^{1,a)}

### Abstract

Recently, Yanay and collaborators [J. Appl. Phys.101, 104911 (2007)] addressed issues regarding the fracture strength of randomly perforated aluminum plates subjected to tensile loads. Based on comprehensive measurements and computational simulations, they formulate statistical predictions for the tensile strength dependence on the hole density but conclude that their data are inadequate for the purpose of deriving the strength distribution function. The primary purpose of this contribution is to demonstrate that, on dividing the totality of applicable data into seven “bins” of comparable population, the strength distribution of perforated plates of similar hole density obeys a conventional two-parameter Weibull model. Furthermore, on examining the fracture stresses as recorded in the vicinity of the percolation threshold, we find that the strength obeys the expression with and . In this light, and taking advantage of percolationtheory, we formulate equations that specify how the two Weibull parameters (characteristic strength and shape factor) depend on the hole density. This enables us to express the failure probability as a function of the tensile stress, over the entire range of hole densities, i.e., up to the percolation threshold.

I wish to thank Yariv Yanay for providing his unpublished data on the tensile strength of unperforated aluminum plates.

I. INTRODUCTION

II. DATA EVALUATION

III. WEIBULL PARAMETERS

IV. FAILURE PROBABILITY

V. CONCLUSIONS

### Key Topics

- Hole density
- 26.0
- Fracture mechanics
- 12.0
- Percolation
- 12.0
- Aluminium
- 10.0
- Cumulative distribution functions
- 7.0

## Figures

Normalized tensile stresses at fracture (mean value plus standard deviation) as a function of the hole density, near the percolation threshold. The solid line represents a weighted least-squares fit to Eq. (2). The broken line illustrates a weighted least-squares fit performed on assuming and values as postulated in Ref. 4.

Normalized tensile stresses at fracture (mean value plus standard deviation) as a function of the hole density, near the percolation threshold. The solid line represents a weighted least-squares fit to Eq. (2). The broken line illustrates a weighted least-squares fit performed on assuming and values as postulated in Ref. 4.

Normalized tensile stresses at fracture as a function of the hole density. This plot includes the totality of data listed in Tables I and II of Ref. 4. The solid line shows a least-squares fit to Eq. (4); the broken lines delineate the 95% *prediction* band, thus identifying three outliers.

Normalized tensile stresses at fracture as a function of the hole density. This plot includes the totality of data listed in Tables I and II of Ref. 4. The solid line shows a least-squares fit to Eq. (4); the broken lines delineate the 95% *prediction* band, thus identifying three outliers.

Weibull statistical analysis of bin-1 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-1 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-2 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-2 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-3 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-3 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-4 data points. The solid lines are least-squares fit; the broken lines delineate the 95% *confidence* bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-4 data points. The solid lines are least-squares fit; the broken lines delineate the 95% *confidence* bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-5 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-5 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-6 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-6 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-7 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Weibull statistical analysis of bin-7 data points. The solid lines are least-squares fit; the broken lines delineate the 95% confidence bands. (a) The “Weibull plot” yields and . (b) The failure probability plot yields Weibull parameters as listed in Table I.

Normalized characteristic strength of perforated aluminum plates pertaining to bins 1–7 (see Table I). Horizontal error bars specify the hole density range of each bin; vertical error bars delimit the 95% confidence band. The solid line illustrates a fit to Eq. (4) with a critical exponent set equal to 0.614, that is, as obtained from fitting the characteristic strength in the vicinity of the percolation threshold (see the inset).

Normalized characteristic strength of perforated aluminum plates pertaining to bins 1–7 (see Table I). Horizontal error bars specify the hole density range of each bin; vertical error bars delimit the 95% confidence band. The solid line illustrates a fit to Eq. (4) with a critical exponent set equal to 0.614, that is, as obtained from fitting the characteristic strength in the vicinity of the percolation threshold (see the inset).

Weibull shape parameter as derived from tensile strength data pertaining to bins 1–7, plotted against the hole density. Horizontal error bars specify the hole density range of each bin; vertical error bars delimit the 95% confidence band. The solid line illustrates a least-squares fit as specified in Eq. (10).

Weibull shape parameter as derived from tensile strength data pertaining to bins 1–7, plotted against the hole density. Horizontal error bars specify the hole density range of each bin; vertical error bars delimit the 95% confidence band. The solid line illustrates a least-squares fit as specified in Eq. (10).

Failure probability of perforated aluminum plates as a function of the tensile stress, for hole densities ranging from 0.02 to 0.60. This plot is based on a conventional two-parameter Weibull model with characteristic strength and shape-parameter values as given by Eqs. (9) and (10). Note how the spread of anticipated fracture strengths increases with the hole density .

Failure probability of perforated aluminum plates as a function of the tensile stress, for hole densities ranging from 0.02 to 0.60. This plot is based on a conventional two-parameter Weibull model with characteristic strength and shape-parameter values as given by Eqs. (9) and (10). Note how the spread of anticipated fracture strengths increases with the hole density .

## Tables

Hole density , population , normalized measured strength , normalized characteristic strength , and shape parameter for each of the seven “bins,” or collections of data points pertaining to plates of similar hole density.

Hole density , population , normalized measured strength , normalized characteristic strength , and shape parameter for each of the seven “bins,” or collections of data points pertaining to plates of similar hole density.

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