A fuzzy set plasticity model for cyclic loading of granular soils

A fuzzy set plasticity model for cyclic loading of granular soils Available online at www sciencedirect com www elsevier com/locate/IJPRT ScienceDirect International Journal of Pavement Research and T[.]

A fuzzy set plasticity model for cyclic loading of granular soils

Cheng Chena,⇑, Lingwei Kongb, Xiaoqing Liua, Zhonghui Hana

a College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China

b State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan

430071, China Received 8 March 2016; received in revised form 26 August 2016; accepted 16 September 2016

Available online 10 October 2016

Abstract

A constitutive model for describing the stress–strain behavior of granular soils subjected to cyclic loading is presented The model is formulated using fuzzy set plasticity theory within the classical incremental plasticity theory framework A special membership function

is introduced to provide an analytical and simple geometrical interpretation to formulate hardening, hysteresis feature, material memory, and kinematic mechanisms without resorting to complicated kinematic hardening formulations The model can accurately describe cyclic loading, dilatancy, material theory and critical state soil mechanics features effects Two series of cyclic drained triaxial tests data are considered The characteristic features of behavior in granular soils subjected to cyclic loading are captured

Ó 2016 Production and hosting by Elsevier B.V on behalf of Chinese Society of Pavement Engineering This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Keywords: Constitutive model; Fuzzy set; Plasticity; Cyclic loading; Hardening hysteresis feature

1 Introduction

Cyclic response of granular materials is complex due to

the pressure and specific volume dependency of the stress–

strain relationship and the highly nonlinear behavior of the

soil matrix Until now, the mechanical behavior of

granu-lar soils has been mainly represented with constitutive

models which need different sets of constitutive parameters

for each density and effective confining pressure In fact,

the study of loading and unloading response in granular

soils and development of relationships for its prediction

in natural formations and engineered materials has been

a major area of research in modern geomechanics

Con-certed effort has been made to develop predictive

capabili-ties associated with topics such as earthquake engineering, soil-structure interaction, soil liquefaction, off-shore engi-neering, etc

Development of constitutive models for a wide range of engineering materials, including soils, has been found extensively for recent decades [1–15] A majority of the models is based on the incremental plasticity theory Within the framework of classical plasticity theory, isotro-pic hardening has been proved sufficient to simulate the stress–strain response of soil subjected to monotonic load-ing while kinematic hardenload-ing and mixed hardenload-ing has been typically used to mimic hysteretic phenomena of soil under cyclic loading

Nowadays, the cyclic behavior of unbound granular materials under traffic loading is another challenging task for geotechnical engineers A typical example is railroad ballast Thus, it is of special interest to determine the over characteristics and constitutive properties of the ballast and to ensure stable and long-lasting properties for such

a material that is not homogeneous The research focuses

on the development of a cyclic constitutive model based

http://dx.doi.org/10.1016/j.ijprt.2016.09.004

1996-6814/ Ó 2016 Production and hosting by Elsevier B.V on behalf of Chinese Society of Pavement Engineering.

This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

⇑ Corresponding author.

E-mail addresses: chencheng401@sina.com (C Chen),

lwkong@whrsm.ac.cn (L Kong), a8583272@qq.com (X Liu),

1427687676@qq.com (Z Han).

Peer review under responsibility of Chinese Society of Pavement

Engineering.

www.elsevier.com/locate/IJPRT

ScienceDirect

International Journal of Pavement Research and Technology 9 (2016) 445–449

on fuzzy set concepts, its numerical integration, and finite

element implementation as well

Unlike convention elasto-plastic hardening models, the

fuzzy set model is physically intuitive and easy to visualize

It provides analytical and simple geometrical

interpreta-tions to formulate hardening, hysteresis features, material

memory, and kinematic mechanisms In this model, based

on fuzzy set plasticity theory, the basic concept rests on the

assumption that there exists a fuzzy surface which in many

ways resemble a bounding surface At each point within the

fuzzy surface, the value of plastic hardening moduli is

defined by the membership function In this view, Bao

et al presented a transparent and accurate

kinematic-cyclic constitutive model to capture the important features

of volume change and pore water pressure build-up related

to soil cyclic mobility[16]

In this study, a cyclic plasticity model based on fuzzy

plasticity theory is presented to model the cyclic behavior

of unbound granular materials under repeated loads The

enhanced fuzzy-set model is built to adapt the simply

for-mat equations of plastic moduli and plastic potential to

simulated the pavement materials deformation problems

particularly related to cyclic mobility Two series of cyclic

drained triaxial tests data are considered The

characteris-tic features of behavior in granular soils subjected to cyclic

loading are captured

2 Preliminaries

2.1 Notation

In the model presented, the material behavior is

assumed isotropic and rate independent in both elastic

nad elastic–plastic response Compression is considered

positive and tension is negative For simplicity, triaxial

stress notation p0 q is adopted throughout;

p0¼ ðr0

3Þ=3 is the mean effective stress and

q¼ r0

3 is the deviator stress, where r01 and r03 are the

axial and radial stresses, respectively The corresponding

work conjugates are volumetric strain ev¼ e1þ 2e3 and

deviatoric strain ed ¼2

3ðe1 e3Þ The pairs of stresses and strains are abbreviated in the vector form as r0

and  ¼ ½ev; edT

The total strain rate is decomposed into elastic and plastic parts according to

where a superimposed dot indicates an increment, and the

superscriptse and p denote the elastic and plastic

compo-nents, respectively

2.2 Elastic behavior

The tangential elastic moduli are calculated assuming

that the slope of unloading/reloading occurs along a j line

in the e lnp0plane The moduli are then defined as

K¼ð1 þ eÞpj 0; G ¼32ð1 þ mÞð1  2mÞð1 þ eÞpj 0 ð2Þ where e is the void ratio and m is the Possion’s ratio 2.3 Membership function

The membership function has been involved in the plas-tic modulus equations When c = 1, the material behaves purely elastically and the corresponding value of the plastic modulus is infinite, while when c¼ 0, the material reaches

a fully plastic state and the plastic modulus is equal to the value on the fuzzy surface, i.e H = H*

With the assistance of the membership function c, we can readily construct reversal plastic loading without resorting to a kinematic hardening rule The basic rules

of kinematic mechanism for the membership function are:

r Plastic loading: _c < 0

r Plastic unloading: _c < 0

r Elastic loading: _c P 0

r Elastic unloading: _c P 0 Although the value of the membership function is 1 at a fully elastic state and 0 at the fully plastic state, the assign-ment of the value in elastoplastic state is deterministic and can be arbitrarily defined as needed A linear variation with respect to stress state was adopted in this study

Fig 1displays an example of the deviatoric stress–strain response and evolution of the membership function for a material subjected to two varied amplitude cyclic loading under a conventional triaxial stress path The unloading– reloading points take place in two different stress levels

q¼ 156 kPa and q ¼ 231 kPa, respectively The two graphs on the left highlight cycle 1 with the loading from

0 to 156 kPa and unloading from 156 to 0 kPa (in solid line) The other two graphs highlight the cycle 2 with the

0 100 200 300

εd

cycle 1

0 100 200 300

εεεεd

cycle 2

0 0.2 0.4 0.6 0.8 1

q

cycle 1

0 0.2 0.4 0.6 0.8 1

q

cycle 2

Fig 1 Deviatoric stress–strain curve and evolution of the membership function c for cycle 1 and cycle 2.

loading from 0 to 231 kPa and unloading from 231 to

0 kPa (in solid line)

2.4 Cone fuzzy surface

Fccone¼ r  a0 a1p a2p2¼ 0

Fe

where r and p are stress invariants, and r¼ q  gðh; vÞ,

p¼ I1=3 The coefficient a1 in the cone fuzzy surface

func-tion for triaxial compression incorporates the concept of

critical state soil mechanics, which is defined as:

a1¼ Mcþ jhwi For loose sand, a1 Mc

Since the critical state line in extension mean stress space

is not well defined and difficult to obtain experimentally, so

it is reasonably assumed that the evolution of coefficient b1

in the cone fuzzy surface for triaxial extension is attained

by keeping the ellipticity v fixed, i.e

2.5 Cone plastic moduli

It should be noted that cone loading surfaces are not

explicitly defined, and one can think that for the current

stress state, there exist cone loading surfaces such that

the cone plastic moduli is defined as follows

H¼ Hþ Mcd

d and M are model parameters that can be determined

from test data It is worth mentioning that cone

member-ship functions have been involved in the cone plastic

mod-ulus equations And the value of the cone membership

function is determined by the second deviatoric stress

invariant and the first stress invariant respectively

2.6 Plastic potential

The plastic potential defines the ratio between the

incre-mental plastic volumetric strain and the increincre-mental plastic

shear strain The most successful and widely used flow rule

in geotechnical engineering is based on Rowe’s

stress-dilatancy relationship[17] Rowe’s original relationship is

modified here to take into account the dependence of

dila-tancy on the state parameter This approach was also used

by Gajo and Muir Wood[9]and is given by

D¼@epv

@ep

d

For loose granular materials, we defined the dilatancy

flow rule as

Dc¼@epv

@ep

d

¼ A½a1B g in triaxial compression

De¼@epv

@ep¼ A½b1B g in triaxial extension ð7Þ

where A and B are constitutive parameters If A¼ B ¼ 1 in

Eq.(6), the flow rule is equivalent to the flow rule of orig-inal Cam Clay model The terms a1and b1of Eq (6) rep-resent the slope of the fuzzy set surface line both on the compression side and the extension side

Plastic flow rules are expressed as

In the fuzzy set plasticity theory, a forth-order tensor is defined in such a way that m  T : n, where the fourth-order tensorT is defined as

T ¼ I 1

-1.5 -1 -0.5 0 0.5 1 1.5

εd

Laboratory Test Numerical Modeling

Fig 2 Deviatoric strain vs stress ratio for drained cyclic test on loose Fuji river sand.

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

η

εv

Laboratory Test Numerical Modeling

Fig 3 Volumetric strain vs stress ratio for drained cyclic test on loose Fuji river sand.

Incremental stress-controlled formulation

_ ¼ _eþ _p_r ¼ De: _r þ 1

Hn : _r

m ¼ De: _r þ 1

Hm  n : _r ¼ Deþ1

Hm  n

_v

_d



¼ ½Dep _p

_q



ð11Þ where

½Dep¼ 1=K 0

þ 1

H

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2þ g2

ð Þ a 2D2þ g2

a1g g2

3 Model validation Performance of the proposed model to simulate the measured behavior of sands under cyclic loading was inves-tigated by comparing numerical simulations with experi-mental results from the literature The results of cyclic loading are shown in Figs 2–7 In all of these figures, model predictions are shown using continuous solid lines and experimental data are shown by discrete symbols 3.1 Drained cyclic test on Fuji river sand

Figs 2–4 show the simulation results for the cyclic drained test conducted by Tatsuoka and Ishihara [18] on loose Fuji river sand with increasing stress amplitude The initial conditions of the test were: p0¼ 196 kPa and

e¼ 0:74 The basic material parameters were: j ¼ 0:001,

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

εd

εv

Laboratory Test

Numerical Modeling

Fig 4 Deviatoric strain vs volumetric strain for drained cyclic test on

loose Fuji river sand.

-1

-0.5

0

0.5

1

1.5

εd

Laboratory Test Numerical Modeling

Fig 5 Deviatoric strain vs stress ratio for drained cyclic constant p0test

on loose Toyoura sand.

0 0.005 0.01 0.015 0.02 0.025 0.03

η

εv

Laboratory Test Numerical Modeling

Fig 6 Volumetric strain vs stress ratio for drained cyclic constant p0test

on loose Toyoura sand.

0.005 0.01 0.015 0.02 0.025 0.03

εd

εv

Laboratory Test Numerical Modeling

Fig 7 Deviatoric strain vs volumetric strain for drained cyclic constant p0 test on loose Toyoura sand.

v ¼ 0:73, k ¼ 0:21, a ¼ 0:8 The plastic parameters were:

M ¼ 10; 450, d ¼ 2:48, a0¼ 110, b0¼ 100, A ¼ 0:17,

B¼ 1:05 Once again, the model simulation matches the

experimentally observed trends The model captures the

contractive responses both during loading and unloading,

and the successive stiffening of the sample with the progress

of cyclic loading

3.2 Dranied cyclic test on Toyoura sand

Pradhan et al [19] executed a series of drained cyclic

tests on Toyoura sand which consisted of mainly quartz

with angular to sub-angular particle shape The triaxial

tests were conducted on 75 mm diameter and 150 mm

high specimens prepared by pulviating air dried samples

Figs 5–7 show results of cyclic, constant p0, drained tests

on loose samples of Toyoura sand The initial conditions

of the test were: p0¼ 98 kPa and e ¼ 0:845 The basic

material parameters were: j ¼ 0:001, m ¼ 0:3,

a1¼ Mc¼ 1:24, b1¼ va1¼ Me¼ 1:08, v ¼ 0:87,

k ¼ 0:24, a ¼ 0:8 The plastic parameters were: M ¼ 8450,

d¼ 2:8, a0¼ 120, b0¼ 60, A ¼ 0:28, B ¼ 1:05 These

tests are also well matched by the model simulation The

model captures the stress–strain response and the

succes-sive stiffening or softening of the sample with the progress

of the cyclic loading

4 Conclusions

A cyclic constitutive model based on fuzzy set concepts

has been developed The cyclic fuzzy set model is physically

intuitive and easy to visualize with the aid of membership

functions The cyclic fuzzy set model provides analytical

and simple geometrical interpretation to formulate

harden-ing, hysteresis features, materials memory, and kinematic

mechanisms without invoking complex analytical

formula-tions In addition the cyclic fuzzy set model accounts for:

realistic stress–strain behavior under repeated load cycles,

nonlinear dilatancy behavior, critical state soil mechanics

concepts, and non-proportional loading The evolution

rule for the fuzzy surface can help simulate the post peak

soil behavior such as strain softening The critical state soil

mechanics concept has been implemented into the fuzzy set

model by linking the fuzzy surface parameter a1to the state

parameter w

Acknowledgments

The support from the State Key Laboratory of

Geome-chanics and Geotechnical engineering is greatly

appreci-ated The financial support from the National Natural

Science Foundation of China (No: 41372314) and Major Subject of The Chinese Academy of Sciences (No: KZZD-EW-05-02) is also acknowledged

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