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.