2013

TectonoMechanics

April 15 -16, 2013 - Ecole Normale Superieure, Paris

Cohesive Critical Coulomb Wedges and Rankine Limit States

Florian K. Lehner

University of Salzburg

2

Abstract

The idea that the outer parts of mountain belts, the foreland fold-and-thrust-belts or accretionarywedges display the large-scale geometry of a taper exists since the early 1900’s (see for exampleE.B. Bailey’s account of the Jura Tectonics in his “Tectonic Essays” of 1935). Major studies byH.P. Laubscher, A.W. Bally, R.A. Price, W.M. Chapple, and others in the 1960’s and 70’s stimu-lated a closer look at the mechanics of these ‘wedges’. Following up on the paper by Davis, Suppe,and Dahlen (1983), Dahlen (1984) produced his well-known exact solution for a critical non-cohesiveCoulomb wedge. This solution was recovered a little later by Lehner (1986) through an alternativemethod, based on the concept of a ‘Rankine limit stress state’ (as familiar from soil mechanics) andthe use of Mohr’s stress circle. An advantage of this method lay in the simplicity in which the crit-ical wedge problem was solved, both graphically and analytically. The Mohr-circle construction alsodisplayed the existence of four critical taper solutions in the compressive (‘passive’) as well as in theextensional (‘active’) regime, results that appear to have escaped the attention of later writers on thesubject.

Here we briefly review the principle steps in the Mohr-circle construction and the notion of passive oractive Rankine limit stress states in an infinite sloping half-space. We then give the exact solution forsuch states of the problem of critical cohesive Coulomb wedges, as analyzed first by Dahlen, Suppe,and Davis (1984) in an approximate manner.

3

Coulomb-plastic limit equilibrium of a half-space (Rankine, 1820 - 1872)

x

yσʼv

σʼhp

g

Figure 1. Vertical and horizontal principal effective stresses σ′v and σ′

h in fluid-saturated half-space with horizontal surface.

Eqs. of equilibrium in effective stress σ′ = σ − p (compression positive):

dσ′hdx

= 0,dσ′vdy− (ρ− ρw)g = 0

4

Integrals, for σ′v(y = 0) = 0

σ′v = (ρ− ρw)gy,

σ′h = f(y).

f(y) for limit equilibrium state from Coulomb criterion (yield condition)

(σ′h + σ′v) sinφ+ 2c cosφ = ±(σ′h − σ′v)

... throughout half-space.

Two solutions for depth-dependent horizontal stress, with ρ′ = ρ− ρw,

σ′h(y) =

kaσ′v(y)− 2c

√ka = kaρ

′gy − 2c√ka

kpσ′v(y) + 2c

√kp = kpρ

′gy + 2c√kp

ka = 1/kp =1− sinφ

1 + sinφcoefficients of ‘active’ and ‘passive’ earth pressure

5

Mohr Circles and Slip Lines for ‘Active’ and ‘Passive’ Rankine Statesof a Coulomb Material

τ

0

σʼv

σʼv = γʼyactivepole Pp

c

σ’n

σ’h,p Y

π/4 - φ/2

π/4 - φ/2φ

passivepole

σʼh,a

Pa

conjugate sets of slip lines

6

Origin of Terms ‘Active’ and ‘Passive’ in Rankine’s Earth Pressure Theory

Ea

Ep

y

H

0

y

H

0

‘Active’ material exerts earth pressure Ea

Material offers ‘passive’resistance Ep

7

Pole Construction

θ

σʼn

τ

0

2θ

X

“pole” P

Y

σʼn σʼx

τxy

τyx

σʼy

τ

τ

σʼn

σʼy

τyxx

y

Q

Given Y on plane y = const., projected into Mohr-plane, find Q for given rotation angle θ(alternatively find θ for given Q)

8

Rankine State in Sloping (Submarine) Half-Space

Sediment

y

x

g

α

Water

σʼy

τyxp

τyx = σʼy tanα

σʼx

z

D

9

Rankine Equilibria in a Sloping Half-Space

Eqs. of equilibrium in effective stresses :

∂σ′x∂x

+∂p

∂x+∂τyx∂y− ρg sinα = 0,

∂τxy∂x

+∂σ′y∂y

+∂p

∂y− ρg cosα = 0,

∂σ′x∂x

=∂τxy∂x

= 0 . . . ‘Rankine State’

Assume pore pressure distribution in half-space: p = ρwgD(x) + ρwg y cosα + u(y); u(y) = excess(above hydrostatic) pressure, with u(y = 0) = 0.

With dD(x)/dx = sinα (cf. above Figure) one finds

dτyxdy− (ρ− ρw)g sinα = 0,

dσ′ydy− (ρ− ρw)g cosα +

du

dy= 0,

and integrating:

σ′x = f(y), τyx = (1− ρw/ρ)ρgy sinα, σ′y = (1− λ)ρgy cosα, λ = ρw/ρ+ u(y)/(ρgy cosα)

f(y) is now determined by invoking a uniform state of failure (Coulomb yield condition) for the entirehalf-space (Rankine limit stress state). - Mohr-circle construction delivers complete solution of criticaltaper problem without explicit computation of f(y)! Consider first two cases for cohesionless slope:

10

Passive Limit Stress State in Cohesionless Normally Pressured or Dry Slope

λ = ρw/ρ (or 0), τyx/σ′y = tanα in both cases!

σʼ1

σʼ3

σʼ1

ψ0

τ

σʼy

Y

M

P

ασʼ3

σʼn

x

yτyxΟ

φ

π - φ/2

Note: Only difference between normally pressured and dry wedges: stress–depth scaling!

11

Passive Limit Stress State in Cohesionless Overpressured Slope

τyx/σ′y = [(1− ρw/ρ)/(1− λ)] tanα = tanα′, λ = ρw/ρ+ const.

σʼ1 σʼ3

ψ0

τ

σʼy

Y

P

σʼn Ο

φ

σʼ3 M

α

τyxαʼ

σʼ1

y

x

12

Construction of Non-cohesive Critical Coulomb Wedge

τb = µbσ’n

α+β

β

α

σ’noriginal halfspace

trench

subducted plate

critical Coulomb wedge in passive Rankine stateplane of weakness

φ

‘Embedding’ of Critical Wedge in Half-Space (Lehner, 1986)

Main idea: Find angle β for which τ/σ′n of Rankine state equals τb/σ′n = µb = tanφb !

13

Determination of Wedge Angle of First Passive (Non-cohesive) Critical Taper

τ=τb ψ0

τ

Ο σʼn

1

2 3

4

βY

Q

M

Pα

N

2ψ0

2ψ1σʼn

ψ1

φ φb

σʼ1σʼ3

α+β

σʼ1

σʼ3

From triangles OYM , OQM , and ONM : ψ0 = 12

[sin−1

(sinαsinφ

)− α

], ψ1 = 1

2

[sin−1

(sinφb

sinφ

)− φb

].

Note: ψ0 = const. 6= 0 possible, because σ′1 → 0 as y → 0. No ‘constraint’ of surface-parallel σ′1(cf. Buiter, 2012), because effective stresses vanish at surface of noncohesive wedges.

14

Wedge angle δ for slope angle β = β1 and ψb = ψ1 :

δ = α + β = ψb − ψ0 =1

2

[sin−1

(sinφbsinφ

)− φb − sin−1

(sinα

sinφ

)+ α

].

For small enough αα = α0 −mβ,

with α0 = sinφ[sin−1(sinφb/ sinφ)− φb]/(1 + sinφ), m = 2 sinφ/(1 + sinφ).

15

4 Passive (Non-cohesive) Critical Wedge Solutions for a Single Value of φb < φ

ψ1

Q1

τ

Ο σʼn

23

4

β1Y

Pα

N

φ φb

σʼ1σʼ3

α+β1

σʼ1

σʼ3Q2

Q4

ψ2

α+β2ψ0

ψ2

β2

β3−α

β4−α

β4

ψ1

Q3

M

β3

1

16

Wedge Angles for Passive (Compressive) Rankine States in Cohesionless Material

From triangles OQiM, (i = 1, . . . , 4), and ONM :

ψ1 =1

2

[sin−1

(sinφbsinφ

)− φb

], ψ2 =

1

2

[π − sin−1

(sinφbsinφ

)− φb

]For a fixed value of µb there exist therefore 4 distinct wedge angles, given by

δ1 = α + β1 = ψ1 − ψ0 =1

2

[sin−1

(sinφbsinφ

)− sin−1

(sinα

sinφ

)+ α− φb

]δ2 = α + β2 = ψ2 − ψ0 =

1

2

[π − sin−1

(sinφbsinφ

)− sin−1

(sinα

sinφ

)+ α− φb

]δ3 = β3 − α = ψ1 + ψ0 =

1

2

[sin−1

(sinφbsinφ

)+ sin−1

(sinα

sinφ

)− α− φb

]δ4 = β4 − α = ψ2 + ψ0 =

1

2

[π − sin−1

(sinφbsinφ

)+ sin−1

(sinα

sinφ

)− α− φb

]

Note: There exist of course any number of quadruple solutions for a corresponding number of distinctbasal friction angles φb < φ. There may therefore be structural settings in which several ‘planes ofweakness’ with distinct friction coefficients fit into a simple Rankine limit state.

17

Cohesive Critical Wedges

Assuming constant cohesive strengths c for bulk material and cb for fault rock, the Coulomb shearstrength of the fault is now

τ = τb = µb σ′n + cb with φb = tan−1 µb as before.

Dahlen (1984, 1990) modified this by introducing an apparent coefficient of friction

µ′b = Λµb and friction angle φ′b = tan−1 µ′b

whereΛ := σ′n,int/σ

′n = (σn − pb)/(σn − p) = 1− (pb − p)/σ′n

is a constant factor that accounts for the apparent weakening effect of the excess fluid pressure pb(y)−p(y) > 0 in an undrained fault zone relative to the fluid pressure immediately outside this zone.

Dahlen takes Λ = (1−λb)/(1−λ) in terms of constant HR-ratios λ and λb (> λ) for the wedge as wholeand the fault zone, respectively. This interpretation of Λ, which implies a complicated dependenceof the excess fluid pressure along the a priori unknown fault, is in fact unnecessary, since all that isrequired in order to preserve the simplicity of a critical wedge solution is a constant value of µ′b, i.e., aconstant ratio (pb− p)/σ′n. This may be motivated more easily by assuming a relationship of the formpb − p = b σ′n between the excess fluid pressure in an undrained fault zone and the effective normalstress acting on it, where b is a constant Skempton-type coefficient.

As a result one hasµ′b = (1− b)µb

where the reference to Skempton’s pore pressure coefficient is meant to suggest a necessary closer lookat the poromechanics of excess fluid pressures in fault zones.

18

Rankine Solution for Cohesive Critical Wedges

M

Q

Tb

2ψbφ’b

r

r/sin φ - c cot φ+ cb cot φ’b

φ

P

M

σ1

σʼn

c

φ’b

r(y)Q

βYα+β

αψb

Tb 0

τ

α’cb

ψ0

r/sin φ - c cot φ c cot φ

T

N

Mα’

Y

0

r2ψ0

r/sin φ - c cot φ

19

From triangle OYM :

ψ0 =1

2

{sin−1

[(1− c cosφ

r

)sinα′

sinφ

]− α′

}From triangle TbQM :

ψb =1

2

{sin−1

[(1− (c cotφ− cb cotφ′b)

sinφ

r

)sinφ′bsinφ

]− φ′b

}With β ≡ β1, ψb ≡ ψ1, the first (out of 4) wedge angles is then given as a function of the radius r(y)of the Mohr circle by

α + β = ψb − ψ0

=1

2

{sin−1

[(1− (c cotφ− cb cotφ′b)

sinφ

r

)sinφ′bsinφ

]− sin−1

[(1− c cosφ

r

)sinα′

sinφ

]+ α′ − φ′b

}with r(y) from the Coulomb criterion

r(y) =1

2(σ′x + σ′y) sinφ+ c cosφ

where σ′y = (1−λ)ρgy cosα and τyx = (1−ρ/ρw)ρgy sinα from equilibrium, and (for passive states)

σ′x = (1 + 2 tan2 φ)σ′y + 2c tanφ+2

cosφ

√(σ′y tanφ+ c)2 − τ 2

yx .

20

Shape of Cohesive Wedges for Constant Topographic Slope Angle α

β

x

y

ρ, φ, c

µʼb, cb, pb

g

ρw

ψ0

σʼy

τyx

pα (α+β)∞

ψbα+β

(α+β)0

σʼ1

σʼ1

σʼx

- Exact Rankine solution for cohesive taper demands upward convex decollement;

- for c cosφ/r(y)� 1 wedge angle α + β approaches constant value of cohesionless taper;

- σ′1 turns surface-parallel (ψ0 → 0) as y → 0.

21

Shape of Cohesive Wedge for Cohesionless Fault Zone Material

β

x

y

ρ, φ, c

pb, µʼb, cb=0

g

ρw ψ0

α

ψbα+β

(α+β)y=0 −> 0σʼ1

σʼ1

For c 6= 0 but cb = 0 one finds that the wedge angle α + β → 0 as y → 0 !

22

Mohr Circle Construction with Large-Depth Asymptotic Solution

α

β(α+β)∞

φσʼn

τ

φb 0

r(y)

α’

P

M∞ r∞ M

α+β

y->∞

y->∞

YP∞

cb

y->∞

c cot φ s(y)

c Q

Q∞

r(y)=[1+c cot φ/s(y)]r∞ --> r∞ for c cot φ/s(y) << 1

23

Mohr Circle for State of Stress at Surface y = 0

σʼn

τ

c

(α+β)0 P

σʼ1

Q

φʼbφ

r(y=0) = ckp

M

cb

Y

Note: ψ0 = 0 for Y = (0, 0), and therefore (α + β)y=0 = ψb → 0 for cb → 0 (Q→ Y )!

24

Conclusions

• In shallow, highly overpressured sediments cohesive strength will sensibly affect fault structures

• ‘Rankine analysis’ yields exact solution of critical cohesive taper problem & associated sliplineswith no restriction to small dip or slope angles

• For every value µb < µ (or µ′b < µ) there exist four ‘passive’ (compressive) and four ‘active’(extensional) critical wedge solutions that fit into a corresponding critical Rankine state

• Exact critical taper solutions for cohesive wedges demand an upward convex decollement

• Cohesive wedge angle approaches cohesionless wedge angle at Mohr circle radius r(y)� c cosφ

• In cohesive wedges principal compressive stress σ1(y = 0) 6= 0 and surface-parallel !

• Cohesive wedge angle vanishes at surface, if fault rock is non-cohesive

• Rankine-state solutions for several coexisting weak fault planes and depth-dependent propertiesremain to be explored.

25

References

Buiter, S.J.H. (2012). A review of brittle compressional wedge models. Tectonophysics 530-531, 1–17.

Davis, D., J. Suppe, and F.A. Dahlen (1983). Mechanics of fold-and-thrust belts and accretionary wedges: cohesiveCoulomb theory. J. Geophys. Res. 88, 1153–1172.

Dahlen, F.A. (1984). Noncohesive critical Coulomb wedges: an exact solution. J. Geophys. Res. 89, 10,125–133.

Dahlen, F.A., J. Suppe, and D. Davis (1984). Mechanics of fold-and-thrust belts and accretionary wedges: cohesiveCoulomb theory. J. Geophys. Res. 89, 10,087–101.

Lehner, F.K. (1986). Comments on ‘Noncohesive critical Coulomb wedges: an exact solution’ by F.A. Dahlen.J. Geophys. Res. 91, 793–797.