Modèle géométrique de plissement constraint par la rhéologie et l équilibre mécanique: Application possibe en régime extensif?

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1 Modèle géométrique de plissement constraint par la rhéologie et l équilibre mécanique: Application possibe en régime extensif? Yves M. Leroy (Laboratoire de Géologie, ENS), Bertrand Maillot (University de Cergy-Pontoise), Nadaya Cubas (ENS), Pauline Souloumiac (Ecole Centrale Paris & ENS) Kristian Krabbenhoft (University of Newcastle, Australia) Nancy, CRPG, 2 et 3 Juillet 2007

Leroy (Laboratoire de Géologie, ENS), Bertrand Maillot (University de Cergy-Pontoise), Nadaya

2 Objectifs Définir des problèmes aux limites simples afin de tester des hypothèses Evolution à court ou long terme? Approche 2D ou 3D, prise en compte des fluides? Eviter les EF et favoriser des approches simplifiées.

3 Geometrical models for folding Limit analysis Application to extension? Conclusion

4 Part I : Geometrical models for folding Geometrical rules, no mechanics Suppe (1986)

5 Part II : Limit Analysis Internal and external approaches Tectonic force Q Kinematically Admissible velocity field (KA) Strength domain (bulk and discontinuities) Upper bound Q Lower bound Q U + min Max among all KA fields Max min among all SA fields L = = Exact solution Q = Q U L Strength domain (bulk and discontinuities) + Statically Admissible stress field (SA) Internal approach External approach

Max among all KA fields Max min among all SA fields L = = Exact solution Q = Q U L Strength domain

6 Part II : Limit Analysis What is new and different? The least upper bound is searched in the space of structures not only velocity fields The structural evolution is based on the least upper bound No discussion of plasticity theory and non-associated plastic flow Lower bound found by Equilibrium Element Method : provides the stress field over the structure optimized with the external approach. Note the non-symmetric application of internal and external approaches

based on the least upper bound No discussion of plasticity theory and non-associated plastic flow Lower bound found

7 Part II : Limit Analysis The strength domain G : example of Coulomb criterion Failure mode : shear band τ σ n Stress tensor in the bulk : σ G if σ e + P sin φ B 2c B cos φ B 0 with σ e = (σ xx σ yy ) 2 + 4σ 2 xy, P = σ xx + σ yy c B cohesion, φ B, friction angle in the bulk Also applicable to inherited discontinuity (normal n, stress vector : T = σ n) T G if τ + tan φσ n c with σ n = n T, τ = τ, and τ = T σ n n

2 xy, P = σ xx + σ yy c B cohesion, φ B, friction angle in the bulk Also applicable to inherited

8 Part II : Limit Analysis The external approach : example of onset of failure, 2D compression p p U 2 e 2 θ Q U 1 e 1 U 1 e 1 H Q L Shear failure Compaction failure KA velocity fields for shear failure : Û1e 1 in region 1 and Û2e 2 in region 2 KA velocity fields for compaction failure : Û1e 1 in region 1 and 0 in region 2

failure KA velocity fields for shear failure : Û1e 1 in region 1 and Û2e 2 in

9 Part II : Limit analysis Application of external approach to shear-compaction failure cos φ tan(θ+φ) Shear failure : Q Hc sin θ cos(θ+φ) + Hp tan θ Select θ = π/4 φ/2 to minimize upper bound. Least upper bound : Q 1 2 Hpk p + 2Hc k p with k p = (1 + sin φ)/(1 sin φ), Rankine passive coefficient. Compaction failure : Q = HP. Q/H Shear failure Transition pressure : p T = 2 k p (P 2c k p ) P* Compaction failure k p 2c kp 2 pt p

Least upper bound : Q 1 2 Hpk p + 2Hc k p with k p = (1 + sin φ)/(1 sin φ), Rankine passive coefficient.

10 Part II : Limit analysis Internal approach Define the space of statically admissible stress space (SA) : div(σ) + ρg = 0 at every point of domain [[σ]] n = 0 over any discontinuity T σ n = T d at the boundary Moreover, σ G, strength domain, at every point of domain. Propose a simple (polynomial) structure to the stress field to obtain analytical expression Since σ is SA, the associated load is a lower bound to the applied force Necessity of a systematic construction of the stress field for complex geometries!

11 Part II : Limit analysis extended to evolution Thrusting sequence : thesis of Nadaya Cubas Time to see the movie!

12 Part II : Limit analysis extended to evolution Thrust with fast erosion : thesis Pauline Souloumiac (2007) Distance to Coulomb criterion Strength domain over décollement : Coulomb (φ D, c D = 0) Strength domain in solid : G = {σ σ e + P sin φ B 2c B φ B 0} avec σ e = (σ xx σ xx ) 2 + 4σxy 2 et P = σ xx + σ xx Distance to Coulomb criterion : π = (σ e + P sin φ B )/2 cos φ B Variation from 0 to 1 for 0.03 of c B Data : c B = 1 MPa φ B = 10 o φ D = 10 o Active ramp, backthrust and décollement.

5 0 0 1 2 3 4 5 Strength domain over décollement : Coulomb (φ D, c D = 0) Strength domain in solid : G = {σ σ e + P sin φ B 2c B φ

13 Part II : Internal approach to provide the stress field Onset of thrusting Distance to Coulomb criterion Data : c B = 1 MPa φ B = 20 o φ D = 15 o Select the position of the new ramp, décollement partly active

5 0 0 1 2 3 4 5 Data : c B = 1 MPa φ B = 20 o φ D = 15 o

14 Part II : Internal approach to provide the stress field Application to thrusting with relief σ 11, cohesion in the bulk = 1 [MPa]

15 Part III : Application to extension? Moretti et al., J. Geodym., 2003

16 Part III : Application to extension? Moretti et al., J. Geodym., 2003

17 Part III : Application to extension? Buck, J. Geophys. Res., 1991

18 Part III : Application to extension? Buck, Tectonics, 1988

19 Part IV : Conclusion Application of internal approach (SA stress fields) -finding the sequences of faults to be activated in the short term - influence of fluid (associated pore fluid diffusion, Terzaghi stress principle) - extension to 3D for fault linkage to the west of the rift Application of external approach (KA velocity fields) -missing kinematics à la Buck for low-angle normal fault -working at the scale of the micro-plate?

principle) - extension to 3D for fault linkage to the west of the rift Application of external approach (KA

4.3 Results... 27 4.3.1 Drained Conditions... 27 4.3.2 Undrained Conditions... 28 4.4 References... 30 4.5 Data Files... 30 5 Undrained Analysis of

4.3 Results... 27 4.3.1 Drained Conditions... 27 4.3.2 Undrained Conditions... 28 4.4 References... 30 4.5 Data Files... 30 5 Undrained Analysis of Table of Contents 1 One Dimensional Compression of a Finite Layer... 3 1.1 Problem Description... 3 1.1.1 Uniform Mesh... 3 1.1.2 Graded Mesh... 5 1.2 Analytical Solution... 6 1.3 Results... 6 1.3.1 Uniform