  The staggered (or imbricated) lamellar "brick-and-mortar" nanostructure of
nacre endows nacre with strength and fracture toughness values exceeding by an
order of magnitude those of the constituents, and inspires the advent of new
robust biomimetic materials. While many deterministic studies clarified these
advantageous features in the mean sense, a closed-form statistical model is
indispensable for determining the tail probability of failure in the range of 1
in a million, which is what is demanded for most engineering applications. In
the authors' preceding study, the so-called `fishnet' statistics, exemplified
by a diagonally pulled fishnet, was conceived to describe the probability
distribution. The fishnet links, representing interlaminar bonds, were
considered to be elastic perfectly-brittle. However, the links may be
quasibrittle or almost ductile, exhibiting gradual postpeak softening in their
stress-strain relation. This paper extends the fishnet statistics to links with
post-peak softening slope of arbitrary steepness. Probabilistic analysis is
enabled by assuming the postpeak softening of a link to occur as a series of
finite drops of stress and stiffness. The maximum load of the structure is
approximated by the strength of the k-th weakest link (k >= 1), and the
distribution of structure strength is expressed as a weighted sum of the
distributions of order statistics. The analytically obtained probabilities are
compared and verified by histograms of strength data obtained by millions of
Monte Carlo simulations for each of many nacreous bodies with different link
softening steepness and with various overall shapes.
