Air Bubbles in Concrete
Prof. Shin-ichi Igarashi's data: a point pattern of the locations, in a cross-section of a concrete body, of the centroids of air bubbles in the cement paste matrix surrounding particles of aggregate.
data("concrete")An object of class "ppp" representing the point pattern
of air bubble centroid locations.
Major scientific interest is focussed on analysing the distribution of the location of the air bubbles in the cement paste matrix. These bubbles are important in assuring frost resistance of the concrete. Each air bubble protects a region around it to a certain distance. To protect an entire concrete object against severe frost attack, it is necessary to cover the whole of the cement paste matrix with subsets of protected regions formed around the air bubbles. It is believed that the protected regions are related to the Dirichlet tessellation of the centroids of the bubbles, and the statistical properties of the protected regions can be determined from those of the Dirichlet tessellation. In this regard, the areas of the tiles are particularly important.
Prof. Shin-ichi Igarashi, of the School of Geoscience and Civil Engineering, Kanazawa University, personal communication.
Natesaiyer, K., Hover, K.C. and Snyder, K.A. (1992). Protected-paste volume of air-entrained cement paste: part 1. Journal of Materials in Civil Engineering 4 No.2, 166 – 184.
Murotani, T., Igarashi, S. and Koto, H. (2019). Distribution analysis and modeling of air voids in concrete as spatial point processes. Cement and Concrete Research 115 124 – 132.
if(require(spatstat.geom)) {
plot(concrete,chars="+",cols="blue",col="yellow")
# The aggregate is in yellow; the cement paste matrix is in white.
# Unit of length: use \mu symbol for micron
unitname(concrete) <- "\u00B5m"
if(interactive()) {
# Compute the Dirichlet tessellation
dtc <- dirichlet(concrete)
plot(dtc,ribbon=FALSE, col=sample(rainbow(dtc$n)))
# Study Dirichlet tile areas
areas <- tile.areas(dtc)
aa <- areas/1000 # Divide by 1000 to avoid numerical instability
# Fit a gamma distribution by the method of moments
mm <- mean(aa)
vv <- var(aa)
shape <- mm^2/vv
rate <- mm/vv
rate <- rate/1000 # Adjust for rescaling
hist(areas,probability=TRUE,ylim=c(0,7.5e-6),
main="Histogram and density estimates for areas",ylab="",xlab="area")
lines(density(areas),col="red")
curve(dgamma(x,shape=shape,rate=rate),add=TRUE,col="blue")
legend("topright",lty=1,col=c("red","blue"),
legend=c("non-parametric","gamma fit"),bty="n")
}
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