I.
Objective
The primary focus of this lab was to perform plane strain and plane stress fracture tests on two compact
tension specimens (CTS). We used the results of these tests to calculate the respective fracture toughness ( K IC
and K C for plane strain and plane stress, respectively) and noted the differences between the two. We then used
our value for K IC to determine the fracture load on cracked and uncracked specimens in 3-point-bending.
Additionally, we also used upper-bound and lower-bound analyses to predict the collapse load for cracked and
uncracked specimens in 3-point-bending, and used these results to formulate a graph of the residual loadcarrying capacity.
II.
Experimental Methods
2.0
Standard Protocol for CTS Testing
In order to comply with ASTM standards for fracture toughness testing, the CTS specimen must be
machined to comply with a set of rigorous dimensional guidelines. These guidelines are outlined in Figure A-1
in the appendices.
2.1
Plane Strain Fracture Toughness Test
In order to determine the plane strain fracture toughness, we began by performing a plane strain tension
test on a specimen of 7075-T6 Aluminum with standard ASTM CTS geometry (a schematic of this geometry
can be found in Figure A-2 in the appendices). Preliminary measurements gave us the following dimensions:
 Thickness (B) = 12.7 mm
 Width (W, horizontal distance from clamp holes to far edge of specimen) = 51 mm
 Crack Length (a, notch + pre-crack) = 25.5 mm
 Ligament (L, W-a) = 25.5 mm
After these dimensions were documented, the specimen was placed in the tensile testing device, and continuallyincreasing load was applied until the specimen ultimately failed. The fracture load was documented in kN.
2.2
Plane Stress Fracture Toughness Test
The plane stress tension was conducted very similarly to the plane strain tension test (7075-T6
Aluminum with ASTM CTS geometry). The only difference was the thickness of the specimen. In order to
achieve the plane stress condition (which will be discussed at-length later in this report), the thickness was
milled down to about 3.5 mm. Additionally, the following dimensions were slightly different than the previous
specimen:
 Width (W) = 51 mm
 Crack Length (a) = 27 mm
 Ligament (L) = 24 mm
We documented these dimensions and placed the specimen in the tensile testing device. A continuallyincreasing tensile load was applied until the specimen failed, and the final fracture load was documented.
2.3
Three-Point Bending Test
The three-point bending test consisted of placing a rectangular specimen of 7075-T6 Aluminum into a
three-point bending situation. We performed this test using two different specimens: a cracked specimen with a
triangular notch and an uncracked specimen with a semi-circular notch (schematics can be found in the
appendices, Figure A-3). The dimensions of these specimens were as follows:
 Depth (B) = 12.7 mm
 Span (s) = 250 mm
 Height (b) = 38 mm
1
For the cracked specimen, the overall crack length (a) was 13 mm. For the uncracked specimen, the radius of the
circular notch was 9 mm.
In order to experimentally obtain values for fracture load, we placed each specimen in a machine and
applied three-point bending (a schematic of the procedure can be found in Figure A-4). A steadily-increasing
center-span load was applied until the specimen experienced failure. We documented the center-span fracture
load for each specimen.
III.
Results
3.1
Plane Strain/Plane Stress Fracture Tests
The CTS specimen that underwent plane strain (thickness = 12.7mm) ended up fracturing at a maximum
tensile load of 4 kN. The CTS specimen that underwent plane stress (thickness = 3.5 mm) failed at a maximum
tensile load of 12 kN. Each of these maximum tensile loads ( P fracture ) can be used to calculate values for the
respective fracture toughness, K IC and K C , of the material. This calculation will be presented in-depth in the
Discussion section of this lab report.
3.2
Three-Point Bending Test
The first specimen tested, which was the uncracked 7075-T6 aluminum beam with the semi-circular
notch, failed at a center-span load of 30 kN. The second specimen, which was the cracked aluminum beam with
a triangular notch, failed at a maximum center-span load of 8.5 kN. These experimental values will be compared
with theoretical analyses, which will be presented in the Discussion section of this report
IV.
Discussion
4.1
Plane Strain Fracture Test
Now that we have experimentally obtained a value for P fracture for the specimen under plane strain, the
fracture toughness of the specimen, K IC , can be calculated. The following general formula will be used for
calculating K IC :
w
w
w
w
w
w
w
w
w
w
w
w
w
K Q   Q Q p a
[1]
where K Q is a “conditional” value for fracture toughness that will be used throughout the bulk of the derivation.
A few of the values in formula [1], namely  Q and Q, must be clarified. The formula for deriving  Q
is as follows:
Q 
Pfffffffff
Q
A
[2]
The value for P Q is the value for the estimated maximum tensile load. This load relates to the maximum tensile
load obtained experimentally in the following way:
Pfffffffffffff
max
< 1.10
PQ
[3]
The experimental value obtained for P max was 12 kN. For the sake of this derivation, we will take the
value of P Q at its minimum possible value based on the above relation. In the minimum case, the value for P Q
is 10.9 kN.
2
The value for Q, which is the correction configuration factor of the experiment, is largely dependent on
the geometry of the specimen. Given the relative crack depth (a/w) of the specimen, the respective value for Q
can be obtained using the following formula:
d
e
d
e2
d
e3
d
e4
afffff
afffff
afffff
afffff
Q  16.7 @104.7
 369.9
@573.8
 360.5
w
w
w
w
[4]
A graph of Q vs. (a/w) can be found in Figure A-5 in the appendices. When the known values of a and w are
substituted into equation [4], we obtain a correction factor of 7.6 for the specimen.
Substituting results from [2] and [4] into equation [1] and solving for K Q (which comes out to be 36.2
w
w
w
w
w
w
w
w
w
MPa p m ), we can calculate a length-dimensioned entity, as follows:
f
LQ a
h
i 2
g K
15
Q
ffffffff j ffffffffff
k
2
y
[5]
If the value obtained from equation [5] is less than the specimen thickness (12.7mm), the crack length
(25.5 mm), and the ligament (25.5 mm), then the “conditional” fracture toughness K Q is simply the plane strain
fracture toughness K IC of the material:
`
a
if LQ < B, a, a @w : K IC  K Q
[6]
In the case of this test, the value for LQ is 12.5 mm (  y = 500 MPa), which holds true to the condition
w
w
w
w
w
w
w
w
w
set forth in equation [6]. Therefore, the value for K IC (when P Q is minimum) is simply 36.2 MPa p m .
w
w
w
w
w
w
w
w
w
Conversely, when P Q is maximum (12 kN), the value for K IC is 39.4 MPa p m . The actual value lies
w
w
w
w
w
w
w
w
w
somewhere between 36.2 and 39.4 MPa p m . For the purpose of this report (and Section 4.3), the average of
w
w
w
w
w
w
w
w
w
these two extremes, 37.8 MPa p m , will be used for the plane strain fracture toughness of 7075-T6 Aluminum.
4.2
Plane Stress Fracture Test
The procedure for calculating K C , which is the plane stress fracture toughness, is very similar to the
procedure discussed in section 4.1. However, in this case, P Q = P max since the specimen does not have to
adhere to the plane strain condition. Using the same procedure outlined in equation [4], and inputting the
dimensions of the plane stress CTS specimen, a correction factor of 8.1 is derived. Additionally, by substituting
in the fracture load ( P max = 4 kN) and the new dimensions for area into equation [2], and substituting this result
w
w
w
w
w
w
w
w
w
into equation [1], a final value of 56 MPa p m is derived for K C .
It becomes evident that the two specimens turned out different values for fracture toughness, despite the
fact that they were made of the same material. This discrepancy is due to the nature of plane strain and plane
stress. Plane Strain occurs when the thickness of the specimen is substantially large, and strains normal to this
dimension, as well as shear strains, are negligible and assumed to be zero. Loads are uniformly distributed with
respect to this large dimension and act perpendicular to it; as a result, strains are only substantial in the direction
of the applied load. Plane Stress occurs when the thickness of the specimen is small, and the stress normal to
this dimension, as well as the shear stresses, are negligibly small. The loads are applied in the plane of this
smaller dimension, resulting in small normal and shear forces with respect to the smaller thickness.
By these definitions for plane stress and plane strain, we can begin to formally analyze the difference in
fracture toughness between the two specimens. In the case of plane strain, strain normal to the direction of
applied load is negligible; therefore, the specimen undergoes negligibly small deformation in this direction. In
the case of plane stress, the thickness of the specimen was small enough that stresses normal to the direction of
3
applied load, as well as shear stresses in this direction, are small enough to be neglected. As a result, the plane
stress specimen experiences more deformation, and less stress resulting from its geometry, than the plane strain
specimen. Consequently, the resistance to fracture is significantly greater.
4.3
Three-Point Bending Test
w
w
w
w
w
w
w
w
w
In Section 4.1, the value for K IC was approximated (37.8 MPa p m ). Using this value, we can predict
the fracture load of the edge-cracked 7075-T6 Aluminum beam and compare this derivation with the
experimental value. The general formula used to predict the fracture load is as follows:
2
4Bb K IC
d e
P fracture  fffffffffffffffffffffffffffffffffffffffffffffff
w
w
w
w
w
w
w
w
w
w
w
w
w
a
6Q fffff s p a
b
[7]
Where B is the depth of the specimen, b is the thickness, and s is the span. The derivation of this formula
(Figure A-6) and a table (Figure A-7) presenting the relative fracture load vs. crack length (from 0.5 mm to 20
mm) can be found in the appendices. For
a crack length of 12 mm, the estimated
value for P fracture using equation [7] is
~7,780 N. Recalling that the
experimental fracture load was 8,500 N,
it is evident that the predicted value is a
somewhat reasonable approximation for
the fracture load, although it is by no
means completely accurate.
It should be reiterated that the
value for Q depends on the relative crack
depth of the specimen used. For 3-pointbending, the relationship between Q and
relative crack depth can be found in
Figure T-1. In the case of the
derivations presented in the Figure A-6,
the following estimations were made:
 For 0 < a/b < .3; Q = 1.1
Figure T-1: Q vs. (a/b) for 3-point bending
 For .3 < a/b < .5, Q = 2(a/b) + .5
where the second case assumes that the slope is constant from (a/b) = .3 to .5. These conditions were used in the
derivation of P fracture .
We can formally estimate the maximum collapse load for the edge-cracked specimen in three-point
bending. The following general formula will be used to derive the collapse load:
`
P collapse
a2
ffffffffffffffffffffffffffffffffffffffff
y b @a B

s
[8]
The bulk of the derivation will be left to the appendices (Figure A-8), and the final result for the collapse load
of the edge-cracked beam under 3-point bending is 16 kN. Notice that this value is much higher than the
experimental value of the collapse load exhibited by the uncracked beam (8.5 kN). This is because equation [8]
does not take into account crack propagation.
Along with the derivation of collapse load, a table of Collapse Load vs. Crack Length (Figure A-9) can
be found in the appendices as well.
4
Now that we have formulated the dependence of P fracture and P collapse on the crack length for a
specimen under 3-point bending, we can calculate the residual load-carrying capacity of the specimen. This is
the expected load that a specimen can bear before it fractures, and is given by
` a
b
` a
` ac
P res a  min P fracture a :P collapse a
[9]
A graph of P collapse , P fracture , and P res vs. crack length (Figure A-10) is available in the appendices.
When we tested the uncracked beam in 3-point bending, we recorded the maximum load that was
effectively the collapse load of the beam, which was 30 kN. Keeping this in mind, we can look at the table in
Figure A-9 and the graph in Figure A-10 to check the validity of our estimates. Since the circular notch had a
radius of 9 mm, the estimated residual load-carrying capacity for an uncracked beam (a = 0) is 36.87 kN, which
is a little bit higher than the experimental value.
In the case of the cracked specimen, recall that the maximum center-span load obtained experimentally
was 8.5 kN. Using Figure A-9, and using 13 mm as the crack length, the theoretical residual load-carrying
capacity for the cracked specimen is ~7.78 kN. This value is significantly lower than the value for P fracture
obtained experimentally. This discrepancy could be due to a couple of factors:
 The value for Q used in equation [7] was approximated
 The crack length (13 mm) was measured beforehand and may not be entirely accurate
w
w
w
w
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w
w
w
 The fracture toughness K IC (37.8 MPa p m ) of the material was averaged; the specimen in reality
could have been at the higher edge of the fracture toughness spectrum
Despite these factors, the approximation for residual load-carrying capacity was relatively close, and was a
conservative estimate of the maximum load for the edge-cracked specimen.
During the 3-point bending test for the edge-cracked specimen, a graph of center span load (P) vs. crosshead displacement (  ) was recorded. In the ideal case, the graph should be linear until the point of fracture.
This is because, in the ideal case, the specimen undergoes no plastic deformation before fracture, since it takes
less applied load to propagate the crack than it does to plastically deform the specimen. However, as was
observed in lab, the curve showed significant deviation from linearity prior to fracture. This discrepancy from
the ideal case is due to small-scale plasticity occurring at the tip of the crack. This holds true with the basic
premises of linear elastic fracture mechanics.
V.
Conclusion
In this lab, it became evident that, although formulas and derivations can provide reasonable
approximations for how specimens will react under different loading conditions, there are several factors that
contribute to discrepancies between theoretical results and experimental results. These discrepancies can be
attributed to human error, experimental setup, specimen geometry, and the material properties of the specimen
(such as how much energy is required to propagate cracks, the presence of deformations, etc.). However,
theoretical equations are effective in coming up with conservative estimates for residual load-carrying capacities
of different specimens under different loading conditions.
5
VI.
Appendices
Figure A-1: Dimensional Guidelines for Compact Tension Specimen
The first step in machining a suitable CTS is to make a guess, based on previous experiments or known values,
for the value of K IC . After doing so, the critical crack-tip zone size must be calculated, via:
f
r IC a
h
i 2
g
K
1 j ffffffffffff
ffffffff
IC k
[10]
y
2
To make sure that only small-scale yielding occurs, the following conditions must hold true:
`
a
a, w @a  15r IC
[11]
Where a is the crack length and (w-a) is the ligament. Additionally, to ensure plane strain, the following
condition must also hold:
B  15r IC
[12]
A CTS must be machined based on the dimensions calculated above. After this is done, a crack must be
introduced at the root of the machined notch by pre-fatiguing the specimen. This is achieved by imposing a
time-varying tensile load on the CTS. The maximum load during pre-cracking should meet the following
condition:
P f max  0.6P Q
[13]
Where P Q is the value of the estimated final fracture load. The fatigue-generated crack should be at least 1.2
mm long. Once all of the aforementioned conditions have been met, the specimen is eligible for fracture
toughness testing.
6
Figure A-2: Schematic of CTS geometry
Figure A-3: Picture of notched and cracked beams used in 3-point bending
7
Figure A-4: Schematic of 3-point bending test
Figure A-5: Graph of Q vs. (a/w) for CTS fracture toughness test
8
Figure A-6: Derivation of Fracture Load for Edge-Cracked Specimen
For a specimen under 3-point bending, the stress intensity factor can be calculated as follows:
d e
w
w
w
w
w
w
w
w
w
w
w
w
w
a
p
K I   a Q ffff
[14]
b
In this case,  is the “no-crack” maximum bending stress in the structure, which can be calculated in the
following way:

6M
fffffffffff
2
Bb
[15]
The maximum moment in the beam occurs at the center where the load is being applied, and this moment can be
calculated by
M
Ps
ffffffff
4
[16]
Putting all of these equations together, we can formulate an equation for the maximum fracture load of a cracked
specimen under 3-point bending:
2
4Bb K IC
d e
P fracture  fffffffffffffffffffffffffffffffffffffffffffffff
w
w
w
w
w
w
w
w
w
w
w
w
w
a
6Q fffff s p a
b
[17]
Inputting all the known values, we obtain a fracture load of 8,106 N for a crack of 13 mm, keeping in mind the
fact that the value of Q depends on the relative crack depth (a/b):
`
a`
a2 b
w
w
w
w
w
w
w
w
wc
4 12.7mm A38mm A37.8MPa p m
fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
P fracture  f d
e
d w
e  7777 N
g
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
`
a w
`w
a
13mm
fffffffffffffffffffff
q
6 2
 0.5 250mm
 13mm
38mm
9
Crack Length
(mm)
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
5.50
6.00
6.50
7.00
7.50
8.00
8.50
9.00
9.50
10.00
10.50
11.00
11.50
12.00
12.50
13.00
13.50
14.00
14.50
15.00
15.50
16.00
16.50
17.00
17.50
18.00
18.50
19.00
19.50
20.00
Q
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.13
1.16
1.18
1.21
1.23
1.26
1.29
1.31
1.34
1.37
1.39
1.42
1.44
1.47
1.50
1.52
1.55
Fracture
Load (kN)
42.62
30.14
24.61
21.31
19.06
17.40
16.11
15.07
14.21
13.48
12.85
12.30
11.82
11.39
11.01
10.66
10.34
10.05
9.78
9.53
9.30
9.09
8.86
8.47
8.11
7.78
7.47
7.18
6.90
6.65
6.41
6.19
5.97
5.78
5.59
5.41
5.24
5.08
4.93
4.78
Figure A-7: Fracture Load vs. Crack Length for 3-point bending
10
Figure A-8: Derivation of Collapse Load for Specimen under 3-Point Bending
For an uncracked specimen, the maximum moment in a beam under 3-point bending is in the center of the beam
where the load is being applied. In light of this, the fully-plastic bending moment for the beam at the center is as
follows:
`
fffffffffffffffffffff
y Bb
4
2
a
Mp a0 
[18]
The estimate for the uncracked collapse load can be calculated using
P collapse
`
fffffffffffffffffffffff
y b B
a0 
s
2
a
[19]
Taking the notch or the crack into account, the effective height, b, is reduced because of the presence of said
crack or notch. With this in mind, we can calculate the collapse load of an edge-cracked specimen:
`
P collapse
a2
ffffffffffffffffffffffffffffffffffffffff
y b @a B

s
[20]
Using known dimensions, as well as the approximation that  y for 7075-T6 Aluminum is 500 MPa, we can
formulate the result for collapse load for the specimen with a crack of 13mm:
`
P collapse 
a`
a2 `
a
500MPa 38mm @13mm 12.7mm
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
250 mm
 15.9 kN
This is much higher than the experimental value, primarily because this analysis doesn’t take into account crack
propagation.
11
Crack Length
(mm)
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
5.50
6.00
6.50
7.00
7.50
8.00
8.50
9.00
9.50
10.00
10.50
11.00
11.50
12.00
12.50
13.00
13.50
14.00
14.50
15.00
15.50
16.00
16.50
17.00
17.50
18.00
18.50
19.00
19.50
20.00
Q
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.10
1.13
1.16
1.18
1.21
1.23
1.26
1.29
1.31
1.34
1.37
1.39
1.42
1.44
1.47
1.50
1.52
1.55
Collapse
Load (kN)
36.87
35.91
34.96
34.02
33.10
32.19
31.29
30.41
29.54
28.68
27.83
26.99
26.17
25.36
24.57
23.78
23.01
22.25
21.51
20.78
20.06
19.35
18.65
17.97
17.30
16.65
16.00
15.37
14.75
14.15
13.55
12.97
12.41
11.85
11.31
10.78
10.26
9.76
9.27
8.79
8.32
Figure A-9: Collapse Load vs. Crack Length for 3-point Bending
12