Chemistry Reference
In-Depth Information
4.11
Fracture Mechanics
This discipline is based on the premise that all materials contain flaws and that
fracture occurs by stress-induced extension of these defects. The theory derives
from the work of A. E. Griffith
[15]
, who attempted to explain the observation
that
initial cross-sectional
area) of fine glass filaments were inversely proportional to the sample diameter.
He assumed that every object contained flaws, that failure is more likely the
larger the defect, and that larger bodies would break at lower tensile stresses
because they contained larger cracks. The basic concept is that a crack will
grow only if the total energy of the body is lowered thereby. That is to say,
the elastic strain energy which is relieved by crack growth must exceed the
energy of the newly created surfaces. It is important to note also that the pres-
ence of a crack or inclusion changes the stress distribution around it, and the
stress may be amplified greatly around the tips of sharp cracks. The relation
that was derived between crack size and failure stress is known as the Griffith
criterion:
the tensile strengths (defined as breaking force
4
1
=
2
2
γ
Y
σ
f
5
(4-79)
π
a
where
σ
f
5
failure stress, based on the initial cross-section,
a
5
crack depth,
Y
5
γ 5
surface energy of the solid material (the
factor 2 is inserted because fracture generates two new surfaces). This equation
applies to completely elastic fractures; all the applied energy is consumed in gen-
erating the fracture surfaces. Real materials are very seldom completely elastic,
however, and a more general application of this concept allows for additional
energy dissipation in a small plastic deformation region near the crack tip. With
this amendment,
Eq. (4-79)
is applicable with the 2
tensile (Young's) modulus, and
term replaced by
G
, the
strain energy release rate
, which includes both plastic and elastic surface work
done in extending a preexisting crack
[15]
:
γ
1
=
2
YG
π
σ
f
5
(4-80)
a
The general equation to describe the applied stress field around a crack tip
is
[16]
:
K
½πa
σ5
(4-81)
1
=
2
where
K
is the stress intensity factor and
is the local stress.
Equation (4-81)
applies at all stresses, but the stress intensity reaches a critical value,
K
c
,atthe
stress level where the crack begins to grow.
K
c
is a material property, called the
fracture toughness
, and the corresponding strain energy release rate becomes the
critical strain energy release rate
,
G
c
.
σ
