Blending stiffness and strength disorder can stabilize fracture

Quasibrittle behavior, where macroscopic failure is preceded by stable damaging and intensive cracking activity, is a desired feature of materials because it makes fracture predictable. Based on a fiber-bundle model with global load sharing we show that blending strength and stiffness disorder of material elements leads to the stabilization of fracture, i.e., samples that are brittle when one source of disorder is present become quasibrittle as a consequence of blending. We derive a condition of quasibrittle behavior in terms of the joint distribution of the two sources of disorder. Breaking bursts have a power-law size distribution of exponent $5/2$ without any crossover to a lower exponent when the amount of disorder is gradually decreased. The results have practical relevance for the design of materials to increase the safety of constructions.

Figure 1

Failure plane of fibers for uniformly distributed threshold values. Each point with parameter values $(E,{\sigma }_{\text{th}})$ inside the rectangle of side lengths $E_{+}-E_{-}$ and ${\sigma }_{+}-{\sigma }_{-}$ represents a fiber of the bundle. The equation of the dashed straight line is ${\sigma }_{\text{th}}=E\varepsilon$ so that fibers above the line fullfill the condition ${\sigma }_{\text{th}}>E\varepsilon$. At a given strain $\varepsilon$ during the loading process those fibers are intact (highlighted with gray color) and keep the external load, which fall above this line of slope $\varepsilon$. The integral in Eq. (3) has to be performed over the domain of intact fibers.

Figure 2

Macroscopic response of a system where both sources of disorder are uniformly distributed with the parameter $W_E=0.5$ for three different values of $W_{\sigma }$. Different symbols and colors are used to highlight the regimes corresponding to different terms of the integral expression Eq. (5): bold line (red), circle (blue), square (green), and triangle (magenta) stand for the contributions of the first, second, third, and fourth terms of Eq. (5).

Figure 3

The fraction of fibers $P_b$, which break before the peak of the constitutive curve is reached during stress controlled loading of the bundle. The surface of $P_b(W_E,W_{\sigma })$ was obtained by numerically evaluating the analytical expression Eq. (7). $P_b$ has a finite value everywhere on the $W_E-W_{\sigma }$ plane except on the two axis.

Figure 4

Probability distribution $h\left({\varepsilon }_{\text{th}}\right)$ of the strain thresholds ${\varepsilon }_{\text{th}}$ of fibers calculated from Eq. (15) for three parameter sets.

Figure 5

Burst-size distributions $p\left(\mathrm{\Delta }\right)$ for several values of strength disorder $W_{\sigma }$ in the absence of stiffness disorder $W_E=0$. Approaching the brittle regime $W_{\sigma }\to 1/2$ the well-known result of the crossover in $\tau$ from $5/2$ to $3/2$ is recovered. The two straight lines represent power laws with exponents $3/2$ and $5/2$. The inset presents the constitutive behavior of the system for three values of $W_{\sigma }$.

Figure 6

(a) Size distribution of bursts for $W_{\sigma }=0.25$ varying the stiffness disorder $W_E$ in a broad range. (b) Scaling collapse of the distributions of (a). Best collapse is achieved with the exponent $\alpha =1/2$. In (a) and (b) the same legend is used. (c) Size distribution of bursts for $W_E=1$ varying the amount of strength disorder $W_{\sigma }$ in a broad range. (d) Scaling collapse of the distributions of (c) using the exponent $\alpha =1/2$. In (c) and (d) the same legend is used.