Research Papers

Fishnet Statistical Size Effect on Strength of Materials With Nacreous Microstructure

[+] Author and Article Information
Wen Luo

Theoretical and Applied Mechanics,
Northwestern University, CEE/A123,
Evanston, IL 60208
e-mail: wenluo2016@u.northwestern.edu

Zdeněk P. Bažant

McCormick Institute Professor,
W.P. Murphy
Professor of Civil and Mechanical
Engineering and Materials Science,
Northwestern University,
Evanston, IL 60208
e-mail: z-bazant@northwestern.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received March 26, 2019; final manuscript received April 26, 2019; published online May 23, 2019. Assoc. Editor: Yonggang Huang.

J. Appl. Mech 86(8), 081006 (May 23, 2019) (10 pages) Paper No: JAM-19-1137; doi: 10.1115/1.4043663 History: Received March 26, 2019; Accepted April 26, 2019

The statistical size effect has generally been explained by the weakest-link model, which is valid if the failure of one representative volume element (RVE) of material, corresponding to one link, suffices to cause failure of the whole structure under the controlled load. As shown by the recent formulation of fishnet statistics, this is not the case for some architectured materials, such as nacre, for which one or several microstructural links must fail before reaching the maximum load or the structure strength limit. Such behavior was shown to bring about major safety advantages. Here, we show that it also alters the size effect on the median nominal strength of geometrically scaled rectangular specimens of a diagonally pulled fishnet. To derive the size effect relation, the geometric scaling of a rectangular fishnet is split into separate transverse and longitudinal scalings, for each of which a simple scaling rule for the median strength is established. Proportional combination of both then yields the two-dimensional geometric scaling and its size effect. Furthermore, a method to infer the material failure probability (or strength) distribution from the median size effect obtained from experiments or Monte Carlo simulations is formulated. Compared to the direct estimation of the histogram, which would require more than ten million test repetitions, the size effect method requires only a few (typically about six) tests for each of three or four structure sizes to obtain a tight upper bound on the failure probability distribution. Finally, comparisons of the model predictions and actual histograms are presented.

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Grahic Jump Location
Fig. 1

Schematic showing four geometrically similar fishnets under uniaxial tension: (a) 8 × 16, (b) 16 × 32, (c) 32 × 64, and (d) 64 × 128 (partially shown). r and s denote the number of fishnet rows and columns, respectively.

Grahic Jump Location
Fig. 2

Typical stress–displacement curves for fishnets of size r × s = 32 × 64 and their evolution of stress field and damage: (a) Kt = −0.1K0 and (b) Kt = −0.5K0 (more brittle). For each case, the number of discrete softening jumps to reach zero stress in each link J = 20.

Grahic Jump Location
Fig. 3

Strength histogram (in the Weibull scale) of various fishnets under longitudinal scaling. The widths r of all fishnets are fixed at 16 rows, while their lengths s vary geometrically from 8 to 64 columns. For each case, the sample size is 104, softening slope Kt = −0.1K0, and number of discrete softening jumps for each link J = 20.

Grahic Jump Location
Fig. 4

Strength histogram (in the Weibull scale) of various fishnets under transverse scaling. Lengths of all fishnets, s, are fixed at 16, while their widths, r, vary in geometric sequence from 8 to 64. For each case, the sample size is 104, the softening slope Kt = −0.1K0, and the number of discrete softening jumps to reach zero stress in each link J = 20.

Grahic Jump Location
Fig. 5

Comparison plot of analytical results of mean load–extension curves of bundles with different fiber brittleness. Fiber strengths are i.i.d. random variables and follow the same distribution G(σ) for all three cases.

Grahic Jump Location
Fig. 6

Strength histograms of fishnets (Kt = −0.5K0) of four typical sizes under uniaxial tension: (a) 8 × 16, (b) 16 × 32, (c) 32 × 64, and (d) 64 × 128. The counts have been normalized to probability density.

Grahic Jump Location
Fig. 7

Strength histograms of fishnets (Kt = −0.1K0) of four typical sizes under uniaxial tension: (a) 8 × 16, (b) 16 × 32, (c) 32 × 64, and (d) 64 × 128. The counts have been normalized to probability density.

Grahic Jump Location
Fig. 8

Optimum fit of the size effect relation (ln σ0.5 versus ln D) using the sample median strength obtained from Monte Carlo simulations. Two different brittleness levels given by Kt = −0.5K0 and −0.1K0 are considered here. The data points are the sample median strengths for the four typical sizes, and the curve is the optimum fit.

Grahic Jump Location
Fig. 9

Comparison of predicted (thick lines) upper bound strength distributions of the fishnet with the actual histograms (discrete markers). Optimum fit by the homogenized two-term fishnet model is shown by dashed curves.

Grahic Jump Location
Fig. 10

Two-dimensional geometrically similar scaling relations of (a) variance and (b) CoV for the weakest-link chain (brittle limit), fishnets, and the ductile fiber bundle (ductile limit). Weibull modulus for the strength distribution, P1(σ), of links is m = 10.



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