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[.]
Available online at www.sciencedirect.com ScienceDirect International Journal of Pavement Research and Technology (2016) 445–449 www.elsevier.com/locate/IJPRT A fuzzy set plasticity model for cyclic loading of granular soils Cheng Chen a,⇑, Lingwei Kong b, Xiaoqing Liu a, Zhonghui Han a b a College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China Received 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 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 granular 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 Concerted effort has been made to develop predictive capabili⇑ 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 ties associated with topics such as earthquake engineering, soil-structure interaction, soil liquefaction, off-shore engineering, 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, isotropic hardening has been proved sufficient to simulate the stress–strain response of soil subjected to monotonic loading while kinematic hardening and mixed hardening 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/) C Chen et al / International Journal of Pavement Research and Technology (2016) 445–449 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; 0 p ẳ r1 ỵ 2r3 ị=3 is the mean eective stress and q ẳ r01 r03 is the deviator stress, where r01 and r03 are the axial and radial stresses, respectively The corresponding work conjugates are volumetric strain ev ẳ e1 ỵ 2e3 and deviatoric strain ed ẳ 23 e1 e3 ị The pairs of stresses and T strains are abbreviated in the vector form as r01 ẳ ẵp0 ; q T and ẳ ẵev ; ed The total strain rate is decomposed into elastic and plastic parts according to e p _ ẳ _ ỵ _ Kẳ ỵ eịp0 31 2mị ỵ eịp0 ; Gẳ 21 ỵ mị j j 2.3 Membership function The membership function has been involved in the plastic modulus equations When c = 1, the material behaves purely elastically and the corresponding value of the plastic modulus is infinite, while when c ¼ 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: Plastic loading: c_ < Plastic unloading: c_ < Elastic loading: c_ P Elastic unloading: c_ P r r r r Although the value of the membership function is at a fully elastic state and at the fully plastic state, the assignment 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 displays 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 q ¼ 156 kPa and q ¼ 231 kPa, respectively The two graphs on the left highlight cycle with the loading from to 156 kPa and unloading from 156 to kPa (in solid line) The other two graphs highlight the cycle with the 300 300 cycle cycle 200 100 ð1Þ 0.02 0.04 200 100 0.06 0.02 εd 0.06 d 1 0.8 0.8 0.6 0.6 γ 0.4 0.4 0.2 0.2 cycle The tangential elastic moduli are calculated assuming that the slope of unloading/reloading occurs along a j line in the e lnp0 plane The moduli are then defined as 0.04 ε γ where a superimposed dot indicates an increment, and the superscripts e and p denote the elastic and plastic components, respectively 2.2 Elastic behavior ð2Þ where e is the void ratio and m is the Possion’s ratio q(kPa) 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 interpretations 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 kinematiccyclic 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 format 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 characteristic features of behavior in granular soils subjected to cyclic loading are captured q(kPa) 446 0 50 100 150 q cycle 200 250 0 50 100 150 200 250 q Fig Deviatoric stress–strain curve and evolution of the membership function c for cycle and cycle C Chen et al / International Journal of Pavement Research and Technology (2016) 445–449 loading from to 231 kPa and unloading from 231 to kPa (in solid line) 2.4 Cone fuzzy surface F ccone ¼ r a0 a1 p a2 p2 ẳ F econe ẳ r ỵ b0 ỵ b1 p ỵ b2 p2 ẳ 3ị where r and p are stress invariants, and r ¼ q gh; vị, p ẳ I =3 The coecient a1 in the cone fuzzy surface function for triaxial compression incorporates the concept of critical state soil mechanics, which is dened as: a1 ẳ M c ỵ jhwi For loose sand, a1 M c 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 b1 ¼ va1 447 where A and B are constitutive parameters If A ¼ B ¼ in Eq (6), the flow rule is equivalent to the flow rule of original Cam Clay model The terms a1 and b1 of Eq (6) represent the slope of the fuzzy set surface line both on the compression side and the extension side Plastic flow rules are expressed as _ p ẳ k_ cone m 8ị In the fuzzy set plasticity theory, a forth-order tensor is defined in such a way that m T : n, where the fourthorder tensor T is defined as T ¼ I ð1 DÞ1 ð9Þ 1.5 ð4Þ 2.5 Cone plastic moduli Mcd H ẳH ỵ cdỵ1 0.5 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 -0.5 ð5Þ -1 d and M are model parameters that can be determined from test data It is worth mentioning that cone membership functions have been involved in the cone plastic modulus equations And the value of the cone membership function is determined by the second deviatoric stress invariant and the first stress invariant respectively Laboratory Test Numerical Modeling -1.5 -0.03 -0.02 -0.01 0.01 0.02 0.03 0.04 εd Fig Deviatoric strain vs stress ratio for drained cyclic test on loose Fuji river sand 2.6 Plastic potential Dẳ @epv ẳ AẵM c ỵ k d wị g @epd 6ị Dc ẳ De ẳ 0.03 0.025 0.02 0.015 0.005 -1.5 ẳ Aẵa1 B g in triaxial compression ẳ Aẵb1 B g in triaxial extension 0.035 0.01 For loose granular materials, we defined the dilatancy flow rule as @epv @epd @epv @epd 0.04 εv The plastic potential defines the ratio between the incremental plastic volumetric strain and the incremental plastic shear strain The most successful and widely used flow rule in geotechnical engineering is based on Rowe’s stressdilatancy relationship [17] Rowe’s original relationship is modified here to take into account the dependence of dilatancy on the state parameter This approach was also used by Gajo and Muir Wood [9] and is given by Laboratory Test Numerical Modeling -1 -0.5 0.5 1.5 η ð7Þ Fig Volumetric strain vs stress ratio for drained cyclic test on loose Fuji river sand 448 C Chen et al / International Journal of Pavement Research and Technology (2016) 445–449 0.04 0.03 Laboratory Test Numerical Modeling 0.035 0.025 0.03 0.02 0.02 εv εv 0.025 0.015 0.015 0.01 0.01 0.005 0.005 Laboratory Test Numerical Modeling -0.04 -0.02 -0.03 -0.01 0.01 0.02 0.03 0.04 -1 -0.5 εd 0.5 1.5 η Fig Deviatoric strain vs volumetric strain for drained cyclic test on loose Fuji river sand Fig Volumetric strain vs stress ratio for drained cyclic constant p0 test on loose Toyoura sand 0.03 Laboratory Test Numerical Modeling 1.5 0.025 0.02 0.015 η εv 0.5 0.01 -0.5 0.005 Laboratory Test Numerical Modeling -1 -0.025 -0.02 -0.015 -0.01 -0.005 0.005 0.01 0.015 0.02 -0.025 -0.02 -0.015 -0.01 -0.005 εd Fig Deviatoric strain vs stress ratio for drained cyclic constant p test on loose Toyoura sand 0.005 0.01 0.015 0.02 0.025 εd 0.025 Fig Deviatoric strain vs volumetric strain for drained cyclic constant p0 test on loose Toyoura sand Model validation Incremental stress-controlled formulation 1 n : r_ m ẳ De : r_ ỵ m n : _ ẳ _ ỵ _ r_ ẳ D : r_ ỵ H H r_ ẳ De ỵ m n : r_ ẳ Dep : r_ ð10Þ H _ _ v ep p ẳ ẵD 11ị q_ _ d e p where 1=K ep ẵD ẳ e 1=3G a1 D 1 ỵ q H a2 ỵ g2 ị a2 D2 þ g2 a1 g 1 a1 gD g2 ð12Þ Performance of the proposed model to simulate the measured behavior of sands under cyclic loading was investigated by comparing numerical simulations with experimental 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: p0 ¼ 196 kPa and e ¼ 0:74 The basic material parameters were: j ¼ 0:001, m ¼ 0:3, a1 ¼ M c ¼ 1:48, b1 ¼ va1 ¼ M e ¼ 1:08, C Chen et al / International Journal of Pavement Research and Technology (2016) 445–449 v ¼ 0:73, k ¼ 0:21, a ¼ 0:8 The plastic parameters were: M ¼ 10; 450, d ¼ 2:48, a0 ¼ 110, b0 ¼ 100, A ¼ 0:17, B ¼ 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: p0 ¼ 98 kPa and e ¼ 0:845 The basic material parameters were: j ¼ 0:001, m ¼ 0:3, a1 ¼ M c ¼ 1:24, b1 ¼ va1 ¼ M e ¼ 1:08, v ¼ 0:87, k ¼ 0:24, a ¼ 0:8 The plastic parameters were: M ¼ 8450, d ¼ 2:8, a0 ¼ 120, b0 ¼ 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 successive stiffening or softening of the sample with the progress of the cyclic loading 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 hardening, hysteresis features, materials memory, and kinematic mechanisms without invoking complex analytical formulations 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 a1 to the state parameter w Acknowledgments The support from the State Key Laboratory of Geomechanics and Geotechnical engineering is greatly appreciated The financial support from the National Natural 449 Science Foundation of China (No: 41372314) and Major Subject of The Chinese Academy of Sciences (No: KZZD-EW-05-02) is also acknowledged References [1] J.H Pre´vost, Plasticity theory for soil stress behavior, J Eng Mech., ASCE 104 (5) (1978) 1177–1194 [2] Z Mroz, V.A Norris, O.C Zienkiewicz, An anisotropic hardening model for soils and its application to cyclic loading, Int J Numer Anal Meth Geomech (1978) 203–221 [3] S Sture, J.C Mould, H.-Y Ko, Elastic-plastic anisotropic hardening constitutive model and prediction of behavior of dry quartz sand, in: G Gudenhus, F Darve, I Vardoulakis (Eds.), Constitutive Relations for Soils, A A Balkema, Rotterdam, Rotterdam, Netherlands, 1982, pp 227–248 [4] J.P Bardet, Bounding surface plasticity model for sands, J Eng Mech 112 (11) (1986) 1198–1217 [5] P Anandarajah, A granular material model based on associated flow rule applied to monotonic loading behavior, Soils Found 34 (3) (1994) 81–98 [6] R.S Crouch, J.P Wolf, Y.F Dafalias, Unified critical-state bounding-surface plasticity model for soil, J Eng Mech 120 (11) (1994) 2251–2270 [7] M.T Manzari, Y.F Dafalias, A critical state two-surface plasticity model for sands, Ge´otechnique 47 (2) (1997) 255–272 [8] A Gajo, D.M Wood, A kinematic hardening constitutive model for sands: the multiaxial formulation, Int J Numer Anal Meth Geomech 23 (1999) 925–965 [9] A Gajo, D.M Wood, Severn-trent sand: a kinematic-hardening constitutive model: the p-q formulation, Ge´otechnique 49 (5) (1999) 595–614 [10] X.S Li, Y.F Dafalias, Constitutive modeling of inherently anisotropic sand behavior, J Geotech Geoenviron Eng 128 (10) (2002) 868–880 [11] C Chen, L Ge, J.S Zhang, Modeling permanent deformation of unbound granular materials under repeated loads, Int J Geomech 10 (6) (2010) 236–241 [12] C Chen, J.S Zhang, Constitutive modeling of loose sands under various stress paths, Int J Geomech 13 (1) (2013) 1–8 [13] A Elgamal, Z Yang, E Parra, A Ragheb, Modeling of cyclic mobility in saturated cohesionless soils, Int J Plast 19 (6) (2003) 883– 905 [14] X.S Li, Y.F Dafalias, A constitutive framework for anisotropic sand including non-proportional loading, Ge´otechnique 54 (1) (2004) 41– 55 [15] N Khalili, M.A Habte, S.A Valliappan, Bounding surface plasticity model for cyclic loading of granular soils, Int J Numer Meth Eng 63 (14) (2005) 1939–1960 [16] Y Bao, S Sture, Numerical modeling of cyclic mobility based on fuzzy-set concepts in plasticity theory, Comput Geotech 38 (3) (2011) 375–382 [17] P.W Rowe, The stress-dilatancy relations for static equilibrium of an assembly of particles in contact, Proc R Soc London, Ser A 269 (1962) 500–527 [18] F Tatsuoka, K Ishihara, Drained deformation of sand under cyclic stresses reversing direction, Soils Found 14 (3) (1974) 51–65 [19] T.B.S Pradhan, F Tatsuoka, Y Sato, Experimental stress-dilatancy relations of sand subjected to cyclic loading 29 (1) (1989) 45–64 ... pavement materials deformation problems particularly related to cyclic mobility Two series of cyclic drained triaxial tests data are considered The characteristic features of behavior in granular soils. .. behavior of unbound granular materials under repeated loads The enhanced fuzzy- set model is built to adapt the simply format equations of plastic moduli and plastic potential to simulated the pavement... function is at a fully elastic state and at the fully plastic state, the assignment of the value in elastoplastic state is deterministic and can be arbitrarily defined as needed A linear variation with