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Critical Plane Analysis of Fatigue Crack Nucleation Under Tension, Shear, and Compression
W. V. Mars, Ph.D, P.E. Endurica LLC
email: wvmars@endurica.com, 1219 West Main Cross, Suite 201, Findlay, Ohio 45840, USA
Presented at the Fall 182nd Technical Meeting of the
Rubber Division of the American Chemical Society, Inc. Cincinatti, OH
October 9-11, 2012
Paper #93
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Agenda
• Context
• Purpose
• Material Behavior – Stress-Strain
– Fatigue Crack Growth
– Microstructure / fatigue precursors
• Mechanics of Tension, Shear and Compression
• Critical plane analysis of crack nucleation
• Lessons learned
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Body with crack precursors
N=0
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Body with crack precursors
N=1000
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Purpose
• How can we estimate fatigue behavior of a rubber part under practical loading scenarios?
• Subject cases:
– Tension
– Shear
– Compression
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Loading Cases / Duty Cycle Definitions
2/1
2/1
)1(00
0)1(0
001
F
100
010
0)1()1(1 1
F
100
010
00)1(
12
F
ma t )cos(
Simple Tension Simple Shear Simple Compression
00.1,75.0,50.0,375.0,25.0 ma
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Applied Duty Cycles in Terms of Nominal Strain
-1
-0.5
0
0.5
1
1.5
2
0 1 2 3 4 5 6 7
Engi
ne
eri
ng
Stra
in
time
Simple Tension
Simple Shear
Simple Compression
100% peak maximum principal strain
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Ogden Hyperelastic Law
i=1 i=2 i=3
µ, MPa 1.0 0.1 0.02
α 2.5 -3.5 8
n
i i
i iiiW1
32123
2
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Stress-Strain Behavior
-0.5
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5
Engi
ne
eri
ng
Stre
ss,
MP
a
Engineering Strain
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2
Engi
ne
eri
ng
She
ar S
tre
ss, M
Pa
Engineering Shear Strain-60
-50
-40
-30
-20
-10
0
-0.8 -0.6 -0.4 -0.2 0
Engi
ne
eri
ng
Stre
ss,
MP
a
Engineering Stress, MPa
Simple Tension Simple Shear Simple Compression
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Applied Duty Cycles in Terms of Nominal Stress
100% peak maximum principal strain
-60
-40
-20
0
20
40
60
-6
-4
-2
0
2
4
6
0 1 2 3 4 5 6 7
Co
mp
ress
ive
Str
ess
, MP
a
Ten
sile
Str
ess
, M
Pa
time
Simple Tension
Simple Shear
Simple Compression
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Applied Duty Cycles in Terms of Strain Energy Density
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7
Stra
in E
ne
rgy
De
nsi
ty,
mJ/
mm
^3
time
Simple Tension
Simple Shear
Simple Compression
100% peak maximum principal strain
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Fatigue Crack Growth Rate Law
cr
cT
F
F
c
cT
Trr
F = 2, Tc = 30 kJ/m2, rc = 1 x 10-3 mm/cyc
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Flaw Size and Damage Law
fc
c Tr
dcN
0 )(
mm1
mm1050 3
0
fc
c
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Critical Plane Analysis
ε
εσ0
),,( rdrtW T
c
),( r)(tij
2( , , ) 2 cT c W c
0
1( , )
( , )
fc
f
c
N dcf T R
( , )dc
f T RdN
-1
-0.5
0
0.5
1
-1-0.8-0.6-0.4-0.200.20.40.60.81
0
0.2
0.4
0.6
0.8
1
min min ,min, , fN
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Cracking Energy Density
ddWc
d
W. V. Mars, Rubber
Chemistry and
Technology, Vol. 75, pp.
1-18, 2002.
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Crack Closure
ddW tnc )(
d
W. V. Mars, Rubber
Chemistry and
Technology, Vol. 75, pp.
1-18, 2002.
0
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Critical Plane Analysis
W. V. Mars, Rubber Chem. Technol., 75, 1 (2002).
W. V. Mars, A. Fatemi, J Mater Sci, 41, 7324 (2006).
W. V. Mars, A. Fatemi, Rubber Chem. Technol., 79, 589 (2006).
N Saintier, G Cailletaud, R Piques, International Journal of Fatigue, 28, 61 (2006)
N. Saintier, G. Cailletaud, R. Piques, International Journal of Fatigue, 28, 530 (2006)
R. J. Harbour, A. Fatemi, W. V. Mars, Int J Fatigue, 30, 1231 (2008).
A. Andriyana, N. Saintier, E. Verron, Int. J. Fatigue, 32, 1627 (2010).
B. G. Ayoub, M. Naït-Abdelaziz, F. Zaïri, J.M. Gloaguen, P. Charrier, Mechanics of Materials, 52, 87 (2012)
0
2
4
6
8
10
12
14
1998 2000 2002 2004 2006 2008 2010 2012 2014
Go
ogl
e S
cho
lar
Pu
blic
atio
ns
Year
~30% growth per year in rate of publications
Endurica LLC founded
Search term: rubber “critical plane”
Endurica fatigue solver technology is available commercially as
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Critical Plane Analysis
1.E+05
1.E+06
0 45 90 135 180
Co
mp
ute
d L
ife
, cyc
les
Plane Orientation
Simple Tension
Simple Shear
Simple Compression
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Damage Sphere
-1
-0.5
0
0.5
1
-1-0.500.51
5.5
6
6.5
7
7.5
8
8.5
9-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1-1 -0.5 0 0.5 1
5.5
6
6.5
7
7.5
8
8.5
9
-1
-0.5
0
0.5
1
-1-0.500.51
-1
0
1
5.5
6
6.5
7
7.5
8
8.5
9
Simple Tension Simple Shear Simple Compression
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Computed CED-Life
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
Cra
ckin
g En
erg
y D
en
sity
, m
J/m
m^
3
Life, cycles
Tension
Shear
Compression
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Computed Strain-Life Curve
5.E-01
5.E+00
1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
No
min
al S
trai
n
Life, cycles
Tension
Shear
Compression
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Computed SED-Life
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
Stra
in E
ne
rgy
De
nsi
ty,
mJ/
mm
^3
Life, cycles
Tension
Shear
Compression
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Computed Stress-Life
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
No
min
al S
tre
ss,
MP
a
Life, cycles
Tension
Shear
Compression
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Conclusion
• Critical plane analysis = considering all possible failure planes – Identify the worst plane – Estimate local experience of crack precursor
• Simple loads with tensile max prin stress = crack orientation normal to prin stress
• Critical plane analysis needed whenever – No principal stress is tensile (ie crack closure) – Nonrelaxing loads – Loads with variable principal directions
• Standard parameters governing fatigue behavior – Stress-strain law – FCG law – Precursor size
• X-N curves differ depending on mode of deformation, but can be calculated using numerical analysis
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The Future
• Duty cycles modeled via FEA
• Variable amplitude, variable mode loading
• Characterization and modeling of fatigue for product development
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Abstract
• When fatigue cracks develop from microscopic precursors in rubber, they tend to
do so on specific planes. The orientation and loading experiences of such planes under the action of a given mechanical duty cycle directly govern the rate at which associated cracks develop, and ultimately the fatigue life. Here we demonstrate the consequences of this principle for three loading scenarios that are common in elastomer components: simple tension, simple shear, and simple compression. For each case, we first identify a prospective failure plane, we consider the associated local mechanical duty cycle, and we estimate the rate of damage accumulation for the prospective plane. After considering all possible planes, we then identify the most critical plane(s) as the one (or several) plane(s) that has(ve) the maximum rate of damage accumulation. The analysis illuminates how fatigue behavior can be expected to differ between tension, shear, and compression, and how these differences derive from the same fundamental material behaviors.
• Keywords: Fatigue, Failure, Mechanics, Multiaxial, Critical Plane Analysis
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