PLASTICITY MODELS AND VARIATIONS OF G=Gmax WITH SHEAR STRAIN FOR SAND UNDER MONOTONIC LOADING

7/23/2019 PLASTICITY MODELS AND VARIATIONS OF G=Gmax WITH SHEAR STRAIN FOR SAND UNDER MONOTONIC LOADING

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plasticity models and the inuences of void ratio (e), strength parameters (cand ) and over-consolidation ratio (OCR) were investigated and compared with those obtained from Harden

and Drnevich (1972) expressions.

THEORYGeneral Formulation

The total strain increment is assumed to be the sum of the elastic strain increment and

plastic strain increment (Chen and Han 1988),

dij =deij+ d

pij (1)

The plastic strain increment is obtained from the associated ow rule,

dpij =d f

ij(2)

where

d

= 0 whenf


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Drucker-Prager Model

Now, for a Drucker-Prager material model, yield condition,f, is given by

f= J2+ I1 k= 0 (10)where,

= 2sin3(3sin) for triaxial condition;

= tan(9+12 tan2 )

for plane strain condition.

and

k

= 6c cos3(3sin) for triaxial condition;

= 3c(9+12 tan2 )

for plane strain condition.

cand being the cohesion and friction angle respectively. From Eqs. (6), (8), and (10), aftersome tensor algebra, it can be shown that the most general form of Drucker-Prager elastic-

perfectly plastic constitutive relationship as,

dij =

2Gimjn +

K

2

3G

ijmn

GJ2

Sij+ 3Kij

G + 9K2

G

J2Smn+ 3Kmn

dmn(11)

Elastic-Plastic Model with Hardening Parameters

The general expression of a yield surface for a work-hardening material has the form,

f(ij , pij , k) = 0 (12)

and hence the consistency condition is assumed as

df= f

ijdij+

f

pijdpij+

f

kdk= 0 (13)

which ensures that in a plastic loading process, the subsequent stress and defromation states

remain on the subsequent yield surface.

Drucker-Prager Isotropic Hardening Model

The general form of the yield function for isotropic-hardening Drucker-Prager material can

be expressed as,

f(ij , p) =(p)I1+

J2 k(p) = 0 (14)

Please note that in Eq. (10), the yield function for a perfectly plastic material has been presented

with =constant andk =constant. Herein, for simplicity, we shall assume that the slope of theloading surface in theI1

J2 space is a constant,(p) =, so that the hardening behavior

of the material can be uniquely determined by a single uniaxial stress-strain relation through

the hardening parameterk(p),

f(ij , p) =I1+

J2 k(p) = 0 (15)

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With the Appendix giving the details of derivation, the general form of Drucker-Prager isotropic-

hardening elastic-plastic constitutive relationship can be written as,

dij =2Gimjn + K 23G ijmn

GJ2Sij+ 3Kij

G + 9K2 +

+ 1

3

2Hp

GJ2 Smn+ 3Kmn dm(16)

which is exactly the same equation obtained for Drucker-Prager elastic-perfectly plastic mate-

rial (Eq. (11)) except the hardening term in the plastic tangent stiffness tensor.

Cam Clay Model

For a cam clay material model, the yield function, f, can be written in terms of stress

invariants p and q as

f=M2p2 M2p0p + q2 = 0 (17)

where

p= I13

q= 3

2oct

M= 6sin

3 sin andp0 is the value ofpat the intersection of the yield cap with the p-axis. Herep0 is treated asa variable which has a unique value for any yield surface and is the hardening parameter.

In terms of stress invariantsI1 and J2, Eq. (17) can be written as

f=M2I21M2I01I1+ 27J2 = 0 (18)

whereI01 is analogous top0 (hardening parameter) and is assumed to depend on the plasticvolumertic strain. Now, at yield as f = f[J2, I1, I01(

pv)], we have the consistency condition

as,

df= f

J2dJ2+

f

I1dI1+

f

pvdpv = 0 (19)

Now, if we dene

Aij = F

ij=

f

J2

J2ij

+

f

I1

I1ij

= f

J2Sij+

f

I1ij

=

2M2I1 M2I01

ij+ 27Sij (20)

using Eqs. (2) (4) (18), it can be shown that the most general form of the cam clay constitutive

relationship can be written as,

dij =

Cijrs

CijklAklAmnCmnrs

AmnCmnrsArs (2MI21 M2I01)M2I1I01(1+e0)

drs (21)

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G/GMAXVERSUS SHEAR STRAIN RELATIONSHIPIt can be observed from any three dimensional elastic-plastic constitutive relationship that

for an initially istropic material the elatic-plastic stiffness (Cepijmn) always composed of two

terms: the elastic stiffness tensor,Cijmn (=2Gimjn + K23G ijmn) and a subractive

term,Cpijmn, called the plastic tangent stiffness, which represents the degradation of the stiff-ness of the material due to plastic ow. The elastic stiffness tensor is dened in terms of two

elastic constants,G andK (orEand ), which can be obtained from triaxial test. The plastictangent stiffness tensor is dened by one or two more constants other thanG and K, depend-ing upon perfectly plastic and work-hardening material model, with the exception of cam clay,

which requires more parameters. Some plasticity parameters can be obtained from strength

parameters of soil,c and , (e.g. and k for Drucker-Prager perfectly plastic model, M forcam clay model) and others require specialized testing (e.g. the plasticity modulus, Hp, forDrucker-Prager isotropic hardening model,and for cam clay model).

These three dimensional plasticity models can be simplied to any particular direction of

interest to model the degradation of modulus with strain. The degradation of shear modulus

with shear starin can be computed by substituting i = m = 1 and j = n= 2 in the general incre-mental elatic-plastic constitutive ralationships() e.g. in the 1-2 direction the cam clay model

reduces to

d12=

G

729G2

2J2

729G + 9K(2M I21 M2I01)2 (2M I21 M2I01)

M2I1I01(1+e0)

S12S12

d12

(22)

whereGis the elastic shear modulus and the term inside the bracket is the elastic-plastic shearmodulus. Eq. (22) clearly shows that the elastic-plastic shear modulus decreases with shear

strain. Similarly elastic-plastic shear modulus can be obtained for other plasticity based models

by substituting i = m = 1 and j = n = 2 in the general forms.

The elastic shear modulus,G, was computed based on Hardin and Blandford (1989) equa-tion as follows,

Gij = OCRaBijF(e)2(1 + )

P1na (i

j)

n/2 (23)

where,Gij is the shear modulus in the shear plane containing principal effective stresses, i

ans j ;Bij is the dimensionless elastic stiffness coefcient; a is the soil constant dependingon the plasticity index of soil;Pais the atmospheric pressure; n is the soil constant equal to 0.5for sand;is the Poissons ratio,F(e) = 0.3 + 0.7e2; andeis the void ratio.

Different plasticity models were used to numerically simulate the soil behavior under K0triaxial condition. K0 loading condition was imposed numerically on a 8-node brick elementand the variations ofG/Gmax with shear strain for different plasticity models were computedusing OpenSees and compared with the hyperbolic stress-strain relationship and reference

strain (r) based model(), described as follows,

G= Gmax1 + r

(24)

where

r = maxGmax

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FIG. 1. Variation of Shear Modulus with Shear Strain for Sands

and

max =

1 + K0

2 vsin

+ c cos 2

1 K02

v

21/2Fig. 1 shows the variation ofG/Gmax with shear strain as predicted by Hardin-Drenvich

equation (Hardin and Drnevich 1972) and by different plasticity models with previously pub-

lished experimental observation (Seed and Idriss 1970). It can be observed that though all the

plasticity based models predicted stiffness degradtion, the cam clay model did it more closely

to the experimental observation. Also all the plasticity based models sensed the small strains

as measured and reported by Vucetic et al. (Lanzo et al. 1997)

CONCLUSIONS AND FUTURE PLAN

Based on the analytical and numerical study the following conclusions can be advanced:

1. Though all the plasticity-based models predicted the reduction of shear modulus with

shear strain the cam clay model predicted the behavior closely with the experimental observa-

tion.

2. The cam clay model did not predict the reduction of shear modulus at small strain

level well. This might be because the yield surface assumed for the cam clay model was for

isotropic consolidation and the experimental data were for K0consolidation. Yield surface forK0 consolidation might work well as it will early yielding and will be studied in the future.

REFERENCES

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Chen, W. and Han, D. (1988). Plasticity for Structural engineers. Springer-Verlag.

Hardin, B. O. (1980). Discussion of anisotropic shear modulus due to stress anisotropy. Jour-

nal of Geotechnical Engineering Division, ASCE, 106(8), 956958.

Hardin, B. O. and Black, W. L. (1968). Vibration modulus of normally consolidated clay.

Journal of Soil Mechanics and Foundations Division, ASCE, 94(2), 353379.Hardin, B. O. and Blandford, G. E. (1989). Elasticity of particulate materials. Journal of

Geotechnical Engineering, 115(6), 788805.

Hardin, B. O. and Drnevich, V. P. (1970). Shear modulus and damping in soils - ii: Design

equations and curves.Technical Report UKY 27-70-CE3, Soil Mechanics Series No. 2, Uni-

versity of Kentucky.

Hardin, B. O. and Drnevich, V. P. (1972). Shear modulus and damping in soils; design equa-

tions and curves.Journal of the Soil Mechanics and Foundation division, ASCE, 98(SM7),

667692.

Lanzo, G., Vucetic, M., and Doroudian, M. (1997). Reduction of shear modulus at small

strains in simple shear. Journal of Geotechnical and Geoenvironmental Engineering,

123(11), 10351042.Roesler, S. K. (1979). Anisotropic shear modulus due to stress anisotropy. Journal of the

Geotechnical Engineering Division, ASCE, 105(GT7), 871880.

Seed, H. B. and Idriss, I. M. (1970). Soil moduli and damping factors for dynamic factors for

dynamic response analysis. Technical Report EERC Report, 70-10, University of California,

Berkeley.

APPENDIX

The increment of the isotropic hardening parameter,dk in Eq. (15) can be obtained as

dk= dk

dpdp (25)

Now, the isotropic hardening parameter,k, can be expressed in the simple form,

dp= C

dpijdpij (26)

and hence,

dk= dk

dpC

f

ij

f

ijd (27)

Now, dpij can be obtained from the associated ow rule (Eq. (2)) and d ij can be obtainedfrom the Hookes law (Eq. (4)) and hence, the consistency condition (Eq. (13)) reduces to the

form

df= fijCijkldkl hd= 0 (28)

where,

h= f

ijCijkl

f

kl

f

pij

f

ij

f

k

dk

dpijC

f

ij

f

ij(29)

From Eq. (28), the scalar function d, can be solved as,

d= 1

h

f

ijCijkldkl =

1

hHkldkl (30)

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where, the second-order tensorHkl associated with the yield function,f, is dened as,

Hkl = f

ijCijkl (31)

Now, for Drucker-Prager yield function for isotropic hardening (Eq. (15)),f

ij=ij+

1

2

J2Sij (32)

Hence, using the elastic stiffness tensor in terms ofG andK (Eq. (5)) we obtain (since theyield function is not expressed as a function ofpij explicitly, we have

fpij

= 0),

Hkl =Cijklf

ij= 3Kkl+

G

J2Skl (33)

Now, to determine the scalar function h dened in Eq. (29), we also need to obtain dkdp and

the parameter C. We can express k as follows (since the effective stress, e, for Drucker-Prager

material can be written ase = 3(I1+J2)1+3

),

k= I1+

J2=1 +

3

3

e (34)

from which we obtain,

dk

dp=

1 +

3

3

dedp

=1 +

3

3

Hp (35)

where,Hp is determined from a uniaxial stress-strain curve, d = Hpd. Now,

dp =dWp

e=

kldpkl

e=Cd

pijd

pij (36)

From Eq. (36), we have the parameter C as,

C=kl

fkl

fst

fst

e

(37)

Substituting Eqs. (5), (32), (35), and (36) into Eqs. (??) we can express, the scalar function,h,as,

h = G + 9K2 +I1+

J2

3k (1 +

3)2Hp

= G + 9K2 + + 132

Hp (38)

Hence, the plastic strain increment , dpij , is given by,

dpij = d f

ij

= 1

h

f

mnCmnst

f

ijdpst

= 1

hHst

f

ijdpst (39)

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and from Hookes law (Eq. (4)), the stress increment,dij , can be determined as

dij = Cijkl

dkl

1

h

f

mnCmnst

f

kldst

= Cijkl

sktl 1h

fmn

Cmnst fkl

dst

=

Cijst

1

h

f

mnCmnstCijkl

f

kl

dst

=

Cijst

1

hHstHij

dst (40)

Hence the most general form of Drucker-Prager isotropic-hardening elastic-plastic consti-

tutive relationship can be written as,

dij =2Gimjn + K 23G ijmn

GJ2Sij+ 3Kij

G + 9K2 +

+ 1

3

2Hp

GJ2 S

mn+ 3Kmn dm

(41)

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PLASTICITY MODELS AND VARIATIONS OF G=Gmax WITH SHEAR STRAIN FOR SAND UNDER MONOTONIC LOADING

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