314 / 352
310
suggests that a certain finite length parameter, determined either by the microstructural or
atomistic considerations, must be introduced into the basic equations underlying the
theory of fracture. Even though various names and symbols have been used in describing
this entity, such as “Neuber particle”, “unit growth step” and “fracture quantum”, the
physical meaning of these length parameters is the same, and it can be accounted for
mathematically by a discretization technique, Wnuk and Yavari /13/.
The interest in understanding fracture on a more fundamental level indicates that some
physics/mechanics coupled problems remain to be resolved. Many researchers have stu-
died the problem at the lattice scale. Thomson, et al /14/ showed that in a very simplified 1D
model for a range of stresses above and below Griffith’s stress crack becomes lattice trapped.
Later Hsieh and Thomson /15/ extended their results to 2D. Esterling /16/, using a lattice
statics method, studied similar problems for three dimensional cracks in a cubic lattice with
nearest-neighbour interactions. Masudajindo, et al /17,18/ studied fracture of crystalline mate-
rials using the lattice Green’s function method. In particular, they showed that mode I and II
crack problems are coupled in the lattice scale. There have been some very recent atomic-
scale fracture studies (Marder and Liu /19/; Marder and Gross /20/; Marder /21, 22/).
Further developments are reported in papers given by Mattoni et al /23/, also Ippolito
et al /2/, Pugno and Ruoff /12/, Cherepanov /24/; Taylor et al /25/, and Cornetti et al /26/.
In this way, the “Quantized Fracture Mechanics” (QFM) was born.
A discretization procedure for the cohesive model of a fractal crack requires that all
pertinent entities describing the influence of the cohesive stress that restrains opening of
the crack, such as effective stress intensity factor, the modulus of cohesion, extent of the
end zone and the opening displacement within the high-strain region adjacent to the crack
tip are re-considered and replaced by certain averages over a finite length referred to as
either “unit growth step”, cf. Wnuk /10/ or “fracture quantum”, cf. Novozhilov /7/ Thus,
two novel aspects of the model enter the theory:
(1) degree of fractality related to the roughness of the newly created surface, and
(2) discrete nature of the propagating crack.
Both variables are shown to increase the effective fracture toughness.
The problem may be solved assuming a fractal crack is represented by a cohesive
model and that propagation of fracture is not continuous but discrete. The present model
yields the results identical to those known in classical fracture mechanics.
Recent work by Ippolito et al
/2/
summarizes the pertinent results of the newly deve-
loped QFM model. This model is based on the hypothesis of discrete nature of crack pro-
pagation and it is used to explain the atomistic simulations data for nano-meter size
cracks in
β
-SiC. The fracture quantum
a
0
is seen to be a fraction of a nanometer. In addi-
tion to fracture quantum, the finite radius of curvature
ρ
found at the tip of a QFM crack
has been considered as another length entity, resembling to blunt crack in linear-elastic
fracture mechanics (LEFM). It turned out that not the fracture quantum alone but rather
the ratio
ρ
/a
0
describes the microstructural behaviour of the material. The third important
variable is the crack length itself, especially so in the nano-scale range, where fracture
resistance is visibly sensitive to the crack length. Comparing the theoretical and atomistic
simulations data Ippolito et al /2/ found that the best fit was obtained for both
a
0
and
ρ