continuum damage mechanics of geomaterials at finite strain

Continuum damage mechanics ofgeomaterials at finite strain

A. Karrech, Research Scientist, CSIRO

K. Regenauer-Lieb, T. Poulet, P. Schaubs, Y, Zhang

29 September 2010

MDU

Outline

1 BackgroundMotivationCurrent approach

2 Elasto-visco-plasticity at finite strain

Multiplicative decompositionConstitutive relations

3 Damage mechanismVoid growth under several control mechanismsThe limit theory approximation

4 Validation / ApplicationValidation of the large transformations modelDamage of a notched plate and effects of pressureChemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs)

5 Summary

Damage at Finite Strain

Instabilities

Large transformations to describe earth systems instabilities

Damage at Finite Strain

Material Softening

•The predicted forces for splitting continents apart are much higher then available from plate tectonics.

•Time and length scales can’t be achieved in the laboratory.

Regenauer-Lieb et al 06, Nature

Outline

1 BackgroundMotivationCurrent approach

2 Elasto-visco-plasticity at finite strain

Multiplicative decompositionConstitutive relations

3 Damage mechanismVoid growth under several control mechanismsThe limit theory approximation

4 Validation / ApplicationValidation of the large transformations modelDamage of a notched plate and effects of pressureChemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs)

5 Summary

Finite strain -- Review

• Additive strain rate decomposition (similar to small deformations): Green Naghdi(65), Mandel (72) , Nemat-Nasser (81)...

• Multiplicative gradient decomposition: Lee and Liu(67), Lee (69)

• Numerical integration: Simo et al. (80s-94), Argyris and Doltsinis(80s), Miehe(90s)

• Several inconsistencies (aberrant oscillations observed by Dienes (79) Simo and Pister (82), K. Regenauer-Lieb and H. Mulhaus (06)…)

• Logarithmic corotational rates: Xiao, Buhrns Meyers (98-06)

• Metallic materials: Lin, Brocks, Betten (02,04,06)

• Formulation + numerical integration for geomaterials: current work

Finite strain – Basic concept

0TT and 1,udu • Small perturbations:

• (+) Well understood + Easy integration

• (-) Limitations in predicting instabilities

Large transformations: 0TT and 1,udu

Finite strain – Oscillations

Source of the figure: www.wikepidia.com

How to formulate thermo-mechanical coupled models for frictional materials in finite strain

How to overcome thesespurious oscillations?

Decomposition

eT

eT

XX

X

X

X

XF

The deformation gradient is:

Hence, the multiplicative decomposition:

FFFFFF ThevpThˆ

XF

ˆ

We consider the measure of athermal strain:

)(Ln2

1 and )(Ln

2

1)(Ln

2

1 et bhFFbh

Objective rates

Objective rates

Dissipation inequality

Helmholtz F. E. and dissipation

Helmholtz F. E. and dissipation

Principle of maximum dissipation

Outline

1 BackgroundMotivationCurrent approach

2 Elasto-visco-plasticity at finite strain

Multiplicative decompositionConstitutive relations

3 Damage mechanismVoid growth under several control mechanismsThe limit theory approximation

4 Validation / ApplicationValidation of the large transformations modelDamage of a notched plate and effects of pressureChemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs)

5 Summary

Micro-scale model

A. C. F. Cocks and M. F. Ashby, progress in materials science, 1982, Vol. 27, pp. 189 to 244

Comparison with other damage models

1 ),f1()f1(f For small f,

Cocks and Ashby models coincide with the descriptions of Kachanov (58) and Lemaitre and chaboche (80s)

f1

1Yf

dt

df0

The comparison highlights what we believe to be certain fundamental weaknesses of the continuum equations:

first, the prediction that the damage-rate is finite even when there is nodamage;

second, the prediction that the damage-rate always accelerates with damage;

Current approach

Assumptions:

• Vacancies within a given RVE are assumed to be within a spacing of min(2d, 2L),

(d and L are distances in the longitudinal and radial directions)

• Voids are assumed to be of small size as compared to the

• Voids are self-similar in terms of shape during the deformation process.

Upper limit (MARTIN, JMPS, 62)

V

ddVWW "'.)'()"( * uTσε

Current approach

After Integration (Karrech el al., ICAMEM Conference 2010)

inng DDD )1()1(

Similarly to Dahar et al (1996), we add a nucleation effect (no justification yet)

1)()1( }1{ YDD n

Integration with respect to the thermodynamic force of damage:

c)Y(Y1)D1(f }1n{D

Outline

1 BackgroundMotivationCurrent approach

2 Elasto-visco-plasticity at finite strain

Multiplicative decompositionConstitutive relations

3 Damage mechanismVoid growth under several control mechanismsThe limit theory approximation

4 Validation / ApplicationValidation of the large transformations modelDamage of a notched plate and effects of pressureChemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs)

5 Summary

Axially loaded sample

Axially loaded sample

Simple Shear

Simple Shear in hyer-elasto-plasticity

Necking problem

Good agreement between the experimental and numerical results

Triaxial test

Damage of a notched plate (Olivine)

Effect of pressure dependency

Effect of pressure dependency

Courtesy of Arcady Dyskin, UWA

Outline

1 BackgroundMotivationCurrent approach

2 Elasto-visco-plasticity at finite strain

Multiplicative decompositionConstitutive relations

3 Damage mechanismVoid growth under several control mechanismsThe limit theory approximation

4 Validation / ApplicationValidation of the large transformations modelDamage of a notched plate and effects of pressureChemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs)

5 Summary

Chemo-thermo-hydro-mechanics

(d) ...2,1 rqc.vc

(c) rqbpM

1

(b) rqT.vCTC

(a) 0bp

i,ii,i

ffi,i

TTi,iii

fp

fp

i,'

j,ij

Chemo-thermo-hydro-mechanics

(d) cq

(c) pq

(b) kTq

(a) uu2

1 with C

i,i

i,f

fi

i,Ti

i,jj,iijepijkl

'ij

Permeability evolution with damage

Chemo-thermo-hydro-mechanics

Fluid flow through damaged zones

Preliminary chemistry

Invitation

I invite you to talk to Thomas Poulet for more details about multi-physics Problems

Outline

1 BackgroundMotivationCurrent approach

2 Elasto-visco-plasticity at finite strain

Multiplicative decompositionConstitutive relations

3 Damage mechanismVoid growth under several control mechanismsThe limit theory approximation

4 Validation / ApplicationValidation of the large transformations modelDamage of a notched plate and effects of pressureChemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs)

5 Summary

Damage & thermo-coupling

• The Late Archaean Yilgarn Craton of Western Australia hosting orogenic gold deposits

• Different loading scenarios

Invitation

I invite you to talk to Peter Schaubs for more details about the field application

Outline

1 BackgroundMotivationCurrent approach

2 Elasto-visco-plasticity at finite strain

Multiplicative decompositionConstitutive relations

3 Damage mechanismVoid growth under several control mechanismsThe limit theory approximation

4 Validation / ApplicationValidation of the large transformations modelDamage of a notched plate and effects of pressureChemo-thermo-hydro-mechanics (See Thomas Poulet) Damage down under (See Peter Schaubs)

5 Conclusions

Outline

• Finite strain for geo-materials based on logarithmic strain measures and corotational rates.

• Solution for the spurious oscillations

• Continuum damage mechanics following based on approximate potential

• Instabilities and localizations are accelerated in such circumstances

• Multi-physics problems in the context of mining

Thank you

Computational Geoscience GroupDr Ali KarrechResearch Scientist @ CSIROAdjunct Associate Professor @ UWA

Phone: +61 8 64 36 86 96 Email: ali.karrech@csiro.auWeb: www.csiro.au