A smooth yield surface consistent with triaxial test data

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/84 $3.00 + .00

Vol. 11(4),281-284, 1984. Printed in the U S A Copyright (c) 1984 Pergamon Press Ltd.

A SMOOTH YIELD SURFACE CONSISTENT WITH TRIAXIAL TEST DATA

R. M. Haythornthwaite Department of Civil Engineering, Temple University, Philadelphia

(Received 26 June 1984; accepted for p u n t 26 June 1984)

Introduction

A recent formulation of the Coulomb yield criterion expressed in terms of invariants [i] is modified to provide a criterion which is non-singular. The resulting convex yield surface possesses a continuously turning tangent, It can be scaled to pass through data points from both triaxial compression and triaxial extension tests whenever these results are consistent with the Coulomb theory of internal friction and, being smooth, is a better fit to data for both sands and clays in other complex stress states. It might prove a useful alternative to the smooth yield surface suggested by Lade and Duncan which cannot be fitted to triaxial test data consistent with the Coulomb theory.

[2]

The Coulomb Yield Criterion

Coulomb based his frictional theory of soils on the concept that failure would occur on a plane where the difference between the shear stress and the material cohesion reached a certain fraction of the normal stress. Unlike some earlier authors and many subsequent ones, he made no attempt to consider the possible effects of lateral stresses and the actual packing of the particles [3]. This theory has been successful in reconciling test data from triaxial compression and triaxial extension tests [4], but less so in reconciling triaxial tests with plane strain tests [4] and with tests in which the intermediate principal stress is controlled [5]. Plotted in the deviatoric plane, the test data trace surfaces which are smoothly curved and lacking corners, in 281

282

R.M.

HAYTHORNTHWAITE

contrast with the piecewise-linear shape predicted by the generalized Coulomb theory

[6], suggesting that the intermediate principal stress may well have

some influence on failure except when it is equal to one of the other principal stresses.

It is the purpose of this paper to present a yield

criterion that models the observed behavior by retaining the triaxial test points consistent with the Coulomb theory and passing a smooth surface through them. In recent work

~],

it was shown that the admissible zone associated with

the Coulomb yield criterion for cohesionless materials can be expressed in terms of the stress invariants and the angle of internal friction,

~, of the

soil as follows: (cosec~)I3-(l+8sec2~)13-1112~O Ii=o1+O2+O3;

I2=-(O102+~2o3+a3~i);

I3=o1o203

(I)

~2 3 I~-lSI I I +I212-413I 13=412-27 1 2 3 1 2 1 3 The invariant I3, first introduced in [7], can also be written I3=8TIT2T3 where ~i=[o2-o31/2,

(2)

etc., so I3 is the product of the three Mohr circle

diameters associated with the stress state.

Non-Sinsular Yield Criteria

The invariant i 3 is zero whenever two principal stresses are equal, as is the case in the triaxial test[8], so suppressing the first term in Eq. i defines a different yield surface, but one consistent with the same triaxial test data: 13

= constant

IiI2 Eq. 3 defines a closed, convex surface for all values of ~, but it also defines outer branches, as illustrated in Fig. i. For the purposes of

(3)

SMOOTH YIELD SURFACE

283

plasticity theory, an admissible zone may be defined by noting that it also must not lie outside a right circular cone passed through the compressive triaxial test points, thus excluding the three outer branches. The admissible zone is then defined by the pair of inequalities: -(I+8sec2~)13-II12~0

FIG. i Intersection of the yield surface, Eq. 3, with the devlatoric plane. Compression axes shown. @~30 o

(4)

(3+sln¢)(l-sln¢)l~ +(3-sin~)212~O

Expressed in polar co-ordinates p,8 in the deviatoric plane, Eq. 4 becomes: 2sln~(3+sin@)o3sin38+3(3+sln2@)O2-(3-sin~)2~O

(5) p-l~O Here p is a non-dimensional radius equal to unity in the compressive triaxial direction and e is measured anti-clockwise from the horizontal direction lying between axes 1 and 2 in Fig. I. Evidently, the first of Eq. 5, a cubic in p possesses either one or two positive real roots, depending on the value of 8. The central branch of the yield surface is shown in greater detail in Fig. 2, where also the smooth yield surface proposed by Lade and Duncan [2]: 13

- constant

(6)

is shown as the dotted line. The latter has been passed through the compressive triaxial data points and it should be noted that it does not then pass through the corresponding extensional triaxial data points consistent with the Coulomb theory. The effective angle of internal friction in the

284

R. M. HAYTHORNTHWAITE

2 triaxial extension test is: 7sin~-9+3A ~'=arcsin ll_5sin~_A (7) A=((9-7sin~)(l+sin~)) ½ For example,

4'=33.63 ° when 4=30 °, as

in Fig. 2. Eq. 6 possesses outer branches, like Eq. 3, a circumstance

that does

3

1

not appear to have been noted previously,

so that it cannot of

itself define an admissible zone as required in the theory of plasticity. The deficiency

is not a major one,

FIG. 2 Deviatoric plane, showing test points consistent with Coulomb theory ( ~ ); Eq. 3 ( .... ); Eq. 6 ( ... ). Compression axes shown. 4=30o

however, because it may be remedied in a fashion which parallels the procedure use~ in the case of Eq. 3.

References

i. 2. 3. 4. 5. 6. 7. 8.

R.M. Haythornthwaite, J. Engrg. Mech. 109, 1016 (1983) P.V. Lade and J.M. Duncan, Proc. ASCE SM99, 793 (1973) J. Heyman, Coulomb's Memoire on Statics, University Press, Cambridge D.H. Cornforth, Geotechnique 14, 143 (1964) H.Y. Ko and R.F. Scott, Proc. ASCE SM94, 883 (1968) R.T. Shield, J. Mech. Phys. Solids ~, i0 (1955) R.M. Haythornthwaite, J. Appl. Mech. 46, 701 (1979) R.M. Haythornthwaite, Proc. ASCE SM86, 35 (1960)

(1972)