Creep rupture of fiber bundles: A molecular dynamics investigation

The creep deformation and eventual breaking of polymeric samples under a constant tensile load $F$ is investigated by molecular dynamics based on a particle representation of the fiber bundle model. The results of the virtual testing of fibrous samples consisting of $40000$ particles arranged on $N_c=400$ chains reproduce characteristic stages seen in the experimental investigations of creep in polymeric materials. A logarithmic plot of the bundle lifetime $\tau$ versus load $F$ displays a marked curvature, ruling out a simple power-law dependence of $\tau$ on $F$. A power law $\tau \sim F^{-4}$, however, is recovered at high load. We discuss the role of reversible bond breaking and formation on the eventual fate of the sample and simulate a different type of creep testing, imposing a constant stress rate on the sample up to its breaking point. Our simulations, relying on a coarse-grained representation of the polymer structure, introduce new features into the standard fiber bundle model, such as real-time dynamics, inertia, and entropy, and open the way to more detailed models, aiming at material science aspects of polymeric fibers, investigated within a sound statistical mechanics framework.

Figure 1

Snapshots from a simulation at $T=0.33$, $F=40$. Black particles: Regular beads, twofold coordinated; blue particles: onefold coordinated beads, marking broken bonds; red particles: chain terminations on the $z=z_a$ and $z=z_b$ planes. For the sake of clarity, only $1/4$ of the chains are displayed. (a) $t/{\tau }_i=0.025$; (b) $t/{\tau }_i=0.5$; (c) $t/{\tau }_i=0.66$; (d) $t/{\tau }_i=0.975$.

Figure 2

Average number of broken bonds as a function of simulation time in samples at $T=0.33$, $F=0$, 20, 40 and 60. Inset: Slope of the linear portion of $n_{bb}\left(t\right)$, estimated by linear fit of the $180\le n_{bb}\left(t\right)\le 220$ range. The range of the linear fit is delimited by the two horizontal dashed lines in the main plot. Results averaged over all the samples simulated at the same load, see text.

Figure 3

Number of broken chains as a function of simulation time in individual samples at $T=0.33$ under the tensile load $F=10$, $F=20$, $F=30$. No averaging has been carried out in this case.

Figure 4

Number of broken chains as a function of scaled time at five values of the tensile load $F$. ${\tau }_i$ is the time to break individual bundles, see text. The results have been averaged over all the samples under the same tensile load. From top to bottom: $F=0$, 10, 20, 40, 60.

Figure 5

Number of broken bonds and of broken chains as a function of time in a sample at $T=0.33$ under load $F=20$. The full line is the prediction of Eq. (23) in the text.

Figure 6

Relative fraction of the bundle lifetime accounted for by the avalanche stage. The avalanche starts at $t_{\mathrm{av}}$, defined as the time at which the time dependent rate of chain breaking equals two times the average rate $N_c/\tau$. Inset: Fraction of chains in the bundle that break during the avalanche stage.

Figure 7

Probability distribution $P\left(\delta t\right)$ for the time interval between two successive chain breaking events. The red curve refers to events taking place when the number of broken chains is less than $50\%$, and the blue dashed curve refers to the remaining $50\%$ of chain breaking events and includes the effect of the final avalanche. Average over 200 bundles of $N_c=400$ chains of $N_b=100$ at $T=0.33$ under load $F=40$. Inset: Detail of $n_{bc}\left(t\right)$ for a single bundle at $T=0.33$ under load $F=40$ in the early stages of stretching. Short bursts of correlated chain breaking events are shown in red (full line), while regular steady segments are shown in blue (dashed line).

Figure 8

Bundle elongation as a function of time at four values of the tensile load. Data in the main panel have been scaled and averaged as in Fig. 4. From bottom to top: $F=0$, 20, 40, 80. Inset: Logarithmic plot of the strain rate $\dot{\gamma }\left(t\right)=d|z_b-z_a|/dt$ for bundles at $T=0.33$ and $F=40$. Dots: Simulation data. Straight line (blue): Linear interpolation of the $0\le t/\tau \le 0.027$ data, corresponding to a power-law dependence $\dot{\gamma }\left(t\right)\sim t^{-\alpha }$; red curve: Interpolation of the $0.95\le t/\tau <1.0$ data with a polelike term $\dot{\gamma }\left(t\right)\sim \mu /(1-t/\tau )$ diverging at $t=\tau$.

Figure 9

Bundle elongation as a function of time at tensile load $F=20$. Results refer to a single bundles; no averaging over samples has been carried out.

Figure 10

Lifetime of fiber bundles as a function of the tensile load. Filled squares: Constant temperature simulations with a Langevin thermostat at $T=0.33$ and $\eta =0.0001$. Full dots: Newtonian simulations (no friction and no random forces) started from samples equilibrated at $T=0.33$. Blue and red lines are a guide to the eye.

Figure 11

Stress $f_1$ versus bundle length $L_z$ in samples under load $F$ from 5 to 40. The definition of $f_1$ contains an extra term accounting for the internal pressure of the bundles, see Eq. (26).

Figure 12

Relative uncertainty on the bundle lifetime $\tau$ as a function of the tension on the bundle. $\mathrm{\Delta }\tau$ is the standard deviation of breaking times. Red dots: Langevin simulation; blue squares: Newtonian simulation. The continuous lines are quadratic fits to the simulation data given as a guide to the eye.

Figure 13

Probability distribution of the stress parameters $\varphi$ (red line) and $\nu$ (blue line) on bonds in a sample under load $F=20$. $\varphi$ is the component of the force ${\mathbf{F}}_{\mathbf{i}+\mathbf{1}}-{\mathbf{F}}_{\mathbf{i}}$ on the $(i,i+1)$ bond directed along the ${\mathbf{r}}_{\mathbf{i}-\mathbf{1}}-{\mathbf{r}}_{\mathbf{i}}$ direction. $\nu$ is defined in a similar way but excludes the contribution from the direct $(i,i+1)$ interaction. Inset: Integral parameters $I_{\nu }^{+}$ and $I_{\nu }^{-}$ [see Eq. (29)] as a function of time for three different values of the applied load. $I^{+}$ and $I^{-}$ represent the fraction (in percentages) of bonds whose $\nu$ parameter exceeds the threshold value (${\nu }^{+}>a_0{D}_0$; ${\nu }_{-}<-a_0{D}_0$) for bond breaking (see text).

Figure 14

Work $W_{90}$ performed by the external load to break $90\%$ of the bundle chains (see text). The dashed line at $W_{90}=2400$ corresponds to the energy $D_0\times N_c$ required to break $N_c$ bonds, thus representing the minimum energy needed to break a bundle. Inset: Ratio of temperatures at the end ($T_f$) and at the beginning ($T_i$) of the simulations based on Newton's equations of motion.

Figure 15

Simulations with bond recombination mechanism (see text): The number of broken chains as a function of time in two samples under tensile load above [(a) $F=50$] and below [(b) $F=40$] the critical load $F^{*}=48$. Inset: Time dependence of the number of broken bonds in three distinct samples under the same (slightly subcritical) load $F=45$.

Figure 16

Stress-strain relation in samples simulated by Langevin dynamics at four values of constant strain rate ($R$) testing.