A comprehensive review of strain localization in elastoplastic soils

Computers and Geotechnics 10 (1990) 163-188

A C O M P R E H E N S I V E R E V I E W OF S T R A I N IN ELASTOPLASTIC

LOCALIZATION

SOILS

J.P. Bardet

Civil Engineering Department University of Southern California Los Angeles CA90089-2531

ABSTRACT The paper reviews the theory of strain localization for elastoplastic soils and relates it to past works on the inclination of shear bands. It outlines and discusses the main theoretical assumptions, describes the localized velocity field, compiles experimental results and compares them to theoretical predictions on shear band orientations. It studies the effect of elastic unloading and examines systematically the influences of friction angle, diJatancy angle, Poisson's ratio and hardening modulus on shear bands. After comparing the predictions of the Mohr-Coulomb and Drucker-Prager models, the study concludes that the application of the strain localization theory to elastoplasticity does not account for the observed shear band orientations in all circumstances. It also recalls that the strain localization theory provides a necessary, but not sufficient, condition for the emergence of shear bands, a theoretical feature which enhances the disagreement between the experimental observations and theoretical predictions on shear band inclinations. INTRODUCTION The deformations within soils are commonly observed to concentrate in narrow zones called shear bands. The theory of strain localization is a well accepted framework in geomechanics to analyze shear banding. The theory was f'n-st proposed by Hadamard [1] and later developed in [2, 3, 4, 5]. It was applied to predict the existence and orientation of 163 Computers and Geotechnics 0266-352X/91/$03-50 @ 1 991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

16a

the emerging shear bands within various types of materials, including elastoplastic materials [6, 7, 8, 9, 10, I1, 12, 13, 14]. The present work reviews comprehensively the theo~' of strain localization for elastoplastic soils and relates it to the past works on the inclination of shear bands in soil mechanics. It outlines and discusses the main theoretical assumptions, interprets physically the localized velocity' field, compiles experimental results and compares them to theoretical predictions. It departs from past analyses [9,13,14] for several reasons. First, it studies the effect of elastic unloading and examines more systematically the influences of the friction angle, dilatancy angle, Poisson's ratio, and hardening modulus on the orientation of shea~bands. Second, it describes the localized velocity field at the inception of shear-banding, compares the predictions of Mohr-Coulomb and Drucker-Prager models and points out some critical limitations of the theory of strain localization and of its application to elastoplasticity. In addition, the present work is not concerned with incremental nonlinearity [15, 16, 171 or thickness of shear bands [18, 19]. Following the introduction, the second section of this paper reviews the conventional theories of soil mechanics that are used to analyze shear bands and then compares theoretical and experimental results. The third section summarizes the theory of strain localization and its application to elastoplastic materials. The fourth section applies the theory of strain localization to Mohr-Coulomb materials and performs a parametric study on its material parameters. The fifth section compares the results obtained with MolarCoulomb and Drucker-Prager materials.

PAST W O R K ON SHEAR BANDS IN SOILS Two theories are traditionally authoritative in Soil Mechanics to calculate the inclination of shear bands in two-dimensional plane strain problems, namely the MohrCoulomb theory and the Roscoe theory. Mohr-Coulomb theory The Mohr-Coulomb theory states that the shear bands are parallel to the surfaces which are subjected to the stress (~, "t) of the failure envelope of Mohr-Coulomb. As shown in Fig.l, the shear band makes an angle 0 with the major principal stress directior,, (51:

165 where

¢is the value of the mobilized friction angle at failure. By definition, the mobilized

(2)

friction angle ¢ is obtained from the major and minor principal stress o I and o3: sine =

°I - °3 o1+ 03

Hereafter, the sign convention of Soil Mechanics is adopted. Stresses are positive in com )ression and negative in tension.

~2

k

/i

a' [Mohr.Coulomb]

d£n

I.osoo. I

I v

I

Figure 1. Orientation of shear bands for Mohr-Coulomb and Roscoe theories. Roscoe

theory

Hansen [21], Roscoe et al. [22] and Roscoe [23] emphasized the importance of strains on failure; they proposed that the shear bands are parallel to the line of no-extension. According to the Mohr representation of incremental strain depicted in Fig. 1, the shear band makes an angle 0 with the major principal strain increment dsl: x x¢

e = $- 2

(3)

where ~ is the value of the dilatancy angle at failure. By definition, the dilatancy angle gt is obtained from the major and minor principal strain increments dE 1 and da3: singt =

d~ I +

de 3

dE 1 - de 3

Adopting the sign convention of soil mechanics, a positive value of gt implies dilatancy, while a negative value of V corresponds to compaction.

(4)

166

T h e o r y versus experiment: scattering of experimental data Figs.2 and 3 compare the orientation of shear bands that are theoretically calculated by using Eqs. 1 and 3 to those experimentally measured by Arthur et al. [24], Vardoulakis [9] and Desrues [20].The experimental data of Figs.2 and 3 is reported in Tables 1 and 2.

40-

A 30-

O O O O O o

13 0 ,A ~ o

Arthur et el. (1977) I Vsrdoulekis (1980) Desrues (1984) Mohr-Coulomb theory

I

13

o

&

O0

&o

20"

=; 30

40

50

0

Friction angle ¢ (deg) Figure 2. Shear band inclination versus friction angle ( Mohr-Coulomb theory and experimental data after [9, 20, 24]). 60"

~

50

0 Vardoulakis (1980) I Desrues (1984) J Desrues (1984) local] Roscoe theory I

.... ,E J

'

" ~

X~

4O

o

30

A A

20

10 -20

(~

2'0

4O

Dilatancy angle ~/ (des) Figure 3. Shear band inclination versus dilatancy angle ( Roscoe theory and experimental data after [9, 20, 24]).

167

Table 1. Summary of experimental results on shear bands after [9, 24]. Test Arthur et al [24] FPSA FPSA FPSA FBC FBC FBC FBC Vardoulakis [9] XV1.21 XV 1.22 VV1.04 VV1.05 VV1.06 VV1.07 VV1.08 VV1.09 VVI.10 VVI.13 VVl.14 VVI.15 VVI.16 VV 1.17 VV1.23 VVl.24

0

0

~g

(de#

(de~)

(de,~)

4 -4 -4

error(%)

(de~)

(~o)

28.0 25.0 25.5 3 1.0 30.0 26.0 26.0

49.0 50.0 45.0 46.0 50.0 51.0 49.0

21.0 30.0 22.5 9.0 19.0 20.0 23.0

27.5 25.0 28.1 31.25 27.75 27.25 27.0

1.78 0.00 -10.29 -0.80 7.50 -4.80 -3.84

31.8 29.8 31.0 33.5 34.0 30.5 26.0 25.5 34.0 30.5 27.5 26.0 24.5 26.0 25.5 27.5

37.5 42.5 39.7 39.1 36.0 44.5 47.5 44,9 37.4 38.6 45.7 47.1 47.6 46.6 46.9 46.3

4.55 9.46 5.54 6.57 3.75 11.93 16.24 16.55 3.46 3.51 16.39 16.84 I6.I5 15.44 18.72 15.95

34.48 32.00 33.69 33.58 35.06 30.89 29.06 29,63 34.78 34.47 29.47 29.01 29.06 29.48 28.59 29.43

-8.45 -7.41 -8.67 -0.23 -3.11 - 1.28 -11.78 -16.22 -2,30 -13.02 -7.18 -11.59 -18.62 - l 3.42 -12.12 -7.03

Table 2. Summary of experimental results on shear bands after [20]. average Test

SHOO SHF00 SHF01 SHF02 SHF03 SHF05 SHF06 SHF07 SHF08 SHF09 SHF10 SHFll SHF12 SHF13 SHFI4 SHF15 SHFI6 SHFI7 SHF18 SHF20

e°

0.65 0.74 0.74 0.90 0.85 0.66 0.66 0.66 0.66 0.66 0.66 0.68 O.66 0.66 0.66 0.66 0.66 0.66 0.66 0.86

o

0

v

v1

(de~;) 32.0

(de~) 42.0

(de]) 4.0

(de~) 5.0

26.0

45.0

I0.0

11.0

21.0 26.0 32.0 22.0 25.0 23.0 23.0 20.0 20.0 20.0 26.0 26.0 20.0 21.0 28.0 22.0 22.0 28.0

45.0 36.0 34.0 49.0 48.0 49.0 48,0 48.0 48.0 48.0 48.0 48.0 47.0 46.0 48.0 48.0 48.0 39.0

14.0 -10.0 -10.0 9.0 I0.0 13.0 15.0 14.0 11.0 10.0 13.0 13.0 14.0 14.0 16.0 11.0 11.0 0.0

23.0 8.0 I0.0 30.0

20.0 15.0

25.0

--

4

4

local x

0

Vl

4

4

4

erTor

eITor

4

(de~) 33.50 31.25 30.25 38.50 39.0O 30.50 30.50 29.50 29.25 29.50 30.25 30.50 29.75 29.75 29.75 3O.O0 29.00 30.25 30.25 35.25

-4.69 -20.19 -44.05 -48.08 -21.88 -38.64 -22.00 -28.26 -27.17 -47.50 -51.25 -52.50 -14.42 -14.42 -48.75 -42.86 -3.57 -37.50 -37.50 -25.8

(%1

33.25 31.00 28.00 34.00 34.00

-3.91 -19.23 -33.33 -30.77 -6.25

25.50

-2.00

28.00 29.25

-40.00 -46.25

26.75

-21.59

168

Fig.2 shows that the Mohr-Coulomb theory (Eq. 1) underestimates the shear band orientation O. Fig.3 indicates that O does not vary. in function of ~ as stipulated by the Roscoe theory (Eq.3). By using stereophotogammetric measurement, Desrues [201 provided local measurement of the dilatancy angle ~1; he estabtished that ~t1within the shear band may be twice as large as the global dilatancy angle V calculated from the global volume change of the samples. Unfomanately, the values of gt l reported in Table 2 improve only slightly the estimation of 8 by the Roscoe theory. In view of the poor performance of the classical theories in explaining the shear band orientation, Arthur et al. [24] proposed the following empirical relation for e: 0

rc ¢ gt =~--~ -~-

(5~

Eq.5 averages the angles predicted by the Mohr-Coulomb (Eq. 1) and Roscoe (Eq.3) theories. Fig.4 shows the straight line predicted by Eq.5 and the experimental data on (3 as a function of ~) + ~. = Arthur et al. (1977) O Verdoulakis (1980) A Desrue= (1984) • Deeruee (1984) Iocll - - A r t h u r e mt ah theory

35 C O " ~

"o

oAa~ o O

O~,,,~

=

A

c

OA ~ , ~

A

•

AA a ~AoOoo

c~

25 ¸

J~

1 5

'

20

4'0

•

60

....

80

Figure 4. Shear band inclination versus 0 + gt ( Arthur et al. theory and experimental data after [9, 20, 24]). In contrast to the Mohr-Coulomb theory, Eq.5 overestimates the shear band inclination O. The use of the local instead of global dilatancy angles slightly improves the prediction of O. The error made in estimating 8 with the help of Eq.5 is reported in Tables

169

1 and 2. It remains smaller than 10% for the data of Arthur et al. [24], reaches 18% for the data of Vardoulakis [9], and exceeds 50% for the data of Desrues [20]. From the comparison of Figs.2, 3 and 4, it is concluded that the theories of Arthur et al. and Mohr-Coulomb give an upper and lower bound, respectively, for the shear band inclination 0 which are more accurate than the predictions of the Roscoe theory. However, neither Eq. t nor Eq.5 accounts for the orientation of shear bands in all circumstances and explains the scatter of data of Figs.2 and 4. Errors on 0 may have been introduced in Fig.4 when estimating the dilatancy angle, as noticed by Desrues [20]. However the local dilatancy angles ~gl reported in Table 2 improve only slightly the estimation of 0 and do not justify the scatter of data in Fig.4. It is therefore necessary to search for a more satisfying explanation by using a rational framework: the theory of strain localization. We shall attempt to review comprehensively this theory that has been disseminated in numerous papers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Our presentation wilt focus on the main assumptions and results relevant to soils. R E V I E W OF T H E T H E O R Y OF S T R A I N L O C A L I Z A T I O N The strain localization theory specifies the conditions for which shear bands emerges within uniformly stressed and strained materials. After detecting the existence of localized strains, the theory gives the shear band orientation and the velocity field inside the shear bands at the onset of localization. It is important to underline that the theory has three major limitations: first, it analyzes only the emergence but not the development of shear band; second, it does not guarantee that the shear band actually emerges; and third, it does not give the thickness of the shear band since there is no length scale in the continuum mechanics problem. T h e o r y of strain localization Consider a material which is homogeneously strained and sustains a unitbrm Cauchy stress (5. It is assumed that the velocity field v ° within the material has a homogeneous spatial gradient L ° with Cartesian coordinates: av ° L ° _

ij ~x. ]

where v ° denotes the velocity components at the position x i. The rate of deformation D ° and spin tensor W ° are:

(6)

170 o 1 o o D i j = 2 C L i j + L j i)

w O I ( L . O _ L o, U z U Ji)

~7;

Strain starts to localize when a velocity field v different from v° emerges in a planar t e n o n , which is referred to hereafter as shear band. The gradient of the velocity field inside the shear band becomes L while the one outside the shear band remains equal to L °. L and L ° are related by: Lij=L; +ALij

(g)

where ALij is a function of the distance across a planar band which vanishes outside the band. ff shear bands are assumed to be stacks of no-extension planes, as it will be justified later, one must select AL such as: ALij= gi nj

or AL = g n

(9)

nj are the components of the unit vector n normal to the shear band. gi is the component of a function g which depends only on the distance across the band and vanishes outside the band. The rate of deformation D and spin tensor W inside the band are: Dij = Du+ADij

and

Wij = Wu+AWij

(10)

1 •Wij = ~ (ginj-gjni)

( 11 I

where: 1 ADij = ~ (ginj+gjni)

and

If the material behavior is idealized by using a rate-type constitutive equation C and C ° A

A

inside and outside the shear band, the Jaumann stress rate o and o ° inside and outside the shear band are: ^o = C ' D

and

oAo= CO: D O

and by definition the Jaumann stress rates are: A A (~0 o = o + o'W - W'o and o °= + o'W ° - W°'o where

o

(12)

(13)

and ~o are the stress rate outside and inside the shear band, respectively.

The equilibrium in the direction n implies that: n ' ( 6 -do)=o

(14)

Eq. 14 states that the equilibrium of stress rate is enforced only in the direction normal to the shear band but not in the directions parallel to the shear band, which implies that the

171

material may become unstable inside the shear band. By means of Eqs. 12 and 13, Eq. 14 becomes: (n • C. n + A ) - g = n . ( C ° - C ) :D O

(15)

where A has for components:

A..= Ij

1

/

\

[-ni(nkCrkj) + (np~pqnq)Sij + (nk(rki)nj - Oij )

(t6)

In general, C and C ° depend on the direction of D and D °. Hereafter it is assumed that both C and C o can take only two different values related to the direction of D and D °. Rice & Rudnicki [7] distinguish continuous and discontinuous localizations depending whether C= C O or C :~ C °.

Continuous bifurcation In the case of continuous bifurcation, the right-hand side of Eq. 15 is equal to zero: (n .C. n + A)-g = 0

(17)

The condition for the emergence of localization is that a solution other than g=0 exists: det(n.C.n

+ A) =0

(t8)

When Eq. 18 is satisfied, Eq. 17 has an infinity of nontrivial solution g at the condition that D resulting from g is compatible with C inside the shear band.

Discontinuous bifurcation In the case of discontinuous bifurcation, the right-hand side of Eq. 15 is different from zero. Unless Eq.18 is satisfied, the matrix n ' C ' n + A is invertible, and the non-trivial solution is: go = ( n ' C ' n + A ) ' I ' [ n ' ( C - C ° ) : D °]

(19)

In contrast to continuous bifurcation, Eq.15 has only one solution. As in the case of continuous bifurcation, D resulting from gd should be compatible with C. P h y s i c a l i n t e r p r e t a t i o n of AL

The strain localization theory assumes that Eq.9 describes the velocity gradient AL within the shear band. Fig.5 illustrates the assumption of Eq.9 by using three parallel planes which are initially equidistant and normal to n.

! 72

x3~

n

•

..............

/4"...--

/

•

)

"

,N)'

V, c,. .

.

X

.

.

.

.

.

.

.

.

2

Figure 5. Relative motion of three fictitious planes within a shear band during plane strain localization. The scalar product x.n, and therefore A L and g, are constant in each plane. The relative velocity of two neighbor points having dx for relative position is: dv : A L dx : g n dx

(120)

Consequently all the particles contained in a given plane normal to n move at the same velocity, which implies that the planes of Fig.5 remain parallel without deforming, stretching or rotating. The planes are no-extension planes. They are only allowed to slide with respect to one another and to get closer or further apart. The relative velocity, between two planes initially separated by a distance dy is: dv = AL.n dy = g dy

(2ti

It has for component normal to the shear band: -': C")

dv : g.n dy n

and m ~ e s the following angle c.xwith the shear band: tanc~ =

g.n

(23J

5/g.g. (g.n) 2 As shown in Fig.5, the angle c~ coincides with the dilatancy angle ~g that was defined for the Roscoe theory. It is worth pointing out that, although the planes do not rotate, the spin tensor W is generally different from zero, which is easily verified, for example, ~hen shearing a deck of cards.

173 Application

to e l a s t o p l a s t i c i t y

The constitutive matrix of the incremental flow theory of plasticity is: (E:P)(Q:E) C = E - h+Q:E:P

(24)

where E is the elastic matrix, h is the plastic modulus, Q and P are the normalized gradients of yield and plastic potential functions, respectively. Eq.24 applies only when the loading criterion is satisfied: 1

^ Q : a _> 0

(25)

Continuous bifurcation By substituting Eq.24 into Eq. 18, Rice [5] shows that the hardening modulus at continuous bifurcation is: h v 2G c _ 2 n . P - Q - n - ( n - P . n ) ( n . Q . n ) - P : Q - (l-v)- ( n . P . n - t r P ) ( n - Q . n - trQ)

(26)

provide, that the magnitudes of stress components are negligible compared to elastic moduli. ] is the elastic shear modulus and v is Poisson's ratio. By using Eq.25, Rice [5] proves that Eq.26 applies when hc satisfies: h

C

+Q:E:P

>0

(27)

i.e. when hc is larger than its minimum admissible value hmin: h . mm

20

=

.

v l-2v

trP t r Q - P : Q

(28)

According to [7], the solution of Eq. 17 is collinear to: gc =

1 (v trP-n'P-n)n+Zn'P

(29)

where gc has for components normal to the shear band: g n = g c "n =

v trP + (1-2v) n'P'n )

(30)

and parallel to the shear band: gt = 2 ~ n ' P ' P - n

-(n-P'n) 2

Hereafter, gn and gt are referred to as the separation velocity and the shear velocity,, respectively. In contrast to gn, gt is independent of Poisson's ratio.

(31)

174 Discontinuous

bifurcation

It is assumed that the incremental response is elastic ou:side the shear band and elastoplastk: reside the shear band: C o = E

with Q:E:D

(E:P)(Q:E) C = E hd+Q:E:p

° < 0

with Q:E:D

(32)

> 0

The case of elastic unloading in the shear band and elastoplastic loading outside the shear band is ruled out on the ground that it has never been observed experimentally. Rice & Rudnicki [7] established that the condition for discontinuous bifurcations is: hd < he

and

hd + Q : E : P > 0

(33)

where h e is given by Eq.26. When E is isotropic, the solution gd of Eq. 19 is collinear to gc specified in Eq.29:

gd-

Q:E:D ° hd-hc gc

(34)

In contrast to the continuous solution which has an undefined amplitude, gd has a finite amplitude.

STRAIN LOCALIZATION WITH MOHR-COULOMB

MODEL

The theory of strain localization may be applied to the soils that have an elastoplastic behavior of the Mohr-Coulomb type. This application conveniently defines the emergence and inclination of shear bands in terms of a limited number of material parameters. The present analysis assumes that the plane of strain localization is normal to the Xl-X3 plane. As shown in Fig.2, the shear band makes the angle (3 with the major principal stress %. The unit vector n normal to the shear band is: n l = sine

t

n2= 0

(35)

n3= COSe

Elastoplastic Mohr-Coulomb materials In the principal stress space (~I,~3), the equation of the Mohr-Coulomb surface is: f(~p~J3) = GI-°3 - sin¢(c:l+~53 +

c )=0 tan¢

(36)

175 where c and ~ are the apparent cohesion and the tangential angle of friction, respectively. In the case of sands, the apparent cohesion is negligible and the tangential and mobilized friction angles are approximately equal. The unit normal Q to the yield surface is obtained by differentiating Eq.36 with respect to stresses cyl and or3: 1-sinq) Q1-

~2(1 +sin2@)

Q2 = 0

(37) l+sin~b

Q3=

42(l+sin~O)

It is also assumed that the components of Q vanishes in the x2-direction and that Q has a unit amplitude:

(38)

Q:Q=I The unit normal P to the plastic potential surface is defined similarly to Q: Pl-

1-sinH a~]2( 1+sin2~t)

P2 = 0 P3 = -

(39) l+sin H ~2(1 +sin2~)

where H is the tangential angle of dilatancy:

sinv=_

P + P3 = . dSp + dEp 1 P1 - P3 d~P- d~p

(40)

The angle H of Eq.39 coincides with the Roscoe angle of Eq.4 when the elastic strains are negligible with respect to plastic strains. Hereafter, ¢ is assumed to vary from 0° to 90 °, while ~ is assumed to vary from -90° to 90°. H is positive for dilatancy and negative for contractancy. Continuous strain localization

By using Eqs.37 and 39 the following results are derived:

176 trP = -

sinV

trQ

"~/2(l+sin2v/)

N/2(l+sin20)

cos20+sin~/ n'P'n

=

cos20+sin9

-

n "Q

"n

3/2( 1+sin'gr) n'P'Q'n =

sinÙ

~' 2( 1+sin20)

( s i n ~ + s i n 0 ) c o s 2 0 +1 + s i n v s i n o '

2

2 ~ ( l + s i n "q)(l+sin'o) p : Q =

1 + singsinO " 2 9 ~ ( l + s i n ~)(l+sin-O)

After substituting Eq.41 into Eq.26, the plastic modulus becomes a function of v, Q, gr and the strain localization angle 0:

h c (sinN-sin~) 2- (2cos20 -sinv/-sinq~) 2 2G2 ") 8(1-v)'S/(l+sin ~g)(l+sin~0)

(42)

Critical hardening modulus Based on the premises that the plastic modulus h decreases gradually during a loading process, the problem of detecting the emergence of continuous strain localization consists at finding the largest value of h c and the corresponding angle 0 which both satisfy Eq.42. Since the numerator of Eq.42 is a difference of squares, the critical modulus hmax is: h re_max _

(sin~-sin~)

2 (43)

9 8(1-v)'Q(l+sin 2v/)(l+sin-0)

2G

The modulus hmax must be larger than its admissible value hmin given by Eq.28, otherwise unloading takes place inside the shear band and prohibits strain localization: h m l.n 2G

=_

I + sin'q/sin0 1-2v

t , 1~-4)

"4f(l+sin2v/)( l+sin20)

The values of 0 and ~ for which strain localization becomes inadmissible are found by equating hrnax and hmin. Fig.6 illustrates that localization cannot take place for all values of 0 and %

177

Q t-

o

~,. . . . . . . .

\",~\0"5

~, impossibie

~\\

O e-

C

N

04

0.3 £3 0.191

I

0

45

Friction angle

90

0~ (deg)

Figure 6. Region of ~:~-Vplane in which strain localization is prohibited due to elastic unloading (Mohr-Coulomb material). Localization is inadmissible in the lower part when v is equal to 0.5. The inadmissible domain gradually shrinks as Poisson's ratio decreases. When v is smaller than 0.191, localization becomes admissible for all values of 0 and ~/. Figs.7 and 8 show the critical modulus hmax as a function o f ~ and gr for two extreme values of Poisson's ratio (v=0 and 0.5).

0. u'~

01

~" ~) m

~

m

o

03 ' 0

0,01 0.05

/ / 0 . 1 5

/ 45

0.2

90

friction angle tp (deg) h Figure 7. Maximum value of~-~ versus (h and V for Mohr-Coulomb material (v =0).

178

The modulus hrna.x is always positive; strain may [ocalize without strain-softening. hrnax is equal to zero independently from the Poisson's ratio only when 0 and w coincide (associative flow rule). Figs.6, 7 and 8 establish that Poisson's ratio consols the admissible range of O and xg and therefore significantly affects the existence of localizado~. Poisson's ratio has a limited influence on hmax. O

m

I

/o,

I

l

o

//

o.o i"

.I-

i-

0 1

tTO C

8r'x5 0

? o

4'5 friction angle

90 Q (deg)

h Figure 8. Maximum value of 97~ versus 0 and ~ for Mohr-Coulomb material (v =0.5). Critical shear band orientation

By using Eq,42, the orientation of shear band O corresponding to the critical modulus hT,::~_ ~ is: cos20 = i sin(~--~-)cos(-~)

(45)

The angle 0 is independent of Poisson's ratio. By assuming that the angles v/and O have close values, Eq.45 may be approximated as in [911: n: - 5~- - ~ 0 e =~-

i461•

Fig.9 shows the error resulting from the approximation of Eq.45 by Eq.46 when 0 and ~ vary; the error increases with the difference 0--~ but remains inferior to 7 °.

179 O

•.~ t'-

I.~

"0.

O

. "~

m

3.

~-I.

O 03

5.

7.

\

I

0

45

friction angle

90

0 (deg)

Figure 9. Error between approximate (Eq.5) and exact (Eq.45) angles of localization for Mohr-Coulomb material. Velocity inside the shear band at critical localization By using Eqs.30, 31 and 41, the separation or contact velocity is obtained: singt + (1-2v)cos20 gn = -

1-v

(47)

where is given by Eq.45. The angle c~ between the relative velocity and the shear band (see Fig.5) is: tanct = g n = _ sin~ + (1-2v)cos20 gt 2(1-v)Isin201

(48)

A negative value of the angle (x means that the shear band compacts while-a positive value c~ indicates that it expands. Note that a is different from the dilatancy angle ~g defined in Eq.39 due to elastic strain. Fig. 10 shows the difference a - ~g as a function of 0 and ~. By using Fig.7, it is noticed that ct tends towards 'q/as the critical plastic modulus tends to zero and the plastic strain becomes much larger than the elastic strain.

180

O

.I _30 j

j

O'1 t~ ~..~201~ .,¢

•

j

~

.f.0

C

lO

20

t'-

3O 4o

O (33

I

0

45 friction angle

9O O (dog)

Figure 10. Difference cc-y between the angle c~ between relative velocity and shear band (Eq.48) and the angle of dilatancy V for Mohr-Coulomb material (v--0).

S T R A I N L O C A L I Z A T I O N WITH D R U C K E R - P R A G E R M O D E L It is interesting to compare the results obtained with Mohr-Coulomb with those of Dmcker-Prager [6] that accounts for the effect of the intermediate principal stress o 2. S u m m a r y of Rudnicki & Rice results Rudnicki & Rice [6] expressed the flow direction P and yield direction Q in terms of two material parameters I-t and 13: p_

1

-

I +~s

(49) Q_

1

.~

-

l+~s

1 +2

where s is the deviator stress: 1

sij

= O'ij - .~ 8ij Okk

(50)

181

and I is the Kronecker symbol: 5ij =

{01 if i=j if i~j

(51)

and J is the second stress invariant: J = "X/~1 s..s..

(52)

U U

As shown in the stress invariant space of Fig.11, ~. is the slope of the yield surface while characterizes the normal to the plastic potential surface.

P \

~

1 i

~,',

yield $urface

p

Figure 11. Representation of Drucker-Prager parameters [3 and g in stress invariant space. Rudnicki & Rice derived that the critical hardening modulus is: l+v . . . .

l + v L . 13+~2

(53) 2G-

.~/([32 ~ - 2 1t.t~(i7_+1)

where N is the ratio between intermediate principal deviatoric stress s2 and second stress invariant J: s2 N=T

(54)

The angle of localization is: ,,~ ] (l+v~N(1-v) -Nmi n tan 0 . . . . . . . . . I ~ Nmax_(l+v ~)-~-Tg+ N(1-v) where:

(55)

182

N .~J 3N" Nmi n = - 95I- --2--

,56; N ~.Nma x = - 5- +

3Nl- - 2 -

The following constraint is to be satisfied in order to have a shear plane normal to the o,-direction: ,-)

max

Comparison between Druker-Prager and Mohr-Coulomb In order to compare Drucker-Prager results with those of Mohr-Coulomb, the parameters 13and ~ are to be expressed in terms of friction and ditatancy angies. There are several ways to match Drucker-Prager and Mohr-Coulomb surfaces, and consequendy several ways to relate the parameters ~ and 0- C h e n & Saleeb [25] recommend the following relationship between p. and ¢, for plane strain condition: bt = -

-,~sin o

-

(58)

~ 3 + sin20 A similar relation pertains between ]3 and ~. In order to have a positive hardening moduI,m as in the case of Mohr-Coulomb, it is assumed that the inten'nediate principal stress obeys: N = - ~ (13+~)

(591

After assuming Eqs.58 and 59, the critical modulus hma~"becomes: h

mix

l+v 18(l_v) (13-F)2

2G -

~6())

The critical orientation of the shear band is independent of Poisson's ratio: cos20 = -

13+~

(6i

(~3+,u.) 2

1-

12

Fig.12 shows the modulus hmax computed from Eq.60 as a function of N and 9 for Drucker-Prager material. The similarity of Figs. 12 and 7 indicates that Mohr-Coulomb and

183

Drucker-Prager materials have approximately identical modulus h ~ tbr all values of friction and dilatancy angles.

0.01

-'o t.o

~001

/~

¢,..) ,,.-.

o '

0

0.05 -~-"--'-----/ 0.1

~ - -

0.15 0.2

45

90

friction angle

~ (deg)

Figure 12. Critical hardening modulus for Drucker-Prager material (v=0.) Fig. 13 shows the difference between the shear band orientations calculated by using Eqs.5 and 61. I

CJ1 :n

10'. o <

m ,~O. oc-to ~ -1. _.~ ~ -:3 '

0

I"

4 5

3.

5.

7.

9.~

90

friction angle ~ (deg) Figure 13. Error between approximate (Eq.5) and exact (Eq.61) angles of localization for Drucker-Prager material.

18~ As in Fig.8 for Mohr-Coulomb, the difference between approximated and exact shear band orientations remains smaller than 9 o. The localization angles that are predicted by Dmcker-Prager and Mohr Couiomb are similar. When the intermediate stress obeys Eq.59 and the parameters a and [3 are Nven by Eq.58, Mohr-Coulomb and Drucker-Prager give similar shear band orientation and critical hardening modulus. These conclusions still pertain when realistic assumptions, however different from Eq.59, are made on intermediate stress and when the material constants or Dmcker-Prager and Mohr-Coulomb materials are fitted in compression or tension instead of plane strain [25]. Remarks

The previous parametric study examined the influence of the parameters of Mohr-Coulomb and Drucker-Prager models on the orientation 0 of shear bands. Unfortunately, these elastoplastic models specify 0 by using Eq.45, which is approximately equal to Eq.4 of Arthur et al. [24]. Therefore, as shown in Fig.4, they do not account for the orientation of shear bands in "all circumstances. The parametric analysis pertains to many other elastoplastic models since the angles Q and gt also specify the directions normal to the yield and plastic potential surfaces and vary in ranges that encompass any shapes of yield and plastic potential surfaces. Assuming that the iocal dilatancy angle gtt is correctly determined, the theoretical predictions of shear band inclinations may be improved when the yield direction Q is characterized by a local friction angle ~ smaller than the mobilized friction angle q) of Eq.2. A more accurate prediction of @in Fig.4 may be obtained by lowering the straight line of Eq.5 by 4 °, which would imply that 0~is 16° smaller than 0 in the average. The disa~eement between the predicted and measured shear band orientations originates from several sources. It comes from the variability of the dilatancy and mobilized friction angles within the sand specimen. It also reflects problems with the applicability of Mohr-Coulomb and Drucker-Prager models to describe the sand behavior. More complex and more realistic elastoplasfic models predict that the orientation of shear bands depends not only on the mobilized friction angle c) and dilatancy angle ",g but also on other constitutive parameters and state variables [ 14]. Another source of disagreement between the predicted and measured shear band orientations is inherent to the incrementally linear structure of elastoplasticity, which prompts the need for incrementally nonlinear constitutive relations, e.g., vertex-like elastoplasticity 16]. The disagreement also resuhs from

185

the strain localization theory that provides a necessary, but not sufficient, condition for the emergence of shear bands. The theory of strain localization stipulates that strain may localize within elastoplasfic soils when h becomes smaller than hmax, but it does not guarantee that strain localizes when h is equal to hraax. This remark is corroborated by the experimental observations of Chambon & Desrues [26]. Their stereophotogrammetric results revealed that the amplitude of the shear strain increments was randomly distributed within loose sands specimens just before a shear band emerged. They pointed out that additional loading was required to obtain shear bands. In the context of elastoplasticity, their remark implies that the hardening modulus h was smaller than lamax when their shear bands emerged. This delayed strain localization may enhance the disagreement between the experimental observations and theoretical predictions of shear bands inclinations. The effects of this delayed strain-localization need to be further investigated. CONCLUSION The application of the strain localization theory to elastoplastic Mohr-Coulomb and Dmcker-Prager models generalizes the past theories of soil mechanics which predict the orientation of shear bands. Mohr-Coulomb and Drucker-Prager models are found to give similar results, provided that appropriate assumptions are made on the effects of the intermediate stress. After examining the effects of friction angle, dilatancy angle, Poisson's ratio and hardening modulus on the existence and orientation of shear bands, these elastoplastic models are found to predict the same orientation as the empirical relation of Arthur et al. (1977). The comprehensive review concludes that the application of the strain localization theory to the elastoplastic Mohr-Coulomb and Drucker-Prager models does not account for the observed shear band orientation in all circumstances. ACKNOWLEDGEMENT The financialsuppo~ of the National Science Foundationisacknowledged(GrantMSM 8657999).

REFERENCES 1.

Hadamard, J., Lec_ons sur la propagation de~ ond¢~ Ct le~ 4quations de l'hydrodvnamique, Librarie scientifique, A. Hermann, Paris (1903).

186

2.

Thomas, T., Plastic flow and fracture in solids. Academic Press, New York ( 196 !

3.

Hill, J. Accelerations waves in solids, J. M~ch. Phys. Solids 10 (1962) i-16.

4.

Mandel, J., Conditions de stabilit4 et postulat de Drucker, I~:TAM symposium on rheotogy and soil mechanics. Grenobl~ (1964) 58-68.

5.

Rice, J.R., The localization of plastic deformation, PrW. 14th int. congress on theoretical and applied mechanics, Delft, Ed. by W.T. Koiter, Vol. 1 (f976) 207-220.

6.

Rudnicki, J.W. & J.R. Rice, Conditions for the localization of the deformation in pressure-sensitive dilatant material, J. Mech. Phys. Solids 23 (1975) 371-394.

7.

Rice, J.R., & J.W. Rudnicki, A Note on some features of the theory of localization of deformation, [n~. J. Solids S¢uct~res 16 (1980) 597-605.

8.

Vardoulakis, I., Bifurcation analysis of the triaxial test on sand samples, Acta M¢chanica 32 (1979) 35-54.

9.

Vardoulakis, I., Shear band inclination and shear modulus of sand in biaxiaI tests. Int. J. Num. Anal. Meth. Geomcch. 4 (1980) 103-119.

10.

Vardoulakis, I., Bifurcation analysis of the plane rectilinear deformation on dry samples, Int. J. $olid~ Stmcture~ 17 (1981) 1085-I 101.

11. Vardoulakis, I., Stability and bifurcation in geomechanics, Proc. Qf numerical meth0ds in g¢0m¢chanic~, Innsl~ruck (1988) 155-168. 12.

Vardoul~is, I., Shear banding and liquefaction in granular materials on the basis of :, Cosserat continuum theory, Ingenieur Archiv 59:2 (1989)t06-113.

13.

Vermeer, P.A., A simple shear band analysis using compliances, IUTAM conference on deformation and failure of ~anular materials, Delft (1982) 493-499.

14.

Molenkamp, F., Comparison of frictional material models with respect to shear band initiation, Geotechnicme 35:2 (t985) 127-143.

15.

Darve, F., An incrementally nonlinear constitutive law of second order and i~s application to localization, Mechanics of Engineering Materials Ed. by C. Desai and R.Gallagher,John Wiley and Sons Ltd, (1984) 179-196.

16.

Chambon, R. & J. Desrues, Quelques remarques sur 1¢ probl~me de la localisation cn bande de cisaillement, Mech. Res. Comm. l 1 (1984) 145-153.

17.

Desrues, J. & R. Chambon, Shear band analysis for ~anular materials: The question of incremental non-linearity, lngenieur Archiv 59:3. (1989) 187-196.

18.

IVItihlhaus, H.B. & I. Vardoulakis, The thickness of shear bands in granular materials, G¢0t¢chniqu~ ~7 (1987) 271-283.

19.

Bazant, Z.P., Softening instability: Part I, - Localization into a planar band, Joumai of Applied Mechanics, ASME. 55 (1988) 517-522.

20.

Desrues, J., La Localisation de la d4formation dans les milieux granulaires, "N!~,~¢de. ~toctorat ~t'~tat, Grenoble (1984). Hansen, B.. Line ruptures regarded as narrow rupture zones: basic equations based on kinematic considerations, Conf. earth pressure problems, Brussels. Vol. 1 (1958) 39-49. Roscoe, K.H., J.R.F. Arthur & R.G. James, Determination of strains in soils b3,' an X-ray method, Civil Engineering & Public Work5 Review (1963) 873-876 and 1009-1112.

21.

22.

187 23. Roscoe, K.H., The influence of strain in soil mechanics, Geotechnique 20:2 (1970) 129-170. 24. Arthur, J.F.R., T. Dunstan, Q.A.J. Assadi & A. Assadi, Plastic deformation and failure in granular material, Geotechniqu¢ 27 (1977) 53-74. 25.

Chert, W.F. & A.F. Saleeb, Constitutive eauations for engineering material~, John Wiley & Sons, (1982)485-486.

26.

Chambon, R. & J. Desrues, Discussion of Molenkamp's paper entitled Comparison of frictional material models with respect to shear band initiation, O~otechnique 35:2 (185) 133-135.

27.

Iwan, W.D.. On a class of models for the yielding behavior of continuous and composite systems, J. Applied Mechanic~, ASME (1967) 612-617.

28.

Mroz, Z., On the description of anisotropic workhardening, 1. Mech. Phv~ Solid~ 15 (1967) 163-175. LIST OF NOTATION

C

constitutive matrix

D

rate of deformation

D°

rate of deformation outside shear band

E

elastic matrix

g

right eigenvector defining localized velocity field (continuous) particular solution for g

gc gn' gt

components of g normal and tangential to shear band right eigenvector defining localized velocity field (discontinuous)

gd h h

plastic modulus plastic modulus at continuous bifurcation

c

plastic modulus at discontinuous bifurcation

hd h

rn ,-Lx

h . mm Lo zXL n

P,P..

critical value of h

c

minimum admissible value of hc velocity gradient outside shear band localized velocity gradient direction cosines of shear plane normalized gradient of plastic potential function

U

Q' Qij V. 1

w Ct

normalized gradient of yield function velocity components with respect to the principal axes spin tensor angle of strain localization with respect to x 1 axis

188

Vi

dilatancy angle local dilatancy angle

0

friction angle

v

Poisson's ratio

o~

angle between the relative velocity vector and shear band normal

angle between shear band and major principal stress direction principal Cauchy stress components ~J' (Jij' (Ji A 0 Jaumann rate of Cauchy stress e

A:B

trA A.n

Double contracted product between two second order tensors (A:B = AijBij ) or between fourth and second order tensors [C:P]ij= CijklPki trace of A (trA=Aii) product ([A.n]i = Aijn j )

rl-ln

scalar product

(n-m = n i mi)

nm

dyadic product ( [n m]ij= n i mj )

Received 24 July 1990; revised version received 26 October 1990; accepted 3 December 1 990