Fishnet statistics for probabilistic strength and scaling of nacreous imbricated lamellar materials

Abstract

Similar to nacre or brick-and-mortar structures, imbricated lamellar structures are widely found in natural and man-made materials and are of interest for biomimetics. These structures are known to be rather insensitive to defects and to have a very high fracture toughness. Their deterministic behavior has been intensely studied, but statistical studies have been rare and is so far there is no undisputed theoretical basis for the probability distribution of strength of nacre. This paper presents a numerical and theoretical study of the PDF of strength and of the corresponding statistical size effect. After a reasonable simplifications of the shear bonds, an axially loaded lamellar shell is statistically modelled as a square fishnet pulled diagonally. A finite element (FE) model is developed and used in Monte Carlo simulations of strength. An analytical model for failure probability of the fishnet is developed and matched to the computed statistical histograms of strength for various sizes. It appears that, with increasing size, the pdf of strength slowly transits from Gaussian to Weilbull distribution but the transition is different from that previously obtained at Northwestern for quasibrittle materials of random heterogeneous mesostructure.

Design of quasibrittle materials and structures to optimize strength and scaling at probability tail: an apercu

“…Daniels [53] in 1945 proved that if the number of fibres (or parallel elements) tends to infinity, the bundle strength converges to the Gaussian (or normal) distribution regardless of the element properties (but if the elements are brittle many parallel fibres are needed to approach the Gaussian). (3) the fishnet model (3 in figure 2), which appeared in 2017 and represents the third analytically tractable model [54][55][56] ( figure 3). It will be discussed later.…”

“…But, for biomimetic engineering applications, one needs a probabilistic analysis establishing the tail failure probabilities. To this end, a new failure model, called the fishnet model ( figure 13b,c), significantly different from the hierarchical model already described, was discovered in 2017 to reflect the 'brick-and-mortar' architecture of nacre [54][55][56] and will now be briefly reviewed.…”

General Fishnet Statistics of Strength: Nacreous, Biomimetic, Concrete, Octet-Truss, and Other Architected or Quasibrittle Materials

“…However, the stochastic aspects have received little attention. Here, we first review our recently proposed fishnet statistical model for nacreous biomimetic materials [6][7][8][9] (as presented at the Century Fracture Mechanics Summit in Singapore on April 8, 2019). Then, we extend this model to architected nanomaterials and show that it may also capture some failure probability features of quasibrittle materials in general, including concrete or fiber composites.…”

“…These simplifications enable the mathematical formulations of their strength probability distribution. Recently [6][7][8], the force transmission in nacreous materials has been idealized as a fishnet with brittle links pulled diagonally, which allowed a mathematical formulation of the fishnet statistics. The fishnet represents the third probabilistic model of strength distribution that is tractable analytically.…”

“…As proposed in Refs. [6,7], the staggered micro-structure of nacre is, for statistical purposes, best idealized as a fishnet (Fig. 1(b)) consisting of brittle links pulled diagonally.…”