Your Name: Ulises Jose Corona
Name of the MAE2160 Materials Science
Course:
Lab Date: January 23, 2009
Report Date: January 26, 2009
LAB REPORT ON “ LAB #2: GLASS FRACTURE TEST”
Abstract:
Background Griffith crack theory
states that for a thin rectangular plate
with a crack perpendicular to the load,
the fracture strength of a brittle
material decreases as the crack length
increases. For different crack lengths,
we have tested the corresponding
glass fracture strengths for glass
slides of identical dimensions.
Results The experiments yielded a
positive feedback as predicted by the
Griffith theory. There is an inverse
relationship between the fracture
strength of the glass and the square of
the given crack length. There was no
sample outside such behavior.
Conclusions The greater is the
size of the crack in the given material
under a load, this is more likely to fail
because of the multiplier effect for
stress when the cross sectional area
of the tested material is reduced. This
trend supports the Hook’s law for axial
loads. As crack lengths are greater,
sectional cross areas decrease, and
then the failure stress is less for the
samples that are more torn out.
Lab Report
I I
NTRODUCTION
In this lab we tested how scratched glass slides behaved as an
increasing amount of load was applied perpendicular to the small
cracks made with sandpaper. We made an attempt to prove and
apply the Griffith crack equation
Sigma = (EG/(pi)*a)^.5
Where E is the elastic modulus, G is the strain energy release rate,
A is the crack length. The experiment yielded a positive feedback
with respect what the equation states. The formula applies for the
simple case of a thin rectangular plate with a crack normal to the
load.
It is necessary to know the effects of fractures as it is possible to
predict the likelihood of the materials to fail functionally, when used
in engineering structures. Preventing wear out and understanding
how the internal stresses distribute inside the body in order to take
the required preventions against failure is useful to prevent
economic and even human losses, in a wide spectrum of
engineering and technology applications.
II E
XPERIMENT PROCEDURE
A) TEST DESCRIPTION
For this test we needed a Griffith test fixture, which consist of a
manual lever press pivoting on the edge of a table platform. In
the middle of this platform, there was the glass support, where
the wedge of the press provided the cutting contact force over
the slide. A back support in the opposite edge of the pivot was
there for constructive convenience, but it is required that the
lever do not support in this point when attempting to increase
the load to the glass slide, as accuracy in the measurement
would be lost.
In order to increase the load to the glass slide, incremental
amounts of weight was added to a bucket hanging on the far
extreme of the lever. The more weight we add to the bucket, the
higher the load applied to the glass slide is. The force equation
that relates Fl (bucket weight) and Sigma-fs (cutting load upon
glass slide) is given by the following relation based on the
geometry of the frame:
Sigma-fs = 3*Fg*L/2bd^2
Where L is the distance between support
b is the width of glass
d is thickness of glass
The distances and geometry is described by the drawing of the
Griffith test fixture below. A counterweight in the lever in the
opposite tip from where the handle is balances the lever from tilting
when the lever is unloaded. When Fl = 0, the weight of the lever
itself do not impose a load on the glass slide.
With this set up, the corresponding scratched glass slides are to be
tested against load in the fixture supporter. To determine
experimentally the critical strength with accuracy, the applied load
should be increased gradually until failure.
B) A DESCRIPTION OF THE GRIFFITH THEORY
As the Griffith theory of fracture mechanics goes
G = pi*Sigma^2*a/E (look for more detail in the intro)
There is a direct relationship between the critical strain
energy release rate and the length of the crack of the tested thin
plate. The energy release rate is the energy rejected as a material
breaks per every new fracture surface unit. The surface area of
fracture will be greater the bigger are the cracks, which implies that
the dissipation rate of energy G will be higher as more surface area
is available from the beginning of the test, and as new surface
emerges as the fracture process goes on.
C) EFFECT OF STRESS RISERS
The stress risers are locations in the given material in which
exists stress concentration, opposed to a uniform distribution
described by a continuous function or a constant value of stress
around a relative extensive area. Cracks behave as stress risers
as they reduce the available surface for load distribution,
tending to localize the stresses in sharp spots that make the
material prone to fail faster. The fatigue strength (failure stress)
will be higher in a material as less localized stressed is solicited
in the given body. Higher strength in a material will be attained if
microfractures do not appear internally in the material. This
quality of integrity of the materials should be taken into account
in machining processes and casting.
D) A THEORETICAL PLOT OF THE FRACTURE STRESS VS.
CRACK LENGTH
E) A PLOT OF THE FRACTURE STRESS VS. THE INVERSE OF
THE ROOT OF THE CRACK LENGTH USING THE DATA
COLLECTED IN YOUR LAB
Comments
(1) There is a direct relationship between Sigma-fs and the inverse square
root of the fracture length a
(2) Unscratched glass slide Sigma-fs cannot be plotted because of
undefined form 1/0
for the given x-value a^-.5; max allowable value for stress ever is Sigma-fs
(inf) = 129.83 MPa
(3) F_L, and a are obtained from logbook, determined by direct
experimentation
(4) All in SI units unless otherwise specified
(5) Use attached picture to reference the dimentions about the stress tester
(CAD drawing)
Given formulas:
F_G = F_L*b/a
Sigma-fs = 3*F_G*L/2bd^2
sqrt(a)
Sigma-fs
(MPa)
Constants
lengths:
x
y
a
0.23
88.73565094
47.59
b
0.69
112.2427067
69.52
L
0.915
125.491161
73.29
d
0.001
162.0083923
92.23
b/a
3
1.5*L/(b*d^2)
1.99E+06
F) YOUR PREDICTION OF THE FRACTURE STRENGTH FOR
CRACK LENGTHS OF 0.5E-03" AND 7.5E-03"
SI units
unless We first find out an average value for Gglass from the experimental
otherwise data. According to the Griffith formula,
specified
G = pi*Sigma^2*a/E (J/m^2) (1)
So for each case in our table we find out:
G1 = 11.3
G2 = 15.07
G3 = 13.4
G4 = 12.72
From where Gavg = 13.12 J/m^2
Rearrange formula (1) to isolate Sigma:
F_L (kg-f)
F_L Sigma = sqrt(EglassGglass/pi*a) (2)
F_G
a Now E = 80E09 (Pa)
a^-.5 G = 13.12 (J/m^2)
Sigma-fs
(1) For a = 7.5E-03’’
Converting to SI units
0.813 a = 7.5*25.4E-06 m
7.97553 Sigma1 = 42 MPa
23.92659(2) For a = .5E-03’’
1.27E-04 Converting to SI units
88.73565094 a = .5*25.4E-06 m
4.76E+07 Sigma2 = 162 MPa
1.1875
11.649375G) A DISCUSSION OF THE TEST RESULTS AND TRENDS
34.94813 INCLUDING ANY OBSERVATIONS OF THE CONDUCTED
7.94E-05 TEST THAT WOULD LEAD TO DISCREPANCIES WITH THE
112.2427067 THEORETICAL RESULTS.
6.95E+07
As a is grater, 1/sqrt(a) is smaller, so we observe an effect of
linear proportionality between Sigma-fs and 1/sqrt(a). The
1.252 opposed we observed in the decreasing function of the graph
12.28212 Sigma-fs vs a. Besides experimental errors accumulated by an
36.84636 improperly settled balance, distortions of the fixture lever, poor
6.35E-05 precision when doing the measurements or incrementing the
125.491161 loads, we know that the experimental value of G given the
7.33E+07 Griffith formula will always be greater theoretically than in
experiments. This is because in reality there are internal cracks
1.5755
15.455655
46.36697
3.81E-05
162.0083923
2.22
21.7782
65.3346
0
inf in the bodies that are subject to induce localized stress not
1.30E+08 considered in the experimentation. The quality of the material
will determine how close the obtained experimental values are
from the Griffith equation.
III. RESULT AND DISCUSSION
4 CONCLUSION
As load is increased in cracked materials, the crack length
propagates inducing a multiplier effect of stress location. The
greater the crack lengths the material happen to have, there will be
points of stress concentration that obey macroscopically with the
hook’s law, and microscopically, with the energy release due to the
unstable arrangement of the atoms that are close at the surface of
the given solid.
5 REFERENCES
Callister, W. (2007). Materials Science and Engineering. John Wiley
& Sons, Inc.
