2005 AGU Fall Meeting: Poster #S43A

Quasi-statically Self-chosenQuasi-statically Self-chosenFaulting Path ModelingFaulting Path Modeling

in Heterogeneous Mediumin Heterogeneous Medium- FEM-b Approach -- FEM-b Approach -

Nobuki KAME*

Dept. of Earth and Planetary Science, Kyushu University. Kenji OGUNI Earthquake Research Institute, University of Tokyo.

1. Abstract1. Abstract We apply FEM-β, a newly proposed Finite Element Method

(Hori, Oguni and Sakaguchi, JMPS, 2005), to quasi-staticallyself-chosen faulting path modeling.

The method, FEM-β, is based on particle discretization of adisplacement field with non-overlapping shape function and itprovides an easy way to express displacement discontinuitiesbetween any two adjacent nodes: this is an advantage of FEM-βfor self-chosen failure path modeling.

FEM-β, originally developed for the analysis on tensile failurewithin a structual material containing local imperfection, is heretested for earthquake shear faulting in strongly heterogeneousmedium to investigate the effect of elastic heterogeneity in thecrust on the formation of geometrically complex fault traces.

2. MOTIVATION2. MOTIVATION Non-planar fault traces in the heterogeneous crustal structure

The effect of elastic heterogeneity on the formation of geometrically complex fault traces→ Quantitative approach by self-chosen failure path modeling

Fig.2: Non-planar fault surfaces anden-echelon surface ruptures

in a layered medium(Bonafede et al., GJI, 2002)

Fig.1: Schematic cross section of the updip portion ofthe Nankai subduction zone: splay fault branching

(Park et al., Science, 2003)

3. Self-chosen faulting path modeling3. Self-chosen faulting path modelingin heterogeneous mediumin heterogeneous medium

(3A) FEM-β : Finite Elemant Method with particle discritization (Hori et al., 2005)

Fig.3. Comparison of discretization forordinary FEM and FEM-β.

Fig.4. Voronoi (thick lines) and Delaunay (thin)tessellation for 2-D domain in FEM-β.

Ordinary FEM ◎different elasticity × discontinuity

FEM-β◎different elasticity◎discontinuity

Smooth but overlapping shape functions

Particle discretization bynon-overlapping functions

Ω1

Ω2

Ω3

Displacement on Volonoi blocks (Ω1, Ω2, Ω3,…)→ Easy expression of failure as separation oftwo adjacent Voronoi blocks→ Enabling self-chosen failure path modeling

(3B) Interpretation of FEM-β as block-spring model

B-matrix of FEM-β is equivalent to the ordinary FEM B-matrix. Particle discretization enables us to interpret FEM as Spring-Block System:

Strain is represented by the displacement gap Δ1=u1-u2 , Δ2 and Δ3. For a gap Δ1=u1-u2, three springs are responsible for the strain energy.

Fig.5. Strain average domain S in FEM-β

!

e11

e22

2e12

"

#

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%

&

' ' '

=1

2S

by1

0

0 (bx1

(bx1

by1

by2

0

0 (bx2

(bx2

by2

by3

0

0 (bx3

(bx3

by3

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' ' '

u1

u2

u3

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' ' '

S

u1

u2

u3

Δ1

Δ2

Δ3b1

b2 b3a1

a2a3

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=1

2S

ay1 " ay

30

0 "(ay1 " ay

3)

"(ay1 " ay

3) ay

1 " ay3

ay2 " ay

10

0 "(ay2 " ay

1)

"(ay2 " ay

1) ay

2 " ay1

ay3 " ay

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2)

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( ( (

u1

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u3

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=1

2S

ay1

0

0 "ax1

"ax1

ay1

ay2

0

0 "ax2

"ax2

ay2

ay3

0

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"ax3

ay3

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% % %

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( ( (

)1

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)3

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( ( (

e = B u

= A Δ

Fig.6. Strain e in terms of u with B-matrix and in terms of Δ with A-matrix in FEM-β.

(3C) Expressing Shear failure in FEM-b

Tensile failure as a displacement gap Δ1: subtract B-matrix componentscorresponding to cutting of the three springs.

Shear failure: subtract B-matrix components responsible only for the shearstrain along a failure plane (no change in the normal strain).

Fig.7. Three springs conecting Ω1 and Ω2 in Δ123

FEM-β, originally developed for tensile failure, is here tested for shear faulting.

Δ1

Δ2

Δ3Δ1

!

J(e) =1

2u1

u2

u3[ ]BT

D B

u1

u2

u3

"

#

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%

&

' ' '

=1

2(1 (2 (3[ ]AT

DAA

(1

(2

(3

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&

' ' '

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J("1) =1

2(A

1T

DAA1)"1"1

+1

2(A

2T

DAA1)"

2"1

+1

2(A

3T

DA

A1)"

3"1

J(e): strain energyK: element stiffness matrixD: elasticity matrix

=K

4.4. Results for Mode II crackResults for Mode II crack Deformation: 2-D plane strain. Simple shear loading up to ε12=0.05. B.C. fixed displacement on y=1,-1, free surface on x=1, -1.

Fig.8. FEM-Fig.8. FEM-ββ mesh used mesh used in the analyses.in the analyses.12918 elements &12918 elements & 6620 blocks.6620 blocks.

Red lines:Red lines:Delaunay Delaunay triangulation of thetriangulation of the domain fordomain forthe stress and strain.the stress and strain.

Blue linesBlue lines::Voronoi Voronoi block boundaries on whichblock boundaries on whichdisplacementdisplacement gap (=failure) is allowed.gap (=failure) is allowed.

(4a) Homogeneous medium

Fig.9. FEM-β: Stress change in σ12 due toa crack in homogeneous medium (mode II)

Fig.10. Analytic solution (mode III)of strress change in σ23

in homogeneous medium(Bonafede et al., 2002)

(4b) Hoterogeneous (Layered) medium: µ1=0.1µ2

Fig.11. FEM-β: Stress change in σ12 due toa crack in layerd medium (mode II).

Fig.12. Analytic solution (mode III) ofstress change in σ23 in layered medium(Bonafede et al., 2002)

(4c) Strains and stresses for different crack positions against the soft layer

Fig.13. FEM-β: Strain (ε12) due to crack in layerd medium (mode II).

Fig.14. FEM-β: Stress (σ12) due to crack in layerd medium (mode II).

5. Summary

Self-chosen faulting path modeling in heterogeneousmedium is considered.

FEM-β, developed for tensile failure, is here tested forshear crack growth.

The stress field in layered medium containing a crack isanalyzed.

Further studies: Analysis of quasi-static crack growth toward stress

concentration direction Criteria for rupture growth in heterogeneous medium. Extension of FEM-β to dynamic analysis