resende-martin+formulation+of+drucker-prager+cap+model.pdf

FORMULATION OF DRUCKER-PRAGER CAP MODEL

By Luis Resende1 and John B. Martin,2 M. ASCE

ABSTRACT: The Drucker-Prager cap and similar models for the constitutive be-havior of geotechnical materials are widely used in finite element stress anal-ysis. They are multisurface plasticity models, used most frequently with an as-sociated flow rule. The cap may harden or soften, and is coupled to the Drucker-Prager yield surface. As a result of this coupling, plastic deformation in pure shear is possible, after some plastic volume change, for any state of stress on the Drucker-Prager surface. This suggests that for full coupling the constitutive equations for the model can be found consistently; however, the model exhibits unstable behavior under certain conditions. To suppress this instability, some modification of the coupling must be made. Two examples of such modifica-tions which appear in the literature are given; each leads to an inconsistent formulation. Numerical examples are used to illustrate differences and conse-quences arising from the different assumptions.

INTRODUCTION

In recent years a number of mathematical models of the constitutive behavior of granular or frictional materials have been proposed. These models originate in the fields of soil and rock mechanics, and in de-scribing the mechanical behavior of concrete, and involve curve fitting, variable moduli, plastic and viscoplastic theories, fracturing theories and endochronic theory [see Nelson (11)]. In this group of material models, one of the most basic and most commonly implemented models in finite element programs is the plasticity model generally referred to as the Drucker-Prager yield condition with a cap.

The Drucker-Prager model (5) is elastic, perfectly plastic, with a yield surface that depends on hydrostatic pressure (in fact a cone in the prin-cipal stress space) and an associated flow rule. The primary shortcom-ings of the model are that it predicts plastic dilatancy that greatly ex-ceeds what is observed experimentally, and that the behavior in hydrostatic compression is poorly represented. To overcome these de-ficiencies, Drucker, Gibson and Henkel (7) introduced a second yield function which hardens and, in the case of a soil, softens; this is the cap, so called because it closes the cone in the principal stress space. The shape of the cap in the principal stress space can be chosen in var-ious ways; models developed by Sandler, et al. (3,13,14,15) use an el-liptically shaped cap, whereas Bathe, et al. (1) allow only for a plane cap.

The constitutive equations for the cap describe behavior in hydrostatic compression, with hardening occurring when plastic deformation takes place. If, however, the Drucker-Prager cone and the cap are coupled,

'Sr. Research Officer, Nonlinear Struct. Mechanics Research Unit, Univ. of Cape Town, Rondebosch, 7700, South Africa.

2Dean of Engrg., Univ. of Cape Town, Rondebosch, 7700, South Africa. Note.Discussion open until December 1, 1985. To extend the closing date

one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on April 3, 1984. This paper is part of the Journal of Engineering Mechanics, Vol. I l l , No. 7, July, 1985. ASCE, ISSN 0733-9399/85/0007-0855/$01.00. Paper No. 19841.

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through the plastic volume strain, the cap softens when plastic volume strain occurs on the cone. When the cap-cone vertex overtakes the stress point, plastic deformation in pure shear becomes possible. The intro-duction of the cap thus overcomes, to some extent, the principal diffi-culties in the Drucker-Prager model.

Our major concern is behavior when yielding occurs simultaneously on the Drucker-Prager cone and the cap. The yield surfaces are coupled, in the sense that the cap position depends on the total plastic volume strain produced on the Drucker-Prager and cap surfaces, among other parameters. The functional form of the yield surfaces, with full coupling and the assumption of an associated flow rule, is sufficient to permit the complete behavior during simultaneous yielding to be derived. How-ever, full coupling is not assumed in the models of Sandler, et al. (3,13,14,15) and Bathe, et al. (1). This is in order to suppress an insta-bility [in the sense that the stability postulate of Drucker (4,6) is not satisfied] that occurs in certain ranges of behavior. Sandler, et al. (3,13,14,15) chose a limited form of coupling, whereas Bathe, et al. (3,13,14,15) chose a limited form of coupling, whereas Bathe, et al. (1) impose additional assumptions on the plastic strain rate vector. Lade (8,9) and Desai, et al. (2), among others, make modifications of the same type.

A consistent treatment of coupled yield surfaces has been set out by Maier (10). In the following sections we apply this process to a fully coupled model, looking only at the case where both the Drucker-Prager and cap surfaces are active. A particular form of the failure surface and the cap are chosen for this illustration, but the general conclusions are not limited to this choice. Stress rates are written in terms of strain rates for all regimes in the shear strain rate and volume strain rate space. Using this framework, we consider the models of Sandler and Rubin (15) and Bathe, et al. (1) (chosen because full details are given in the re-spective papers) and show that they are not fully consistent, for different reasons.

STRUCTURE OF CONSTITUTIVE EQUATIONS

Plasticity models provide inviscid relations between stress rate &tj and strain rate e,-,. We assume that the total strain rate e,) can be written as the sum of an elastic and a plastic component:

** = *# + *? (1) and that the elastic behavior is isotropic. The elastic equations are writ-ten as

1 1 = * ; el= s^ (2)

in which K and G = bulk and shear moduli , respectively; and e,-, and Sjj = the deviatoric components of e,y and d,-,.

The plastic strain rate ejj is given as the sum of contributions from the associated flow rate for n active yield surfaces:

856


dFa tf = K (3)

dff,y

in which a = 1, 2, . . . , n and the summation rule applies. Fa = yield functions, and X = non-negative multipliers, with

Xa > 0 if Fa = 0 and Fa = 0;

Xa = 0 if Fa = 0 and Fa < 0 or Fa < 0 (4) In the Drucker-Prager cap model the yield surfaces are assumed to de-pend on the first and second invariants of the stress tensor. For our purposes, we shall choose these invariants as the mean hydrostatic ten-sion cr, and an effective shear stress s, where

1 / l vm = - cr t t; s = yl -SijStj (5)

Eq. 3 thus becomes

' dF dcr, dF ds \ d t r m d(Tjj ds d(Jjj/

By standard manipulations, we see that

da, 1 ds 1 ^ = - 8, ; = s, (7)

From these results, it follows that the plastic volume strain rate is

*'*=* (8) dff,

and that the deviatoric plastic strain rate is

^m;s< (8t) Initially, when basic ideas will be developed, it is convenient to sim-

plify these equations. In particular, it is convenient to sketch the yield surfaces and the plastic constitutive relations in a two-dimensional space of the invariants cr, and s. To be able to do this, we must define effective strain rate quantities which are conjugate to the stress invariants. The first of these is simple to define, and is the volume strain rate which we will now denote by kv:

eP = e te (9) The effective shear strain rate we shall define as

1 e = - Sijdij (10)

This definition gives a scalar strain rate of degree zero in the stress com-ponents, and

se = Sijiij (11)

857


We may break e into elastic and plastic components ee, ep without diffi-culty. The shear stress rate s is obtained by differentiating Eq. 5, and is

1 s = SijStj (12)

Using these definitions, we may now cast the constitutive equations in a very simple form. We have

e = ee + e; e = k% + eg (13)

with the elastic relations, from Eq. 2, given by

Tension vertex

TENSION F = CUT- -p r

(+ve)

'A/7, Compression vertex

movable cap

0 (-ve)

FIG. 1.Yield Surface of Plasticity Model

(-ve)

ej (-ve)

FIG. 2.Nonlinear Hardening Rule for Cap Yield Surface

FIG. 3.Stress Space Behavior of Compression and Tension Vertices

859


F2 = -(a, -

e dF1 dF-

Xj - + X2 = Xa + 2R2s\2 (22a) as 3s

e = Xj + X2 = aXi - X2 (22b) do-,,, d 0 and X2 a 0. The total strain rates are

e = - + \ 1 + 2R2s\2; iv = + a\1-\2 (23)

and thus, inverting, we have

s = G(e - Xj - 2R2sX2); 0 and X2 a 0 provides the conditions for loading. We now solve for X] and X2 fom Eq. 24, using Eq. 27; after some arithmetic, we find

\2R2sGK 1 \ l + 2 2 s a /

1 ~ 1 + 2R2sa6 {GK - HG + 2GKasR2) *" '

( a2HK [GK + -a V l + 2 R 2 s a /

X2 = e e (28) (l + 2R 2 sa) (GK - HG + 2GKasR2) '

861


The conditions for loading on both yield surfaces at the vertex are X1 a 0 and X2 s 0 in Eq. 28. The expressions can be simplified with extensive algebraic manipulation (which we shall not give in detail). Thus, the constitutive equations for Ft = 0, F2 = 0 and F3 < 0, and Fj = 0 and F2 = 0 are given by

-aKH H-K(l + 2R2sa)

KH H-K(l + 2R2sa).

for

and

(4:R4s2GKa + 2R2sGK - aKH) . (GK -GH + 2R2saGK)

(GK + 2R2saGK + HKa2) e - ; -' iv > 0

a(GK-GH+2R2saGK)

(29a)

(2%)

(29c)

Note that the stress invariant rates depend only on the volume strain rate, and not on the shear strain rate. Of course, the shear strain rate is not zero; it is given in Eq. 23, and it can be seen that the plastic shear strain rate will be non-negative for the condition X] s 0 and X2 0- If the volume strain rate is zero, the stress rates are both zero, and plastic shear deformation may take place at constant stress. If the total volume strain rate is negative, the stress point moves along the Drucker-Prager line in Fig. 3 to the right, pushing the cap ahead of it. If, on the other hand, the total volume strain rate is positive, the stress point moves along the Drucker-Prager line to the left, pulling the cap behind it. In the latter case the relations are not necessarily stable, since

se + d,e = KHjil - aeve)

H-K(l +2R2sa)' (30)

is not necessarily non-negative when eB > 0 and e > 0 , Before commenting further on these relations, we shall complete the

set of equations for the response at the vertex. First, we treat case 3 where yielding takes place on the Drucker-Prager yield surface only, i.e., fI = 0, F2 = 0, and F3 < 0, and F1 = 0 and F2 < 0. Nonzero plastic strain rates are possible, with

dfi dF1 ep = \ i = k1; eg = Xi - = a \ j

oS ocr, The condition for loading (i.e., Xj s 0) is F] = a&, + s = 0 The toal strain rates are

(31)

(32)

e = H Xi; G

= + aXj K

which give the stress rates as

s = G(e-X1); &, = K(ev - aXj)

(33)

(34) 862


These equations are now substituted into Eq. 32 to give \1: G aK

e + G + a2K G + a2K " (35)

Since it is required that X i > 0, Eq. 35 also gives the condition for load-ing. Thus, on substituting Eq. 35 into Eq. 34 for Xa > 0, we get, for F1 = 0, F2 = 0 and F3 < 0, and for = 0 and F2 < 0:

aGK ' G + a2K

K2a2

G + a2K-

G + a2K aGK

L G + a2K for Ge + aKi > 0 . . .

K (36a)

(36b) (GK + 2R2saGK + HKa2)

and e - eB < 0, (F2 < 0) (36c) a(GK - GH + 2R2saGK)

Secondly, consider case 2, where yielding is associated with the cap, i.e., Fj = 0, F2 = 0 and F3 < 0, and < 0, andF2 = 0.

From Eq. 14b the plastic strain rates are

3F2 , dF2 ds ' dv

with X2 0. The total strain rates are then

e = - + 2R2sk2; eB = G X and thus, on inverting, we have s = G(e-2R2s\2); &, = K(iv + X2) The condition for loading is F, = - 0

K- K + iRis2G-HJ

(42a)

(42b) 863


(+ve)

e | -ve)

(4ab)

FIG. 4.Total Strain Rate Space Behavior of Compression Vertex

and (4R VGICa + 2R2sGK - aKH)

e

e{+ve)

(36c) with R=0 >v

^v Gap

Case'3 fsNs^

D - p \y

/ Casel / Elastic

^ Y (43c) (43b) with R=0 1

(45b) (45a)

D-P & Cap / Case 4 / Gap -*^

^iTCase 2 / Cap

(42b) with R=0

with R=0

^(42c) with R=0

ev(-ve)

FIG. S.Bathe, et al. (1) Modification: Total Strain Rate Space Behavior of Compression Vertex

2R2sG K \ 2 = e n e (44a)

(K + 4 R V G - H) (K + 42? V G - H) The stress rate, strain rate equations then effectively come out to be made up of the terms in Eqs. 36 and 42:

G - -4R4s2G2 -o tGK

G + a2K K + 4 R V G - H G + a2K 2R2sCK

-aGK 2R2sGK G + a2K K + 4Ris2G-H

K-

K + 4 R V G -K2

H K'a2

G + a2K K + 4 R 4 s 2 G - H J

(44b)

It can be seen immediately from these equations that, for e > 0 and e = 0, the stress rates are not zero. A modified flow rule of the von Mises form is then introduced, permitting ev > 0 and e = 0, and there-fore s = cr, = 0 for e > 0 and e = 0.

The additional assumption of a von Mises flow rule is not consistent with the form of the yield function, since it has been seen that no further assumptions are necessary. Eqs. 29a and 44b are substantially different in many respects, and could potentially lead to very different responses. One of the most important aspects is the suppression in Eq. 44b of any possible instability, since the matrix is positive definite.

The conditions which separate the various loading and unloading cases for the vertex behavior can be derived for the model of Bathe, et al. (1) in the same manner as that used for the present model. We have done this for the case of R = 0 (plane cap of Bathe, et al.) and the result is shown diagrammatically in Fig. 5. It can be seen immediately that the loading-unloading conditions for cases 1-3 are identical to the ones de-rived for the present model. However, the loading conditions for case 4 (yielding on both Drucker-Prager and cap yield surfaces) are given by ^ > 0 and \ 2 s 0, where \1 and \ 2 are those given in Eq. 44fl, i.e.

Ge + aKi.v > 0 (45a) 865


(49b)

\ V \ a p

( 3 6 c ) \

C a s e 3

D-p

( 3 6 / H 3 b ,

e (+ve)

- P S C a p t a s e 4

E?>0

Cas

E l a

(47b)

D-P s C a p

C a s e 4

sU

-Kev > 0 (R = 0) (45b) Comparing Eq. 36c to Eq, 45b, and Eq. 42c to Eq. 45a, it is clear that they do not match and that there are two gaps in the total strain rate space. This shows that the model of Bathe, et al. does not provide solutions for certain strain rate paths, due to the additional assumptions about the behavior of the plastic strain rate vector at the vertex, and due to the inconsistent calculation of the plastic multipliers, \x and X2/ in the case when both yield surfaces are active.

Let us now apply the vertex behavior suggested by Sandler and Rubin (15) to the framework of the present work. While Sandler and Rubin use an elliptical cap and a curved failure surface, the essential difference at the vertex is that Eq. 18 is modified by putting

0 if Fr = 0, F2 = 0 and e > 0;

otherwise ef (46) This condition is introduced to prevent the cap from acting as a soft-ening yield surface; it essentially prevents the cap-Drucker-Prager vertex from moving to the left in Fig. 3 with the stress point, and thus is in-

Coefficients for Equations 51 and 67 22 (5) 0

K a

G + a2K

K2

K + 4 R 4 s 2 G - F f

K

K

KH

H - K ( l + 2R2sa)

Conditions (6)

Fi < 0, F2 < 0, F3 < 0 Fi = 0, F2 < 0, F3 < 0, Gi + aKk-, < 0 F, < 0, F2 = 0, F3 < 0, 2R2sGi - Kir < 0 Fi < 0, F2 < 0, F3 = 0, e,. < 0 Fi = 0, F2 < 0, F3 = 0, Gi + o K e , < 0, iv < 0 Fi = 0, F2 = 0, F3 < 0, Gi + aKk,. < 0, 2R2sGe - Kt , < 0 Fi = 0, F2 = 0, F3 = 0, 2R2sGe - Ke r < 0, e,, < 0 Fj = 0, F2 < 0, F3 < 0, Gi + aKkr 2 0 Fi = 0, F2 < 0, F3 = 0, Gi + aKk,. a 0, -ui + e r < 0 Fi = 0, F2 = 0, F3 < 0, Gi + aKk., > 0,

(GK + 2R2saGK + HKo2) i : fc,. < 0

a ( G K - G H + 2R2saGK) Fi < 0, F2 = 0, F3 < 0, 2R2sGe - Kkv == 0 Ft = 0, F2 = 0, F3 < 0, 2R2sGe - Kit. a 0,

(4R4s2GKc< + 2R2sGK - aKH) i + ; r < 0

( G K - G H + 2 R 2 s a G K ) Fi = 0, F2 = 0, F3 = 0, 2R2sGe - Kk-, a 0,

(4R4s2GKa + 2R2sGK - aKH) (GfC- GH+ 2R2saGK)

Fi < 0, F2 < 0, F3 = 0, ,. 2 0 Fi = 0, F2 < 0, F3 = 0, i < 0, e,. =: 0 h = 0, F2 = 0, F3 = 0, c > 0, ,, 2 0 Fi = 0, F2 < 0, F3 = 0, i == 0, -ai + e, > 0 Fi = 0, F2 = 0, F3 = 0, i 2 0, kv < 0

F, = 0, F2 = 0, F3 < 0, (4R4s2GKa + 2R2sGK - aKH)

e + ; e,. =- 0, [GK -GH + 2R2saGK) (GK + 2R2saGK + HKa 2 )

e = 6,. 2 0 a (GJC-GH + 2R2saGK)

F, = 0, F2 = 0, F3 = 0, ,, < 0 (4R4s2GKo + 2R2sGK - aKH)

e + ; 6 , 2 0 ( G K - GH + 2R2saGK)

867


tended to suppress the possible instability apparent in Eq. 30 w h e n e > 0.

Since we are now dealing with a discontinuous evolution equation (Eq. 46), it is necessary to satisfy the constraints on eg in addition to the normal loading-unloading constraints. The conditions that separate the various loading and unloading cases can be derived for this modified model, and are shown in Fig. 6. The conditions for cases 1-3 are iden-tical to those obtained earlier, as would be expected. For case 4, Eq. 29b, Xi a 0 and the condition ipv < 0 bound the range of total strain rates for which evolution Eqs. 18 and 46 give the same results. The volume plastic strain rate constraint can be expressed in terms of total volume strain using Eqs. 22b and 28. Thus

(l + 2R2sa)GK e? = a \ , - X2 = v r . ~ r v V* * " ( 4 7 f l )

GK - HG + 2GKasR which implies that i < 0 (47b) This means that, for Xj and X2 0 and zero or negative total volume strain rates, there is no difference between the Sandier-Rubin modifi-cation and Eq. 29; however, for \1 and X2 > 0 and positive total volume strain rates e > 0, eg = 0), the equations will be different. In this case, using the conditions Fi = F2 = 0, and following the same procedures that were used previously, the constitutive equations are found to be s = or = 0 (48) and the associated values of Xj and X2 are

Ki =

n , 2 x

ten in invariant form as

G - a - 2 1

~12 K - fl22

(51)

in which the coefficients au, a12, a2\ and a^ depend on the current state. The values of the coefficients are given in Table 1. In all cases except one, the coefficient matrix is symmetric, with au = #21 / a n d semipositive definite, in the sense that se + CT,6 > 0. The exceptional case was out-lined in some detail in the foregoing.

CONSTITUTIVE EQUATIONS FOR PLANE PROBLEMS

The constitutive equations for the cap model have been given in in-variant form, which permits a simple two-dimensional representation. For finite element implementation, however, a more general matrix form is required. For completeness, we present a generalization for the case of axisymmetric problems that can be reduced to plane stress and plane strain without difficulty.

The nonzero components of the total stress and strain are written in vector form as

or = (o-n 0-220-12 0-33 )T (52a) = (en e22 7i2633)T (52b) in which yX2 = 2e12 (52c) It is also convenient to identify the deviatoric components of stress and strain, in vector form, as s = (ss22s12)T (53a) e = (ee22e12)T (53b) We make use of the fact that s + s22 + s33 = 0, and e + e22 + e33 = 0 (54) Simple transformations provide us with the total stress and strain com-ponents in terms of the deviatoric stress and strain vectors, the mean hydrostatic stress um and the volume strain e.v:

C22

0-12

.

CT33.

and e Av_

en

e22

in _ e B _

1 1 0 1

S l l

s22

S12

_

c r" i _

= C s (55a)

0

0

- 1 3

- 1 3

l l

e22

7 1 2

Ie33I

= Ce (55b)

869


The deviatoric invariants can be written in terms of s and e. The in-variant shear strain rate is

1 S jj ji

S

in which n

Similarly, it can be shown that 1/2 /., \ 1/2

"2 1 0

1 0~| 2 0 0 2_

1 1 _ s = SijSjj = s ns 2s 2s

(56a)

(56b)

(57 a)

(57b)

The plastic shear strain rate components comprise contributions from yielding on Fi, F2 and F3 (not simultaneously, of course). We can thus put

3Fi 3Fo dp3 \i dF\ \2 dF2 X3 flFq ep = X, + X 2 + \ 3 = J s + J s + J s 3s 3s 3s 2s ds 2s ds 2s ds Substituting for F1, F2 and F3 (Eqs. 16, 17 and 20), this becomes

*-&+**)

(58)

(59a)

(5%) whereas k.% = (a\1 - \2 + A.3) The stress rates are then

s = 2 G ( e - e f ) = 2Ge - 2 G ( + R2\2) s;

d, = K(6B - eg) = KeB - K(a\j - X2 + X3) (60) For any particular state, we substitute the appropriate expressions for Ki, \ 2 and \ 3 , and write Eq. 60 as

s = 2Ge flu . a 12 . se sev s s

(61a)

(616) ffra = Kev - a2\i - a22e Premultiplying Eq. 61a by (s rn/2s), Eqs. 61a and 61b become s = (G - an)e - auev; &, = -a21e + (K - a22)kv (62) which is identical to Eq. 51. Thus, the coefficients an, a12, a21 and a22 can be taken directly from our examination of the invariant equations. This leaves us with the task of transforming Eqs. 61a and 61b into stress and strain rates ir and e.

Substituting from Eq. 56a, Eqs. 61a and 61b can be written in matrix

870


form as

flu T fli 2 G I - - ^ s s T n ^

s s

K-a7

e

e J = D e

LeJ (63)

in which I = a 3 x 3 unit matrix. We note that we may partition the matrices C and C (Eq. 55) and write

I h - h T 1 ; C =

3 h T 1

in which h T = (1 1 0) It is also convenient to define

1 . 1 s = - s; t = - n s = n s .

s s

Using these relationships, we then form

& = C

in which

= C D C e = D*e

D * = I h h T 1

2GI - flnstT a 12 -a7A' K-a 22_

n 1 h 3

h T 1

In multiplying out Eq. 65b, we note several simple relationships:

n _ 1 t = s; - t T h = s r h ; h T n 3

r- - ' - - h r .

3

(64a)

(64b)

(64c)

(65a)

(65b)

(66)

Carrying through the multiplications, and simplifying through the use of Eq. 66, we find that

D* =

In'1 - f l s s T

- ( f l i 2 sh T + a 2 ihs T )

+ ( K - a 2 2 ) h h T 2

G h T + a n h T s s 7 ' 3

G h + flnssTh 3

-fl12 + a 2 i h s T h

+ (K- a22)h

G + K ~ f l ( h T s ) 2 4 (67)

+ fl12hThT - azlsT +(i2 + 2i)hT - a22

. + ( K - a 2 2 ) h r

The matrix D * is thus given explicitly. It reduces to the elasticity matrix

871


when there is no plastic deformation, i.e., when an = al2 = a21 = a22 = 0. Careful inspection will show that D* is symmetric and semipositive definite when a12 = a2l, but D* is not symmetric and not necessarily positive definite when a12 ^ an . The exceptional behavior for loading on the Drucker-Prager surface and the cap simultaneously is thus pre-served in the full set of equations.

The expressions for the plastic multipliers, \ l t \2 and \ 3 , developed for the invariant case, are preserved in the generalized equations. These are necessary for the calculation of the plastic strain rates.

Summarizing, Eqs. 65a and 67, with Table 1, provide the complete constitutive equations for the cap model in the case of axisymmetric problems. The generalization of the invariant case is essentially straight-forward, and the behavior noted in the invariant case is qualitatively preserved.

NUMERICAL ILLUSTRATIONS

The simplest way to illustrate the differences between the model de-veloped in this paper and previously formulated cap models is to com-pare the output of numerical analysis in which the models are incor-porated. The model described in this paper has been implemented in a finite element code for the solution of plane stress, plane strain and axi-symmetric problems, and we have chosen to compare these results with the model described by Bathe, et al. (1), for which we have the software available. The major interest in the comparison is the behavior when both the Drucker-Prager yield surface and the cap are active, and the various possibilities that arise when the stress point moves along the Drucker-Prager yield surface in the direction of increasing hydrostatic tension. There are other differences between the writers' model and the Bathe, et al. model, but these are simply the result of different formu-lations and are of no consequence in the present context.

In order to provide insight into the major differences between the two models, the simple four-element plane strain model shown in Fig. 7(b) was subjected to a variety of loading paths. The stress and strain fields were uniform, and the block was first loaded so that the stress point was forced onto the Drucker-Prager cap vertex. [In Fig. 7(b), E = 100 ksi; v = 0.25; a = 0.05; k = 0.1 ksi; W = -0.066; D = 0.78 ksi-1; and R = 0.] The way in which this was done is not important; significant are the subsequent loading paths, shown in Fig. 7(a). The strain increments are imposed, and fall within the boundaries defined by the various con-straints shown in Fig. 4. The paths imposed are thus categorized as:

1. Path ALoading on both yield surfaces with the cap moving out. 2. Path BLoading on both yield surfaces with the cap stationary. 3. Path CLoading on both yield surfaces with the cap moving in.

This path involves potentially unstable behavior. 4. Path DLoading on the Drucker-Prager yield surface, but unload-

ing from the cap, with e = 0. The cap does not keep up with the stress point.

The results of the various analyses, run both with the writers' model

872


e(+ve)

(-VE)

i I3

3=-

tr 3 x 3 Gauss integ.

Prescribed

Ell

Prescribed

E22

^jfc-&312r~ nfr fffr rm iffr

(b) -5 i n

FIG. 7.Strain Paths Imposed on Model when Stress Point Is at Intersection of Cap and Drucker-Prager Model

and the Bathe, et al. model, are shown in Fig. 8 (plot of s against e), Fig. 9 (plot of cr, against e) and Fig. 10 (plot a, against e). For con-venience, in examining these results, we shall refer to the absolute mag-nitude of

*__

Path A Path B Path C

Path D

I (a) _ l '

.27 .29 .31 .33 .35 .37

33 . 3 5

FIG. 8.Deviator Stress versus Deviator Strain: (a) Writers' Model; (b) Bathe, et al. Model (1)

874

.21

.20

.19

.18

.17

.16

-

(b) i

P a t h A

I 1 I 1

- . 4 2 - . 4 4 - . 4 6 - . 4 8 - . 5 0 - . 5 2

FIG. 9.Hydrostatic Stress versus Volume Strain: (a) Writers' Model; (b) Bathe, et al. Model (1)

875


.175 - . 1 8 0

a m

21

20

19

18

17

16

-

-

(b)

S

1

Path A

Path C Path D

1

1

i

f

t

t I .

.170 - . 1 7 5 - . 1 8 0 - . 1 8 5 - . 1 9 0 - . 1 9 5 e p

FIG. 10.Hydrostatic Stress versus Plastic Volume Strain: (a) Writers' Model; (h) Bathe, et al. Model (1)

876


Loading

Unloading

Reloading

FIG. 11 .Cap Movement and Its Consequences on Loading-Unloading-Reloading Cycle (Path E): (a) Writers' Model; (b) Bathe, et al. Model (1)

ferent. The slopes in the s-e plot and the a-,-e plot are not the same, and there is substantial disagreement in the plot of cr, and e.

Path D is treated by the Bathe, et al. model in the same way as path C, whereas the writers' model predicts separation of the stress point and the cap, with plastic volume strain associated with plastic deformation on the Drucker-Prager yield surface. The differences are again most dis-tinct in Fig. 10.

We see thus that differences occur in the two models when the stress path follows the Drucker-Prager yield surface with s decreasing. This difference has further consequences, which can be seen in the loading, unloading, and reloading path shown in Fig. 11. In the second part of the path [shown as E2 in Fig. 7(a)], the writers' model predicts sepa-ration from the cap, and thus subsequent reloading is made up of linear elastic behavior followed by yielding on the cap and elastic plastic hard-ening behavior. The Bathe, et al. model, on the other hand, gives in-consistent behavior on unloading (i.e., no plastic volume strain accom-panied by cap movement), and yielding on the cap occurs immediately on reloading. These differences are shown in Fig. 12, where s and e are plotted against e.

Note also that while path C falls into a potentially unstable regime, the particular path chosen is such that se + crmev is positive. Whether or not the path is unstable will depend on the previous loading history; for this particular case, unstable behavior occurs only for 0 : e/e s 33,

877


s

16

.14

12

10

08

06

04

02

0

04

08

12

16

V

-

-

-

1 .04

1 .08

1 .12

1 .16

1 .20

^^ ^ Present mode 1 Bathe et al model

El E2

p v

E3 1 1 1

.24 .28 .32 e

^/.. y FIG. 12.Path E Responses

878


Loading Constraint

unstable behaviour region

FIG. 13.Unstable Behavior Region

whereas the domain of potentially unstable behavior is 0 s This is shown diagrammatically in Fig. 13.

v/e

CONCLUSIONS

The fully coupled Drucker-Prager cap model leads to a complete and fully consistent set of constitutive equations, as we have shown. These equations contain a regime within which the stability postulates are not necessarily satisfied, and the suppression of the unstable regime be-comes a matter of interest.

However, it is shown that an arbitrary change in the coupling rules is not acceptable, since such a change can lead to an incomplete for-mulation. This suggests that the question of what coupling rules are per-missible is open: Further work is necessary in order to ascertain whether the goal of a fully stable Drucker-Prager cap model is possible, and un-der just what coupling rules.

ACKNOWLEDGMENT

The support of the Council for Scientific and Industrial Research of South Africa, is acknowledged.

APPENDIX I.SECOND-ORDER WORK

The constitutive model in this paper is formulated in terms of the in-variants, s and

Sij^ij $ ij S,y SijSij A.a Sc (/U^

s,ye,y = Siji'j + Sijilj (69a) se = se' + sep (69b) Using Eqs. 8b, 12 and 14b, we see that

Is ds 2s ' ' 5s Further, from Eqs. 2 and 14a:

1 s>4 = ^ s (71fl)

1 . 1 (S/;SK)(SMSW) whereas see = - s2 = v ; "K ' (71b)

It is evident from simple geometric arguments that

see^Sije\ (72) and thus s'e + &,ev o-^ e,-, (73)

APPENDIX II.REFERENCES

1. Bathe, K. J., Snyder, M. D., Cimento, A. P., and Rolph, W. D., "On Some Current Procedures and Difficulties in Finite Element Analysis of Elasto-Plas-tic Response," Computers and Structures, Vol. 12, 1980, pp. 607-624.

2. Desai, C. S., Phan, H. V., and Sture, S., "Procedure, Selection and Appli-cation of Plasticity Models for a Soil," International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 5, 1981, pp. 295-311.

3. Dimaggio, F. L., and Sandler, I. S., "Material Model for Granular Soils," Journal of the Engineering Mechanics Division, ASCE, Vol. 97, No. EM3, 1971, pp. 935-950.

4. Drucker, D. C , "Some Implications of Work-Hardening and Ideal Plastic-ity," Quarterly of Applied Mathematics, Vol. 7, 1950, pp. 411-418.

5. Drucker, D. C., and Prager, W., "Soil Mechanics and Plastic Analysis or Lim-ite Design," Quarterly of Applied Mathematics, Vol. 10, 1952, pp. 157-175.

6. Drucker, D. C., "On Uniqueness in the Theory of Plasticity," Quarterly of Applied Mathematics, Vol. 14, 1956, pp. 35-42.

7. Drucker, D. C , Gibson, R. E., and Henkel, D. J., "Soil Mechanics and Work-Hardening Theories of Plasticity," Transactions, American Society of Civil En-gineering, Vol. 122, 1957, pp. 338-346.

8. Lade, P. V., "ElastcPlastic Stress-Strain Theory for Cohesionless Soil with Curved Yield Surfaces," International Journal of Solids and Structures, Vol. 13, 1977, pp. 1019-1035.

9. Lade, P. V., "Three-Dimensional Behaviour and Parameter Evaluation for an Elasto-Plastic Soil Model," Proceedings, International Symposium on Numer-ical Models in Geomechanics, R. Dungar, G. N. Pande, and J. A. Studer, eds., A. A. Balkema, 1982.

10. Maier, G., "Linear Flow Laws of Elastoplasticity: A Unified General Ap-proach," Ace. Naz. Lincei, Rendic. Sci. Se. 8, Vol. 67, No. 5, 1969.

11. Nelson, I., "Constitutive Models for Use in Numerical Computations," Pro-ceedings, Conference on Dynamical Methods in Soil and Rock Mechanics, G. Gudehus, ed., A. A., Balkema, 1978.

12. Resende, L., and Martin, J. B., "A Consistent Drucker-Prager Cap Model for Geotechnical Materials," Technical Report No. 15, Nonlinear Structural Me-chanics Research Unit, University of Cape Town, Rondebosch, South Africa, May, 1982.

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13. Sandler, I. S., Dimaggio, F. L., and Baladi, G. Y., "Generalized Cap Model for Geological Materials," Journal of the Geotechnical Engineering Division, ASCE, Vol. 102, No. GT7, 1976, pp. 638-699.

14. Sandler, I. S., and Baron, M. L., "Recent Developments in the Constitutive Modelling of Geological Materials," Proceedings, 3rd International Conference on Numerical Methods in Geomechanics, W. Wittke, ed., A. A., Balkema, 1979.

15. Sandler, I. S., and Rubin, D., "An Algorithm and a Modular Subroutine for the Cap Model," International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 3, 1979, pp. 173-186.

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