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AlogarithmicstressstrainfunctionforrocksandsoilsARTICLEinGOTECHNIQUEJANUARY1996ImpactFactor:1.67DOI:10.1680/geot.1996.46.1.157

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TECHNICAL NOTE

A logarithmic stressstrain function for rocks and soils

A. M. PUZRIN and J. B. BURLAND

KEYWORDS: constitutive relations; laboratory tests;numerical modelling and analysis; stiffness.

INTRODUCTION

Numerical methods of solution of many boundaryvalue problems in rock and soil mechanics requirean analytical simulation of the relevant stressstrainrelationship through the entire range of strains,from zero to strains at and beyond peak strength. Inproblems of settlement and displacement of struc-tures, the contribution of zones with very smallstrains to boundary displacements can be largerthan that of zones of contained failure (see, forexample, Burland (1989)). Therefore, an accuratesimulation is required for both small strain andclose-to-failure regions. A solution is usuallyimplemented by means of a non-linear elasticformulation, but other approaches may be used.This technical note is concerned not with theseprocedures but with the simplest possible mathe-matical representation of the experimental stressstrain curve.

A vast amount of data on small strain behaviourof soils and rocks has accumulated since theintroduction of local strain-measuring techniques(strain gauges, electrolevels, proximity and localdeformation transducers). However, in most of thestandard commercial tests, the resolution of strainmeasurements is not sufciently high to givereliable data in the small strain region. Therefore,the problem of stressstrain behaviour simulationby an analytical function should be treateddifferently in the following two cases

(a) case 1: small strain data are not availablethe function should give acceptable accuracyover the whole strain range using a minimumnumber of parameters available from thestandard test

(b) case 2: small strain data are availablethe

function should be versatile enough to t thesedata, not sacricing accuracy at large strains.

The key requirements for a stressstrain func-tion are

(a) the lowest possible number of constantsconsistent with accuracy

(b) that the constants should have physical mean-ing

(c) that the constants should be easy to derive.

GENERAL CONDITIONS FOR NORMALIZED

STRESSSTRAIN CURVE

Most of the published analytical stressstraincurves were generated to t normalized experi-mental curves from different types of tests(triaxial, simple shear, plane strain and so on).Hardin & Drnevich (1972) proposed the followingexpressions for normalized stresses and strains

y q=qux =rr qu=Emax

where q is a stress, is a strain, qu is the ultimate(peak) stress, r is a reference strain and Emax is theinitial stiffness modulus. Then expressions fornormalized secant and tangent stiffness are

Es=Emax y=xEt=Emax dy=dxGrifths & Prevost (1990) formulated ve

general conditions that a normalized stressstraincurve should satisfy (Fig. 1).

At the strain origin x = 0

condition 1: y 0condition 2: dy=dx 1At the ultimate strain (corresponding to the

ultimate stress) x = xu

condition 3: y 1condition 4a: dy=dx 0

In some circumstances it has proved convenient togeneralize conditions 3 and 4 such that at a limitingstress qL the normalized stressstrain curve has a

Puzrin, A. M. & Burland, J. B. (1996). Geotechnique 46, No. 1, 157164

Article number = 625

157

Manuscript received 12 December 1994; revised manu-script accepted 6 June 1995.Discussion on this technical note closes 3 June 1996; forfurther details see p. ii. Imperial College of Science, Technology and Medicine,London.

positive slope. Replacing qu by qL and xu by thenormalized limiting strain xL, condition 3 remainsunchanged while condition 4a is modied inaccordance with Fig. 1

condition 4b: dy=dx tan (1 c)=xLIn the particular case when the limiting stress ischosen to be equal to the ultimate (peak) stress, = 0 (c = 1) and conditions 4a and 4b are identical.

Inside the strain interval 0 , x , xL

condition 5: dy=dx > 0

d2y=dx2 < 0

To improve the accuracy of the prediction at smallstrains, additional conditions should be imposed onthe curve at the linear elastic strain limit x = xe.Seven tests on different soils and rocks under

various types of loading (Table 1) are used toillustrate these additional conditions. All the testsused local deformation measurement techniques.The rst four tests are reproduced from Tatsuoka &Shibuya (1991).

Figure 2 shows the normalized secant modulusof these soils and rocks plotted against x at smallstrains. It is seen that, for the soils, the secantmodulus at x = xe decreases very rapidly, so that aninnite derivative of secant modulus can beassumed

condition 6a:d(y=x)

dx 1

For rocks, the rate of decrease of the secantmodulus at x = xe is very low, so that the conditionof zero derivative at x = xe is assumed

condition 6b:d(y=x)

dx 0

APPRAISAL OF EXISTING CURVE-FITTING

FUNCTIONS

Seven widely used stressstrain functions areexamined in the light of the above conditions(Table 2). The number at the end of a function'sabbreviation indicates the number of parameters itrequires in normalized stressstrain space. H0indicates the original hyperbolic model, H1 its

x = xlimx = 100

c

1

Y

Fig. 1. Normalized stressstrain curve

0 0.2 0.4 0.6 0.8 1

x

0

0.2

0.4

0.6

0.8

1

Kimachi sandstoneOya tuffShizuoka mudstone

Toyora sandLondon clayPentre siltHPF4

E s/E

max

Fig. 2. Small strain secant modulus for rocks and soils

158 PUZRIN AND BURLAND

Table 1. Description of tests

Material tested

Kimachi Oya tuff Shizuoka Toyora Articial London Pentresandstone mudstone sand silt HPF4 clay silt

Testingconditions

Uniaxialcompression

Uniaxialcompression

Uniaxialcompression

Drained planestraincompression

Undrained HCrotation of 19, 39directions

Undrained triaxialcompression

Undrained RC + HCsimple shear

q Axial stress Axial stress Axial stress Stress deviator Shear stress inhorizontal plane

Differencebetween currentand initialstress deviators

Shear stress inhorizontal plane

Axial strain Axial strain Axial strain Strain deviator Shear strain Axial strain Shear strain

E Young's modulus Young's modulus Young's modulus Shear modulus Shear modulus UndrainedYoung's modulus

Shear modulus

qu: kPa 27 600 1440 140 168 78 372 87

Emax: MPa 6580 5216 707 672 1476 620 7525

xe 025 02 01 0008 0006 0016 00003

xu 191 15 19 535 84 39 47

Reference Noma & Ishii(1986)

Noma, Waku,Kadota &Murakami(1987)

Kim, Ochi &Tatsuoka (1990)

Tatsuoka &Shibuya (1991)

Zdravkovic (1994) Standing (1994) Porovic (1994)

HC hollow cylinder; RC resonant column.

LO

GA

RIT

HM

ICS

TR

ES

SS

TR

AIN

FU

NC

TIO

N1

59

modication with a correction coefcient forstrength, and H2 with correction coefcients forboth strength and initial stiffness (Kondner, 1963).HD2, GP1 and MH5 are modied hyperbolicmodels proposed by Hardin & Drnevich (1972),Grifths & Prevost (1990) and Tatsuoka & Shibuya(1991) respectively. J4 is the model proposed byJardine, Potts, Fourie & Burland (1986). For easeof analysis, GP1 and J4 have been transformedinto the conventional normalized form. It is seenthat none of the curves satises all the conditionsspecied above for soils and rocks.

A NEW STRESSSTRAIN CURVE FOR CASE 1 (L1)

As no data for the small strain region areavailable in case 1, the linear elastic region has tobe neglected (xe = 0). The value of Emax (fornormalization) may be derived from dynamic tests.An alternative approach is to estimate Emax usingexisting knowledge about the inuence of variousfactors (mean consolidation stress, porosity, OCR,and so on) on initial stiffness (e.g. Tatsuoka &Shibuya, 1991; Jardine, 1994).

The following normalized relationship (L1) isproposed

y x x[ln (1 x)]R (1)Equations for the normalized secant and tangentmoduli are obtained from equation (1)

Es

Emax 1 [ln (1 x)]R (2a)

Et

Emax 1 [ln (1 x)]R

Rx(1 x) [ln (1 x)]

R1 (2b)

The coefcients R and are dened fromconditions 3 and 4b

R c(1 xL) ln (1 xL)xL(xL 1) (3)

xL 1xL[ln (1 xL)]R

(4)

The normalized limiting strain is never less thanunity, therefore both coefcients are positive.Equation (1) requires only one parametereitherc for the chosen value of xL or xL = xu for c = 1 atthe peak strength. For xL = 1, curve L1 simulateslinear elastic behaviour. It can be shown thatEquation (1) satises conditions 15 for allxL . 1. Equation (1) also satises conditions 6aand 6b for values of limiting strain xL greater thanand less than the `critical' value xLc respectively.This critical value of limiting strain xLc is found bysolving equation (3) for R = 1 (e.g. for c = 1,xuc = 281449). Figure 3 indicates a family of curvesL1 for c = 1 (limiting stress equal to the peakstrength) and different values of xu.

For xu , xuc (typical for intact rocks, reachingtheir peak strength at relatively small strains),R . 1 and condition 6b is satised. In Fig. 4 theresults of the tests on rock samples are comparedwith model L1. The accuracy of simulation isremarkable considering that only one parameter isinvolved. The fact that condition 6b is satisedresults in a very slow change in stiffness, thereforetaking xe = 0 does not inuence the accuracy atsmall strains. The L1 relationship also enables apartial simulation of post-peak behaviour.

For xu . xuc (typical for soils), R , 1 and

Table 2. Conditions satised by various stressstrainmodels

Model Conditions

1 2 3 4 5 6a 6b

H0 + + 2 2 + 2 2H1 + + 2 2 + 2 2H2 + 2 + 2 + 2 2HD2 + + 2 2 + 2 2GP1 + + + + + + 2MH5 + + + 2 + 2 2J4 + + 2 2 + 2 +L1 + + + + + + +L4 + + + + + + +

Condition not satised at x = 0, but only at the linearelastic strain limit x = xe.

xu = 1

xu = 1.5

xu = 2

xu = xuc

xu = 5

xu = 15

xu = 45

E s/E

max

x0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Fig. 3. Family of L1 curves

160 PUZRIN AND BURLAND

condition 6a is satised. The results of thesimulation of the test on Toyora sand are shownin Figs 5(a) and 5(b) for the entire strain(0 , x , xu) and small strain (0 , x , 3) rangesrespectively. The accuracy of simulation of this testis comparable with that provided by the sevenfunctions already listed. The coefcients for GP1and J4 were calculated using procedures given inthe corresponding papers, whereas the coefcientsfor the rest of the models were taken fromTatsuoka & Shibuya (1991). As a quantitativecriterion for accuracy, the following parameter(increasing with increasing accuracy of prediction)was chosen

a

1

n 1Xni1

(yip yie)2

1=2(5)

where yip is predicted normalized stress at x = xi,

yie is experimental normalized stress at x = xi, and

n is the total number of points compared.It can be seen from Fig. 6(a) that, for the entire

strain range, only MH5 provides higher accuracythan L1. For small strains (x , 3), it can be seen

from Fig. 6(b) that the curve L1 (using a singlelarge strain parameter) is bettered only by func-tions J4 and MH5, which use parameters derivedfrom the small strain data.

A NEW STRESSSTRAIN FUNCTION FOR CASE 2

(L4)

When small strain data are available, the strainrange beyond the linear elastic limit is treated astwo intervals (for smaller and larger strains)separated by the strain x1 (where x1 = xe + e 2 1;e = 2718281828 . . .). The stressstrain curve (L4)is then represented by two functions that haveequal values and rst derivatives (i.e. continuousnormalized stress, secant and tangent moduli) atx = x1.

In the smaller strain region xe , x , x1

y xe x9 sx9[ln (1 x9)]Rs (6)where x9 = x 2 xe. This function requires threeparameters, which have to be derived from thenormalized experimental stressstrain curve atsmall strains (Fig. 7): xe is a linear elastic strainlimit, y1 is a normalized stress at x1 = xe + e 2 1,and ys is a normalized stress at an arbitrary valuex = xs inside the small strain interval. It has beenfound that setting xs = 05 usually gives a good tfor small strains.

Test

L1

(c)x

0 0.5 1 1.5 2 2.50

0.5

1

0

0.5

1

0

0.5

1

0 0.4 0.8 1.2 1.6(b)

(a)0 1 2 3

y

Fig. 4. Uniaxial compression of: (a) Kimachi sand-stone; (b) Oya tuff; (c) Shizuoka mudstone

1

0.5

0

1

0.5

0

0 20 40 60

10 2 3

(a)

(b)x

yy

L1

L3

Test

Fig. 5. Plane strain compression of Toyora sand: (a)large strains; (b) small strains

LOGARITHMIC STRESSSTRAIN FUNCTION 161

Then coefcients s and Rs are denedfrom the conditions that the function shouldpass through the points (x1, y1) and (xs, ys)respectively

s x1 y1e 1 (7)

Rs ln (xs ys) ln [s(xs xe)]ln [ln (1 xs xe)] (8)

In the larger strain region x1 , x , x1y y11 y1

x99 Lx99[ln (1 x99)]RL (9)

where

x99 x x11 y1

g; xL99 xL x11 y1

g

g dydx

xx1 1 s

1 Rs Rs

e

Coefcients RL and L are dened from conditions3 and 4b

RL [c (1 c)(x1=xL) y1]1 y1

(1 xL99) ln (1 xL99)xL99(xL99 1) (10)

L xl99 1xL99[ln (1 xL99)]RL

(11)

Equation (9) requires one additional large strainparameter: either c for the chosen value of xL orxL = xu for c = 1 at the peak strength. RelationshipL4 satises conditions 15 always and either 6a or6b depending on the value of Rs. Relationship L4can be simplied, assuming xe = 0 and choosing alimiting stress equal to the peak strength. Thismodication is presented as a solid line in Fig. 5,simulating the test on Toyora sand as L3 with onlythree parameters. The original function L4 gives abetter prediction, but L3 was used in order to allowa comparison with other functions, which alsoneglect the elastic region in simulating this test. Itis seen from Fig. 6 that L3 ts both the large andsmall strain test data with the highest accuracy ofall the models, including those with four and veparameters.

Relationships L1 and L4 were used to simulatevarious types of tests on London clay, Pentre siltand articial silt HPF4 as indicated in Figs 810respectively. The accuracy of simulation isseen to be remarkable for L4 and very acceptablefor L1 over both the small and entire strainregions.

CONCLUSIONS

The conditions for a typical non-linear stressstrain function for soils and rocks are specied,and existing functions are shown to be decient inmany of them. A new logarithmic function isproposed which is shown to satisfy all the generalconditions.

The function can be tted to normalizedexperimental stressstrain data using one, threeor four parameters depending on the availabilityof small strain measurements. All the parametershave physical signicance and are easy to derive.

Comparison with the results of different types oftests on a wide range of rocks and soils showshigh accuracy of simulation over the entire strain

120

80

40

0

120

80

40

0

160

H0 H1 H2 HD2 GP1 J4 MH5 L1 L3

H0 H1 H2 HD2 GP1 J4 MH5 L1 L3

(a)

(b)

aa

Fig. 6. Appraisal of the accuracy of various models forsimulating stressstrain behaviour of Toyora sand: (a)for large strains; (b) for small strains

1c

y1

00

ys

xe

y

xe xs xe = xe + e 1 xlim

Fig. 7. Parameters of the L4 curve

162 PUZRIN AND BURLAND

range. Even in its single parameter form thefunction gives remarkably good simulation.

ACKNOWLEDGEMENTS

The work was supported by the UK Foreign andCommonwealth OfceClore Foundation Schemethrough the British Council.

NOTATIONc parameter dened in Fig. 1

Emax initial stiffness modulusn number of points comparedq stress

qL limiting stressqu ultimate (peak) stressR coefcient dened in equation (3)xe linear elastic strain limitxL normalized limiting strainx1c critical limiting strainyi

e experimental normalized stressyi

p predicted normalized stressyL normalized stress at xL = xe + e 2 1ys normalized stress at x = xs coefcient dened in equation (4) parameter dened in Fig. 1 strainr reference strain

0

1

0.8

0.6

0.4

0.2

05 10 15 20 25 30

0.4

0.3

0.2

0.1

0

(a)

(b)

x

Log x0.001 0.01 0.1 1

yy

L1TestL3

Fig. 8. Triaxial compression of London clay: (a) largestrains; (b) small strains

0 10 20 30 40x

0.8

0.4

0

0.6

0.2

1

0.3

0.4

0

0.2

0.1

(a)

(b)Log x

0.0001 0.001 0.01 0.1 1

yy

L1

TestL4

Fig. 9. Simple shear of Pentre silt: (a) large strains;(b) small strains

0 2 4 6 8x(a)

0.8

0.6

0.4

0.2

0

1

Log x(b)

0.4

0.3

0.2

0.1

0

0.5

0.001 0.01 0.1 1

L1TestL4

Fig. 10. Hollow cylinder test on articial silt (HPF4):(a) large strains; (b) small strains

LOGARITHMIC STRESSSTRAIN FUNCTION 163

REFERENCESBurland, J. B. (1989). Small is beautifulthe stiffness of

soil at small strains. Can. Geotech. J. 16, No. 4, 499516.

Grifths, D. V. & Prevost, J. H. (1990). Stressstraincurve generation from simple triaxial parameters. Int.J. Numer. Analyt. Methods Geomech. 14, 587594.

Hardin, B. O. & Drnevich, V. P. (1972). Shear modulusand damping in soils: design equations and curves.J. Soil Mech. Fdn Engng Div. Am. Soc. Civ. Engrs 98,No. 7, 667692.

Jardine, R. J. (1994). One perspective of the prefailuredeformation characteristics of some geomaterials.Proceedings of international symposium on pre-failuredeformation characteristics of geomaterials, Sapporo.pp. 151182. Rotterdam: Balkema.

Jardine, R. J., Potts, D. M., Fourie, A. B. & Burland, J. B.(1986). Studies of the inuence of non-linear stressstrain characteristics in soilstructure interaction.Geotechnique 36, No. 3, 377396.

Kim, Y.-S., Ochi, K. & Tatsuoka, F. (1990). Strengthand deformation properties of mudstones in triaxial

compression (in Japanese). Proc. 8th Jap. Symp. RockMech., 357362.

Kondner, R. L. (1963). Hyperbolic stressstrain response:cohesive soils. J. Soil Mech. Fdn Engng Div. Am.Soc. Civ. Engrs 89, No. 1, 115143.

Noma, T. & Ishii, T. (1986). On measuring method ofaxial strain in triaxial test on rock (in Japanese).Proc. 41st Annual Conf. Jap. Soc. Civ. Engrs 3, 657658.

Noma, T., Waku, A., Kadota, S. & Murakami, H. (1987).On measuring method of axial strain of rock speci-men by means of proximeter transducers (in Japa-nese). Proc. 20th Jap. Symp. Rock Mech. 6165.

Porovic, E. (1995). Investigation of soil behaviour usinga resonant column torsional shear hollow cylinderapparatus. PhD thesis, University of London.

Standing, J. R. (1994). Private communication.Tatsuoka, F. & Shibuya, S. (1991). Deformation charac-

teristics of soils and rocks from eld and laboratorytests, Proc. 9th Asian Conf. Soil Mech., Bangkok 2,101170.

Zdravkovic, L. (1994). Private communication.

164 PUZRIN AND BURLAND