Critical Plane Analysis of Fatigue Crack Nucleation
Under Tension, Shear, and Compression
W. V. Mars, Ph.D, P.E.
Endurica LLC
email: [email protected],
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
   a cos(t )   m
Simple Tension
1  
F   0
 0
0
(1   ) 1/ 2
0


0

1 / 2
(1   ) 
0
 a   m  0.25, 0.375, 0.50, 0.75,1.00
Simple Shear
1 (1   )  (1   )

F  0
1
0
0

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1
Simple Compression
0

0
1
 1
 (1   ) 2

F 0
 0


0 

1 
0 
0 1  

0
Applied Duty Cycles in Terms of
Nominal Strain
2
Simple Tension
Simple Shear
1.5
Engineering Strain
Simple Compression
1
0.5
0
0
1
2
3
4
5
6
-0.5
-1
time
100% peak maximum principal strain
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7
Ogden Hyperelastic Law
n
W 
i 1
2i

2
i

i
1

 2 i  3 i  3
i=1
i=2
i=3
µ, MPa
1.0
0.1
0.02
α
2.5
-3.5
8
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Stress-Strain Behavior
Simple Tension
Simple Shear
3
Engineering Stress, MPa
2.5
2
1.5
1
0.5
0
0
-0.5
0.5
1
Engineering Strain
1.5
0
-0.8
-0.6
-0.4
-0.2
2.5
0
-10
Engineering Stress, MPa
Engineering Shear Stress, MPa
3
Simple Compression
2
1.5
1
0.5
-20
-30
-40
-50
0
0
0.5
1
1.5
Engineering Shear Strain
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2
Engineering Stress, MPa
-60
Applied Duty Cycles in Terms of
Nominal Stress
60
Simple Tension
Simple Shear
4
40
Tensile Stress, MPa
Simple Compression
2
20
0
0
0
1
2
3
4
5
6
7
-2
-20
-4
-40
-6
time
100% peak maximum principal strain
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-60
Compressive Stress, MPa
6
Applied Duty Cycles in Terms of Strain
Energy Density
6
Strain Energy Density, mJ/mm^3
Simple Tension
5
Simple Shear
Simple Compression
4
3
2
1
0
0
1
2
3 time 4
5
6
100% peak maximum principal strain
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7
Fatigue Crack Growth Rate Law
rc
T 
r  rc  
 Tc 
F
Tc
F = 2, Tc = 30 kJ/m2, rc = 1 x 10-3 mm/cyc
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F
Flaw Size and Damage Law
N 
cf
c0
dc
r (T )
3
c0  50 10 mm
c f  1 mm
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Critical Plane Analysis

r ( ,  )
 ij (t )
ε


Wc ( ,  , t )   r T σ dε r
dc
 f (T , R)
dN
0
T2 ( ,  , c)  2 Wc c
cf
N f ( ,  ) 

c0
1
dc
f (T , R)
1
0.8
1
 min , min , N f ,min
0.6
0.5
0.4
0
0.2
-0.5
0
1
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0.8
0.6
-1
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Cracking Energy Density

d




dWc    d
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W. V. Mars, Rubber
Chemistry and
Technology, Vol. 75, pp.
1-18, 2002.
Crack Closure

d





dWc  ( n   t )  d
0
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W. V. Mars, Rubber
Chemistry and
Technology, Vol. 75, pp.
1-18, 2002.
Critical Plane Analysis
Google Scholar Publications
14
12
Search term:
rubber “critical plane”
10
~30% growth per
year in rate of
publications
8
6
Endurica LLC
founded
4
2
0
1998
2000
2002
2004
2006
2008
2010
2012
Year
2014
Endurica fatigue
solver technology
is available
commercially as
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)
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Critical Plane Analysis
Computed Life, cycles
1.E+06
Simple Tension
Simple Shear
Simple Compression
1.E+05
0
45
90
Plane Orientation
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135
180
Damage Sphere
Simple Tension
Simple Shear
9
Simple Compression
9
9
8.5
8.5
-1
1
8.5
-0.8
-0.6
0.5
8
7.5
0
-0.2
-1
7.5
7.5
-0.5
0
7
7
7
0.2
-0.5
8
8
-0.4
6.5
0
0.4
6.5
1
6
0
0.6
6
0.8
-1
5.5
1
0.5
0
-0.5
-1
0.5
6.5
6
5.5
1
-1
-0.5
0
0.5
1
5.5
-1
1
1
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0.5
0
-0.5
-1
Computed CED-Life
Cracking Energy Density, mJ/mm^3
1.E+02
Tension
Shear
Compression
1.E+01
1.E+00
1.E-01
1.E+03
1.E+04
1.E+05
Life, cycles
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1.E+06
1.E+07
Computed Strain-Life Curve
5.E+00
Nominal Strain
Tension
Shear
Compression
5.E-01
1.E+03
1.E+04
1.E+05
Life, cycles
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1.E+06
1.E+07
Computed SED-Life
Strain Energy Density, mJ/mm^3
1.E+02
Tension
Shear
Compression
1.E+01
1.E+00
1.E-01
1.E+03
1.E+04
1.E+05
Life, cycles
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1.E+06
1.E+07
Computed Stress-Life
Nominal Stress, MPa
1.E+04
Tension
Shear
Compression
1.E+03
1.E+02
1.E+01
1.E+00
1.E+03
1.E+04
1.E+05
Life, cycles
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1.E+06
1.E+07
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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