A Strain-Hardening Elastoplastic Model For

8/2/2019 A Strain-Hardening Elastoplastic Model For

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M A T H E M A T I C A LC O M P U T E RM O D E L L I N G

PERGAMON M a t h e m a t i c a l a n d C o m p u t e r M o d e l l in g 3 7 ( 2 0 0 3 ) 6 2 3 -6 3 0 www.e l sev i e r .n l / l oca t e /mcm

A S t r a i n -H a r d e n i n g E l a s t o p l a s t i c M o d e l fo rS a n d -S t r u c t u r e I n t e r fa c e u n d e r M o n o t o n i ca n d C y c li c L o a d i n g

M. BOULONUn iversitZ: oseph F ourier de GrenobleBP 53-38041 Gren oble Cedex 9, Fr an ce

marc.boulon(Dhmg.inpg.frV. N. GH IONN A AND G. MORTARA

Fa culty of E ngineerin g, Universit y of Reggio Cala briaVia Gra ziella, Feo di Vito, 89060 Reggio Calabria , Ita ly

@ing.unir c.it

A b s t r a c t - -F r i c t i on a l i n t e r a c t i on b e t w e e n g r a n u l a r m a t e r i a l s a n d s o li d s u r f a c e s i s e n c o u n t e r e di n m a n y g e o t e c h n i c a l a p p l i c a t i on s . S u c h a n i n t e r a c t i on d e v e l o p s t h r o u g h a v e r y t h i n l a y e r o f s o ilc a l l ed i n t e r f a c e . T h i s p a p e r d e a l s w i t h t h e f or m u l a t i o n o f a n o n a s s o c ia t e d e l a s t o p l a s t i c m o d e l f o ri n t e r f a c e b e h a v i o u r u n d e r m o n o t o n i c a n d c y cl i c c on d i t i o n s . T w o p la s t i c s u r f a c e s a r e e m p l o y e d : a ne x t e r n a l s u r f a c e s u b j e ct e d t o i s ot r o p ic h a r d e n i n g a n d a n in t e r n a l on e s u b je c t e d t o a p u r e l y k i n e m a t i cr o t a t i o n a l h a r d e n i n g . F l o w r u l e i s s i m i l a r t o t h a t u s e d i n C a m C l a y m o d e l b u t d i ff e r s if h a r d e n i n g o rs o ft e n i n g c on d i t i o n s a r e i n v o l ve d . @ 2 0 0 3 E l s e v i e r S c ie n c e L t d . A l l r i g h t s r e s e r v e d .

Keywords-I+iction, Soil-structure interface, Plasticity, Cyclic loading.

1 . I N T R O D U C T I O NThe study of the frictional characteristics of the interfaces between granular materials and struc-tures is fundamental for understanding the behaviour of many geotechnical work s. This interac-tion develops inside a thin layer of sand called interface who se behaviour principally depends onsand characteristics and roughness of the inclusion surface. Furtherm ore, this behaviour is verydifferent when static or cyclic conditions are examined.These features can be observed through experimental tests perform ed with the modified shearbox apparatus, that allow s one to perform tests at constant normal load (CN L) and tests atconstant normal stiffness (CN S). In the latter tests the stress acting in the direction normal tothe shear surface is allowed to vary (linearly) with the displacement normal to the interface,according to a constant stiffness co efficient K,

wher e on and u are the stress and the displacement normal to the interface, respectively. In thisway, the volumetric behaviour of the interface defines the sliding resistance of the inclusion in a

0 8 9 5 -7 1 7 7 /0 3 /$ - s e e f r on t m a t t e r @ 2 0 03 E l s e v i e r S ci e n c e L t d . A l l r i g h t s r e s e r v e d . Typ ese t by A@-QXP I I : S O895- 7177( 03) 00056- 6


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decisive manner. The conceptual model that defines the aforementioned is shown in Figure 1 [I!The aim of this paper is to explain the mechanical behaviour of the interface through a nonassociated constitutive elastoplastic two surface model based on isotropic and rotational hardeningThe model is able to predict interface behaviour under static and cyclic conditions, where t lwrole of dilatancy and sand densification explain the increase and decrease of the shear resistam (,of the inclusion, respectively.

s z

grouted body adjacent soil(rough surface)I (idealised by spring)

initial state

axial displace-ment of anchor

t Traction

.amount ofdilation Vr< 5 -a*- = compression

d springstrained state

Figure 1. Restraining effect of the soil on the interface dilation [I].

Boulon and Nova [2] observe the similarities between triaxial and direct shear tests on sandand, as a consequence of these analogies, adapt for the interfaces an elastoplastic model thatwas previously formulated for the triaxial behaviour of sands [3], substituting opportunely thevariables involved in the model. The present work follows this approa ch and is formulated interms of normal and shear stresses and displacements, that is cn, r, u, and w.

2. P LA S T IC FU N C T I O N S A N D H A R D EN I N G LA W SThe isotropic plastic function of the model is based on the definition of the normalized dis-

placement w, defined by the equation [4]wn = J 2, d t , (2 )t

wher e ti, is the ratio between the plastic shear displacement rate tip and the plastic displacementat failure wpr corresponding to the peak value of the stress ratio np,

tipti,=--.W P

From definition (2) it follows that th e locus of the stresses at failure correspond s to the plasticfunction for w, = 1 (Figure 2). The equation of the family of the plastic functions of the modelis

f = 0 - Q1un = 0, (4 )


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S t r a i n - H a r d e n i n g E l s s t o p l a s t i c M o d e l 6 25

experimental data at failure

0 I I I0 20 0 4 0 0 6 0 0

normal stress CL (kPa)F i g u r e 2 . E x p e r i m e n t a l d a t a a t f a i lu r e a n d p l a s t i c fu n c t i on f or W , = 1 ( d a t a f r o mM o r t a r a [ 4 ] ) .

wher e ~1: s a function of the normalized displacement that defines the rule of expansion andcontraction of the plastic function. Function CY epresents the slope of the stress envelopes fordifferent values of wn.

Figure 3 show s the comparison between experimental cr values and function cr(w,) utilized tointerpolate oeXP values. The function a(~,) is described by the equation

4%) = o, [(ww, + 1) - 23 exp { -Lw,M} + oCr (5)where CQ, w, and $ are parameters of the model and L and M are derived according to theconditions

a(w, = 1) = CQ , &(w_= 1) = 0.7lIn (6), or, is the maximum value of the a(~,) function. The extension of the model to cyclic

conditions is obtained by inserting a cyclic surface subjected to kinematic rotational hardening

1 .0 10.8

g 0.6._I!128 0.4

a.np -0.2 ab.) = = =

I I I1 .0 2.0 3.0normalized displacement w.

I4.0

F i g u r e 3 . C om p a r i s on b e t w e e n e x p e r im e n t a l a n d m o d e l c x v a l u e s


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M. BOULON et ~11.

inside the isotropicsurface is described

*n o r m a l s t r e s s 4

Figure 4. Plastic surfaces of the tiodel.

plastic surface. Figure 4 show s the plastic surfaces of the model. Cyclicby equation

f(J = v@ - Q()dn = 0. (7 )Stresses 8, and 7 are the stresses in the solid reference system with the surface (7) rotated byan angle 8 with respect to axis T = 0,

8, =c~cos~+Ts~~I~,7 = -o ,s ine+7.c0 se. (8)

Constant CY O etermines the size of the cyclic surface and is assumed as a constant proportion ofthe maximum amplitude of the isotropic surface

arctan o0 = C, arctan cyp,wher e C, is a paramete r of the model. The angle 0,. shown in Figure 4 is given by

(9)

The hardening modulus for plastic state inside the external surface is given by an interpolationfunction which is formally the same as in the Dafalias and Popov model [5]

c er = arctano - sgn arctan L.cn (10)

6.H=H,+h------ , eTo - 8, (11)where H , is the hardening modulus relative to the isotropic mechanism and 6& c represents thevalue of the angle 6 $, when a load inversion occurs. Function h is given by

h= (12)wher e y, b, n, and Xh are param eters of the model while se is an arbitrary reference displacement.Function (12) includes the dependency of the hardening modulus on normalized displacement,densification of sand, and relative position of the two surfaces. Quantity R a e is

R ae = 2(arctancr - arctanac)e (13)TO


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S t r a i n - H a r d e n i n g E l a s t o p l a e t i c M o d e l 6 2 7

Rae is important in the simulation of tests with unsymm etric cycles and condition Rae # 1follows only if reloading occurs w ithout that th e external surface is activated in the previouscycle.

3. THE PLASTIC POTENTIALThe relation between the components of the plastic incremental displacement vector and the

components of the stress vector is given byagP=j(-_-a f f , ,P=ia9 a71 (14

wher e UP and tip are the plastic rate of the normal and shear displacements and i is the plasticmultiplier. The plastic potential g is obtained through the relation between the stress ratio 77and dilatancy d which are defined, respectively, as

Figure 5 show s the stress-dilatancy relation for an interface test. As one can observe, the flowrule can be considered as bilinear but is not unique as for triaxial tests. Flow rule shown inFigure 5 can be described for hardening and softening conditions, respectively, as

t5;s:0Gix

0.8

0.6

0.4

*00.2 0 *

(16)

I-0.3

0* 0


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S t r a i n - H a r d e n i n g E l a s t o p l a s t i c M o d e l 6 2 9

-80 - ,0 2 4 6 8 0 2 4 6 8

shear displacement (mm) shear displacement ( m m )( a ) T o y ou r a s a n d ( D R = 8 5 % ) -r o u g h a l u m i n i u m ( b ) T o y ou r a s a n d (D R = 8 5 % ) - r o u g h a l u m i n i u mp l a t e , K = 5 0 0 k P a / m m . p l a t e K = 5 0 0 k P a / m m .

160 - 160 -120 - 4/ 1 2 0 -

-8 0 I I x I I I -8 0 I I I I -I0 4 0 8 0 1 2 0 1 6 0 2 0 0 0 4 0 8 0 1 2 0 1 6 0 2 0 0

normal stress (kPa) normal stress (kPa)

Cc) (4F i g u r e 8 . C o m p a r i s o n b e t w e e n c y c li c e x p e r i m e n t a l t e s t s a n d m o d e l p r e d i c t i on s ( d a t af r om Vi t a [ 7 ]) .

the gradient of the plastic potential. Under general conditions, therefore, the gradient v of theplastic potential is given by

dv =

[ 1afo .( > (201z4. MOD EL PREDICTION S

The param eters of the model are derived only from constant normal load stress tests. Theelastic behaviour is assum ed as linear, and the normal and shear elastic stiffnesses are identifiedto be K ; = lOPa/m and K z = O.lKz. Figure 7 shows the comparison between experimentaldata and model predictions for three static C NS tests K = 1 0 0 0 kPa/m m) with different initialnormal s tresses g,-,c. Figure 8 show s instead the comparison between experimental data andmodel predictions for two cyclic CN S tests. Stress path s in Figure 8 evidences the importance ofshear stress degradation for interface behaviour during cyclic loading (first part of the test) anda remaining capacity of dilatancy resulting from monotonic shearing (end of the test).

5. CONCLUSIONSThis paper ha s dealt with the formulation of a sand-structure interface model. The model is

based on isotropic and kinematic hardening and utilizes a flow rule different according to wheth erhardening or softening conditions are involved. The model is able to predict interface behaviour


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6 3 0 M . B O ~ J L O N t a l.

un der both sta tic an d cyclic conditions, an d compa risons with experiment al test s show a goodaccuracy level.

1 .2 .3 .4 .5 .6 .

7 .

REFERENCESE . W e r n i c k , S k i n fr i c t i on o f c y li n d r i c a l a n c h o r s i n n o n - c oh e s i v e s o il s , I n S y m p o s z u m o n S o i l Reinforcing andSta b&sing Techniques in Engineering Practice, S y d n e y , A u s t r a l ia , O c t o b e r 1 6 -1 9 , 1 9 7 8 , p p . 2 0 1 -2 1 9 .M . B o u l o n a n d R . N o v a , M o d e l li n g o f s o i l- s t r u c t u r e i n t e r f a c e b e h a v i o u r , a c om p a r i s o n b e t w e e n e l a s t o p l a s t i ca n d r a t e t y p e l a w s , C o m p u t e r s a n d G e ot e c h n i c s 9 ( l/ 2 ), 2 1 -4 6 , ( 1 9 9 0 ).R . N o v a a n d D . M . W o o d , A c on s t i t u t i v e m o d e l fo r s a n d i n t r i a x i a l c om p r e s s i o n , I n t e r n a t i o n a l J o u r n a l f orN u m e r i c a l a n d A n a l y t i c a l M e t h o d s i n G e o m e c h a n i c s 3 ( 3 ), 2 5 5 -2 7 8 , ( 1 9 79 ) .G . M o r t a r a , A n e l a s t o p l a s t i c m o d e l fo r s a n d - s t r u c t u r e s i n t e r f a c e b e h a v i ou r u n d e r m o n o t o n i c a n d c y cl i cl oa d i n g , P h . D . T h e s i s , D i p a r t i m e n t o d i I n g e g n e r i a S t r u t t u r a l e e G e o t e c n i c a , P o l i t e cn i c o d i T o r i n o , ( 2 0 0 1 ).Y .F . D a f a l i a s a n d E . P . P o p o v , P l a s t i c i n t e r n a l v a r i a b l e s f or m a l i s m o f c y cl i c p l a s t i c it y , A S M E , J o u r n a l o fApplied Mech an ics 43 ( 4 ), 645 - 651 , ( 1976) .M . P a s t o r , O . C . Z i e n k i e w i c x a n d K . H . L e u n g , S i m p l e m o d e l f o r t r a n s i e n t s o il l o a d i n g i n e a r t h q u a k e a n a l -y s i s . I I . N o n - a s s o c ia t i v e m o d e l f or s a n d s , I n t e r n a t i o n a E J o u r n a l for Numerical a n d Analytical Methods inGeom echa n ics 9 ( 5 ), 477 - 498 , ( 1985) .G . P . V i t a , C o m p o r t a m e n t o d e l l e i n t e r f a c c e t r a t e r r e n i s a b b i o s i e d i n c l u s i on i s o li d e in c a m p o d i n a m i c o , D e g r e eT h e s i s , D i p a r t i m e n t o d i M e c c a n i c a e M a t e r i a l i , U n i v e r s i t & d i R e g g i o C a l a b r i a , ( 1 9 98 ) .

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A Strain-Hardening Elastoplastic Model For