Dynamic Shear Failure of WeakDynamic Shear Failure of Weak Planes in Materials
Demirkan CokerOklahoma State University
March 27, 2009Middle East Technical University Ankara TurkeyMiddle East Technical University, Ankara, Turkey
Department of Aerospace Engineering
OutlineOutline
1. Examples of dynamic failure along weak planes
2. Fracture: Shear failure of coherent interfaces
1. Dynamic fracture experiments
2. How fast can cracks propagate in materials with p p gweak planes?
3. Friction: Shear failure of incoherent interfaces
1. Rate‐State friction laws and the finite element model
2. What are the frictional sliding modes?
4. Summary
Dynamic fracture and friction: Shear failure along weak planes or interfaces
Coherent interface: Fracture
Incoherent interface:Friction
V i (~km/s)
σo
Vtip (~km/s)Vtip ( km/s) tip ( / )
Vimpact
(~10 m/s)σo
Vtip (~km/s) ‐ ‐ ‐ propagation velocity of discontinuity tip
V i i d ( j il i )Vimpact ‐ ‐ ‐ ‐ ‐ ‐ Driving speed (e.g. projectile impact)
Aircraft Hardening (FAA/Boeing)
i k i i i i d h
Aircraft Hardening (FAA/Boeing)
• Dynamic crack initiation and growth criteria in ductile metals.
Damage sustained by aircraft fuselageduring explosive loading experiment.
• Formulation of local/global methodology to predict dynamic crack initiation and growth from pre‐existing fatigue cracks thereby quantify the susceptibility of fuselage structures to global d l ddynamic loading
• Evaluation of existing and future structural design concepts for their resistance to internal explosive ploading (aircraft hardening).
Composite Fan Blades (LANL/GE/Boeing)
Five foot long composite fan bladesof the GE‐90 engine used in Boeing 777.
• Incorporation of dynamic fracture criteria and dynamic crack growth toughness values into elaborate 3‐D numerical codes
• Codes utilized to model composite fan blade bird impact test for FAA certification.
Navy composite hull structuresNavy composite hull structures
∙High strain rate effects and properties of thermoset composites
∙Fracture mechanics (joints, strain energy release rates,…)
Blast or Shock Response
Impact Response
∙Material failure models/Complex stress statesFatigue
DYNAMIC DEFORMATION & FAILURE OF COMPOSITE LAMINATES
M d l S t
Cracks running along a weak plane in a multi‐layered material system under dynamic loading
Unidirectional Graphite/Epoxy composite laminates
Model System
under dynamic loading.p
Interface Failure in EngineeringInterface Failure in Engineering
Site of shear dominated failure
Lightweight Tomahawk Missile Capsule Co‐molded Hybrid FRP ‐ Steel JointLightweight Tomahawk Missile CapsuleSteel/S‐Glass Composite Joint
Co‐molded Hybrid FRP ‐ Steel Joint
The integrity of structures are often limited by failure at interfaces.
San Andreas fault as an example of crack growth along a weak plane
Part I. FractureCrack Growth
Vtip (~km/s)Vtip ( /s)
Vimpact
(~10 m/s)
Modes of crack growthModes of crack growth
M d I (O i ) M d II (Sh )Mode-I (Opening) Mode-II (Shear)
• Crack growth in homogeneous materials can only occur by Mode‐I
• In homogeneous materials, Mode‐II cracks will change direction such that crack tip locally becomes mode‐I.
• To grow Mode‐II cracks, we need a weak plane that will trap it and force it to grow – an interface.
Stresses near a crack tipStresses near a crack‐tip
LINEAR‐ELASTIC FRACTURE MECHANICSStationary and Growing cracks
Stationary Crack:
S fi ld KStress field: KI,II: Stress Intensity Factor)(2
,, θπ
σ ijIIIIII
ij fr
K=
Failure Criterion:Experiments)(),( MaterialKaQK IcI =Elasticity
Energy release rate: KG I2
Energy release rate:
Growing Cracks:
EG I
I =
Growing Cracks:
Equations for slow crack growth is the same except a velocity dependence is added.
Dynamic crack growth criteriony g
( ) ( ) fDdI
dI ttvKvatQKtK >== for)()( ( ) ( ) fDII ttvKvatQKtK >== for ,),()(
…. Dynamic Crack Growth Toughness
(Crack tip driving force)
D d l l i h h k i l i l
)(vK D
•Depends on local strain rate through crack tip velocity only.
St d t t i l t fi ld f b i ll i k iSteady-state singular stress field for a subsonically growing crack in orthotropic materials:
(Liu, Rosakis, Stout and Ellis (1996))l d di
⎟⎟⎠
⎞⎜⎜⎝
⎛−= )2/cos(
),()2/cos(
),(22
)(),,,( 22/12
112/1
1
12111 θθ
πσ
r
vbB
r
vbAtKtvxx ijijdI
Scaled Coordinates
⎟⎟⎞
⎜⎜⎛
=
+=
− 21
22
221
tan x
xxr
α
αα
μθ
μ
⎟⎟⎠
⎞⎜⎜⎝
⎛−= )2/cos(
),()2/cos(
),(22
)(),,,( 22/12
212/1
1
22122 θθ
πσ
r
vbB
r
vbAtKtvxx ijijdI
⎟⎟⎠
⎜⎜⎝
=1
tanxαθ
),( vbijαα μμ =
QuestionQuestion
• Can a crack travel faster than any of the characteristic waves in the material?
• Initial Answer: NO!
Mach wave for a disturbance traveling faster than the characteristic speed
b S iFLUIDS
LC0
Subsonic SupersonicFLUIDS
Vη1
Vη1
*βSinCV S=
β∗
β∗
V
η2
β∗
β∗
V
η2
Sub‐Shear Intersonic Supersonic
SOLIDS
RC SC LC0 √2 cs
Fiber reinforced unidirectional graphite/epoxy composite laminate
Fiber Direction
x1
x3 HomogenizedElastic Characteristic
Wave SpeedsDirection
x2 Properties Wave Speeds
E1 80 GPa cl// 7500 m/s
E2 8.9 GPa cl 2700 m/s
ν12 0.25 cs 1560 m/s
μ 3 6 GPa c 1548 m/s
50 μm
μ12 3.6 GPa cR 1548 m/s
Experimental set‐up for dynamic fracture testing using coherent gradient sensing (CGS) optical techniquecoherent gradient sensing (CGS) optical technique
Gas Gun
Grating 1Grating 2Grating 2
Lens
ApertureAperture
R t ti Mi t hi h
Collimated Laser beam (50 mm diameter)
Rotating Mirror type high‐speed camera2x106 frames/sec
Mode‐I Opening CrackMode‐I Opening Crack
• CGS fringe pattern Surface Deformation
( )h ( )223211313 2σσ bbhu +=
EXPERIMENTAL SET‐UP FOR DYNAMIC FRACTURE
Camera
TEST USING OPTICAL TECHNIQUE OF CGS
CameraSpecimenGratings
Gas Gun IR Camera
Mode I (Opening) crack propagationMode‐I (Opening) crack propagation
5 mm
-0.6 μs
1.2 μs
Crack‐tip speeds for dynamic mode‐I crack propagation in Gr/Ep unidirectional composites
Homogeneous material with aHomogeneous material with a weak plane
(Washabaugh & Knauss, 1994)
HOMALITE
HOMALITE
MODE-IMODE ICR Cs Cl
Experimental CGS Interferogram of a fast moving shear crack
Shear dominated intersonic crack growth in aunidirectional graphite‐epoxy composite laminate
Intersonic shear crack propagationin unidirectional composites
6000
7000
8000
s)
cl
vc
3000
4000
5000
rack
tip
spee
d (m
/s
0
1000
2000
0 2 4 6 8
C
cR
Time (μs)
Fiber Direction
Field of View
50 mm
Crack‐tip speeds for mode‐I and mode‐II dynamiccrack propagation in unidirectional composites
8000
9000
c lpl- σ
6000
7000
(m/s
)
v c
4000
5000
tip S
peed
Mode-II
2000
3000
Cra
ck
c R =0 99 c s
0
1000 Mode-I
c R 0.99 c s
00.0 20.0 40.0 60.0 80.0
Crack Extension (mm)
Crack tip stress singularities for intersonically growing cracks in orthotropic materialsorthotropic materials
1.0 Mode-IHuang, Wang, Liu, Rosakis; 1998
• Energy needed for Mode-I fracture:
),/,()( ijsvqI ccvf
rA
I
θσ αβαβ =0.8
0.9 qIqI(v)
gyGI = -∞ for cs < v < cl
0 5
0.6
0.7
s/mc)(
Ev sc 65801 1212
1 =ν+μ
=
)/(II ccvfA θσ0 3
0.4
0.5
qIIqII(v)Mode-II
• Energy needed for Mode-II fracture:G = 0 for c < v < c
),/,()( ijsvqII ccvf
r II
θσ αβαβ =
0.1
0.2
0.3
v c / c s = 4.472
GII 0 for cs < v < cl
GII = finite for v = vc only0.00.0 1.0 2.0 3.0 4.0 5.0 6.0
v/csV/c s
•Stable and unstable intersonic crack growth is possible under shear (mode-II) conditions only.•Stable intersonic growth is possibly at v=vc.
V/cs
Steady state crack tip speed for intersonic shear crack growth
Orthotropic Materials: = 6600 m/s)()(
12122211
1=
−=
Ecccvcp)()( 122212 1 νρρ ++ ccc
Isotropic Materials (Freund, 1979):
Sc CEv 221
==+
=ρμ
νρ )(
Fracture along a weak plane: experimentsFracture along a weak plane: experiments
Collimated Laser beamG G Collimated Laser beam (50 mm diameter)
Projectile
Gas Gun
Circular
125 mm
Homalite‐100
Specimen
ProjectilePolarizer
150 mm
50φ
125 mm
CircularPolarizer
L
50φ
150 mm
50φ
Lens
150 mm
150 mmRotating Mirror type high-speed camera
Homalite‐100high-speed camera
Intersonic shear crack growth i h t i l ith k lin a homogeneous materials with a weak plane
Isochromatic Fringe Patterns
Homalite
Homalite
Homalite
28 m/s
HomaliteHomalite
Rosakis, Samudrala, Coker; SCIENCE, 1999
Intersonic Mode‐II Crack Propagation
125 mm125 mm
Homalite
150 mm
Homalite
150 mm
50Φ50Φ
ExperimentRosakis, Samudrala & Coker ‘99
TheoryFreund ‘79
Homalite
150 mm
Homalite
150 mm
HomaliteHomalite
GIMPDaphalapurkar, Lu, Coker, Komanduri ‘07
MD SimulationsAbraham ‘04
Field evidence of intersonic rupture during the 1999 Izmit d D h k i T kand Duzce earthquakes in Turkey
M. Bouchon, M. Bouin, H. Karabulet, M. Toksöz, M. Dietrich and A. Rosakis, Geophysical Research Letters, 2001
V = √2 CS = 4.9 km/sV = CR
S
QuestionQuestion
• Can a crack travel faster than any of the characteristic waves in the material?
• Initial Answer: Depends!
F M d I k NO• For Mode‐I cracks: NO
• For Mode‐II cracks: YES.
Part II FrictionPart II. Friction
"God made solids but surfaces were the work of the devil"
‐Wolfgang Pauli
Tribological investigations of LIGA i t tLIGA microstructuresT. Bieger, U. Wallrabe
Frictional sliding:Homogeneous and Heterogeneous Slip
Davis and Reynolds, Structural Geology of Rocks and Regions
Earthquakesq
Earthquakes can be viewed as frictional sliding of tectonic plates at two time scales: Stick‐slip at
San Andreas Fault in California
100‐1000 year time‐scale and dynamic frictional sliding at 100 seconds time‐scale.
(Heaton, 1990)
Years Seconds
Frictional sliding in composite materials
Frictional sliding is an important toughening mechanism duringFrictional sliding is an important toughening mechanism during fiber pull‐out in composite materials
(Tsai & Kim, 1996)
Effect of history and sliding speed on frictionEffect of history and sliding speed on friction
τ = μ σAmontons‐Coulomb Law
μ
oμs
μ
τ μ σ
μd
V
37
ModelingModeling
• Continuum mechanics
• Elastic material properties
• Existence of an interface or weak plane– Mathematically straight interface
C h i d l d f h i f / k l• Cohesive zone models used for the interface/weak plane– Cohesive law for fracture simulations
– Rate and state dependent friction law for friction simulations– Rate‐ and state‐dependent friction law for friction simulations
• No contact model is used
Isochromatic Fringe Patterns during Frictional Sliding showing shear Mach Waves
Σo =6 MPa, Vimp=2 m/s
Isochromatic Fringe Patterns during Frictional Sliding showing shear Mach Waves Periodic Slip Pulses
Σ =10 MPa V =20 m/sΣo =10 MPa, Vimp=20 m/s
Molecular Dynamic Simulations of SlidingJ Ma H Lu B Wang R Hornung A Wissink and R Komanduri 2006J. Ma, H. Lu, B. Wang, R. Hornung, A. Wissink, and R. Komanduri, 2006
(a) t = 56 (b) t = 64
(c) t = 72(c) t 72
SummarySummary
• Sliding and Fracture of Interfaces show similar characteristics– Discontinuity tip travels at speeds faster than the shear wave speed
– Shear Mach Waves are observed through optical techniques
l l d h f f l l lf h l• Frictional sliding in the form of multiple self‐healing pulses traveling at intersonic speeds are observed
• These characteristics are observed at different length scales from the atomic to tectonic.