Simulation of Dynamic Crack Propagation in ...

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Simulation of Dynamic Crack Propagation in Heterogeneous Media Falk K. Wittel and Hans Herrmann

Physical Aspects of Fracture Scaling and Size Effects Symposium at Monte Verità (Ascona, Switzerland) March 9 - 14, 2008

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Simulation of Dynamic Crack Propagation in Heterogeneous Media Falk K. Wittel and Hans Herrmann

Physical Aspects of Fracture Scaling and Size Effects Symposium at Monte Verità (Ascona, Switzerland) March 9 - 14, 2008

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

outline • Dynamic fracture …. …in experiments …from continuum perspective …bottom up with DEM

• DEM simulations … … model validation … revolving simulation cell

• Some results

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Dynamic fracture in experiments

FINEBERG&MARDER 1999

Crack instability  Finite velocity F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

branch length [m]

Dynamic fracture in experiments

y  0.2 x 0.7

Critical velocity vc < cR Universal power law for branch shape F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Dynamic fracture in experiments - fragmentation • • • • •

Sudden energy increase “sudden” system disintegration Power laws for fragment mass distributions With universal exponent depending on the dimensionality Branching / merging mechanism

 dynamic fracture propagation

KUN, WITTEL, KRÖPLIN, HERRMANN, MALOY: PRL 96 (2005)

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Dynamic fracture from the continuum perspective d (U e ) dU o  da da dW dU dT G   dA dA dA

GRIFFITH-criterion

MOTT 1948

kinetic energy

Fracture mechanics does not explain…    

Why RAYLEIGH-wave speed cR remains unreached Why instabilities occur LEMAITRE 1990 Why cracks branch and show rough surfaces What really happens inside the process zone and how this effects G F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Dynamic fracture from the continuum perspective Yoffe-instability: The Moving Griffith Crack Crack stress field interacts with stress waves emitted by crack propagation • stress maximum (hoop stresses) shifts from 0° to 60° as the crack accelerates • cracks are intrinsically instable at speeds > 66% cR Deformation field controlled instability

E. Yoffe: Phil. Mag.Verità Lond.-42 (1951)F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte 03/2008

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

The Problem remains: Discrepancies theory-experiment-simulation • Observed branching velocities 30%cR much lower than those predicted by Yoffe (66%cR)  formation of micro cracks ahead of main crack  interactions of microcracks among themselves and with the main crack (Ravi-Chandar, Knauss 1984) • Observed branching angles smaller than 60°  Origin of crack tip instability not explained by continuum solution • Solids fail through the propagation of cracks, whose speed is controlled by instabilities at the small scales Attempts of explanation: Fineberg&Marder(lattice models, 1992-2000), Abraham (lattice vibration, 1994), Gao (purely hyper elastic, 1996), Sander& Ghaisas (thermal noise, 2008)

Everything depends on crack speeds and disorder, …. …but in finite systems crack speeds are never constant

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Dynamic fracture from the bottom up perspective Requirements for simulation models:   

   

Dynamic simulation scheme “Correct” cracking properties Representation of inherent disorder Continuum properties Crack-crack interactions Natural representation of size effects Processes in the failure zone DEM refers to any computational modeling framework which allows for finite displacements and rotations of discrete bodies, including complete detachment and recognizes new contacts automatically as the calculation progresses. CUNDALL 1989

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

DEM framework

Model construction Particle Cohesive element

• Follow

time evolution of all particles (Gear Predictor-Corrector)

Predictor step Relative Velocity, pos. Orient. Boundary conditions Force calculation

• Explicit

solution for many body systems • Particle interaction via arbitrary rheological elements

bonds contact Volumetric forces Corrector step corrected State variables Damage criterion Updates Element orientations neighbors

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Model construction Beam elongation:

elo F j  Eb Ab rij

Bending force:



1  rij  l0 l0





z,b b b 1 z z  Q j  3E I 2 (i  j )ey L

Bending momentum:



z,b z,b b b b 1 z z    M j  E I (i  j )ez  Qi  rij ex L



   max( i ,  j )  1?    th   th  2

Particles: Discs, Polygons, Spheres, Polyhedra, Combinations Cohesive elements: Springs, Beams, arbitrary rheological behaviour disorder; failure criteria F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Model construction – topological disorder

Model construction – strength disorder Weibull distribution (0,0,m)

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

DEM quasi static with disorder

E  1.15E *  t

• Continuum properties approved • Moderate dependency on a

G 

l

E 2(1  )

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

DEM dynamic    

Longitudinal – Transverse waves excited by impulse Agreement with continuum solution Surface wave speed = 0.93*ct Week dependency on disorder

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

DEM strength with disorder

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

DEM with revolving simulation cell

• •

Study morphology/velocity/elastic wave interaction As Function of loading/material/simulation technique

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

ENERGIE

Dynamic crack propagation: observations

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

DISORDER

Dynamic crack propagation: observations m=20 a=0.4 =2

m=10 a=0.4 =2

m=5 a=0.4 =2

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

DISORDER

Dynamic crack propagation: observations m=10 a=0.8 =2

m=10 a=0.6 =2

m=10 a=0.4 =2

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Dynamic crack propagation: branch length Average branch length

Branch geometry - mirror – mist – hackle -

0.33cR F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Dynamic crack propagation: branch geometry

y  0.2 x 0.7

• •

Universal shape of branches, with respect to disorder in breaking thresholds Cascading are Aspects observed for low materials F.K. cracks Wittel - Physical of Fracture Scaling anddisorder Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Dynamic crack propagation: crack roughness • Hurst exponent H

h( x )  

H

h( x )

• Evaluation with wavelet analysis Simonsen, Hansen, Nes 1998

• 0.73 < H < 0.93  needs larger systems

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Conclusions • Measurements limited to 0.4-0.8 v/cR • Instability onset in correlation with experiments for low disorder • Mirror – mist – hackle regimes observable • Branch geometry with universal shape • Roughness seems to be not universal with respect to topological disorder Revolving simulation cell promising tool or measurements of dynamic fracture propagation

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Thank you for your attention F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -

INSTITUTE FOR BUILDING MATERIALS … computational physics for engineering materials

Dynamic crack propagation

• Revolving cell also in vertical direction • Burst statistics • Elastic-plastic elements • More detailed study on crack velocity oscillations  correlation with AE or magnetic noise (?)

F.K. Wittel - Physical Aspects of Fracture Scaling and Size Effects - Symposium at Monte Verità - 03/2008 -