Stress Transmission and Failure in Disordered Porous Media

By means of extensive lattice-element simulations, we investigate stress transmission and its relation with failure properties in increasingly disordered porous systems. We observe a non-Gaussian broadening of stress probability density functions under tensile loading with increasing porosity and disorder, revealing a gradual transition from a state governed by single-pore stress concentration to a state controlled by multipore interactions and metric disorder. This effect is captured by the excess kurtosis of stress distributions and shown to be nicely correlated with the second moment of local porosity fluctuations, which appears thus as a (dis)order parameter for the system. By generating statistical ensembles of porous textures with varying porosity and disorder, we derive a general expression for the fracture stress as a decreasing function of porosity and disorder. Focusing on critical sites where the local stress is above the global fracture threshold, we also analyze the transition to failure in terms of a coarse-graining length. These findings provide a general framework which can also be more generally applied to multiphase and structural heterogeneous materials.

Figure 1

Maps of the vertical tensile stress field in four different samples of porosity $\varphi =0.37$ and increasing disorder: (a) $S_1$ with disorder index $I_d=0.008$, (b) $S_2$ with $I_d=0.087$, (c) $S_3$ with $I_d=0.120$, and (d) $S_4$ with $I_d=0.150$. The scale shows the range of normalized stresses varying from 0 and smaller values to 10 and larger values.

Figure 2

Probability density functions (PDFs) of normalized vertical stresses for the four samples of Fig. 1. (Inset) The excess kurtosis $\kappa$ of tensile stresses as a function of disorder index $I_d$ for $\varphi =0.2$ (green squares), $\varphi =0.4$ (blue disks), and $\varphi =0.5$ (yellow diamonds). The solid line is a quadratic fit.

Figure 3

(a) Tensile strength of all porous systems normalized by the matrix tensile strength ${\sigma }_c^m$ as a function of porosity $\varphi$ for all values of disorder index $I_d$. The arrow shows the direction of increasing values of $I_d$. The dashed line is a fit by Eq. (2) and black markers correspond to ordered systems. (b) The same data normalized by the tensile strength ${\sigma }_c^r(\varphi )$ of reference ordered systems as a function of the disorder index. The red symbols represent the averages and the standard variations of the data over ten intervals of $I_d$. The dashed line is a fit by Eq. (3).

Figure 4

(a),(b) Critical sites and their equivalent probe sizes encoded in color levels on a scale varying from blue (black) to yellow (light shade). The two black circles are the probes centered on the most critical sites at two instants of loading. (c) The diameter $d_{\max }$ of the most critical domain normalized by the system size $L$ as a function of vertical stress normalized by the system tensile strength for the four samples of Fig. 1. The solid lines are exponential fits. (Inset) The same data on a log-log scale.