  With the fundamental objective of establishing the universality of the
Griffith energy competition to describe the growth of large cracks in solids
\emph{not} just under monotonic but under general loading conditions, this
paper puts forth a generalization of the classical Griffith energy competition
in nominally elastic brittle materials to arbitrary \emph{non-monotonic}
quasistatic loading conditions, which include monotonic and cyclic loadings as
special cases. Centered around experimental observations, the idea consists in:
$i$) viewing the critical energy release rate $\mathcal{G}_c$ \emph{not} as a
material constant but rather as a material function of both space $\textbf{X}$
and time $t$, $ii$) one that decreases in value as the loading progresses, this
solely within a small region $\Omega_\ell(t)$ around crack fronts, with the
characteristic size $\ell$ of such a region being material specific, and $iii$)
with the decrease in value of $\mathcal{G}_c$ being dependent on the history of
the elastic fields in $\Omega_\ell(t)$. By construction, the proposed Griffith
formulation is able to describe any Paris-law behavior of the growth of large
cracks in nominally elastic brittle materials for the limiting case when the
loading is cyclic. For the opposite limiting case when the loading is
monotonic, the formulation reduces to the classical Griffith formulation.
Additional properties of the proposed formulation are illustrated via a
parametric analysis and direct comparisons with representative fatigue fracture
experiments on a ceramic, mortar, and PMMA.
