Damage-Cluster Distributions and Size Effect on Strength in Compressive Failure

Abstract

We investigate compressive failure of heterogeneous materials on the basis of a continuous progressive damage model. The model explicitely accounts for tensile and shear local damage and reproduces the main features of compressive failure of brittle materials like rocks or ice. We show that the size distribution of damage-clusters, as well as the evolution of an order parameter, the size of the largest damage-cluster, argue for a critical interpretation of fracture. The compressive failure strength follows a normal distribution with a very small size effect on the mean strength, in good agreement with experiments.Understanding how materials break is a fundamental problem that has both theoretical and practical relevance. The topic has received considerable renewed attention during the last few decades because of the limitations of the classical Griffith theory for heterogeneous media [1][2][3]. The practical applications are numerous, from material and structural design, to the important problem of size effects on strength (e.g. [4]). The two key components that make material failure challenging to understand are long-range interactions and material disorder.Traditionally, Weibull and Gumbel distributions associated with the weakest-link approach have been widely used to describe the strength of brittle materials. These distributions naturally arise from extreme-value statistics of initial defect distributions based on the assumptions that [5] (i) defects do not interact with one another, (ii) failure of the whole system is dictated by the activation of the largest flaw (the weakest-link hypothesis), and finally (iii) the material strength can be related to the critical defect size. These assumptions are reasonable for materials with relatively weak disorder loaded under tension, but do not hold for heterogeneous materials with broad distribution of initial disorder or for loading conditions stabilizing crack propagation, such as compression. In these cases there is experimental evidence that a considerable amount of damage can be sustained before failure [6]. Under these conditions, failure is the culmination of a complex process involving the nucleation, propagation, interaction and coalescence of many microcracks [7]. Stress states observed under various natural conditions, ranging from soil and rock mechanics to earthquake physics, suggest the importance of compressive failure.A cornerstone for the understanding of breakdown of disordered media has been lattice models of fracture in * lucas.girard@geo.uzh.ch which networks with prescribed bond failure thresholds are subject to increasing external loads [2]. In these models, failure is described on a qualitative level as the interplay between disorder and elasticity. When strong disorder is considered, these models suggest that fracture strength does indeed not follow a Weibull or a Gumbel distribution but a log-normal distribution [8,9]. Similar strength distributions have been obtained for different model types (fuses and springs), in 2D and 3D, suggesting ...

Rupture Cascades in a Discrete Element Model of a Porous Sedimentary Rock

“…Typically, only a very small proportion of the microcracks revealed by destructive thin sectioning after the test result in detectable acoustic emissions [11]. As a consequence, experimental data provide only limited insight into the complexity of the microscopic processes at work prior to failure, notably the probability distributions of the relevant parameters, their scaling properties, and their population dynamics.Theoretical approaches to the dynamics and statistics of rupture cascades have typically been based on stochastic fracture models comprising lattices of springs [12], beams [13,14], fuses [15,16], or fibers [17][18][19]. However, such lattice models involve a strong simplification of the material microstructure and the inhomogeneous stress field.…”

“…Theoretical approaches to the dynamics and statistics of rupture cascades have typically been based on stochastic fracture models comprising lattices of springs [12], beams [13,14], fuses [15,16], or fibers [17][18][19]. However, such lattice models involve a strong simplification of the material microstructure and the inhomogeneous stress field.…”

Damage cluster distributions in numerical concrete at the mesoscale

“…A power-law distribution of crack sizes was also observed with a scalar spring model, and it was reported that the distribution was exponential for the low-strain loadings and approaches to a power law as the loading increases [36]. Girard et al [12] saw an identical trend of the damage cluster size distribution for their finite-element model, applying a random distribution of disorder. In their case, the size distribution of the damage clusters follows a power law with an exponential cutoff at the peak load under compressive loading.…”

“…The exponent obtained for the fiber bundle models is 1.86 [38], while scalar spring models yield τ = 2 [36]. Progressive damage finite-element models under the compressive loading give different exponents with different levels of heterogeneity as the reported values range from 2.6 to 3.6 [12]. It is known that the percolation theory calculates the Fisher's exponent in two dimensions as τ 2D = 187/91 ≈ 2.05 and τ 3D = 2.18 in three dimensions [17].…”

“…There are different classes of models depending on the type and properties of the failure process of interest. These models include, but are not limited to, random fuse models [2][3][4], central force models [5], elastic beam models [6], fiber bundle models [7][8][9][10], molecular dynamics models [11], and finite-element models with progressive damage [12]. For broader reviews of the models mentioned above, refer to Refs.…”