Transcript Fracture Mechanics modeling of subsurface crack propagation

Analytical Modeling of Surface and
Subsurface Initiated Fretting Wear
Arnab Ghosh
Ph.D. Research Assistant
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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Outline
• Motivation & Background
• Surface Initiated Fretting Wear
– Simulation of fretting
– Stress based wear model (Damage Mechanics)
– Effect of friction, hardness and Young’s modulus
• Subsurface Initiated Fretting Wear
–
–
–
–
Use of Linear Elastic Fracture Mechanics (LEFM)
Crack propagation criteria
Crack paths and life calculations
Effect of friction and normal load on life
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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Motivation and Background
Cracks caused by alteration of the
friction forces acting on surfaces of
actual contact (Hirano & Goto, 1967)
Intergranular fracture of ball bearing
steel due to hydrogen embrittlement
(Scott, 1968)
Surface Crack Initiation
Alternating tensile and compressive stresses induce fatigue
crack formation around the regions of surface contact. The
direction of propagation of these cracks is clearly associated
with the direction of the contact stresses.
Cross section of specimen showing
surface cracks. (Nishioka & Hirakawa,
1969)
Crack formation underneath the wear track of annealed copper
(Suh, 1973)
Subsurface crack initiation
- Ductile fracture initiated by formation of microcracks at
interface between precipitates
- Subsequent removal of material in fretting wear happens
due to delamination (Waterhouse, 1977)
Fracture surface showing crack extension by alternating shear
(wavy slip region) – (Pelloux, 1970)
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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Simulation of Fretting in FEA
Partial Slip
A 2 dimensional Hertzian line contact
with plain strain condition is simulated
in FEA to study the stress states at
different stages of fretting.
Gross Slip
Von Misses stress and fretting loops at the interface
It can be observed that high contact stresses are observed in the slip regions and therefore,
surface damage (wear) can be related to these stresses.
Each Voronoi cell
is divided into
Constant Strain
Triangle elements
Steel microstructure
Voronoi Tessellation
FEA mesh
2D Voronoi tessellations incorporate randomness in the microstructure and geometrically simulate the grain morphology
observed in reality.
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November 14, 2013
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Stress based Wear Model
ENERGY BASED WEAR EQUATION
DAMAGE EVOLUTION
𝐹𝑓 = 𝐹𝑠 + 𝐹𝑝
(E Rabinowicz)
Fs : Shear Force
Fp=Ploughing Force
Generalized damage equation:
 

dD
 

dN
 r 1  D  
𝐹𝑓 = 𝑆𝑢𝑠 𝐴𝑟 + 𝐹𝑝
Fp = 0 (for surfaces with similar
hardness and roughness)
𝐴𝑟 ≅ 𝐹𝑁 /𝐻
𝐹𝑠 ≅ 𝐹𝑁 𝑆𝑢𝑠 /𝐻
𝜇 = 𝐹𝑠 /𝐹𝑁 ≅ 𝑆𝑢𝑠 /𝐻 (Amonton)
𝑉 ∝ 𝜇𝐹𝑁 𝑠
(Fouvry et al)
𝑆𝑢𝑠 𝐹𝑁 𝑠
𝑉=
𝐸𝐻
m
Damage Law derived for Wear
equation:
𝒅𝑫
𝑺𝒖𝒔 ∆𝝉
=
𝒅𝑵 𝑬𝑯(𝟏 − 𝑫)
∆𝜏 ∶ (max − min) shear stress
𝑆𝑢𝑠 : Ultimate strength in shear (0.8𝑆𝑢𝑡 )
INTERGRANULAR CRACK PROPAGATION
Grain removal
(Crack
surrounds a
grain)
D  Dc
Crack at grain
boundary
Crack Propagtes along the
grain boundary in CCW
direction
Simulating wear by
removing grains at
the contact interface
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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Wear Propagation
Evolution of contact pressure as wear progresses
Vw  Vwr N  Vwo
Archard’s Law:
VAR 
k AR FN 4 N
H
VAR k AR FN 4

N
H
From the Damage Mechanics model:
Vwr 
VAR kGS FN 4

N
H
kGS 
Vwr H
FN 4
The coefficient kGS thus obtained is compared to
Archard’s wear coefficients found in literature
Comparison of wear scars with experiments
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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Effect of Coefficient of Friction
2000
1500
1000
0.25
0.5
0.75
1.0
500
0
0
500
1000
Number of Cycles
H=4GPa, E=200 GPa
1500
2500
22.928,-4018
30.911,-4812
33.492,-4547.7
33.255,-4504.3
3000
2500
2
2500
3500
9.6586,-4317.2
7.3796,-2516.8
13.379,-4528.9
13.612,-4631.9
3000
2
Wear Volume (m /m)
2
Wear Volume (m /m)
3000
3500
5.1226,-4199.4
8.5199,-5385.9
8.4701,-4716.2
6.7386,-3586
Wear Volume (m /m)
3500
2000
1500
1000
0.25
0.5
0.75
1.0
500
0
0
500
1000
Number of Cycles
2000
1500
1000
0.25
0.5
0.75
1.0
500
1500
H=2.5GPa, E=200 GPa
0
0
500
1000
Number of Cycles
1500
H=1GPa, E=200 GPa
𝑉𝑤 = 𝑉𝑤𝑜 + 𝑉𝑤𝑟 𝑁
H (GPa) E (GPa)
4
200
4
200
4
200
4
200
µ
0.25
0.5
0.75
1
Vwr
5.1226
8.52
8.47
6.7
Vwo V(@10000)
k
4199.4
47027
1.14E-02
5385.9
79814
1.89E-02
4716.2
79984
1.88E-02
3586
63414
1.49E-02
A critical value of µ was observed between 0.25
and 0.5 for the mentioned input parameters.
Increasing µ beyond 0.5 doesn’t change wear rate
considerably.
Wear rate vs Coefficient of Friction
V@10,000 : Wear Volume after 10000 cycles
calculated using the equation
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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3500
3000
3000
3000
2500
2
2500
2
Wear Volume (m /m)
3500
Wear Volume (m /m)
3500
2
Wear Volume (m /m)
Effect of Hardness
2000
1500
8.5199,-5385.9
14.352,-5734
30.911,-4812
1000
4 GPa
2.5 GPa
1GPa
500
0
0
500
1000
Number of Cycles
1500
µ=0.5, E=200 GPa
2000
8.4701,-4716.2
13.379,-4528.9
33.492,-4547.7
1500
1000
4 GPa
2.5 GPa
1GPa
500
0
0
500
1000
Number of Cycles
1500
2500
2000
1500
1000
4 GPa
2.5 GPa
1GPa
500
0
0
µ=0.75, E=200 GPa
H (GPa) E (GPa)
4
200
4
200
4
200
2.5
200
2.5
200
2.5
200
1
200
1
200
1
200
6.7386,-3586
13.612,-4631.9
33.255,-4504.3
500
1000
Number of Cycles
µ=1.0, E=200 GPa
µ
0.5
0.75
1
0.5
0.75
1
0.5
0.75
1
Vwr
8.52
8.47
6.7
13
13.38
13.61
30.91
33.49
33.255
Vwo V(@10000N)
k
5385.9
79814
1.89E-02
4716.2
79984
1.88E-02
3586
63414
1.49E-02
5734
124266
1.81E-02
4529
129271
1.86E-02
4632
131468
1.89E-02
4812
304288
1.72E-02
4547.7
330352
1.86E-02
4504.3
328046
1.85E-02
Wear rate vs Hardness
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
1500
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Effect of Young’s Modulus
3500
14.352,-5734
8.0862,-2980.9
3000
6.4052,-2362.3
1500
1000
200 GPa
300 GPa
400 GPa
500
0
0
500
1000
Number of Cycles
1500
2500
2
2500
2
2000
30.911,-4812
20.25,-2987.1
16.329,-2414.1
3000
Wear Volume (m /m)
Wear Volume (m /m)
2500
2
Wear Volume (m /m)
3000
3500
3500
8.5199,-5385.9
5.2344,-3131.5
4.138,-2451.1
2000
1500
1000
200 GPa
300 GPa
400 GPa
500
0
0
µ=0.5, H=4 GPa
500
1000
Number of Cycles
2000
1500
1000
200 GPa
300 GPa
400 GPa
500
1500
0
0
µ=0.5, H=2.5 GPa
H (GPa) E (GPa)
4
200
4
300
4
400
2.5
200
2.5
300
2.5
400
1
200
1
300
1
400
Wear rate vs Young’s Modulus
500
1000
Number of Cycles
1500
µ=0.5, H=1 GPa
µ
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
Vwr
8.52
5.23
4.14
14.35
8.08
6.4
30.9
20.25
16.33
Vwo V(@10000N)
k
5385.9
79814
1.89E-02
3131.5
49169
1.16E-02
2451.1
38949
9.20E-03
5734
137766
1.99E-02
2981
77819
1.12E-02
2362.3
61638
8.89E-03
4812
304188
1.72E-02
2987
199513
1.13E-02
2414.1
160886
9.07E-03
It has been shown that for low cycle fatigue wear of dry and
smooth contacts , the wear coefficients are of the order of 10-3 to
10-2 (Challen & Oxley, 1986)
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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Subsurface Crack Propagation
1600
•
•
•
Linear Elastic Fracture Mechanics is used to propagate
subsurface cracks
An initial crack of length 5 𝜇𝑚 is created at depth of 10 𝜇𝑚
from the contact surface
Alternating shear stress is observed at both the crack tips
A mode II fracture mechanism is assumed
1500
LEFT CRACK TIP
1400
 xy (MPa)
•
1300
1200
RIGHT CRACK TIP
1100
1000
2
2.2
2.4
2.6
2.8
3
3.2
Time (s)
3.4
3.6
3.8
4
Shear stress reversal at the 2 crack tips
Detailed view of the Left crack tip
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
Use of Linear Elastic Fracture Mechanics
(Mode II)
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Under compressive load (Hertzian Pressure), Mode
I growth is suppressed and Mode II growth is more
predominant. Linear Elastic Fracture Mechanics
(LEFM) can be used to find the direction of crack
growth
Check for LEFM assumption
The plastic zone size:
𝜎𝑌
CYCLIC PLASTIC
ZONE
𝟏 𝑲𝑰𝑪
𝑹𝑷 =
𝟐𝟒𝝅 𝝈∗𝒀
𝟐
= 𝟐. 𝟖𝟖 𝝁𝒎
The crack extension, Δa therefore needs to be a
value higher than 𝑅𝑃
𝐾𝐼𝐶 is the fracture toughness and 𝜎𝑌∗ for plane strain
is given by:
𝝈𝒀
𝝈∗𝒀 =
𝟏 − 𝟐𝝂
CRACK
MONOTONIC
PLASTIC ZONE
−𝜎𝑌
𝜎𝑌 is the yield stress of the material
52100 steel properties are used:
𝜎𝑌 = 1220 𝑀𝑃𝑎, 𝐾𝐼𝐶 = 18 𝑀𝑃𝑎√𝑚
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
12
Crack Propagation Direction
The radial shear stress at the crack tip is given by:
𝟏
𝜽
𝝉𝒓 =
𝐜𝐨𝐬
𝑲𝟏 𝒔𝒊𝒏𝜽 − 𝑲𝑰𝑰 𝟑𝒄𝒐𝒔𝜽 − 𝟏
Stress Intensity Factors (SIFs)
~ Modified Crack Closure Technique
𝟐 𝟐𝝅𝒓
𝟐
The alternating shear stress is given by 𝚫𝝉𝒓 =
𝟏
𝜽
𝐜𝐨𝐬
𝚫𝑲𝟏 𝒔𝒊𝒏𝜽 − 𝚫𝑲𝑰𝑰 𝟑𝒄𝒐𝒔𝜽 − 𝟏
𝟐 𝟐𝝅𝒓
𝑮𝑰 =
𝟏
𝑭 𝒄 𝒖𝒚𝒂 − 𝒖𝒚𝒃
𝟐𝚫𝒂 𝒚
𝑮𝑰𝑰 =
𝟏
𝑭𝒙𝒄
𝟐𝚫𝒂
𝟐
𝒅𝚫𝝉𝒓
=𝟎
𝒅𝜽
𝒖𝒙𝒂 − 𝒖𝒙𝒃
For plane strain,
𝑲=
𝑮𝑬
𝟏 − 𝝂𝟐
Possible crack paths
𝛥𝑎
Valid for 𝑎 < 0.05, but Crack Tip Opening Displacement is
restricted under compressive load and MCCT can be used
for higher 𝛥𝑎.
In the current model, 𝛥𝑎 = 3𝜇𝑚
The crack propagates in the direction of maximum
alternating shear stress
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November 14, 2013
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Crack Tip Mesh Refinement
•
•
•
Crack Growth showing adaptive meshing
around crack tip
The mesh around the crack tip is refined within a radius
of 𝑟 > Δ𝑎
A refined mesh around the crack tip is required to
– Obtain a more accurate singular stress field
– Propagate crack within the refined region.
An automatic adaptive mesh refinement is used which
moves with the crack tip and maintains the same level of
refinement as the crack grows.
Rp= 2.88 𝝁𝒎
CYCLIC PLASTIC ZONE
REFINED MESH ZONE
(r=5 𝝁𝒎)
CRACK
Crack Tip mesh refinement and the von Mises
stress field
𝚫𝐚= 3 𝝁𝒎
Regions around crack tip and crack extension
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
14
Crack Paths and Life
𝝁=0.1
•
10 𝝁𝒎
5 𝝁𝒎
•
𝝁=0.3
𝝁=0.6
•
Growth of Initial Crack for different values of Coefficient of Friciton
PH=0.5 GPa
•
•
PH=1 GPa
PH=2 GPa
•
Growth of Initial Crack for different values of Hertzian Pressure
Initial Crack Dimensions:
– Length: 5𝜇m
– Depth from Contact Surface: 10 𝜇𝑚
As coefficient of friction increases the span
of the crack when it reaches the surface
decreases. This is due to decrease in slip
with increase in 𝜇.
As Hertzian Pressure increases, the span of
the crack as it reaches the surface decreases.
As the applied load increases, the slip
decreases
Life is defined as the number of cycles till the
crack reaches surface, after which material will
be removed.
In the current model, life is calculated using the
Paris’ Law:
𝒅𝒂
= 𝑪 𝚫𝑲 𝒎
𝒅𝑵
For Martensitic steel,
– C=1.36 x 10-10 m/cycle MPa√m
– m=2.25
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
15
Effect of Different variables on Life
PH
PH=0.5 GPa
Approaching partial slip
Life decreases
𝜇
PH=1 GPa
PH=2 GPa
Life vs applied pressure at different values of
coefficient of friction
•
•
•
•
•
Life decreases with increase in applied load
Effect of 𝜇 at higher loads is negligible
Life decreases with increase in 𝜇.
Shear Force, Q=𝜇P
Life Decreases significantly with increase in
Shear Force.
6.5
Log (N) = - 1.39*Log(Q) + 11.3
6
5.5
Log (N)
5
4.5
4
3.5
3
2.5
3.5
4
4.5
5
5.5
6
Log (Q)
Life vs Shear Force (Q)
Log-log plot of Life vs Shear Force
Log(N) = -1.39 Log (Q) +11.3
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
16
Summary
•
Surface initiated fretting wear can be modeled by damage mechanics using only standard material
properties
–
–
–
•
Wear rate decreases with increase in Hardness and Young’s modulus
Increasing coefficient of friction beyond 0.5 doesn’t impact wear rate
The wear coefficients obtained from the model are comparable to Archard’s wear coefficient
Sub surface initiated fretting wear can be modeled by Linear Elastic Fracture Mechanics
–
–
–
–
Alternating shear stress at crack tips drives crack propagation. Crack direction is calculated using a
Mode II criteria
Crack path is studied for different combinations of variables
Paris’s Law is used to calculate the Life
Life decreases with increase in applied load and coefficient of friction
Future Work
•
•
•
•
•
Incorporate plasticity effects and model hardness in the stress based damage mechanics model
Study the effect of grain size and surface roughness
Extend the LEFM model to study cracks at different depths from the contact surface
Model stress risers (inclusions, void) in the domain and study its effect on crack path
Combine Damage Mechanics and LEFM: Subsurface crack initiation using damage mechanics and
propagation using LEFM
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013