A mathematical model for soils

Math/ Comput. Modelling, Vol.14, Printed inGreatBritain

pp. 915-920,

1990

0895-7177/90 s3.c0+o.oo Pergamon Press plc

A MATHEMATICAL MODEL FOR SOILS

By Erman Evgin and Ameir Altaee Department of Civil Engineering University of Ottawa, Ottawa, Ontario, Canada KlN 6N5

Abstract. Two mathematical models developed previously by Lade for the stress, strain, and strength behaviour of soils are combined in this paper. The first model, which has only an open ended conical yield surface, is modified by adding a cap yield surface. With this addition, the predictions for improved along soil behavior in laboratory experiments are the stress paths remaining within the current conical yield surface. The capabilities of the present version of the and model are evaluated by comparing the calculated measured response of soil in proportional loading, isotropic compression, and reloading in a triaxial stress path. The differences between the two versions of the work hardening model with and without a cap are illustrated. Key words. elasticity, model., yielding.

plasticity,

soil,

stress-strain

that zone. In order to overcome this limitation, the cap type yield surface, formulated by Lade (1977) for his second model (Model-II), is attached in this paper to the open end of the conical yield surface of the work hardening model, Model-I (Lade and Duncan, 1975).

INTRODUCTION Lade (1975,1977) has developed two elasto-plastic models for soils. These models are different from each other in their formulation of yield surfaces, flow rules, work hardening rules, and failure criteria. The first model (Model-I) is a work hardening type and uses a conical yield surface as shown in Fig. 1. Model-I accounts for the important most aspects of stress, strain, and strength characteristics of cohesionless soils. For example, it is capable of modelling nonlinearity, shear dilatancy, inelasticity, stress path dependency, and the influence of the intermediate principal stress on yielding and failure. Coincidence of strain increment and stress increment low axes at stress levels, with transition to coincidence of strain increment and stress axes at high stress levels is also modelled. However, Model-I has a number of limitations. One of these limitations is unsatisfactory in modelling for proportional loading. Similarly, all stress paths remaining within the current conical yield the surface, model predicts only recoverable strains which is in contradiction with observed soil behaviour. Plastic strains may take place for some stress paths in

LADE'S WORK HARDENING MODEL: MODEL-I that the strain increIt is assumed can be divided into an ments, dE. ij' P P plastic, dE. elastic, and dcij, ij‘ parts. P + dE.. de.. = dEe (1) ij 13 13 increments are elastic strain The calculated from the generalized Hooke's unloading-reloading law, using the modulus defined by Duncan and Chang (1970). For the calculation of plastic strain increments, a conical yield surface, a flow rule, and an nonassociative work determined experimentally As shown in hardening rule are used. Fig. 1, the yield surface is assumed to 915

Proc. 7th Int. Conf. on Mathematical and Computer Modelling

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be cone shaped, with the apex at the principal the centre of coordinate as a stress space. It is expressed function of the first and third stress invariants. fps = (11)3 / I3

(2)

The parameter fps denotes the stress between 27, at varies level. It and a stress conditions, hydrostatic With increasing value KI at failure. surface fps, the yield values of idenbecoming continuously, expands tical with the failure surface at its outermost position. A nonassociative flow rule is employed to determine the direction of plastic strain increments. The plastic potential function, gps, is expressed as

gps = (I,)3 - K2 I3 where value 02,~~

K2

b = 1 / (fps - ft)ult

(5)

The initial slope of the curve repre(fPS- ft) versus senting the Wps the reciprocal of relation is the II a I, value of The parameter a.

(8)

In the theory of plasticity, the relation between plastic strain increments and the plastic potential function is expressed by Ps

The work-hardening rule adopted for the model is an experimentally determined relation between the plastic work, Wps, and stress level, fps. Some of the test data used in the development of the hardening rule are shown in Fig. 3. These results indicate that the plastic work is very small for the range of fps 27, at a hydrostatic starting with state of stress up to the threshold stress level, ft. In fitting curves to the experimental data, it is assumed that for values of fps between 27 and no plastic strains occur, and no ft' The relation plastic work is done. between Wps and (fps- ft), shown in Fig. 4, is approximated by a hyperbola (Lade and Eq. 5 is used for which Duncan, 1975).

(7)

Because the value of (fPS- ft)ult, determined by a curve-fitting procedure, is always larger than the value of (fps- ft) at fail ure for finite values of wps, a new parameter called rf is introduced to relate the asymptotic value of (fPS- ft) to its value at failure, defined by

dE

series of describes a Equation 3 of surfaces normal to the direction increments. Fig. 2 strain plastic illustrates the shape of the curves of the produced by the intersection with surface the potential plastic triaxial plane.

(fps - f,) = Wps / (a + b Wps)

O3 1 (6) (-) Pa where Pa is the atmospheric pressure and M and 1 are dimensionless numbers. The parameter, b , in Eq. 5 is the ;@pr;;.al of the ultimate value of

(4)

in which A is a material constant.

and

a = MPa

rf = (Kl - ft) / (fPS - ft)ult

constant for a given a,", is calculated from

= A fps + 27 (1-A)

increases with confining pressure, this variation is expressed as

=

XP”

-

agPs

ij

(9)

goi, The dermination of the proportionality constant, Aps, follows the development outlined by Hill (1950):

),ps = dWPs ,

3

gps

(10)

in which the increments in plastic work is given by a dfPs dWPs =

(11) fps - f t )2 (1 - rf K1 - ft

Thus, the elastic and plastic parts of strain increments are fully described. CAP

YIELD

SURFACE

In this study, the cap yield surface of Model-II was chosen without any change in its formulation as the desired cap for the work hardening model. Plastic collapse strains The plastic strain increments related to the cap yield surface are called the plastic collapse strains. The development of the stress-strain relation is as follows. Yield Criterion The cap yield surface used by Lade (1977) is a sphere with the center in

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Con& on Mathematical

origin of the principal space as shown in Fig. 5 and described by the function. the

stress it is

and Computer

Modelling

hp” = dWPc /

2

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gpc

(16)

where fPC

=

(I,)2

+

(12)

I2

2

where I1 and I2 are the first an second If a stress invariant8 respectively. stress increment results in an increase in the value of the yield function fpc, will soil dfpc>O , then the i.e., deformation elasto-plastic undergo This while work hardening takes place. type of yielding does not result in The type of yielding eventual failure. which causes the conical yield surface for soil is responsible to expend failure. Flow Rule An associative flow rule is used for collapse of plastic the calculation plastic increments. The strain potential function is identical to the yield function and it is expressed as:

gPC

=

(II)2

I2

+ 2

(13)

plastic the relation between The and increments the strain collapse plastic potential function is given as:

PC

de. ,

11

=

xpc

agpc (14)

au ij

xpc

is the proportionality and its value is determined work hardening rule.

constant from the

Work Hardeninq Rule The work hardening rule required for the calculation of the magnitude of plastic strain increments is expressed a5 an experimentally determined relation between the total plastic work. Wpc , due to collapse strain and the value of the yield function, fpc . Wpc =Cp(

fpc -) Pa2

p (15)

where C and p are constants which can be determined form the (WPc/Pa) versus scale as in log-log (fpc/Pa2 1 plot shown in Fig. 6. It is assumed that the work hardening relation is independent of the stress path. The value of A'" can be determined the following expression.

from

fpc pa2 (l-P)a( awpC = c Pa p (-1 2) Pa fpc derivation The Lade (1977).

of Eq.

(17)

16 is given by

By substituting Eq. 16 into Eq. 14 and calculating the partial derivatives of the plastic potential function, the stress-strain relation for the plastic collapse strains is obtained. NEW VERSION OF LADE'S MODELS A new version of Lade's models, which is presented in this paper, makes use of the equations related to the plastic strains in Model-I and the collapse strains in Model-II. The conical yield surface is used in the new version the same way as described above in the section entitled "LADE'S WORK HARDENING MODEL: MODEL-I". When the stress increments stress levelcaypsJanI Einc?a:z u%d % . calculate the plastic strain increments. However, the conditions under which the plastic collapse strains are calculated in the present development are not the same as those given by Lade (1977). In Model-II, plastic strain increments are calculated a5 the 5um of plastic collapse and plastic expansive parts (Plastic expansive strain5 are related to the expansion of the cone yield surface). It is assumed that each part can be calculated separately and then Any stress change causing superposed. the yield cap to expand produces collapse plastic strain increments without any requirement on where the conical yield surface is located during this stress increment. For example, the stress paths from A to B and from P to D of Fig. 7 and 8 respectively, require that the plastic collapse strains be calculated and superposed on the other parts of strain. The significance of the stress path from A to B is that it causes only the cap yield surface to expand while the conical yield surface On the stays where it was before. other hand, the stress path from P to D of both yield results in expansion surfaces. In the new version of Lade's Models, it is that assumed Model-I correctly simulates the soil behaviour when the conical yield surface (Eq. 2) expands, and therefore, the plastic collapse strains are already accounted for by Eq. 9. For this reason, the plastic collapse strains are not calculated for the stress path5 similar to the path (P to D) shown in Fig. 8. Plastic collapse

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when the strains are calculated only and surface expands the cap yield surface remains yield conical For example, the stress stationary. increment shown in Fig. 7 causes only the cap yield surface to expand, and therefore, the plastic collapse strains are calculated using Eq. 14. Predictions

of New Version

Predictions have been made for the soil different stress behaviour in three order show the to paths. In improvements that are achieved by the development, the predictions present made by the new version and Model-I are compared. to the first case is related The Sand Ottawa under behaviour of loading conditions. The hydrostatic stress-strain experimental results and curves as predicted by the new version and Model-I are plotted in Fig. 9. In the second case, predictions were behaviour of loose for the made Monterey No. 0 Sand subjected to a proportional loading. Figure 10 gives the comparisons. In the third case, the behaviour of the Monterey No. 0 sand was predicted for a load increment along the stress path shown in Fig. 11. The sand sample was a loading and originally subjected to unloading cycle before it was reloaded deviator increasing with an again stress while the confining pressure was kept constant. Improvement was achieved the predictions of only in the volumetric strains.

DISCUSSION The second soil model of Lade, Modelfor some additional accounts II, features of soil behaviour such as the strain softening and the curvature of envelope. Further, it the failure differs entirely from the first model in its formulation, and appears to be more elaborate than the new version. These additional features may well be very important for the analysis of some However, not in engineering problems. every geotechnical engineering project the strain softening and curvature of are essential failure envelope the Usually, requirements in the analysis. brings its own better model a For example, Model-II complications. requires additional parameters and the convergence becomes more difficult in element With analysis. the finite in mind, the these considerations of Lade has been model first complemented with a cap yield surface taken from his second model in the present study.

CONCLUSIONS Two models of Lade are combined in this With paper. this modification, the work hardening model can be used to calculate plastic strains produced by the stress increments remaining within the cone yield surface. The following conclusions are drawn. 1. Soil behaviour in any proportional loading condition, including the isotropic compression, can now be modelled more accurately as compared to Model-I. 2. Plastic strains developed in the soil, before the treshold stress level is reached, can now be modelled using the new version of Lade's Models. 3. Plastic strain increments a0 not have to be divided into two types of plastic Strains for all stress paths. Because of this reason, the finite element implementation of the new version of Lade's Models will be simpler to than use some other plasticity models with two yield surfaces. REFERENCES Duncan, J.M., and Chang, C-Y. (1970). Nonlinear analysis of stress and strain in soils. ASCE Journal of the Soil Mechanics and Foundations Division, 96 (SM 51, 1629-1653. Evgin, E. and Eisenstein, 2. (1980). Re-evaluation of work hardening model. Proceedings, ASCE Symposium on Limit Equilibrium, Plasticity and Generalized Stress-Strain Applications in Geotechnical Engineering, Edited by R.N. Yong and E.T. Selig., 226-239. Hill, R. The (1950). Mathematical Theory of Plasticity. Oxford Univ. Press, London. Lade, P.V. (1977). Elasto-plastic stress-strain theory for cohesionless solid with curved yield surfaces. Int. J. Solids and Structures. 13:1019-1035. Lade, P.V., and Duncan, J.M. (1975), Stress-path dependent behavior of soil. cohesionless Journal of Geotechnical Engrn. Div., ASCE, vol. 101, GTl, 51-68.

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Proc. 7th Int. Conf: on Mathematical and Computer Modelling

Asymptote _--_------_-------r@ &

YIELD

SUEZFACES

m ~E _------------

Threshold value

O3

= constant

Total plastic Fig.

Fig.

Yield and Model-I.

1

failure

surfaces

4

work

Mathematical representation of work versus stress plastic level.

of conical yield surface

*w” 2

"plastic

collapse strain

increment

vectors

b"

Fig.

Fig.

2

Trace surface

of plastic on triaxial

100 -.

2?

,’

80--

,” II

0

o

0

cap

yield

surfaces.

0

6

8

10

12

WPS, kPa 3

and

u3 = 60 kPa

so-,

4

Fig.

Conical

potential plane.

0

0

o

0

5

Plastic work level (Reproduced Duncan, 1975).

versus from

fpC/ Pa2 stress Lade and

Fig.

6

Plastic collapse work stress level (Replotted Lade, 1977).

versus from

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and Computer

Mode&g

cap yield surface

stress increment

b-

Fig. I

Stress increment which plastic collapse strain.

causes Fig. 10

Predictions for the behaviour of loose Monterey No. 0 Sand in proportional loading.

Fig. 11

Predictions for the behaviour of loose Monterey No. 0 Sand triaxial along in reloading stress path.

D stress increment

b-

Fig. 8

Stress increment which does not cause plastic collapse strain.

Volumetricstrain, TV (X)

Fig. 9

Predictions for the behaviour of Ottawa sand in hydrostatic compression test.