times for the largest stress value for which yielding does not occur. Crossed indicate the yielding time $\tau_{\rm y}$ for all stress values for which yielding does occur. b) Differentiated creep curves of strain rate versus time. c) Parametric plot of strain rate versus strain. d) Fraction of filaments broken as a function of strain. Filament breakage threshold $b = 0.1585$ and initial network connectivity $Z = 4.5$ and in all panels.