Experimental Characterization of Damage in Random Short Glass

Transcription

Experimental Characterization of
Damage in Random Short Glass
Fiber Reinforced Composites
MARIE-LAURE DANO* AND GUY GENDRON
Department of Mechanical Engineering
Laval University, Quebec City
Quebec, G1K 7P4, Canada
FRANÇOIS MAILLETTE AND BENOIˆT BISSONNETTE
Department of Civil Engineering
Laval University, Quebec City
Quebec, G1K 7P4, Canada
ABSTRACT: This article presents the results of an experimental program carried
out to characterize the mechanical behavior of random short glass fiber reinforced
composites. Tensile, compressive, and shear tests are first performed. The results
show that the material is characterized by in-plane isotropy and that it exhibits
a damageable elastic behavior in tension and a brittle linear elastic behavior
in compression. Then, a series of tests are conducted to evaluate the elastic stiffness tensor of the damaged material. The experimental results reveal that damage
induces anisotropy. The results of the experimental program are used to identify
and validate a continuum damage mechanics model that has been developed to
predict the material mechanical behavior.
KEY WORDS: damage characterization, short glass fiber composites, testing,
anisotropy, cracks, behavior.
INTRODUCTION
S
composites have become an attractive
material for many industrial applications. For example, short glass
HORT FIBER REINFORCED
*Author to whom correspondence should be addressed. E-mail: Marie-Laure.Dano@gmc.ulaval.ca
Journal of THERMOPLASTIC COMPOSITE MATERIALS, Vol. 19—January 2006
0892-7057/06/01 0079–18 $10.00/0
DOI: 10.1177/0892705706055447
ß 2006 SAGE Publications
Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015
79
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M.-L. DANO
ET AL.
fiber reinforced polymers are being used to make bus body panels,
urban train seats, and recreative vehicle parts. To advantageously use
these materials and optimize the design process, a reliable prediction of their
mechanical behavior is essential.
A continuum damage mechanics model was proposed [1] to predict
the mechanical behavior of random short fiber reinforced composites. The
model development is discussed in detail in [1], but for purposes of completeness, the main equations are briefly presented here. The model uses two
phenomenological internal damage variables, D1 and D2, to define the
elastic compliance tensor of the damaged material:
2
Cd
1=Eð1 D1 Þ2
1 6
¼6
4 =Eð1 D1 Þð1 D2 Þ
3
=Eð1 D1 Þð1 D2 Þ
0
1=Eð1 D2 Þ2
0
0
1=Gð1 D1 Þð1 D2 Þ
ð1Þ
0
7
7
5
The residual strains and the unilateral behavior relative to crack closure
are taken into account. The residual strains are assumed to be a function of
the two damage variables and are expressed as:
(
"p11 ¼ D1 D2
"p22 ¼ D1 þ D2
ð2Þ
where and are two material parameters associated with the permanent
strain effects. The evolution laws of the damage variables D1 and D2 are
established within a classical thermodynamic framework using the associated thermodynamic forces Y1 and Y2. These forces are determined by
derivation of the thermodynamic potential with respect to the associated
damage variable:
Yi ¼
@U
@Di
i ¼ 1, 2
ð3Þ
The evolution laws are assumed to be coupled to account for a possible
interaction between the two principal damage directions. Therefore, for
quasi-static loading, the evolution laws for the two damage variables are
written as:
Di ¼ f YSi ðtÞ
i ¼ 1, 2
ð4Þ
Experimental Characterization of Damage in Composites
81
where f is a function intrinsic to the material and YSi ðtÞ is defined as:
YSi ðtÞ ¼ max Y0 , supt ðYi þ bYj Þ
i ¼ 1, 2 i 6¼ j
ð5Þ
where b is an additional material parameter accounting for coupling effects
between the two principal damage directions. It is further assumed that the
function f is equal to:
f YS ðtÞ ¼ a YSi Y0
ð6Þ
where a and Y0 are two additional material parameters. In summary, the
model uses a total of five material parameters: three parameters (a, b, Y0)
govern the damage evolution laws and two parameters (, ) govern the
permanent strain effects. To identify these parameters and validate the
model, a series of specific tests have to be conducted on the material.
The mechanical behavior of short fiber reinforced composites, especially
sheet molding compound (SMC) materials, has been studied experimentally by several researchers [2–6]. Denton [2] characterized the mechanical
properties of an SMC-R50 composite, a structural grade SMC with a
50% fiber content in weight, as a function of temperature under static
and fatigue loading. Wang and Chim [3] studied fatigue degradation in a
random SMC composite and identified various forms of damage mechanisms. Hour and Sehitoglu [4] studied damage evolution in SMC specimens
by determining the damage volumetric strain. More recently, Berthaud
et al. [5] analyzed the degradation mechanisms of a short fiber reinforced
vinyl ester composite from tensile tests and microscopic observations. The
material sensitivity to notches or holes was assessed. The effect of biaxial
loading was also investigated.
Although the mechanical behavior of short fiber composites has been
studied for two decades, few attempts have been made to characterize
the behavior of short fiber composites, once damage has occurred. Perreux
and Siqueira [6] analyzed the influence of damage on a virgin SMC material.
A specimen (SMC sheet) was tested in tension to induce predamage.
The predamaged sheets were then sampled in the 90 and 45 directions with
respect to the loading axis. Tensile tests were performed on the specimens
to study the behavior of the damaged material. However, no attempt was
made to measure the apparent elastic properties of the damaged material
and determine the components of the compliance tensor.
This article reports the results of an experimental program carried
out to determine the mechanical behavior of a short glass fiber reinforced composite. Specifically, tests were conducted to (1) characterize the
material behavior under tensile, compressive, and shear stresses, (2) study
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the evolution of damage, and (3) measure the apparent elastic properties
of the damaged material to determine its compliance tensor components.
Specific objectives of the study were also to identify the five material
parameters of the theoretical model developed in [1] and to validate its
predictions.
In the sections to follow, a description of the material investigated is first
presented. Then, the results of the tests performed to study the material
mechanical behavior are discussed. In particular, the results of a progressive repeated tensile loading test are presented and used to identify the
model parameters. Finally, the experiments to evaluate the apparent
elastic properties of the damaged material are described and the results
presented.
MATERIAL DESCRIPTION
The material investigated in this study is a short fiber reinforced composite fabricated using a spray-up open-mold technology. Chopped E-glass
fiber strands and catalyzed polyester resin are sprayed on the mold surface
(with a chopper/spray gun). Manual rollers are then used to remove the
entrapped air. This process presents the advantages of being both economical and well adapted to high volume production. The end product contains
30% fiber in weight. The mean diameter and length of the fibers are about
10 mm and 30 mm, respectively. Given that each strand contains about 300
fibers, the length-to-diameter ratio of a strand is 175. Microscopic
observations of the composite [7] still revealed the presence of a significant
number of entrapped air bubbles, and confirmed the random orientation
of the fibers through the thickness. Therefore, the material was expected to
initially exhibit an in-plane isotropic behavior.
MATERIAL MECHANICAL BEHAVIOR
The material was tested in tension, compression, and shear for mechanical behavior characterization. The stress–strain curves shown in Figures 1,
2, 4, and 5 summarize the experimental results.
Tensile Behavior
The specimen configuration used for the tensile tests is shown in
Figure 1(a). A total of ten specimens were cut from 60 cm by 60 cm
sheet panels provided by the material supplier. To assess the isotropic
behavior of the composite, specimens were cut along two perpendicular
83
100
50 mm
200 mm
60 mm
18 mm
Stress σ1 (MPa)
90
80
x1
70
60
50
40
30
20
10
0
-0.010 -0.005
x2
24 mm
Thickness = 4±0.4 mm
0.000
0.005
0.010
Strain (ε)
ε2
(a)
0.015
0.020
0.025
ε1
(b)
Figure 1. Monotonic tensile behavior: (a) specimen configuration and (b) stress–strain
relation.
σ1
90
Stress σ1 (MPa)
80
70
60
50
d
E1
d
E1
40
x1
d
30
ν12 =−(ε1/ε2)
20
x2
10
0
-0.010
-0.005
ε2
0.000
0.005
Strain (ε)
0.010
ε1
0.015
0.020
σ1
Figure 2. Repeated tensile loading test.
directions. The rather important specimen thickness variation (0.4 mm)
is due to the manufacturing process, which does not allow for tighter
tolerances.
The specimens were tested in tension at a testing speed of 2 mm/min
using a universal testing machine. The longitudinal (x1-direction) and transverse (x2-direction) strains were recorded using an axial and a transverse
extensometer.
The stress–strain curves obtained for the ten specimens are illustrated
in Figure 1(b). No significant differences were visible between the results
obtained in the two perpendicular directions. This confirms that the
material initially has an in-plane isotropic behavior. The stress–strain curve
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M.-L. DANO
ET AL.
is quite linear at the beginning. But, as the applied stress reaches a value
of about 30 MPa, the curve becomes nonlinear. The tensile modulus was
determined from the slope of the initial linear portion of the stress–strain
curve, and, accordingly, the Poisson’s ratio was evaluated from the slope
of the initial portion of the transverse versus longitudinal strain curve.
The average results for elastic modulus, Poisson’s ratio, and tensile strength
were 7.08 GPa, 0.35, and 91.6 MPa, respectively. During testing, before
the curve becomes nonlinear, noise signaling crack initiation is clearly heard.
As the load was increasing, the number of cracks kept increasing
at an accelerating pace up to failure. From these observations, it seems
that damage initiates as a critical stress level is reached and that it increases
gradually until failure occurs.
Repeated tensile loading tests with progressively increasing maximum
stress were performed on a series of ten specimens to study damage initiation and evolution. The specimens were instrumented with an axial
and a transverse extensometer. A typical stress–strain curve is presented
in Figure 2. Up to a stress of approximately 30 MPa, the material shows
a linear elastic behavior. Beyond this critical stress, the elastic modulus
decreases progressively. After unloading, residual strains are observed,
though they remain rather small. Such a behavior strongly suggests the
apparition and development of damage in the material. Similar observations were reported by other investigators [5,6]. Based on these results,
the material behavior can be considered as damageable elastic. For each
cycle, the elastic modulus E d1 , Poisson’s ratio d12 , and the residual strains
"p11 and "p22 were determined. From the elastic modulus measurements,
D1, the damage variable in the direction of loading (x1-direction) could be
quantified using Equation (1):
E d1 ¼ Eð1 D1 Þ2 ) D1 ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffi
E d1 =E
ð7Þ
In theory, D2, the damage variable in the direction perpendicular to loading
(x2-direction) can be evaluated from Poisson’s ratio measurements since
from Equation (1):
d12 ¼ 1 D1
) D2 ¼ 1 d ð1 D1 Þ
1 D2
12
ð8Þ
However, it turned out that Poisson’s ratio measurements were not precise
enough to correctly evaluate D2.
Table 1 presents all the experimental data obtained from the repeated
tensile test. As can be observed, as the applied stress increases, elastic
85
Table 1. Experimental data from the repeated tensile test.
Maximum
stress
level (MPa)
24.50
39.43
51.41
61.34
70.27
77.30
Elastic
modulus
Ed1 (GPa)
Poisson’s
ratio d12
Permanent
axial strain
ep11 (km/m)
Permanent
transverse
strain ep22 (mm/m)
Damage in the
direction of
loading D1
7.016
6.843
6.591
6.281
5.947
5.643
0.332
0.333
0.331
0.325
0.315
0.299
72.24
240.8
337.1
550.7
722.4
987.2
126.7
267.6
352.1
450.6
549.2
647.8
0.0134649
0.025708
0.043811
0.066579
0.091765
0.115231
Table 2. Numerical values of the unknown parameters (from [1]).
Y0 (MPa)
a (MPa1)
b
0.06819
0.07884
0.5878
0.010071
0.008371
modulus progressively decreases while the permanent strains increase. From
the elastic modulus, variable D1 could be evaluated using Equation (7).
Poisson’s ratios are also indicated in Table 1. However, the measurements could not be used to accurately evaluate damage variable D2. At the
exception of the Poisson’s ratios, the test data of Table 1 were used to
identify the five unknown parameters of the model using a constrained
optimization technique [1]. The results are reported in Table 2. Note
that parameter b is nonzero, which means that applying stresses in the
x1-direction has induced damage not only in the x1-direction, but also in
the x2-direction.
Figure 3 shows typical damage in the material after being tested in
direct tension. The micrographs were obtained from scanning electron
microscopic observations of a polished sample cut from a specimen tested
up to failure. As illustrated in Figure 3(a), large matrix cracks form primarily in the direction perpendicular to the tensile stresses. Their development
undergoes different stages. First, cracks initiate at fiber ends and around
air bubbles entrapped in the material [7]. As the stress is further increased,
the microcracks grow in the material. As illustrated in Figure 3(b), cracks
propagate from the bulk matrix towards the matrix–fiber interface, and
finally across the fibers. From these microscopic observations, it is expected
that damage will induce anisotropy in the material. The elastic properties
of the damaged material should be more affected in the direction parallel
to the applied load than in the transverse direction.
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σ1
x1
x2
σ1
(a)
(b)
Figure 3. Scanning electron micrographs of a tension specimen: (a) overall view and
(b) close-up view.
200
Back-to-back gages
180
20 mm
x1
x2
Stress σ1 (MPa)
160
140
120
100
80
60
40
20
12.7 mm
Thickness=5.7 mm
(a)
0
-0.024
-0.021
-0.018
-0.015
-0.012
-0.009
-0.006
-0.003
0.000
Strain (ε1)
(b)
Figure 4. Monotonic compressive behavior: (a) specimen configuration and (b) stress–strain
relationship.
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Compressive Behavior
The specimen configuration used for the compressive tests is shown in
Figure 4(a). Two back-to-back strain gages were mounted on the specimen
surface. A total of five specimens were tested in compression at a testing
speed of 1 mm/min.
Typical compressive test results are presented in Figure 4(b). The stress–
strain curve is clearly linear until the specimen failure, without any stiffness loss nor residual strain. The compressive modulus was found to be
equal to 8.33 GPa, which is slightly higher than in tension. The compressive
strength is 153 MPa, which is 70% larger than the tensile strength. These
results are consistent with previously reported data [3,4] showing that
short fiber SMC composites are stronger in compression than in tension.
Examination of fractured specimens showed that failure was due to matrix
shear failure.
From the test results, it is clear that the material has different mechanical behaviors in tension and in compression. The material exhibits a
damageable-elastic behavior in tension and a brittle linear elastic behavior
in compression. Therefore, the model developed to predict the material
response [1] has to take into account this so-called ‘unilateral’ behavior.
In-plane Shear Behavior
The in-plane shear mechanical behavior of the material was determined
by applying edgewise shear loads using a three-rail device. As shown
in Figure 5(a), specimens are rectangular plates with (0 /90 ) strain gage
(0°/90°) Strain gage
152 mm
Stress τ (MPa)
80
60
40
20
0
0.000
0.010
0.020
137 mm
Strain (γ)
(a)
(b)
0.030
0.040
Figure 5. Monotonic shear behavior: (a) specimen configuration and (b) stress–strain
relation.
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Stress τ (MPa)
60
50
40
30
20
10
0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
Strain (γ )
Figure 6. Repeated shear loading test.
rosettes, oriented at 45 to the main axis, mounted in the center of both
test sections. The specimens were tested to failure at a testing speed of
1.5 mm/min.
A typical stress–strain curve obtained in monotonic shear is shown
in Figure 5(b). The material has an average shear strength of 68.8 MPa
and an average shear modulus of 2.63 GPa. It is interesting to note
that the elastic properties of the intact material satisfy the isotropic
condition:
G ¼ E=2ð1 þ Þ ¼ 7:08=2ð1 þ 0:35Þ ¼ 2:62 GPa
ð9Þ
Therefore, the test results confirm that the intact material is isotropic.
Repeated shear loading tests with progressively increasing maximum
load were performed to study the material damage response. As shown in
Figure 6, the shear modulus decreases progressively as the stress increases
and a small permanent strain develops. Therefore, the in-plane shear behavior may also be considered as damageable-elastic.
Samples cut from specimens tested to failure were examined using
a scanning electron microscope to study shear damage. As revealed by
the micrographs in Figure 7, matrix microcracks develop at a 45 angle
with respect to the direction of the applied shear load. As a matter of fact,
the microcracks are oriented perpendicular to the principal tensile stress
direction.
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Figure 7. Scanning electron micrographs of a shear specimen showing 45 matrix
microcracks: (a) overall view and (b) close-up view.
ELASTIC PROPERTIES OF THE DAMAGED MATERIAL
Test Description
A series of tests were performed to determine the elastic properties of
the damaged material. Material sheets were preloaded in uniaxial tension
to produce various degrees of damage. Afterwards, specimens cut from
the damaged sheets were tested to determine their residual engineering
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elastic properties, i.e., the effective Young’s moduli, Poisson’s ratios, and
shear moduli. Similar tests were performed in [8] to evaluate the degradation
of the elastic properties of steel.
Methodology
The geometry of the material sheets that were pre-loaded in tension is
presented in Figure 8. The specimens were 4 mm-thick, 900 mm-long
and 200 mm-wide with tabs at both ends. They were loaded in tension
up to four levels of stress (30, 50, 60, and 70 MPa) at a testing speed of
1.5 mm/min. For each stress level, two plates had to be tested to provide
a sufficient amount of tensile and shear specimens. Therefore, a total of
eight rectangular sheets were tested.
Figure 8 shows how the specimens were cut from the sheets. Dog
bone shaped specimens (12) were cut along three directions (0, 90, and 45
with respect to the initial loading direction). Rectangular specimens (2) were
Plate P1
Figure 8. Sheet dimensions and specimen configurations.
Plate P2
91
cut along the longitudinal and transverse directions. Tensile tests were
performed on the dog bone specimens to determine the effective Young’s
modulus and Poisson’s ratio in the 0, 90, and 45 directions, referred to
respectively, as E d1 and d12 , E d2 and d21 , and E d(45) and d(45). Shear tests
were carried out on the rectangular specimens to measure the effective
shear moduli Gd12 and Gd21 . Due to symmetry of the elastic tensor, it could
be expected to obtain identical results for the two shear moduli.
The strains were recorded using the same equipment as for the characterization tests. A longitudinal and a transverse extensometer were used for
the tensile specimens and two (0 /90 ) strain gage rosettes were mounted on
the shear specimens.
Experimental Results
Figures 9–11 show the evolution of the various elastic properties as a
function of the pre-stress applied to the material. For each pre-stress level,
several data points are plotted. Each point was obtained from the test result
of a single specimen. Data scatter is important for most properties except
for shear modulus.
The experimental results reveal that all three effective Young’s moduli,
E d1 , E d2 , and E d(45) decrease as the pre-stress increases (Figure 9). Modulus
E d1 decreases at the highest rate, which was expected since it is measured
in the pre-stress direction. Modulus E d2 exhibits a trend similar to the one
of modulus E d(45). The reduction of elastic modulus E d2 shows that damage
develops also in the direction transverse to the load, but to a lesser degree
than in the loading direction. These results are consistent with scanning
electron microscopic observations made on the damaged material.
An interesting comparison can be made with somewhat similar tests
performed by Perreux and Siqueira [6] on SMC specimens. The investigated
material was a random short fiber reinforced sheet molding compound
with 26 wt% 25 mm long E-glass fibers in a polyester resin containing
calcium carbonate filler. It was also found that the elastic modulus of the
material is affected in the 45 -direction. However, tensile tests performed
in the 90 -direction showed that the elastic modulus of the material was
unchanged. A micrographic study revealed that only small matrix microcracks perpendicular to the load develop under uniaxial tension. Therefore,
although the SMC material investigated by Perreux and Siqueira had
the same resin type and the same reinforcement, the two materials did not
undergo the same damage mechanisms. A deeper analysis would be required
to explain the different results, but it is beyond the scope of this study.
Figure 10 shows that the effective Poisson’s ratios d12 , d21 , and d(45)
are not affected significantly by the pre-stress, irrespective of its level.
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8
Young's modulus (GPa)
(a)
7
6
5
4
Data from Plate P1
3
Data from Plate P2
2
Initial modulus
1
0
0
20
40
60
80
Pre-stress (MPa)
8
(b)
7
6
5
4
3
2
Data from Plate P1
1
Data from Plate P2
0
0
20
40
60
80
Pre-stress (MPa)
8
(c)
7
6
5
4
3
Data from Plate P1
2
1
0
0
20
40
60
80
Pre-stress (MPa)
Figure 9. Degradation of effective Young’s modulus caused by initial pre-stress: (a) effective
Young’s modulus E d1 vs pre-stress; (b) effective Young’s modulus E d2 vs pre-stress; and
(c) effective Young’s modulus E d(45) vs pre-stress.
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0.40
(a)
Poisson's ratio
0.35
0.30
0.25
0.20
Data from Plate P1
Data from Plate P2
Initial value
0.15
0.10
0.05
0.00
0
20
40
60
80
Pre-stress (MPa)
(b)
0.40
Poisson's ratio
0.35
0.30
0.25
0.20
0.15
Data from Plate P1
Data from Plate P2
0.10
0.05
0.00
0
20
40
60
80
Pre-stress (MPa)
(c)
0.40
Poisson's ratio
0.35
0.30
0.25
0.20
0.15
Data from plate P1
0.10
0.05
0.00
0
20
40
60
80
Pre-stress (MPa)
Figure 10. Degradation of effective Poisson’s ratios caused by initial pre-stress: (a) effective
Poisson’s ratio d12 vs pre-stress; (b) effective Poisson’s ratio d21 vs pre-stress; and
(c) effective Poisson’s ratio d(45) vs pre-stress.
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Shear modulus (GPa)
3
2
d
G 12
1
d
G 21
Initial modulus
0
0
20
40
60
80
Pre-stress (MPa)
Figure 11. Effective shear modulus vs initial pre-stress.
Finally, Figure 11 presents the results obtained for the shear tests.
For each level of pre-stress, the average effective shear moduli Gd12 and Gd21
obtained from the two specimens are very close. As for the tensile modulus,
the shear modulus decreases slightly as damage develops.
The experimental results presented in Figures 9–11 were used to validate
the choice for the elastic tensor of the damaged material and verify the
reliability of the identified set of parameters. The reduction of the effective
Young’s modulus E d2 (Figure 9(b)) revealed that damage was induced in the
direction transverse to the load, which is consistent with the value found for
parameter b in the identification process. The evolution of shear modulus
Gd12 and Poisson’s ratios d12 and d21 were used to verify the expressions
derived for the elastic properties of the damaged material. Finally, the
results for the tensile tests along the 45 -direction allowed a further
validation of the model. Comparisons between the experimental and
simulated results can be found in [1]. Overall, the predictions agree
quite satisfactorily with the test data.
CONCLUSIONS
An experimental program was conducted to characterize the mechanical
behavior of a sprayed random short fiber reinforced material. The results
of tensile, compressive, and shear tests showed that the material is initially
characterized by an in-plane isotropic behavior, a damageable-elastic
behavior in tension, and a brittle elastic behavior in compression.
The residual elastic properties of the damaged material were also determined. The experimental results revealed that damage causes the elastic
95
stiffness of the material to decrease. However, damage does not develop
in an isotropic fashion. The Young’s modulus in the loading direction is
more affected than in the direction perpendicular to the applied load.
Part of the experimental program was used to identify the parameters of a
continuum damage mechanics constitutive model. A total of five parameters
defining the constitutive laws for damage and plasticity were evaluated.
Other results obtained from the experimental program were used to partially
validate the predictions of the model. Additional experiments, in particular
biaxial tests, will be required to complete the model validation.
NOMENCLATURE
E ¼ Young’s modulus of the undamaged material
E d1 ¼ Young’s modulus of the damaged material in the x1-direction
E d2 ¼ Young’s modulus of the damaged material in the x2-direction
d
E (45) ¼ Young’s modulus of the damaged material in the 45 -direction
G ¼ shear modulus of the undamaged material
Gd12 ¼ shear modulus of the damaged material in the 1–2 plane
Gd21 ¼ shear modulus of the damaged material in the 2–1 plane
¼ Poisson’s ratio of the undamaged material
d12 ¼ Poisson’s ratio of the damaged material for uniaxial stress in the
x1-direction
d21 ¼ Poisson’s ratio of the damaged material for uniaxial stress in the
x2-direction
d(45) ¼ Poisson’s ratio of the damaged material for uniaxial stress in the
45 -direction
1 ¼ uniaxial stress applied in the x1-direction
¼ shear stress applied in the 1–2 plane
ACKNOWLEDGMENTS
The authors would like to thank NSERC, ADS Groupe Composites,
Bombardier and Prevost Car for their financial support.
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European Journal of Mechanics A-Solids, 19(2): 187–208.

Experimental Characterization of Damage in Random Short Glass

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