A Model for the Plastic 

Flow of Landslides 

U.S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 1385 

A Model for the Plastic 

Flow of Landslides 

By WILLIAM Z. SAVAGE and WILLIAM K. SMITH 

U.S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 1385 

An application of the theory of Coulomb 

plasticity to the flow of landslides 

LANDSLIDE INFORMA1ION CEWIEW 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON 1986 
DEPARTMENT OF THE INTERIOR 

DONALD PAUL HODEL, Secretary 

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director 

Library of Congress Cataloging in Publication Data 

Savage, William Z. 

A model for the plastic flow of landslides. 

(U.S. Geological Survey professional paper ; 1385) 

"An application of the theory of Coulomb plasticity to the flow of landslides." 

Bibliography: p. 

Supt. of Docs. no.: I 19.16:1385 

1. Landslides-Mathematical models. 2. Plasticity-Mathematical models. I. Smith William K. II. Title. 

III. Series: Geological Survey professional paper 1385. 

QE599.A2S28 1986 551.3 85-600052 

For sale by the 

Books and Open-File Reports Section 

U.S. Geological Survey 

Federal Center 

Box 25425 

Denver, CO 80225 
CONTENTS 

Page Page 

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The Landslide Model-Continued 

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Determination of the velocity field . . . . . . . . . . . . . . . . . . 21 

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 

Coulomb plastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Similarities and differences between Nye's alpine glacier 

Stress field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 model and our landslide model . . . . . . . . . . . . . . . . . . 30 

Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The assumption of associated flow . . . . . . . . . . . . . . . . . . 31 

The landslide model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 

Determination of the stress field . . . . . . . . . . . . . . . . . . . . 10 References cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 

ILLUSTRATIONS 

Page 

FIGURE 1. Sketch showing Coulomb failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

2. Sketch showing Cartesian coordinate system inclined at an angle (J to the horizontal . . . . . . . . . . . . . . . . . . . . . . . . . . 4 

3. Sketch showing semi-infinite mass of soil inclined at an angle (J to the horizontal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 

4-13. Graphs showing: 

pgy* - 4 __ versus fJ for various values of P and cf> .........•.•...................•••...............••••.... . k 

12 

5. Bounds on Pas a function of slope angle (J for various angles of internal friction cf> .......•..••.•.......... 13 

6. Variation of u:x'• u:xy' and u~ with y/y* for cf> = 30°, fJ = 20°, and P = 0.5 in extending and compressive flow . . 15 

7. Variation of u:xy at y = y* with (J for various values of P and cf> • . . . . . . . . . . • • . . . . . . . . . . . . . . . . • . . • • • • . . . . 16 

8. Variation of u~ at y = y* with (J for various values of~ and cf> . . . . . • . . . . . . . . . . . • . . . • . . . . . . . . . . . . . . . . . . . 17 

9. Variation of u~ at y = y* with (J for various values of P and cf> . . . . . . • . . . . . . . . . . . . . • . • . . . . . . . . . . . . . . . . . • 18 

10. Variation of u~ at the surface, y = 0, with cf> . . . . . . . . • • . . . . . . . . . • • . . . . . . . . . . . • • • . . . . . . . . . . . . . . . . . . • • • • 19 

11. Dimensionless depth yly* to which u~ is tensile in extending flow as a function of (J for various values of P and cf> 20 

12. Dimensionless depth yly* of maximum values of u~in compressive flow as a function of (J for various values of cf> 

and P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 

13. Maximum value of u~ in compressive flow as a function of (J for various values of cf> and P . . . . . . . . . . . . . . . . . . 22 

14. Sketches showing stress characteristics for various slope angles fJ, angles of internal friction cp, and dimensionless 

pore pressures P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

15. Graphs showing variation of dimensionless dilatancy rate di!rl with dimensionless depth yly* for various values of 

fJ, c/>, and P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 

16. Graphs showing dimensionless velocities (u-u0 )/ry* and v/ry* as functions of dimensionless depth yly* for compressive 

and extending flow and various values of fJ, cp, and P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 

17. Sketches showing fault surfaces produced by propagation of discontinuities along {3 _ slip lines in A, compressive flow 

and B, extending flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 

18. Graph showing magnitude of resultant velocity jump on the surface as a function of angle of internal friction cf> for 

extending and compressive flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

19. Sketch of an idealized long, shallow landslide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 

III 
IV A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

Integration constants 

Determinant 

SYMBOLS 

Recurring terms in denominators 

Dilation rate 

ex, ey, exy Coordinate deformation rates . . . 

el' e2' e3 Principal deformation rates 

F [<f>, 8

• P] Complicated expressions used for brevity in equations G [<f>, 8, P] 

F(y) 

f 

g 

H 

h 

k 

P,P 

uo 

ul' u2 

vl' v2 

u:I, u:2 

vl, v2 

v 

X, y 

y* 

y' 

a 

, 

p 

u' 

(]X' (Jy 

(Jl ' (]2 ' Ug 

uxy 

T 

Function of y 

Yield function 

Gravitational acceleration 

Material constant 

Depth 

Cohesion 

Pore pressure 

Constant longitudinal deformation rate 

Curvilinear coordinates 

Stress invariant 

Cartesian velocity components 

Basal sliding velocity 

Velocity components parallel to (j _ and (j + characteristics 

Velocity components normal to (j _ and (j + characteristics 

Velocity discontinuity components parallel to (j _ and (j + characteristics 

Velocity discontinuity components normal to (j _ and (j + characteristics 

Resultant velocity jump vector 

Cartesian coordinates 

Depth of landslide 

Dimensionless y coordinate ( = yly *) 

Angle that greatest principal stress makes with positive x axis 

Characteristics 

(8- <f>) 

(2a + 8) 

Inclination of landslide surface 

Undetermined multiplier 

(45° - <f>/2) 

Angle of dilation 

3.14159 ... 

Density 

Stress invariant ( = S' +H) 

Coordinate normal stresses 

Principal normal stresses 

Coordinate shear stresses 

Shear stress 

Maximum shear stress 

Angle of internal friction 

Symbols may be further modified by subscripts or primes. Generally, letter subscripts refer to 

coordinate directions or directions normal to given planes, whereas numeric subscripts refer to principal directions. Exceptions are Dl' D2, F1[<t>, 8, P], G2[<t>, 8, P], etc., where the subscripts merely 

distinguish between similar expressions. Effective stresses, the difference between total normal 

stresses and the pore pressure, are denoted by primes. 
A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

By WILLIAM Z. SAVAGE and WILLIAM K. SMITH 

ABSTRACT 

To further the understanding of the mechanics of landslide flow, 

we present a model that predicts many of the observed attributes of 

landslides. The model is based on an integration of the hyperbolic differential equations for stress and velocity fields in a two-dimensional, 

inclined, semi-infinite half-space of Coulomb plastic material under 

elevated pore pressure and gravity. 

Our landslide model predicts commonly observed features. For example, compressive (passive), plug, or extending (active) flow will occur under appropriate longitudinal strain rates. Also, the model 

predicts that longitudinal stresses increase elliptically with depth to 

the basal slide plane, and that stress and velocity characteristics, surfaces along which discontinuities in stress and velocity are propagated, 

are coincident. Finally, the model shows how thrust and normal faults 

develop at the landslide surface in compressive and extending flow. 

INTRODUCTION 

To further the understanding of landslide mechanics 

we have developed a model that describes many of the 

observed attributes of landslides. This model, based on 

the flow theory of Coulomb plastic materials, is similar 

in many respects to a previous model for alpine glaciers 

developed by Nye (1951). In this section, we review the 

principal results of Nye's model and other work that 

influenced ours, explain how the report is laid out, and 

give our principal results. 

N ye ( 1951) successfully applied the flow theory of a 

perfectly plastic incompressible Mises material (Hill, 

1950; Varnes, 1962) to the flow of alpine glaciers. His 

model predicted many of the flow patterns observed in 

alpine glaciers. For example, he predicted that either 

active (extending), passive (compressive), or plug flow 

would occur depending on the longitudinal deformation 

· rate. Nye based his predictions on the assumptions that 

shear stresses and that vertical normal stresses varied 

linearly with depth, and found that longitudinal stresses 

varied elliptically with depth in the glacier. Nye also 

found that, in active (extending) flow, the longitudinal 

stresses were tensile, which explains the crevasses commonly observed in parts of glaciers undergoing extending flow. Nye found that the predicted slip-line field 

explained the thrust surfaces seen in compressiveflow regions of glaciers and normal faults observed 

in extending-flow regions. For the extending- and 

compressive-flow states, Nye's model predicted the 

longitudinal velocity was greatest at the surface and 

decreased elliptically toward the base. Finally, his model 

predicted the glacier to simply slide on its base as a rigid 

body during plug flow. 

Landslides have many of the same features as Nye 

modeled in alpine glaciers. For example, extending, compressive, and plug flows occur, and tensile regions and 

normal fault scarps develop near the surface in parts 

of the landslide undergoing extending flow. Also, thrust 

surfaces and thrust-fault scarps occur in regions of compressive flow. The~e similarities are not surprising as 

soils in general behave as materials with plastic states 

characterized by Coulomb's law, which is a generalization of the Mises criterion for plastic material (Hill, 

1950). 

Following Nye's lead in modeling the flow of an alpine 

glacier, we demonstrate that, as with Nye's glacier 

model, our landslide model predicts that either compressive, extending, or plug flow can occur-depending 

on the longitudinal deformation rate. The shear stresses 

and vertical normal stresses vary linearly from the surface to the limiting depth of the landslide's slip-line field, 

and the longitudinal stress increases in an elliptic manner over the same depth. Both the slope angle, and the 

density and cohesion of ice, control the thickness of the 

glacier slip-line field. However, the thickness of the slipline field in a landslide in homogeneous and isotropic 

soil is controlled by the slope angle, by the pore pressure, 

and by the density, cohesion, and angle of internal friction of the soil. As in the alpine glacier model, longitudinal stresses near the surface of the landslide will be 

tensile during extending flow, and the slip-line field predicted for the idealized landslide will explain the thrust 

surfaces seen in compressive-flow regions of landslides. 

Also, as in the glacier model, the landslide model 

predicts that the longitudinal velocity is greatest at the 

surface and varies in an elliptic manner down to the 

basal slip plane of the landslide. Finally, like N ye' s 

glacier-flow model, the landslide model predicts that, 

during plug flow, the landslide simply slides as a rigid 

body on its basal slip plane. 

Besides Nye's important work, our model for landslide 

flow will draw heavily on other work. Frontard (1922) 

1 
2 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

derived equations for the slip-line fields in extending 

and compressive flow for a semi-infinite mass of 

Coulomb material whose surface is inclined at an angle 

greater than the angle of internal friction. (See Terzaghi, 

1943, p. 40-41.) Sokolovskii (1960, p. 38-40; 1965, 

p. 77-80) gave similar solutions for soils, as did Johnson 

(1965) for debris flows. In each of these solutions, the 

slope angle, cohesion, and angle of internal friction of 

the soil making up the slide control the depth to which 

the slip-line field can extend. More recently, Crans and 

Mandl (1981) applied the theory of plastic flow of 

Coulomb materials to create a model of the development 

of growth faults in sediment layers. Like Frontard 

(1922) and Sokolovskii (1960, 1965), Crans and Mandl 

(1981) found that the slip-line field was bounded at a 

finite depth when the slope angle exceeds the angle of 

internal friction. Crans and Mandl (1981) considered the 

effect of an elevated pore pressure and found that under 

certain conditions this pressure led to a bounded slipline field, even when the slope angle was less than the 

angle of internal friction. 

In this report, we first develop the general differential equations for stress and velocity fields in twodimensional Coulomb plastic materials under an 

elevated pore pressure. Our results show that these partial differential equations are hyperbolic and, in general, 

integrable by a method that uses special curves in an 

x-y coordinate plane called characteristics, along which 

the solution of the partial differential equations is reduced to integration of ordinary differential equations. 

The characteristics are also surfaces across which first 

derivatives of stress and velocity may be discontinuous. 

In general, two separate sets of characteristics may occur: one for velocities and one for stresses. However, 

for the Coulomb plastic material in our model, the stress 

and velocity characteristics are coincident. The coincidence of the stress and velocity characteristics, that 

is, the slip lines, is important because these surfaces 

represent rupture surfaces along which velocity or 

deformation rate discontinuities occur in landslides. 

Finally, we see that in a Coulomb material, discontinuities in the tangential-velocity component across a 

characteristic will be accompanied by a discontinuity 

in the velocity component normal to the characteristic. 

We then proceed to the landslide model, where, 

because of the assumptions regarding this model, we 

find that the governing hyperbolic differential equations 

can be integrated directly. First, we discuss the stresses 

and the stress characteristics for the idealized landslide, 

and we give the stress distribution and configuration 

of the stress characteristics for various values of pore 

pressure, slope angle, cohesion, and internal friction in 

dimensionless form for extending and compressive flow. 

Second, we derive the velocity field and give the 

distribution of longitudinal and normal velocities for 

various values of pore pressure, slope angle, cohesion, 

and internal friction in dimensionless form for extending and compressive flow. Third, we consider the special case of a purely cohesive material-a Mises 

material-arriving at a solution similar to Nye's (1951) 

solution for alpine glacier flow. This solution emphasizes 

that our model for landslide flow is a more general treatment of the gravitational flow of plastic materials for 

which Nye's solution for alpine-glacier flow is a special 

case. 

Finally, we compare our model of landslide flow with 

Nye's (1951) model for glacier flow and discuss the applicability of this landslide model to actual landslides. 

Here, we emphasize the restrictiveness of the assumptions made in developing the landslide model and the 

main conclusions derived from the model. 

DEDICATION 

This work is dedicated to the memory of George M. 

Sowers. 

COULOMB PLASTIC MATERIALS 

STRESS FIELD 

It is known from experiment that at a point in a soil 

at incipient shear failure there occur two intersecting 

planes along which the shear stress, denoted Tn, is 

given by 

(1) 

where k is the cohesion, ¢ is the .angle of internal friction, and u~ is the effective normal stress acting on the 

planes. (Throughout this report effective stresses, the 

difference between total normal stresses and the pore 

pressure, are denoted by primes.) The two intersecting 

planes are equally inclined at angles of 45 o - ¢/2 to the 

most compressive principal effective stress, a~, and the 

line of intersection of these two planes is parallel to the 

intermediate principal effective stress, u~. Equation (1) 

was first suggested by Coulomb (1773) for total stresses 

and is known as the Coulomb failure criterion. 

Equation (1) plots as two straight lines in twodimensional Mohr stress space (Terzaghi, 1943, 

p. 15-24), as shown in figure 1. When ¢ = 0, the two 

straight lines in figure 1 are parallel, and the soil is 

termed purely cohesive or a Mises material. These lines 

also separate the Mohr stress space into two parts: a 

stable part and an unstable part. If a Mohr circle 
COULOMB PLASTIC MATERIALS 

T 

FIGURE 1.-Sketch showing Coulomb failure criterion. Normal and shear stresses are represented by u and T, 

respectively. Here, T max is the maximum shear stress, T n is the shear stress acting on planes inclined at 

an angle a to ui (the most compressive principal effective stress), u3 is the least compressive principal effective stress, </> is the angle of internal friction, and k is the cohesion. Other symbols are defined in the text. 

constructed from u~ and u~ (the least compressive principal stress at a point) lies between the Coulomb failure 

lines, the soil behaves elastically. However, if a Mohr 

circle becomes tangent to these lines, failure is imminent and Coulomb plastic flow will ensue. This state of 

incipient failure is termed a state of limiting equilibrium. 

No Mohr circle can lie beyond these limiting lines 

because the shear stress cannot exceed the yield 

strength if the body is to remain in equilibrium. 

Certain other relationships in a region at limiting 

equilibrium can be derived from figure 1. For example, 

where 

(1~ + (1~ S'= . 2 

Defining 

u' = S' + H 

we can write equation (3) as 

Tmax = u' Sin {j>. 

3 

(4) 

(5) 

(6) 

H=kcot(j>, 

and 

(2) 

Also, Coulomb's failure criterion can be written by using equations (1) and (2) as 

(7) 

(11 - (13 

T max = = (S, + H) sin (j> ' 

2 

(3) 

By using the equations for stress transformation 
4 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

and 

(8c) 

we can write expressions for the three Cartesian components of stress in a region undergoing Coulomb 

plastic flow as 

a~} = S' ± Tmax cos 2a =a' (1 ± sin <P cos 2a) -H, (9a) ay 

Here, a; and a; are the normal effective stresses that 

are positive in compression; axy is the shear stress; and 

a is the angle between the direction of the maximum 

compressive principal effective stress a~ and the positive x axis. The magnitudes of a;, a;, and axy' as given 

by equations (8) and (9), are represented in Mohr stress 

space in figure 1. The angle a is reckoned positive in 

a counterclockwise direction as shown in figures 1 and 2. 

The yield criterion can be written in terms of a;, a;, 

and axy by using equation (3) or 

a1 - a3 (a~ + a~) • . T max= = sm <P + H srn <P ' 

2 2 

(10) 

and the relations (Sokolovskii, 1965, p. 12) 

(11) 

2 2 

and and 

axy = Tmax Sin 2a = a' Sin</> sin 2a. (9b) --- = - (a - a )

2 + a 2 • a1 - a3 [ 1 ]1/2 

2 4 x y xy 

FIGURE 2.-Sketch showing Cartesian coordinate system inclined at an angle(} to the horizontal. Also shown 

are the angle a between the positive x axis and the direction of qi; the two stress characteristics {3+and 

{3_; the tangents to the two stress characteristics at a point; the velocity components ul' u2, vl' and 

v 2 on each characteristic; and the resultant velocity vectors ti 1 and v2

• 

(12) 
COULOMB PLASTIC MATERIALS 5 

The yield criterion in terms of u;, u;, and uxy then 

becomes 

In addition to satisfying the yield criterion, stresses 

in regions of incipient Coulomb plastic flow must also 

satisfy the differential equations of plane equilibrium 

and 

au; auxy . aP 

- + -- =pg sm fJ - - ax ay ax 

a~y a~ aP -- + - =pg cos (} - - . ax ay ay 

(14a) 

(14b) 

Here, p is the density, fJ is the inclination of the x axis 

from the horizontal (fig. 2), and P represents the porewater pressure. 

Provided that the pore-pressure distribution is 

known, the three equations (13), (14a), and (14b) contain 

three unknowns u;, u;, and uxy' and can in principle be 

solved for the stress field if the boundary conditions are 

expressed in terms of stresses-that is, if the system 

is statically determinate. The method for solving (13) 

and (14a, b) for the stress field, the method of characteristics, will now be discussed. 

Equations (9a, b), which identically satisfy equation 

(13), the Coulomb yield condition, are substituted in the 

equilibrium equations (14a, b), and result in 

au' aa 

[1 + sin <P cos 2a] - - 2u' sin <P sin 2a - ax ax 

au' aa 

+ sin <P sin 2a - + 2u' sin <P cos 2a -

and 

ay ay 

aP 

= pg sin fJ--

ax 

au' au' 

sin <P sin 2a- + 2u' sin <P cos 2aax ax 

au' au' 

+ [1 - sin <P cos 2a] - + 2u' sin <P sin 2a - ay ay 

aP 

= pg cos (} - - . ay 

(15a) 

(15b) 

This result yields two equations for the four derivau' au' aa aa atives --, --, --, and--. We can add two more 

ax ay ax ay 

equations, the equations of the total derivatives 

au' au' 

du' = -dx + -dy ax ay 

(16a) 

and 

aa aa 

da = - dx + - dy , ax ay 

(16b) 

giving a system of four equations for the four unknown 

derivatives. In matrix form, 

[ l+sinlj>cos2a sin 4> sin 2a -2u' sin 4> sin 2a 2u' sin 4> cos 2a l sin 4> sin 2a 1-sinq,cos2a 2u' sin 4> cos 2a 2u' sin 4> sin 2a 

dx dy 0 0 

0 0 dx dy 

au· 

ax 

aP au' pgsin6-a; 

ay aP (17) X pg cos 6- ay a a 

ax du' 

da a a 

ay 

Following the procedures of matrix algebra, we can invert equation ( 17) and obtain a unique solution for the 

four unknown derivatives, provided that the determinant of the square matrix 

D =[cos 2a +sin <~>l(!f- 2 sin 2a : 

(18) 

- [cos 2a - sin <P] 

is nonzero. However, if D = 0, the result is a quadratic 

form in dy with the two roots dx 

l

dy I - dx ± 

or 

sin 2a ± [sin2 2a + cos2 2a - sin2 <PJ1'2 

cos 2a + sin <P 

(19) 

=....;...._ sin ____ 2a ± cos <P ..;... 

cos 2a + sin <P 
6 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

l

dy l += sin 2a ±sin [90°- cp) 

dx - cos 2a + cos [90° - cp) 

(20) 

Th f kn d . t• a a' a a' a a d e our un own er1va 1ves, -,-,-, an 

aa ax ay ax a y , are then no longer unique along two curves in aregion of Coulomb plastic flow with tangents at every 

point given by equation (20). Curves having the slopes I: ] + and I: ]_everywhere are called character· 

istics. And the partial differential equations (15a, b) are 

called hyperbolic (Crandall, 1956, p. 354) because there 

are two real characteristics at a given point. The stress 

characteristics labeled {3 + and {3 _ are shown in figure 2. 

The stress characteristics intersect at every point at 

the angle 90° - cJ> (fig. 2) and are loci along which the 

shear stress is given by equation (1). For a plastically 

deforming soil, the characteristics represent the potential paths for propagation of discontinuities in stresses 

and stress gradients. Also, from equation (20), we see 

that when a = 45 o - cp/2, the slopes I dy ] are either indx ± 

finite or zero. Lines of discontinuity, lines along which 

I~ I± are either infinite or zero, represent boundaries 

between plastic regions or between plastic and nonplastic regions. Stresses can undergo finite jumps, and 

stress gradients can be infinite across lines of discontinuity. (See, for example, Prager and Hodge, 1951.) 

Finally, lines of discontinuity may form on stress 

characteristics or form an envelope around a system of 

stress characteristics (Sokolovskii, 1960, 1965). 

In addition, for a compatible system of hyperbolic differential equations, the determinant formed by replacing any column on the left of equation (17) with the 

column on the right must also vanish (Abbott, 1966). 

For example, 

1 + sinl/>cos2a sin 1/> sin 2a -2u' sin 1/> sin 2a pg sin 9- aP 

ax 

D= sin 1/> sin 2a 1-sinl/>cos2a 2o' sin 1/> cos 2a pg cos 9- aP 

ay 

dx dy 0 do' 

0 0 dx da 

= cos' <I> du' + 2u' sin <I> [(cos 2a + sin</>) 

(:) - sin 2a] da 

+ l(pg sin 8- ::) sin <1> sin 2a- (pg cos 8- :~) 

(1 + sin <1> cos 2a)] (~) 

-l(pg sin 8 - ~:) (1-sin <I> cos 2a) - (pg cos 8 - :~) 

sin <1> sin 2a] dx = 0 . (21) 

Substituting [ dy I given by equation (20) in equation dx ± 

(21), we find 

da' ± 2a' tan cp da = 

co!'<!> [(pg sin 8 - ~~ ) (1-sin q, cos 2a) 

-(pg cos 8- ~:)sin <I> sin 2a] dx 

- co!'<!> [(pg sin 8- ~~) (sin <1> sin 2a) 

-(pg cos 8-~:) (1 + sin <1> cos 2a)] dy. (22) 

If the pore-pressure distribution is known, equations 

(20) and (22) represent a pair of ordinary differential 

equations for a and a' along each characteristic. Presuming that they can be solved for a' and a, we find the 

stresses ax', uY', and axy along each characteristic from 

equations (9a, b). In general, this system must be solved 

numerically (Sokolovskii, 1960, 1965). 

VELOCITY FIELD 

Let us adopt the hypothesis of associated plasticity 

(Hill, 1950; Drucker and Prager, 1952; Shield, 1955) 

which states that the principal plastic-deformation rates 

e1' e2' and e3 are proportional to the gradients of the 

yield function with respect to the principal stresses u~, 

u~ , and a; . Although the associated flow-rule hypothesis is adopted for mathematical simplicity, it leads to 

unrealistically large dilation rates. (More will be said 

about this problem later.) 

Writing the yield function (equation 10) as 

f = i(a;- a;)- i(a~ + a;) sin cJ>- H sin cJ> = 0, (23) 

and taking the appropriate derivatives, we find that 

e = A ~ = ~ (1 - sin ,~.) 1 a a' 2 'II ' 1 

(24a) 
COULOMB PLASTIC MATERIALS 7 

(24b) 

and 

. '" at -A . e = " - = - (1 - sm ,~,.) 3 au' 2 'f' ' 3 

(24c) 

where A is an undetermined parameter that is eliminated 

in the following equations. 

Equation (24b) is a result of assuming that flow occurs only in the plane of u1 and u3• Adding (24a) and 

(24c), we find the mean deformation rate 

(25) 

Subtracting (24a) from (24c) yields the maximum shear 

deformation rate 

(26) 

Eliminating A from equations (25) and (26) yields the 

equation of continuity 

Then, with these equations, the condition of isotropy 

(equation 2S) becomes 

av +au 

a X a y = tan 2a . 

au av 

(30) 

ax- ay 

Using the deformation-rate transformation equations, 

(31a) 

(31b) 

and 

. I e1 - e3 1 e xy = 2 sin 2a ' (31c) 

which are of the same form as equations (Sa, b, c) for 

stress transformation, and equations (25) and (26), we 

arrive at 

-au -A 

e =-= [sin cP - cos 2a] , X ax 2 (32a) 

-av -A 

When cJ> is greater than zero, the volume increases 

during flow. When cJ> = 0, equation (27) becomes the 

equation of continuity for plane incompressible flow. and 

We also assume isotropy-that the principal axes of 

deformation rate and stress coincide. Then, from stress 

transformation equations (Sa, b, c), we have 

e =-= 

y ay 2 [sin cP + cos 2a], 

e = -!_I a v +au I = ~ sin 2a . xy 2 ax ay 2 

(32b) 

(32c) 

As with equation (11) for the stresses, the sum of the 

(2S) principal deformation rates is an invariant, or 

By taking u and v to be the velocity components in 

the x andy directions, respectively, and by taking contracting deformation rates as positive, we can define the 

deformation rates by 

. -au 

ex= ax ' (29a) 

. -av 

ey = ay ' (29b) 

and 

e = - _!_ ( av +au\ . xy 2 ax ayf (29c) 

. . . . au av 

el + e3 = ex + ey = - ax - ay . (33) 

The difference of principal deformation rates from equations (31a, b) is 

e, - e, = - ~~ - ~~ . e1 - e3 = cos 2a cos 2a (34) 

We can rewrite equation (27), the equation of continuity, as 

l

au ~v - au - av - ax- ay sin cP = 0 ' 

ax ay cos 2a (35) 
8 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

or 

::(cos 2a + sin cp) + :~(cos 2a - sin cp) = 0. (36) 

Equation (28), the condition of isotropy, is rewritten 

as 

au au av au -tan 2a --- - - -tan 2a = 0 . (37) 

ax ay ax ay 

Equations (36) and (37), together with the equations 

for the total derivatives 

du = au dx + au dy 

ax ay 

(38a) 

and 

au au dv = -dx +- dy ax ay ' (38b) 

yield a system of four equations for the four derivatives 

_, 

au _, _, au av au and _, or 

ax ay ax ay 

cos 2a +sin cp 0 0 cos 2a-sin cp 

au 

- 0 

ax 

tan 2a -1 -1 -tan 2a au 

- 0 

ay = .(39) 

dx dy 0 0 au 

- du 

ax 

0 0 dx dy au du 

ay 

The determinant of the square matrix is given by the 

quadratic form 

(

dy)2 dy D = (cos 2a + sin cp) dx - 2 sin 2a dx 

(40) 

- (cos 2a - sin cp) • 

Then, following the procedure outlined for determining 

the stress characteristics, we find the velocity 

characteristics 

or 

[

dy] =sin 2a ± [sin2 2a + cos2 2a - sin2 q,)1t2 

dx ± cos 2a + sin cp 

(41) 

+ sin 2a ± cos cp 

cos 2a + sin cp 

[

dy l _ sin 2a ± sin [90°- cp) 

dx ± cos 2a + cos [90° - cp) (42) 

Comparing equations (20) and (42), we see that the 

stress and velocity characteristics coincide. For a plastically deforming soil, the characteristics then represent 

potential paths for propagating discontinuities in velocity and deformation rates (equations 32a, b, c), as well 

as stresses and stress gradients. Because these coincident surfaces (the stress and velocity characteristics) 

are potential paths for propagating velocity discontinuities, they are called slip surfaces. Like stresses, 

velocities can also undergo finite jumps and shear- and 

normal-deformation rates can be infinite across lines of 

discontinuity-that is, at boundaries between two 

plastic regions or between plastic and nonplastic 

regions. Finally, lines of discontinuity can form along 

velocity characteristics or form an envelope to a system 

of velocity characteristics. 

Like stresses, a compatible system of hyperbolic equations for the velocities requires that the determinant 

formed by replacing the last column of the square 

matrix in equation (39) by the column on the right of 

the equals sign must also vanish. Carrying out this 

operation, we find that 

. D= 

cos 2a sin cp 

tan 2a 

dx 

0 

0 0 0 

-1 -1 0 

dy 0 du 

0 dx dy 

(43) 

= du dx + dv dy = du + du : = 0 . 

Substituting equations (42) in (43) yields the differential relations between du and dv along the characteristics {3 + and {3 _ 

du + tan [a ± l45° -t )] dv = 0 . (44) 

Here, a is known from the stress solution. 

To interpret equation (44), we rewrite it in the 

canonical forms, 

au av 

- + tan [a - p.] - = 0 

as1 aS1 

(45a) 

and 
COULOMB PLASTIC MATERIALS 9 

au au -+tan [a +JL]- = 0, a82 a82 

(45b) 

where 8 1 is a curvilinear coordinate coincident with the 

{3 _ characteristic, 82 is a curvilinear coordinate along 

the {3 + characteristic, and JL = 45 o - : • Next, we define the velocity components u1 and u2, which are the 

velocities respectively parallel to the {3 _ and {3 + 

characteristics. The relationships between these velocities and the Cartesian velocity components u and v are 

given by 

u1 = u cos (a - JL) + v sin (a - JL) (46a) 

and 

u2 = u cos (a + JL) + v sin (a + JL), (46b) 

which have the inverses 

u1 sin (a + JL) - u2 u = __..;. _____ ____;;.__ sin ___ (a - JL) _ cos c/> 

(47a) 

and 

u1 cos (a + JL) + u2 v = ____;; ______ _:_ ___ cos (a -....;._ JL) (47b) 

cos c/> 

Substituting equations (47a, b) in (45a, b), we find the 

canonical equations describing the variation of u1 and 

u2 along the 81 and 82 curvilinear coordinates, the {3_ 

and {3 + characteristics: 

au1 ( u2 l aa -+ u tanct>--- -=0 a~ 1 c~ct> a~ (48a) 

and 

au2 ( u1 j aa -- u tanct>- -- --o a 8 2 

2 cos ct> a 8 2 - • 

(48b) 

Equations (48a, b) represent the extension rates along 

the two characteristics (Salencon, 1977) and show that 

the extension rate along each characteristic vanishes. 

In addition, we can write equations (48a, b) as the ordinary differential equations 

du1 ( u -+ u tanct>---2 ~ =0 da 1 cos c/> (49a) 

and 

du2 ( U -- u tan¢---1 ~ =0. 

da 2 cos c/> (49b) 

Equations (48a, b) become the Geiringer equations of 

perfect plasticity when c/> = 0 (Davis, 1968, p. 367). 

Suppose that a discontinuity in velocity occurs across 

a {3 _ characteristic, with the sense of relative displacement shown by the arrows in figure 2. If this discontinuity, symbolized by u1, is to continue in an 

unimpeded manner along the {3 _ characteristic, any 

discontinuity in u2, symbolized by u2, must vanish 

where the two characteristics intersect. Thus, for the 

propagation of the discontinuity u1 along 81' we have 

from equation (49a) 

du1 _ 

-- + u1 tan c/> = 0 . da (50) 

By a similar argument for the {3 + characteristic, we 

find that for the propagation of the discontinuity u2 

along 82, 

du2 _ 

-- + u2 tan c/> =0 . da 

(51) 

Integrating equations (50) and (51) yields 

(52) 

for propagating tangential velocity discontinuities 

along a {3 _ characteristic and 

for propagating tangential velocity discontinuities 

along a {3 + characteristic. Here, u1 and u2 are known 

tangential velocity discontinuitie~ at some a = ao 

along 81 or 82. 

In Coulomb materials (c/> =F 0), discontinuities in velocity can also occur in directions normal to the characteristics {3_ and {3+. Defining the normal velocity 

components v 1 and v 2 along 8 1 and 82, we have for the 

relationships between these velocity components and 

the Cartesian velocity components u and v 

v 1 = -u sin (a - JL) + v cos (a - JL) (54a) 

and 

v 2 = -u sin (a + JL) + v cos (a + JL) . (54b) 

Inverting the pair of equations (46a) and (54a), we arrive at 

(55 a) 

and 
10 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

v = u1 sin (a - JA.) + v1 cos (a - JA.) (55 b) 

for the {3_ characteristic. Inverting equations (46b) and 

(54b), we find 

u = u2 cos (a + JA.) - v 2 sin (a + JA.) (56a) 

and 

v = u2 sin (a + JA.) + v 2 cos (a + JA.) (56b) 

for the {3 + characteristic. 

Substituting equations (55a, b) in equation (45a), and 

equations (56a, b) in equation (45b), we find the canonical equations describing the relationships between 

the tangential and normal velocities along each 

characteristic: 

(57 a) 

and 

(57b) 

We can also write equations (57 a, b) as the ordinary differential equations 

and 

du1 v= --

1 da 

v= 2 

(58 a) 

(58 b) 

If a discontinuity in ul' represented by ul' occurs 

along the {3 _ characteristic, there will then be a corresponding discontinuity in vl' represented by vl' which 

is given by substituting equation (50) in equation (58a), 

or 

v 1 = - u1 tan cp • (59) 

Similarly, substituting equation (50) in equation (58b) 

yields 

(60) 

Then, from equations (52) and (53), we determine that 

v1 = - tan cp u1 exp [- (a - a ) tan c/>] (61) 0 0 

and 

(62) 

which are the relationships for propagating normal 

velocity discontinuities along the {3_ and {3+ 

characteristics. For a material where cf> = 0, we see that 

v1 = v2 = 0 and only tangential-velocity discontinuities u1 and u2 are propagated along the {3_ and {3+ 

characteristics. Finally, from equations (59) and (60), we 

see that the resultant jump in velocity on either 

characteristic is at an angle cf> to the characteristic at 

every point. 

Equations (52), (53), (59), (60), (61), and (62) are 

originally due to Shield (1953), and were discussed by 

Pariseau (1970) and by Houlsby and Wroth (1980). 

Physically, velocity discontinuities are manifested as 

zones of intense shear along which volume expansion 

(dilatancy) and slip occur (Drucker and Prager, 1952). 

The normal component of velocity discontinuity controls the amount of dilatancy (equations (61) and (62)). 

In general, velocity discontinuities originate at boundaries between two plastic regions, or between plastic 

and nonplastic regions-that is, along lines of 

discontinuity-and propagate into the plastically deforming mass along the velocity characteristics in an 

exponential manner (equations (52), (53), (61), and (62)). 

We are now in a position to proceed to the landslide 

model, but, before doing so, let us summarize some of 

the salient points about Coulomb plastic materials. We 

have developed general differential equations for stress 

and velocity fields in two-dimensional Coulomb plastic 

materials and found them to be hyperbolic and integrable by the method of characteristics. Also, we have 

found that the surfaces along which discontinuities in 

velocities and deformation rates may occur (velocity 

characteristics) and the surfaces along which derivatives in stress may be discontinuous (stress characteristics) are coincident. Also, we have seen that, if cf> =I= 0, 

the material must dilate. Finally, because these surfaces 

are potential paths for propagation of velocity discontinuities, we have termed them slip surfaces and found 

that velocity discontinuities grow and decay along them 

exponentially with changes in the angle a between the 

maximum compressive principal stress and the x axis. 

THE LANDSLIDE MODEL 

DETERMINATION OF THE STRESS FIELD 

Our idealized landslide is assumed to form with a 

definite basal slip plane at depth y*, in the semi-infinite, 

isotropic, and homogeneous mass of soil inclined at 

an angle 8 to the horizontal as shown in figure 3. The 

soil mass, acted upon by gravity, fails according to 
THE LANDSLIDE MODEL 11 

X 

FIGURE 3.-Sketch showing semi-infinite mass of soil inclined at an angle 9 to the horizontal. The linearly 

varying pore pressure P and the predicted slide depth y* are also shown. 

Coulomb's criterion (equation (1)), and a pore pressure 

P is present, which varies linearly from a depth y = h 

to the surface y = 0, or 

P = P0 y/h. (63) 

The depth h is considered to be equal to, or greater than, 

the depth of the basal slip plane of the landslide. 

Under these conditions, the stresses can be taken as 

functions of y only, and the equations of equilibrium 

(equations (14a, b)) can be directly integrated to give 

(Jxy = pgy sin 8 (64a) 

and 

u; = pgy (cos 8 - P) , (64b) 

where P = Pipgh is the dimensionless pore pressure. 

1 - sin cf> cos 2a -pg (cos 8-i>)] [ Yu'l = [ H

0 

] 

sin cf> sin 2a -pg sin 8 · (

66) 

Carrying out the inversion, we have 

H sin 8 

u'= --------------------~~-------- sin 8 - sin cf> sin (2a + 8) + P sin cf> sin 2a 

and 

H sin cf> sin 2a 

(67) 

y = . (68) 

pg [sin 8- sin cf> sin (2a + 8) + P sin cpsin 2a] 

For the sake of brevity in future equations, we shall 

define the recurring terms 11 and D1: 

11 = (2a + 8) (69) 

and 

For the soil in the inclined half-space to be in a state D1 = sin 8- sin cf> sin 11 + P sin cf> sin 2a. (70) 

of limiting equilibrium, we have from equations (9a, b) 

uxy = pgy sin 8 = u' sin cp sin 2a 

and 

The validity of the solutions (67) and (68) is limited. 

(65a) For equilibrium of the soil medium, we see from figure 1 

that 

u; = pgy (cos 8 - P) = u' (1 - sin cf> cos 2a) - H, (65b) Tn ~ (u~ + H) tan cp (71) 

from which we can solve for u' andy by inverting 

or, in this case, where Tn = uxy = pgy sin 8, and 

u~ = u; = pgy (cos 8 - P), that 
12 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

pgy [ sin () - (cos () - P) tan 4>] ~ H tan ¢. (72) 

Since k = H tan 4> by equation (2), relation (72) can 

be written 

criterion is satisfied-represents the limit to the validity of solutions (67) and (68). _ 

Thus, when sin (8 - cf>) + P sin cf> > 0, limiting 

equilibrium is _possible only in the strip 0 ~ y~y *. If 

k cos 4> 

y ~ . (73) 

pg [sin (() - cf>) + P sin 4>] 

The upper limit 

k cos 4> 

y= y*= - ' (74) 

pg [sin (8- cf>) + P sin cf>] 

where uxy = k + u; tan cf>-that is, where Coulomb's 

sin (8- cf>) + P sin 4> = 0, then y*- oo. Stress fields 

and characteristics for y * - oo were discussed by Terzaghi (1943) and Sokolovskii (1960, 1965). Equation (74) 

clearly show~ that y* will occur at a finite dept~ when 

() = cf> and P is positive. When sin (() - cf>) + P sin cf> 

< 0 a stable slope occurs, as the slide plane is predicted 

to be above the slope surface. 

In figure 4, the depth y * from equation (7 4) is plotted in dimensionless form pgy */k as a function of the 

30~~--------~------~------~--------~-------------------------------, 

25 

20 

* 

~~~ 15 

10 

5 

P=O.O 

ct>=30° 

OL---------~L---------~L---------~----------~----------~----------~ 

0 10 20 30 40 50 60 

e, IN DEGREES 

FIGURE 4.-Graph showing pgy* versus 9 for various values of P and cf>. 

k 
THE LANDSLIDE MODEL 13 

slope angle 8 for various values of P and tj>. From equation (7 4), we see that y * increases with increasing cohesion k and decreases with increasing density p. The 

gravitational acceleration g is assu~ed constant. From 

figure 4, we see that variations in P have no effect on 

y* when tJ> = 0. In this purely cohesive case, y* depends 

only on 8, p, and k. When tJ> and 8 are constant (tJ> > 0), 

the depth of the lan_!islide y * decreases as the dimensionless pore pressure P increases. This dependence on pore 

pressure is expected from Coulomb's criterion and the 

effective stress law, as a reduction in normal stress u n 

requires a corresponding reduction in the maximum 

shear stress uxy = pgy*_ sin 8, and hence a reduction of 

y *. Also, for constant P and 8, y * increases as tJ> is increased. Finally, as 8 increases, the curv~s in figure 4 

converge, the dependence of y * on tJ> and P lessens, and 

the depth of the landslide approaches that for pure 

cohesion. 

For limiting equilibrium to be possible _in the strip 

0 ~ y ~y *, the dimensionless pore pressure Pin equation 

(7 4) must be restricted. Let us rewrite equation (7 4) as 

k cot tJ> 

y* = ----------~=--

pg [sin 8 cot tJ> - cos 8 + P] 

(75) 

If P = cos 8 - sin 8 cot tj>, then y*- 00• If.? = cos 8, 

then from equation (64b) u; = pgy (cos 8 - P) = 0, and 

there will be no normal stress on any plane parallel to 

y* in 0 ~ y ~ y*. If P >cos 8, there will be tensile normal stresses on all planes parallel to y*, a case not 

covered by the Coulomb criterion. Thus, the limits on 

P for tJ> > 0 are given by 

cos 8 - sin 8 cot tJ> ~ P ~ cos 8 . (76) 

These bounds on P as functions of 8 and tJ> are shown 

in figure 5. For tJ> = 0, P is taken to have the range 

0 ~ p ~cos 8. 

If we rewrite equation (7 4) as 

kcostj> - --= y * [sin (8 - tj>) + P sin tJ>] , (77) 

pg 

OOL-----------+----------~10----------~~--------~2~0----------~2~5----------~ 

FIGURE 5.-Graph showing bounds on Pas a function of slope angle(} for various angles of internal friction t/1. 
14 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

equation (68) becomes 

y= 

y* [sin (8 - ¢) + P sin ¢] sin 2a 

D1 

(78) 

If a = 45 o - : is substituted in (78), it reduces to 

y = y *; and, hence the line y = y * represents a line of 

discontinuity along which the slopes I Z I± given by 

equation (20) are respectively infinite and zero. Because 

stresses undergo finite jumps and stress gradients are 

infinite as we cross a line of discontinuity, the equations 

describing limiting equilibrium in the soil mass are valid 

only when 0 ~ y ~ y *. The line y = y * represents the 

basal slip plane of the landslide. 

We have equations (65a, b) for uxy and u; in our 

idealized landslide. We can find the expression for u; 

by substituting equation (67) into equation (9a), which 

yields 

H sin 8 ( 1 + sin ¢ cos 2a) u; = -H. 

D1 

(79) 

and 

U 1 (cos 8 - P) sin 2a 

U1 = -= = + 2 cos 2a ; X k (83c) 

sin 8 

and equation (78) reduces to 

y = y* sin 2a (84a) 

or 

sin 2a = yly*, (84b) 

from which 

1/2 cos 2a = ± [1 - (yly *)2] • (84c) 

The stress field for the purely cohesive case can then 

be given in the alternate dimensionless forms 

u = yty* xy ' (85a) 

(cos () - P) -a~= (yly*), Y sin () (85b) 

For future convenience, we shall write equations and 

(65a, b) and (79) in the dimensionless forms 

_ uxy pgy . sin 8 cos ¢ sin 2a uxy = - = - Sin 8 = (80) k k D 1 ' 

_ u; pgy - cos ¢ (cos () - P) sin 2a U 1 =-=-(cos()- P) = (81) Y k k D 1 ' 

and 

_ u; cos ¢[(cos()- P) sin 2a + 2 cos 2a sin 8] U 1 

=- = (82) X k D ' 1 

which are valid for 0 ~ yl ~ 1, where yl = yly*-that 

is, between the ground surface and the basal slip plane 

of the landslide. 

When ¢ = 0, equations (80), (81), and (82) reduce to 

(J xy pgy sin 8 

u = - = = sin 2a , xy k k (83a) 

- - _ u; pgy (cos () - P) [cos () - P] . ul = - = = sm 2a (83b) Y k k sin() ' 

(cos () - P) 11

2 

-a~ = (yly*) ± 2 [1 - (y/y*)2J , x sin () (85c) 

where the upper sign in the last equation is taken for 

compressive flow. These equations for u;, u;, and uxy 

are, with notational and coordinate axis differences, 

similar to those presented by Nye (1951, equations 8, 

p. "558) for stresses in a purely cohesive alpine glacier. 

There are two sets of limits on a, the angle that the 

greatest principal effective stress u~ makes with the x 

axis. The first set is for compressive (passive) flow when 

a = 0 is at the surface and u~ is oriented parallel to the 

x axis at y = 0. The limits on a for 0 ~ y 1 ~ 1 are then 

0 ~ a ~ 45 ° - ¢/2 , (86) 

where the upper limit is the value of a on the line of 

discontinuity at y = y*. The second set is for extending 

(active) flow when a at the surface is 90 o and u; is normal to the ground surface at y = 0. The limits on a for 

extending flow, 0 ~ y I ~ 1 are then 

¢ 

45 ° - 2 ~ a ~ 90 ° , (87) 

where the lower limit is the value of a at y = y * . Note 
THE LANDSLIDE MODEL 15 

that these limits apply in the purely cohesive case when 

Q> is set to zero in equations (86) and (87). 

Compressive (passive) flow would be expected where 

the ground surface is concave upward (where the slope 

is decreasing). Extending (active) flow would be expected where the ground surface is convex upward (as 

over a local topographic high on the slope). 

Figure 6 shows the variation of u', u', and u with x y xy 

yly * in compr~ssive and extending flow for 8 = 20 °, 

Q> = 30°, and P = 0.5. For various values of Q>, 8, and 

P, the stress distribution will have this general appearance so it will suffice to investigate some of its predominant features. 

As we can see from the first forms of equations (80) 

and (81), the shear stress ux and the vertical normal 

effective stress u; increase linearly with depth to a 

maximum at the basal slip plane of the landslide (y = 

y*). Figures 7, 8, and 9 show the variation of u , u', 

'!lld u; on y = y * with 8 for various values of%~ W:d 

P. These curves, obtained from the second forms of 

equations (80) and (81) and equation (82), show that the 

shear and normal stresses at the land_!Jlide 's basal slip 

plane decrease as the pore pressure P increases for a 

given 8 and Q>. This decrease is expected from the effective stress law. Also, we see that increasing Q> increases 

each stress component for a given value of P and 8. This 

0.2 

0.4 

0.6 

0.8 

increase happens because the stresses depend on the 

depth of the landslide, which increases as Q> is increased 

(fig. 4). Also, for the same reason, the stresses on 

y = y *increase without limit as 8 approaches the value 

that makes y * jnfinite (equation (7 4)) for particular 

values of Q> and P. Finally, as the slope angle increases, 

the stresses on the basal slip plane decrease, become 

less dependent on Q> and P, and approach those in the 

purely cohesive case. The stresses on the basal slip plane 

decrease because the landslide gets thinner as the slope 

angle increases (fig. 4). As the landslide gets thinner, 

the mean stress on y == y * is reduced, and by the 

Coulomb criterion (fig. 1) the maximum allowable shear 

stress must be reduced. However, note that in the purely cohesive case, the shear stress a xy on y = y * is 

always equal to the cohesion k. 

When y = 0, equation (82) reduces to 

u' = % 

2 cos Q> 

1 - sin Q> 

for compressive flow (a = 0 when y = 0), and to 

- 2 cos Q> u'=----

:1: 1 + sin <1> 

(88) 

(89) 

1.0L--------i--------~-------LL---~--~--------~--~--~--------~ -2 0 2 4 6 8 12 

NONDIMENSIONAL STRESSES 

FIGURE 6.-Graph showing variation of u', u , and u' with y/y* for q, = 30°, 8 = 20°, and P = 0.5 in extending % :~:y y 

and compressive flow. 
16 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

25.-----~~--------~---------------------------------------, 

~ lb 

20 

15 

10 

5 

.0=0.5 

cf>=15° .0=0.5 

cf>=30° 

0~----------~--------~----------~----------~----------~ 

0 10 20 30 40 50 

e, IN DEGREES 

FIGURE 7.-Graph showing variation of oxy at y = y* with fJ for various values of P and q,. 

for extending flow (a = 90 o when y = 0). Thus, when 

y = 0, u; depends only on c/> and becomes more 

compressive (less tensile) as c/> increases, as shown in 

figure 10. 

Figure 6 shows that u; is tensile to a certain depth 

in extending flow. By solving equation (82) for ex when 

u; = 0, we find that ex is given by 

1 1 I cos (} - p l 

ex = 45 ° + 2 tan- . , 2 sm 0 (90) 

where ex lies within the limits given by relation (87). The 

value of a computed from equation (90) is then 

substituted in equation (78) to give the depth yly* to 

which u; is tensile in extending flow. Figure 11 s!_lows 

this depth as a function of 0 for various values of P and 

c/>. From figure 11, we see that the depth to which u; is 

tensile in extending flow increases with slope angle and 

pore pressure but decreases as internal friction 

increases. 

Figure 6 shows that u; is a maximum at a certain 

depth in compressive flow. By differentiating equation 

(82), setting the result equal to zero, and solving for ex 

we find that 

_1 I 2 sin 0 

ex = tan ---------

(P- cos 0) (1 - sin e/>) 

(91) 

_ [4 sin2! + cos2 c/> (P- cos 0)2 - ] 1/21 , 

(P- cos 0 ) (1 - sin e/>) 

subject to relation (86), for u; to be a maximum in compressive flow. This value of a is then substituted in 

equation (78) to give the depth where u; is a maximum 

and substituted in equation (82) to give the magnitude 

of u; at this same depth. Figure 12 shows the depth of 

the maximum values of u; in compressive flow as a 
THE LANDSLIDE MODEL 17 

50~----~~--------~----------------------------------------, 

40 

e, IN DEGREES 

FIGURE 8.-Graph showing variation of o~ at y = y* with 8 for various values of P and</>. 

function of 0 for various values of cp and P. Note that 

this depth decreases with increasing slope angle and 

pore pressure, and increases with increasing internal 

friction. Figure 13 shows the maximum value of u; in 

co~pressive flow as a function of 0 for various values 

of P and cp. Like the other stress values shown in figures 

7' 8, and 9, the maximum values of u; in compressive 

flow decrease with increasing pore pressure and slope 

angle, and become larger with increasing internal friction. Note that here, u; also becomes infinite as 0 approaches the value that makes y * infinite and the 

dependence of the maximum u; on cp and p decreases 

as 0 increases. 

In general then, all dimensionless stresses other than 

u; at the surface (which depends only on cp, as shown 

in fig. 10) decrease with increasing pore pressure, increasing slope angle, and decreasing internal friction. 

Of course, the actual stresses (equations (80), (81), and 

(82)) also increase in direct proportion to the cohesion k. 

Stress characteristics are determined by integrating 

equation (20), which can be written in the canonical form 

ay I ( cJ> )j ax - = tan a± 45° -- -. aa 2 aa 

(92) 

Taking the derivative of equation (78) with respect to 

a, we find 

ay 2y* [sin (0- cp) + P sin cp] sin 0 (cos 2a- sin cp) 

- = .(93) 

aa D~ 

We now define the angle 

0 = (0 - cp). (94) 

From equation (92), we find 
18 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

e, IN DEGREES 

FIGURE 9.-Graph showing variation of u~ at y = y* with 8 for various values of P and cp. 

or 

- = ax cot I a ± ~ 45° - - - c1>) l ay aa 2 aa 

(95) 

()X - 2y* (sin 0 + p sin e/>) Sin 8 (sin 2a + COS e/>) 

- = ,(96) 

8a D~ 

which is integrated (Gradshteyn and Ryzhik, 1980, 

p. 149) to give 

_ [sin c1> (P - cos 8) ± cos c1> sin 8] 

G[e/>, 8, P] = (D2)st2 

(99) 

I (sin 8 + sin c1> sin 8) tan a + sin c1> (P - cos 8) ) tan -I , 

(D2)1/2 

and 

D 2 = sin2 8 - sin2 c1> sin2 8 - sin2 c1> (P - cos 8)2

• (100) 

x = X 0 + 2y* (sin o + P sin e/>] sin 8 

{F [e/>, 8, P) + G [e/>, 8, PJ}, 

(

97) Equation (97) represents the x coordinates of the first 

(upper signs in the functions F and G) and second (lower 

signs in F and G) characteristic directions and is valid 

only when where 

F(cJ>, 8, P] = 

-sin c1> sin 8 

+[sin 8 ±cos c1> sin c/> (P-cos 8)] cos 2a 

± cos c1> sin c1> sin 8 sin 2a 

2D2 [sin 8 -_sin q, sin 8 cos 2a 

+ sin c1> (P- cos 8) sin 2a] 

sin2 8 > sin2 cJ> sin2 8 + sin2 cJ> (F- cos 8)2 • (101) 

(98) 

Relation (101) can be written 

(.P - cos 8)2 > sin2 8 cot2 c1> , (102) 
or 

THE LANDSLIDE MODEL 19 

5~----------------------------------------------------------. 

2 

0 

cl>, IN DEGREES 

FIGURE 10.-Graph showing variation of o~ at the surface, y = 0, with f/J. 

cos () - sin () cot ¢ < P < cos () 

+ sin () cot ¢ , 

was made on the grounds that the normal stress u Y 

was to be compressive only. 

When ¢ = 0, equations (78) and (97) for the stress 

(103) characteristics reduce to 

which is similar to relation (76) if the upper limit on y = y* sin 2a (104) 

Pis restricted to cos () (equation (64b)). This restriction and 
20 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

1.0,----------------------------------, 

0.8 

0.6 

0.4 

P=0.5 

cj>=30° 

30 40 50 

9, IN DEGREES 

FIGURE 11.-Graph showing dimensionless depth yly* to which!~ is tensile in extending flow as a function of 6 for various values of P and cp. 

x = X 0 + y* cos 2a ± 2y*a . (105) 

Relation (101) shows that equations (104) and (105) are 

valid when the slope angle() is greater than zero. 

Equations (78) and (97) give the y and x coordinates 

of the stress characteristics. Figure 14 gives a few examples of stress characteristics for various values of(), 

¢, and P in compressive and extending flow. Comparing figures 14A and 14E, we see that increasing ¢ for 

a constant () and P causes the characteristics to become 

less curved. We see the same effect as the slope angle 

() is decreased (compare figs. 14D and 14E or figs. 14A 

and 14C). The stress characteristics are also flattened 

asP is decreased (compare figs. 148 and 14C). The flattening of the stress characteristics is consistent with 

our observation (fig. 4) that increasing ¢ or decreasing 

() or P increases the depth y* of the model landslide. In 

fact, as y* increases without bound, the stress 

characteristics approach straight lines at depth (Terzaghi, 1943, p. 38). Finally, only some of the stress characteristics for each value of(), ¢, and Pare shown in 

figure 14; but, because of the arbitrary constant of integration x 0 in equation (97), there are actually an infinite number of stress characteristics within the 

deforming regions shown. 

As we mentioned before, the stress characteristics in 

figure 14 intersect at every point at fixed angles dependent on¢. The characteristics are loci along which the 

shear stress is the limiting shear stress given by equation (1), and loci along which potential discontinuities 

in stress and stress gradient will be propagated. Also, 

the basal slip planes in figure 14A through 14F form 

envelopes to the second set of stress characteristics and, 

hence, are lines of discontinuity. 
THE LANDSLIDE MODEL 

1.0~~--------------~"----------------------------------------, 

P=0.5 

<!>=30° 

0.8 

0.6 P=O.a 

<!>=30° 

~ * 

:.;; 

0.4 

0.2 

OL-----------L-----------~--------~~----------~--------~ 0 10 20 30 40 50 

e. IN DEGREES 

FIGURE 12.-Graph showing dimensionless depth yly* of maximum valu~s of u~ in compressive flow as 

a function of 8 for various values oft/> and P. 

DETERMINATION OF THE VELOCITY FIELD or 

u = 'rx + F(y) + U0 , 

where rand u0 are constants and 

au 

-=r. 

ax 

21 

(107) 

(108) 

To determine the velocity field in our idealized landslide, we use equations (32a, b, c) for the deformation 

rates, equation (36) for continuity, and equation (37) for 

isotropy. We assume that v, the vertical velocity, is independent of x. Such an assumption is reasonable for 

an infinitely long slope. Then, from equation (36) for continuity we have 

_a_2v-= o 

ax ay (106a) 

Note that in equation (108) r represents a constant 

longitudinal deformation rate, or from equation (32a) 

that 

and 

. au 

e =- -= -r. X ax (109) 

= 0' (106b) Under these conditions, equation (36) for continuity 

becomes 
22 

)( 

ib 

A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

~~----~~--------~-------------------------------------, 

40 

30 

20 

10 

P=0.5 

4>=30° 

0~---------L----------~----------~--------~~--------~ 0 10 20 30 40 ~ 

e, IN DEGREES 

FIGURE 13.-Graph showing maximum value of u~ in comeressive flow as a function of (J for various 

values of q, and P. 

av = r I sin tP + cos 2al 

ay sin tP- cos 2a ' (110) basal slip plane of the landslide. Such infinities occur 

because of rapid velocity changes across a narrow zone 

on the base of a landslide. 

and equation (37) for isotropy becomes The discontinuity in the vertical deformation rate 

a v relates to the volume dilatancy rate. A nonzero volau lau avj a Y = ax - a Y tan 2a . (111) 

Substituting equations (108) and (110) in equation (111), 

we arrive at 

_a_u = . -2r sin 2a = F' (y) . 

a y sm tP - cos 2a (112) 

Since a = 45° - !!!_ on y = y*, we see that infinities oc2 

cur in a v (equation 110) and au (equation 112) on the 

ay ay 

~e dilatancy rate occurs if the soil making up the landslide is frictional-that is, if f/> :f: 0. Expressions for 

volume dilatancy rates in compressive and extending 

flow are easily derived from equation (33) for the 

sum of principal deformation rates, equation (108) for 

au , and equation (110) for a v . Combining these equa- ax ay tions, we obtain the dilatancy rate 

d = 2r sin t/> 

cos 2a- sin cJ> 

(113) 

as a function of a, the angle that u; makes with the 
THE LANDSLIDE MODEL 23 

COMPRESSIVE FLOW EXTENDING FLOW 

B~ 

c~ 4>=15° 8=10° .0=0.5 

D~ 

4>=30° 8=20° p =0.5 

FIGURE 14.-Sketches showing stress characteristics for various slope angles 8, angles of internal friction t/l, and dimensionless 

pore pressures P. Values of 8, q,, and P used for each case are given below the lower right-hand comer of each sketch. 

Arrows indicate the relative sense of slip across the characteristics and the dashed lines separate regions of compressive 

and extending flow. 

positive x axis. Here, r is negative for compressive 

flow and positive for extending flow. Using equation 

(78) the volume dilatancy rate d can be obtained as 

a function of yly*. The results are shown in figures 15A 

and 15B for some _values of(), c/>, and P. Note that the 

negative sign on d represents volume expansion. The 

dilatancy rate in every case where cJ> is nonzero is infinite 

on the landslide base because of the discontinuity in 

a v on y = y*. When y = 0, the dilatancy rate 

ay 

increases with cJ> for extending and compressive flow. 

In general, the dilatancy rate is greater at a given depth 

in compressive flow than in extending flow._ The dilatancy rate increases as cJ> increases or as P or () decreases-a consequence of increasing shear stress in 

each of these cases. Finally, as we can see from equation (113), the dilatancy rate vanishes when cJ> = 0; thus 

flow in the purely cohesive case is incompressible. 

Returning to equations (110) and (112), we next integrate these equations to find the x andy components 

of velocity, u and v, of the landslide as functions of a. 

Since vis a function of y only, from equation (110) for 

au 

ay we have 
24 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

-16~---------------------------------------------------------------,n 

A 

-14 

-12 

-10 

-.::-

-~ -8 

QQL------------0L.2------------0~.-4------------0~.6------------0~.8~--------~1.0 

Y/y* 

FIGURE 15 (above and facing page).-Graphs showing varia!ion of dimensionless dilatancy rate d/lrl with 

dimensionless depth y/y* for various values of 8, cp, and P. A, Compressive flow. B, Extending flow. 
THE LANDSLIDE MODEL 25 

-16.---------------------------------------------------------------... B 

-14 

-12 

-10 

-6 

-4 

-2 

0~-----------L------------~----------~L_ __________ _L __________ ~ 

0 0.2 0.4 0.6 0.8 1.0 

!'l"y* 
26 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

v = r J [ s~n cp + cos 2a l dy + B 

sm cp- cos 2a 

= 2y*r [sin o + P sin cp] sin 8 

J [sin cp + cos 2a] da 

[sin 8 - sin cp (sin 2a cos 8 + cos 2a sin cp) 

+ P sin cp sin 2a]2 

+ B , (114) 

where the differential of equation (78) with respect to 

a replaces dy. Then, by integrating equation (114) (Gradshteyn and Ryzhik, 1980, p. 149), we get 

v - - = -[sin o + P sin ¢] sin 8 {F1[c/>, 8, P] 

Y*r 

where 

and 

sin cp (P - cos 8) (1 + sin cp cos 2a) 

+ sin 8 (1 + sin 2 cp) sin 2a 

D 2 [sin 8 - sin cp sin 8 cos 2a 

+ sin cp (P - cos 8) sin 2a] 

4 sin cp sin 8 

(115) 

(116) 

~ 

u-u - - = [sin o + P sin ¢] sin 8 [F2 [¢, 8, P] 

y r - + G2 [¢, 8, P]] + C, 

where 

2 sin 8 (sin cp - cos 2a) 

F2 [¢, 8, P] = ----------

D2 [sin 8 - sin cp sin 8 cos 2a 

+ sin cp (P- cos 8) sin 2a] 

and 

4 sin cp (P - cos 8) G2

[c/>, 8, P] = __ .;...._ __ __,_ (D2)3t2 

(119) 

(120) 

(121) 

[ 

sin 8 (1 + sin 8) tan a + sin cp (P - cos 8) ]· tan-1 

(D2)1'2 

Equation (119) is also subject to the conditidn given by 

relation (101), and since we are only interested in how 

u varies with a on an arbitrary cross section, we ignore 

the term in x given in equation (107). The term U 0 

represents the sliding velocity when y =y*. For u to 

equal U0 when y = y*, we must have 

C = 4 (sin o + P sin cp) sin 8 sin cp (P - cos 8) 

(D2)3t2 

tan-1[ sino + p sin cp I· 

(D2)1'2 

(122) 

G1[c/>, 8, P]= ___ ;.._.__ 

(D2)3/2 When cp = 0, equations (115) and (119) for the 

(117) velocities reduce to 

[ 

sin 8 (1 + sin cp) tan a + sin cp (P - cos 8) ]· 

tan-1 (D2)112 

Equation (115) is subject to the condition given by relation (101), and since v = 0 when y =y*, we have 

B= (sin o + P sin ¢)2 sin 8 (1 + sin2 cp) 

D 2 (sin 8 - P sin cp cos cp - sin cp cos o) 

+ 4 (sin o + P sin cp) sin cp sin2 8 (118) 

(D2)3/2 

tan-l sino + P sin cp 

(D2)112 

By following a similar procedure for equation (112), 

we find that 

and 

u-uo 

y*r 

__!!__ = 1 - sin 2a = 1 - Y 

y*r y* 

(123) 

-cos 2a = + 2 [1 - (y/y*)2)112 . (124) 

Here, the relationships yly * = sin 2a and cos 2a = 2 1/2 • ± [1 - (yly*)] , which apply when cp = 0, have been 

invoked. These expressions are, with notational and 

coordinate axis differences, the same as those presented 

by Nye (1951, equation 9, p. 559) for velocities in an 

alpine glacier. 

The dimensionless velocities [(u - U 0 )/ry*) and vlry* 

are shown as functions of yly* (where yly* has been 

obtained from equation (78)) for compressive and 

extending flow and various values of¢, 8, and Pin 

figure 16. 
0 

0.2 

0.4 

* 

~ 

0.6 

0.8 

1.0 

0 

0 

0.2 

0.4 

~ * 

0.6 

0.8 

1.~4 

0 

0.2 

0.4 

* 

~ 

0.6 

0.8 

1.0 -6 

2 

-5 

THE LANDSLIDE MODEL 

3 

u-uo fry* 

4 5 

-1 

A 

6 

B 

P=O 

2 

FIGURE 16.-Graphs showing dimensionless velocities (u- u0 )/ry* an':! v/ry* as functions of dimensionless depth yly* 

for compressive and extending flow and various values of 8, q,, and P. A, Dimensionless longitudinal velocity for compressive and extending flow. B, Dimensionless normal velocity for extending flow. C, Dimensionless normal velocity 

for compressive flow. 

27 
28 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

Figure 16 shows that the dimensionless longitudinal 

velocity [(u - U0 )/ry *) is the same in compressive and 

extending flow. This velocity decreases as() increases 

and as (j> decreases. The_ dimensionless longitudinal 

velocity also decreases as P increases. Note that, in the 

purely cohesive case, the velocities are independent of 

P, beca~se y* in equations (123) and (124) is independent of P when (j>=O. (See equation (7 4).) These results 

for (u - U0 )/ry * are consistent with the fact that the 

allowable shear stress, and hence the shear strain rate 

(equation (112)) within the slide, is increased as (j> is increased or asP and () are reduced. Finally, although the 

observation that the dimensionless longitudinal velocity 

decreases with increasing slope angle seems contradictory, recall that (u- U 0 )/ry* given by equation (119) occurs in response to deformation within the landslide 

mass and this velocity is relative to the basal slide 

velocity U 0 , which increases with slope angle. 

Since the dimensionless y component of velocity vlry * 

is positive in the positive direction of they axis (fig. 3), 

vlry * is more negative in compressive than in extending 

flow as figure 16 shows. We see a similar effect as () is 

reduced and as (j> is increased. Again, reducing P causes 

a more negative vlry *. Whether the flow is compressive 

or extending, a more negative value of v means a greater 

dilatancy rate in the landslide. This result is consistent 

with the observation made previously that increasing 

(j>, and decreasing P or 8, increases the shear stress and 

enhances the dilatancy rate as seen in figure 15. 

So far, we have discussed the velocity distribution 

within the idealized landslide for extending and 

compressive flow. There is a third case where r, the 

longitudinal deformation rate in equation (108), vanishes. When r vanishes the longitudinal velocity u given 

by equation (119) decreases to the basal sliding velocity U0 , and the y component of velocity v given by 

equation (115) vanishes. The landslide then slides as a 

rigid body on the basal slip plane, and the flow is termed 

plug flow (Nye, 1951). 

As mentioned earlier, the velocity and stress characteristics for our Coulomb material coincide. Thus, 

potential discontinuities in velocities and strain rates 

will be propagated along the stress characteristics (slip 

surfaces) shown in figure 14. Also, we found that discontinuities in velocity originating at, say, the line of 

discontinuity y = y * will be propagated in an exponential manner with change in the angle a along the slip 

lines (equations (52), (53), (61), and (62)). 

Suppose a local irregularity at a point Q (fig. 17A) 

on the basal slip plane of a landslide (y = y *) causes 

a discontinuity in tangential velocity u1 • This, in 0 

turn, propagates a discontinuity of tangential velocity 

ul and a discontinuity of normal velocity v 1 along a 

(3_ characteristic with the sense of relative velocity 

shown in figure 17 A. The magnitude of the tangential 

velocity at a point on (3_ is given by equation (52) and 

the magnitude of normal velocity is given by equation 

(59). 

y 

x------------------------------------~--------~----~----------~ 

Q' 

B 

y 

FIGURE 17.-Sketches showing fault surfaces produced by propagation of discontinuities along /3_ slip lines 

in A, compressive flow and B, extending flow. The directions of the velocity jump components and the 

resultant velocity jump vectors are also shown. Arrows show sense of relative slip across the fault surfaces. 
THE LANDSLIDE MODEL 29 

Since a0 in this case is 45 o - : , the magnitude of the 

tangential velocity jump on the surface in compressive 

flow (where a = 0) is given in dimensionless form as 

~:. = exp I( : - : ) tan ~>] , 

(125) 

where ¢ is in radians, and the normal velocity jump 

when y = 0 is given in dimensionless form as 

- - vl ul 

=- = - =- tan ¢ . (126) 

ulo ulo 

The negative sign indicates that this component acts 

upwards. As mentioned earlier, the resultant velocity 

... jump vector vl is inclined at an angle¢ to the {3_ slip 

line, and its magnitude is an increasing function of ¢ 

as seen in figure 18. 

Similarly, if basal slip is impeded at a point Q' 

(fig. 17 B) when y = y* during extending flow, discontinuities in ul and v 1 will propagate along a {3-

characteristic with the sense of relative velocity shown 

in figure 17 B. The dimensionless magnitude of the 

tangential velocity jump ul on y = 0 is given here (a = 

90° when y = 0) by 

~: = exp 1- ( ; -f) tan ~>] , (127) 

0 

where ¢ is in radians, and the dimensionless magnitude 

2.2,-----------------------------------------------------~ 

FIGURE 18.-Graph showing magnitude of resultant velocity jump on the surface as a function of 

angle of internal friction t/J for extending and compressive flow. 
30 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

of the normal velocity jump v1 when y = 0 is given 

by equation (126). Again, v1 is directed toward the 

ground surface. The resultant velocity jump vector 

-

v1 is inclined at an angle ct> to the {3_ slip line, and its 

magnitude is a decreasing function of ct> (fig. 18). 

Physically, the jump in the tangential component of 

velocity on the surface leads to a thrust fault scarp 

in compressive flow and a normal fault scarp in extending flow (fig. 18). In compressive flow, the height of 

the scarp increases as ct> increases; the opposite is true 

in extending flow. When c/> = 0, the scarp height is 

equal to the magnitude of the discontinuity on the base 

of the landslide, in both compressive and extending 

flow. 

The discontinuity in the normal component of velocity leads to a zone of disturbance along a {3 _ slip line 

that is proportional in thickness to the normal velocity 

discontinuity. Physically, one might expect a disordering of the particle-packing arrangement in this 

zone-that is, dilatancy .. In compressive flow, the 

thickness of the dilatant zone increases with c/>. For 

extending flow, the thickness of the dilatant zone 

increases with ct> to a maximum at ct>::::: 37 °, thereafter, 

decreasing in thickness with increasing internal friction. When ct> = 0, the normal velocity discontinuity 

vanishes, and no dilatancy accompanies the propagation of velocity discontinuities on the slip lines. N ye 

(1951) discussed the propagation of discontinuities 

in alpine glacier flow-that is, in the purely cohesive 

case. 

DISCUSSION 

SIMILARITIES AND DIFFERENCES BETWEEN 

NYE'S ALPINE GLACIER MODEL AND OUR 

LANDSLIDE MODEL 

The flow of landslides is similar to that of alpine 

glaciers. These similarities occur because soils within 

landslides can be described as Coulomb plastic 

materials, a generalization of Mises plastic materials. 

Like Nye's (1951) Mises plastic model for alpine glaciers, 

our model for landslides predicts, depending on the 

longitudinal deformation rate, that a landslide can 

undergo compressive, extending, or plug flow. For both 

models, shear and normal stresses increase linearly with 

depth, and longitudinal stresses increase in an elliptic 

manner with depth. Both models predict tensile 

longitudinal stresses and normal fault scarps at the surface in extending flow, and thrust surfaces and thrust 

fault scarps at the surface in compressive flow. 

Longitudinal velocities caused by internal deformation 

in both models are greatest near the surface and 

decrease elliptically to the basal slip plane where they 

are equal to the basal slide velocity. Also, like Nye's 

glacier-flow model, our model predicts that in plug flow 

a landslide moves as a rigid body at the basal slide 

velocity. Finally, in both models, the thickness of the 

plastically flowing region is controlled by the slope 

angle, the density, and the cohesion. This thickness increases with increasing cohesion and decreases with increasing density and slope angle. 

However, the landslide model and glacier model are 

in some ways dissimilar. For example, unlike the alpine 

glacier model, our model is affected by pore pressure 

and internal friction as well as slope angle, density, and 

cohesion. Increasing the pore pressure or decreasing the 

angle of internal friction for a given slope angle and 

cohesion decreases the thickness of the plastically flowing region. 

The models also differ because stresses in the landslide are influenced by pore pressure and internal friction. In extending flow, all stresses are smaller and the 

depth to which longitudinal stresses are tensile is 

greater when pore pressure is increased or when internal friction is decreased for a given slope angle, density, and cohesion. 

Another difference is that the stress characteristics 

are influenced by pore pressure and internal friction in 

t_he landslide model. They flatten as ct> is increased or 

P is decreased because of the increase in depth of the 

basal slide plane with increasing internal friction or 

decreasing pore pressure. An excess pore pressure also 

leads to a bounded slip line field even where the slope 

angle is less than the angle of internal friction, a point 

made by Crans and Mandl (1981). 

As in N ye 's ( 1951) model for alpine glacier flow, 

longitudinal velocities occurring in response to deformation in the landslide are the same in both extending and compressive flow. However, in the landslide 

model, internal deformation and hence longitudinal 

velocities increase as the pore pressure is reduced or as 

internal friction is increased. Also, in both models the 

sense of they component of velocity is consistent with 

vertical expansion in compressive flow and vertical 

contraction in extending flow. But, in the landslide 

model, the vertical expansion in compressive flow is 

amplified by dilatancy and the vertical contraction in 

extending flow is reduced by dilatancy. Dilatancy 

vanishes for c/> = 0. 

Dilatancy also influences the propagation of velocity discontinuities in the landslide model as the resultant velocity jump vector is inclined at the angle of 

internal friction to the slip lines. As we have noted, the 

normal component of velocity discontinuity vanishes 

when c/> = 0. 
CONCLUDING REMARKS 31 

THE ASSUMPTION OF ASSOCIATED FLOW 

The dilatancy predicted by our model is a consequence 

of assuming an associated flow rule. Such a flow rule 

assumes that at any smooth point on a yield surface the 

plastic strain rate vector is normal to the yield surface 

(Drucker, 1956; Davis, 1968). The advantages of this 

assumption are (1) velocity and stress characteristics 

coincide, and (2) simple and useful limit theorems apply only to such materials (Davis, 1968). The principal 

disadvantage of assuming associated flow is that it leads 

to an unrealistically large dilatancy for Coulomb plastic 

materials, exemplified by the occurrence of infinite 

dilatancy rates on the basal slip plane in our landslide 

model (fig. 15). 

The implication that Coulomb plastic soils expand indefinitely and yet retain shear strength is physically implausible, and in fact, no known material behaves in this 

way (Houlsby and Wroth, 1980). Houlsby and Wroth 

(1980) pointed out that soils, when sheared monotonically, first dilate and then approach a critical state in which 

deformation occurs at constant volume. 

An alternative (Davis, 1968; Salencon, 1977; Houlsby 

and Wroth, 1980), then, is to replace the angle of internal friction in equations (32a, b, c) with a variable angle 

of dilation, v, which when deformation begins, equals the 

angle of internal friction q,, and with continued deformation drops to zero. Theoretically, the angle v could be 

larger than c/>. However, this would imply volumetric expansion under hydrostatic compression. 

Using a variable angle of dilation would add considerable complexity to the analysis unless we assu1ne 

that the deformation being modeled has progressed to 

the point where v = 0. Since q, + v, the flow rule would 

be nonassociated-that is, stress and velocity characteristics would no longer be coincident, and the simple limit 

theorems referred to above would no longer apply. 

Region of extending flow 

In any case, velocity characteristics initially intersecting at an angle of 90° - q, would, in the course of 

continued flow, come to intersect at 90 ° and would be 

equivalent to the q, = 0 examples given earlier for the 

landslide modeL Thus velocities, velocity characteristics, 

and velocity jump relations given for q, = 0 in the 

landslide model apply when the angle of dilation v is 

zero. 

Another approach to the initiation and growth of slip 

surfaces is the Rice (1973) and Rudnicki and Rice (1975) 

proposal that the physical mechanisms involved in dilatant materials lead to vertices on the yield surfaces and 

consequently to nonassociated flow rules. Flows described by such flow rules are unstable and lead to localized 

planar zones of nonuniform deformation in otherwise 

homogeneously deforming materials. Of course, resolving the issue of whether an associated or nonassociated 

flow rule applies in a given landslide depends on careful 

field and laboratory observations. 

CONCLUDING REMARKS 

The model for landslide flow presented here is by no 

means a complete solution. Landslides are complicated 

by variable geology and topography, factors obviously 

missing in our idealized isotropic, homogeneous, inclined 

half-space. However, like Nye (1951), we have simplified 

our model so that some essential features of the 

mechanics of landslide flow can be highlighted. Any field 

test of our model should be carried out under conditions 

that best approximate our assumptions. Such field tests 

would be best carried out in relatively homogeneous and 

isotropic landslides that are long and shallow, and in 

which regions of compressive, extending, and plug flow 

are clearly identified. An idealized sketch of such a field 

situation is shown in figure 19. 

Region of compressive flow 

FIGURE 19.-Sketch of an idealized long, shallow landslide. Arrows indicate relative senses of slip along failure surfaces in extending and compressive flow. 
32 A MODEL FOR THE PLASTIC FLOW OF LANDSLIDES 

REFERENCES CITED 

Abbott, M. B., 1966, An introduction to the method of characteristics: 

New York, Elsevier, 243 p. 

Coulomb, C. A., 1773 [1776], Essai sur une application des regles de 

maximis & minimis a quelques problemes de statique, relatifs a 

1' architecture: Memoires de Mathematique & de physique, 

presentes a I' Academie Royale des Sciences (Savants Etrang.), 

v. 7, p. 343-382. [Reprinted and translated in Heyman, Jacques, 

1972, Coulomb's memoir on statics: London, Cambridge University Press, 212 p.] 

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