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Comparison between Duncan and Chang’s EB Model and the Generalized Plasticity Model in the Analysis of a High Earth-Rockfill Dam

Comparison between Duncan and Chang’s EB Model and the Generalized Plasticity Model in the... Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 709430, 12 pages http://dx.doi.org/10.1155/2013/709430 Research Article Comparison between Duncan and Chang’s EB Model and the Generalized Plasticity Model in the Analysis of a High Earth-Rockfill Dam Weixin Dong, Liming Hu, Yu Zhen Yu, and He Lv State Key Laboratory of Hydro-Science and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China Correspondence should be addressed to Yu Zhen Yu; yuyuzhen@tsinghua.edu.cn Received 4 June 2013; Revised 19 August 2013; Accepted 20 August 2013 Academic Editor: Fayun Liang Copyright © 2013 Weixin Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Nonlinear elastic model and elastoplastic model are two main kinds of constitutive models of soil, which are widely used in the numerical analyses of soil structure. In this study, Duncan and Chang’s EB model and the generalized plasticity model proposed by Pastor, Zienkiewicz, and Chan was discussed and applied to describe the stress-strain relationship of rockfill materials. The two models were validated using the results of triaxial shear tests under different confining pressures. The comparisons between the tfi tings of models and test data showed that the modified generalized plasticity model is capable of simulating the mechanical behaviours of rockfill materials. The modified generalized plastici ty model was implemented into a finite element code to carry out static analyses of a high earth-rockfill dam in China. Nonlinear elastic analyses were also performed with Duncan and Chang’s EB model in the same program framework. eTh comparisons of FEM results and in situ monitoring data showed that the modified PZ-III model can give a better description of deformation of the earth-rockfill dam than Duncan and Chang’s EB model. 1. Introduction mainly grouped in two categories: nonlinear elastic models and elastoplastic models. The constitutive model of soil is the keystone in the ni fi te For nonlinear elastic model, the nonlinear characteristic element analyses of geotechnical structures. A suitable con- of soil stress-strain relationship is considered by sectionalized stitutive model can simulate the stress-strain relationships of linearization. A typical nonlinear elastic model is Duncan soils under static or dynamic conditions. Numerical analysis, and Chang’s Model [5, 6], which has been widely used in especially for ni fi te element method incorporated with soil the numerical analyses of earth-rockfill dams, as the model constitutive models, has played a very important role in parameters are quite easy to be determined from conven- geotechnical analyses which always include complex bound- tional triaxial tests. And, a lot of experience of application has ary conditions, nonlinearity of material, and geometry [1]. been accumulated for this model. However, nonlinear elastic Biot presented the famous three-dimensional consolida- models also have some inherent limitations to represent the tion theory based on the eecti ff ve stress theory, equilibrium stress-strain characteristics of soils, such as shear-induced equation, and continuity condition [2]. However, it is quite dilatancy and stress path dependency. dicffi ulttogivethe theoreticalsolutionofBiot’sconsolidation Elastoplastic models would be very adequate in describ- theory except for few simple problems. Up to the 1960s, ing many key features of soils. Classical elastoplastic models with the rapid development of electronic computer and are based on the plastic incremental theory composed of yield constitutive models of soils, Biot’s consolidation theory was condition, flow rule, and hardening law. In the 1950s, Drucker successfully implemented in finite element codes to study the et al. (1957) [7] suggested a cap yield surface controlled by behavior of geotechnical structures [3, 4]. So far, thousands volumetric strain. Roscoe et al. [8, 9]proposedthe concepts of constitutive models have been proposed, which can be of critical statelineand stateboundarysurface,and then 2 Journal of Applied Mathematics they built the Original Cam Clay Model based on triaxial Consider tests. Burland [10]suggested adieff rentenergyequation andthenestablished theModiefi d CamClayModel.Since 𝜎 −𝜎 = , 1 3 (1) 𝑎+𝑏𝜀 the establishment of Cam Clay Model, some other types of elastoplastic constitutive models have also achieved great in which𝑎 and𝑏 are model constants. development [11–18]. Among these models, the generalized In this constitutive model, the tangential Young’s modu- plasticity model [16, 19, 20] can simulate the static and lus𝐸 and tangential bulk modulus𝐵 areusedtosimulatethe 𝑡 𝑡 dynamicmechanicalbehaviorsofclaysandsands.Thismodel nonlinear elastic response of soils, which are assumed to be is very flexible and convenient to extend, as the complicated yield or plastic potential surfaces need not to be specified 𝑛 𝜎 2 explicitly. And the model has been used successfully in the 𝐸 =𝐾𝑃 ( ) (1−𝑅 𝑆 ), 𝑡 𝑎 𝑓 𝑙 static or dynamic analyses of some geotechnical structures (2) [21–24]. Furthermore, based on the framework of generalized plasticity theory [16], some limitations of the original model 𝐵 =𝐾 𝑃 ( ) , 𝑡 𝑏 𝑎 have been solved [25–28], such as pressure dependency, den- sification under cyclic loading. The details of the generalized where𝑃 is the atmospheric pressure,𝐾 and𝐾 are modulus 𝑎 𝑏 plasticity theory and the original and proposed modified numbers, 𝑛 and 𝑚 are exponents determining the rate of Pastor-Zienkiewicz-Chan’s models will be introduced in the variation of moduli with confining pressure, and 𝑅 is the sections below. failure ratio with a invariable value less than 1. However, little experience has as yet been accumulated in The Mohr-Coulomb failure criterion is adopted in the applying the generalized plasticity model to the simulation model, and 𝑆 is a factor defined as shear stress level given of rockfill materials. And we know that rockfill material is by quite different from sands in mechanical properties [ 29–31]. eTh rockfillmaterialhas largeparticlesizeand sharpedges (1−sin𝜙)(𝜎 −𝜎 ) 1 3 𝑆 = . (3) and corners, which can result in remarkable particle breakage 2𝑐⋅ cos𝜙+2𝜎 ⋅ sin𝜙 andchangethe shear-induceddilation[32, 33]. On the other hand, though the generalized plasticity model has gained In theunloading andreloading stage, thetangential great success in the modeling of soils, the application of this Young’s modulus is defined as model in the large-scale ni fi te element analyses of earth dams was less reported. 𝐸 =𝐾 𝑃 ( ) . (4) 𝑢𝑟 𝑢𝑟 𝑎 In this study, the original generalized plasticity model was modified to consider the stress-strain relationships of rockfill materials, as most of previous studies focused on sands So far, the model has 8 parameters, 𝑐 , 𝜑 , 𝐾 , 𝐾 , 𝑛 , 𝑅 , 𝑢𝑟 𝑓 and clays. en, Th based on conventional triaxial test data, 𝐾 , 𝑚 . es Th e parameters can be determined with a set of conventional triaxial tests. the model parameters for dam materials of the Nuozhadu high earth-rockfill dam in Southwest China are determined. In general, a curved Mohr-Coulomb failure envelop is Finally, the static simulation of this dam is carried out by adopted by setting𝑐=0 and letting 𝜑 vary with confining using a finite element code incorporating with Duncan and pressure according to Chang’s EB model and the modified generalize plasticity model. The comparison of numerical results and in situ mon- 3 𝜑=𝜑 −Δ𝜑 log( ). (5) itoring data illustrates the advantages of modified generalized plasticity model in the simulation of earth-rockfill dams. eTh n parameters 𝑐 and𝜑 are replaced by𝜑 andΔ𝜑 . Although Duncan and Chang’s EB constitutive model is 2. Constitutive Model Descriptions quite simple, it has gained significant success in geotechnical engineering. On one hand, it is easy to obtain the model Two constitutive models of soils were used in the n fi ite parameters; on the other hand, much experience has been element analyses. One is the Duncan and Chang’s EB model accumulated. Nevertheless, it cannot incorporate dilatancy belonging to nonlinear elastic model, the other one is the which has an important inu fl ence in the mechanical behavior generalized plasticity model. of soils. And furthermore, it can only consider unloading process in a crude way. 2.1. Duncan and Chang’s Model. Duncan and Chang’s model [5] is a nonlinear elastic model, which has been widely used 2.2. Generalized Plasticity eTh ory and Its Original in the geotechnical engineering, especially in the numerical Constitutive Model analyses of earth dams. It is attributed to Kondner [34] who proposed the hyperbolic stress-strain function below to 2.2.1. Basic eTh ory. The generalized plasticity theory was describe the deviatoric stress-axial strain curve obtained from proposed by Zienkiewicz and Mroz (1984) [16]tomodel the triaxial tests. behaviors of sand under monotonic and cyclic loading. The Journal of Applied Mathematics 3 key futures of this theory are that neither yield surface nor In order to determine the plastic stiffness tensor, variables plastic potential surface needs to be defined explicitly, and n , n,and 𝐻 need to be den fi ed. n and n are /𝑈 𝐿/𝑈 /𝑈 consistency law is not required to determine plastic modulus. expressed as follows: In the theory, the total strain increment is divided into elastic and plastic components. 𝑔 1 Consider n =( , ) , 𝑔𝐿 2 2 1+𝑑 1+𝑑 √ √ 𝑔 𝑔 𝑒 𝑝 = 𝜀𝑑 +𝑑𝜀 , (6) (12) 𝑒 𝑝 where and = elastic and plastic strain increments, 𝑓 1 n=( , ) . respectively. 2 2 1+𝑑 1+𝑑 √ √ 𝑓 𝑓 eTh relationship between strain and stress increments is expressed as The dilatancy 𝑑 and stress ratio𝜂=𝑞/𝑝 are related as follows: = D :𝑑𝜀, (7) where D is the elastoplastic stiffness tensor given as 𝑑 = = (1+𝛼 )(𝑀 −𝜂 ). (13) 𝑔 𝑝 𝑔 𝑔 𝑒 𝑇 𝑒 D : n : n : D /𝑈 D = D − , (8) And𝑑 has a similar expression as 𝑇 𝑒 𝐻 + n : D : n 𝐿/𝑈 /𝑈 𝑑 =(1+𝛼 )(𝑀 −𝜂), (14) 𝑓 𝑓 𝑓 where D , n , n,and 𝐻 are elastic stiffness tensor, /𝑈 𝐿/𝑈 plastic flow direction vector, loading direction vector, and where 𝛼 , 𝛼 are model parameters and 𝑀 /𝑀 is equal plastic modulus under loading or unloading conditions, 𝑓 𝑔 𝑔 𝑓 to relative density. If 𝑑 = 𝑑 , associated flow rule is used, respectively. 𝑓 𝑔 otherwisenonassociated flowruleisused. The loading direction vector n is used to judge the loading In the case of unloading, the unloading plastic flow and unloading conditions: direction vector n is defined as 𝑔𝑈 ⋅ n>0 loading, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑑 󵄨 󵄨 󵄨 1 ⋅ n=0 neutral loading, (9) 󵄨 󵄨 󵄨 󵄨 (15) n =(− , ) . 𝑔𝑈 󵄨 󵄨 󵄨 󵄨 󵄨 2󵄨 2 󵄨 1+𝑑 󵄨 1+𝑑 √ √ 󵄨 𝑔 󵄨 𝑔 󵄨 󵄨 ⋅ n<0 unloading. The loading plastic modulus 𝐻 is proposed as Then, the elastoplastic stiffness tensor D can be obtained corresponding to the loading and unloading con- 𝐻 =𝐻 𝑝 𝐻 (𝐻 +𝐻 )𝐻 , (16) ditions. 𝐿 0 𝑓 V 𝑠 In the framework of generalized plasticity theory, all the components of the elastoplastic constitutive matrix are deter- where𝐻 =(1−𝜂/𝜂 ) limits the possible state and𝜂 =(1+ 𝑓 𝑓 𝑓 mined by the current state of stress and loading/unloading 1/𝛼 )𝑀 ,𝐻 =1−𝜂/𝑀 accounts for phase transformation; 𝑓 𝑓 V 𝑔 condition. 𝐻 =𝛽 𝛽 exp(−𝛽 𝜉) considers soil degradation and 𝜉 is the 𝑠 0 1 0 accumulated plastic shear strain;𝐻 =(𝜍 /𝜍) accounts MAX (−1/𝛼) 2.2.2. Pastor-Zienkiewicz-Chan Model. This model was pre- for past history and𝜍=𝑝[1−𝛼 𝜂/(1+𝛼 )/𝑀 ] which 𝑓 𝑓 𝑓 sented by Pastor et al. [19]. The relationships between elastic is the mobilized stress function; and 𝐻 , 𝛽 , 𝛽 , 𝛾 are model 0 0 1 volumetric and shear strain increments and stress increments parameters. are defined as Under unloading condition, the plastic modulus is defined as 󸀠 𝑒 𝑒 =𝐾 ,𝑑𝑞=3𝐺 , (10) 𝑒 V 𝑒𝑠 V 𝑠 𝑀 𝑀 𝑔 𝑔 𝐻 =𝐻 ( ) , >1, where𝐾 , 𝐺 are tangential bulk and shear moduli, respec- 𝑈 𝑢0 𝑒 V 𝑒𝑠 𝜂 𝜂 tively, and they are assumed to be (17) 󸀠 󸀠 𝐻 =𝐻 , ≤1, 𝑈 𝑢0 𝑝 𝑝 (11) 𝐾 =𝐾 ,𝐺 =𝐺 , 𝑒 V 𝑒𝑠𝑜 𝑒𝑠 𝑒𝑠𝑜 𝑝 𝑝 𝑜 𝑜 respectively, where𝐻 ,𝛾 are model parameters and𝜂 is the 𝑢0 𝑢 𝑢 where𝐾 ,𝐺 ,and𝑝 are model parameters. stress ratio from which unloading takes place. 𝑒𝑠𝑜 𝑒𝑠𝑜 𝑜 𝑑𝜀 𝑑𝜀 𝑑𝑝 𝐷𝑀 𝐷𝑀 𝑒𝑝 𝑑𝜎 𝑑𝜎 𝑑𝜎 𝑔𝐿 𝑔𝐿 𝑒𝑝 𝑔𝐿 𝑑𝜀 𝑑𝜀 𝑒𝑝 𝑑𝜎 𝑒𝑝 𝑑𝜀 𝑑𝜀 𝑑𝜀 𝑔𝐿 𝑔𝐿 4 Journal of Applied Mathematics 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =300 kPa 𝜎 =300 kPa 3 3 𝜎 =700 kPa 𝜎 =700 kPa 3 3 𝜎 = 1200 kPa 𝜎 = 1200 kPa 3 3 (a) (b) Figure 1: Simulation of stress-strain relationships for Original PZ-III model. 2.2.3. Modified Model. The Pastor-Zienkiewicz-Chan model where 𝑚 is a model parameter and 𝐼 =𝑝/𝑝 in which 𝑝 𝑝 𝑝 𝑐 𝑐 (PZ-III for short) has gained considerable success in describ- is the mean pressure at critical state. The critical state line is ing the behavior of sands and clays under monotonic and given by cyclic loadings. But it still has some shortcomings to predict the static or dynamic responds of sands, especially for rockfill 𝑒 =Γ−𝜆 log(𝑝 ). (20) 𝑐 𝑐 materials which are widely used in earth-rockfill dams. eTh Original PZ-III model has serious limitation in reflecting pressure dependency of soils. 3. Nuozhadu Hydropower Project Figure 1 shows the stress-strain relationships of a rockfill material under drained conventional triaxial tests using a Nuozhadu hydropower project is located in the Lancang set of parameters under different confining pressures, but River which is also named Mekong River in the down- PZ-III model gives the same 𝜀 -𝜀 curve, where 𝜀 , 𝜀 are stream in Yunnan Province, Southwest China, as shown in 1 V 1 V axial strain and volumetric strain, respectively. As confining Figure 2(a). eTh installed capacity of the powerstation is pressure ranges from 0 kPa to several MPa for a rockfill dam 5850 MW.ThemostimportantpartofNuozhaduhydropower with height of 200–300 m, the original PZ-III model cannot project is the high earth-rockfill dam with a maximum height be used to describe the mechanical behavior of rockfill dams. of 261.5 m, which is the highest one with the same type in Some relations of the original model are modified to take China and the fourth highest in the world. eTh reservoir has 8 3 into account the influence of confining pressure as astorage capacity of 237.0 × 10 m ,withthe normal storage water level of 812.5 m and dead water level of 765 m. Figure 3 shows the material zoning and construction 𝑚 𝑛 󸀠 󸀠 stages of the maximum cross-section. The elevation of the 𝑝 𝑝 𝐾 =𝐾 𝑝 ( ),𝐺 =𝐺 𝑝 ( ) , earthcorebottomand thecrest of thedam are562.6mand 𝑒 V 𝑒0 𝑎 𝑒𝑠 𝑒0 𝑎 𝑝 𝑝 𝑎 𝑎 824.1 m, respectively. The dam crest has a longitudinal length (18) 󸀠 of 630 m with a width of 18 m. eTh upstream and downstream 𝐻 =𝐻 𝑝 ( ) 𝐻 (𝐻 +𝐻 )𝐻 , slopes are at 1.9 : 1 and 1.8 : 1, respectively. eTh dam body is 𝐿 0 𝑎 𝑓 V 𝑠 composed of several different types of materials. The shells of upstream and downstream are composed of decomposed rock materials. Anti-seepage material in the earth core is clay where 𝐾 and 𝐺 are elastic constants, 𝑚 and 𝑛 are model 𝑒0 𝑒0 mixed with gravel. Adding gravel to the clay can improve the parameters to consider the eeff ct of pressure dependency. strength of clay and reduce the arching effect between shells As sand behavior is dependent on densities or void ratio, and earth core. eTh gravel material consists of fresh crushed a state pressure index, 𝐼 ,proposedbyWangetal. [35]was 𝑝 stone of breccia and granite with a maximum diameter of introduced in the PZ-III model and (13)was modiefi das 150 mm. In addition to these, the ne fi rockfill and filter materials are filled against the earth core to prevent the fine particle from being washed away. 𝑚 The dam construction was started in 2008 and was 𝑑 = = (1+𝛼 )(𝑀 𝐼 −𝜂 ), (19) 𝑔 𝑝 𝑔 𝑔 𝑝 completedatthe endof2012. Figure 2(c) shows the dam −𝜎 (kPa) 1 3 (%) 𝑑𝜀 𝑑𝜀 𝐷𝑀 Journal of Applied Mathematics 5 China Burma Laos Thailand Vietnam (a) (b) (c) (d) Figure 2: Nuozhadu dam. (a) Nuozhadu dam location, (b) project blueprint, (c) Nuozhadu dam under construction, and (d) dam site geomorphology. Table 1: Material parameters of Duncan and Chang’s EB model. under construction. Figure 3(b) demonstrates the practical construction process. Material Rockfill I Rockfill II Mixed gravel clay 𝜑 / 55.82 54.33 39.30 Δ𝜑 / 12.29 12.07 9.80 4. Experimental Validation of 𝑅 0.73 0.74 0.77 Model Parameters 𝑓 𝐾 1450 1360 520 The modified PZ-III model was implemented in a finite 𝐾 550 600 250 element code which has been successfully used to analyze 𝐾 2800 2500 900 𝑢𝑟 earth dams with Duncan and Chang’s EB model and some 𝑛 0.30 0.43 0.42 other constitutive models. A set of triaxial test data was used 𝑚 0.13 0.08 0.25 to make sure that the model has been incorporated into the FEM code accurately. eTh proposed generalized plasticity model totally needs 17 parameters. eTh model parameters used in the computa- tion of the earth-rockfill dam were obtained by tt fi ing the behavior of rockfill materials and mixed gravel clay, especially triaxial test results. Drained triaxial tests under different con- for dilatancy. With the reduction of conn fi ing pressure, n fi ing pressures were conducted to test the rockfill materials the rockfill materials tend to dilate as the experimental and mixed gravel clay, which are the main parts of the dam volumetric strain curve shows. Especially for the rockfill body. materials under low confining pressure, negative volumetric Duncan and Chang’s EB model parameters are shown in strain rapidly develops aeft r a short stage of volumetric Table 1 and the modified PZ-III model parameters in Table 2. contraction. Due to the intrinsic limitation, Duncan and As shown in Figures 4, 5, 6, 7, 8,and 9, the modified PZ- Chang’s EB model cannot simulate the dilatancy which is a III model presents a better ability to simulate the mechanics crucial feature of rockfill materials. 6 Journal of Applied Mathematics Upstream Downstream 824.1 RU1 RD3 RU3 F2 F2 F1 RD2 F1 RU2 Cofferdam 600 ED RD1 F1/F2: filter material zone I/II RU1/RD1: upstream/downstream rockfill zone I RU2/RD2: upstream/downstream rockfill zone II ED: clay mixed gravel RU3/RD3: upstream/downstream fine rockfill : electromagnetism type settlement gauges (a) 2012.12.31 812.5 2011.05.31∼2012.05.31 2010.05.31∼2011.05.31 2009.05.31∼2010.05.31 2008.02.15∼2008.05.31 2008.05.31∼2009.05.31 (b) Figure 3: eTh maximum cross-section. (a) Material zoning and (b) construction stage. −4 −3 −2 6000 −1 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 1 300 kPa 900 kPa 300 kPa 900 kPa 2500 kPa 2500 kPa 1500 kPa 1500 kPa Duncan-Chang EB Duncan-Chang EB (a) (b) Figure 4: Comparison between tfi tings of Duncan and Chang’s EB model and experimental triaxial tests results for rockfill material I. −𝜎 (kPa) 1 3 (%) Journal of Applied Mathematics 7 10000 −4 −3 −2 −1 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 1 1 300 kPa 900 kPa 300 kPa 900 kPa 2500 kPa 2500 kPa 1500 kPa 1500 kPa Modified PZ Modified PZ (a) (b) Figure 5: Comparison between tfi tings of the modified PZ-III model and experimental triaxial tests results for rockfill material I. −3 −2 −1 0 3 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =900 kPa 𝜎 =900 kPa 𝜎 =300 kPa 𝜎 =300 kPa 3 3 3 𝜎 = 1500 kPa 𝜎 = 2500 kPa 𝜎 = 1500 kPa 𝜎 = 2500 kPa 3 3 3 3 Duncan-Chang EB Duncan-Chang EB (a) (b) Figure 6: Comparison between tfi tings of Duncan and Chang’s EB model and experimental triaxial tests results for rockfill material II. −3 −2 −1 0 3 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =900 kPa 𝜎 =900 kPa 𝜎 =300 kPa 𝜎 =300 kPa 3 3 3 𝜎 = 1500 kPa 𝜎 = 1500 kPa 𝜎 = 2500 kPa 𝜎 = 2500 kPa 3 3 3 Modified PZ Modified PZ (a) (b) Figure 7: Comparison between tfi tings of the modified PZ-III model and experimental triaxial tests results for rockfill material II. −𝜎 (kPa) −𝜎 (kPa) −𝜎 (kPa) 1 3 1 3 1 3 (%) (%) (%) 8 Journal of Applied Mathematics 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =300 kPa 𝜎 =900 kPa 𝜎 =300 kPa 𝜎 =900 kPa 3 3 3 3 𝜎 = 1500 kPa 𝜎 = 2500 kPa 𝜎 = 1500 kPa 𝜎 = 2500 kPa 3 3 3 3 Duncan-Chang EB Duncan-Chang EB (a) (b) Figure 8: Comparison between tfi tings of Duncan and Chang’s EB model and experimental triaxial tests results for clay. 6000 0 4000 1 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =900 kPa 𝜎 =300 kPa 𝜎 =900 kPa 𝜎 =300 kPa 3 3 3 3 𝜎 = 1500 kPa 𝜎 = 1500 kPa 𝜎 = 2500 kPa 𝜎 = 2500 kPa 3 3 3 Modified PZ Modified PZ (a) (b) Figure 9: Comparison between tfi tings of the modified PZ-III model and experimental triaxial tests results for clay. First, the 2D ni fi te element mesh of the maximum cross- section of the dam was discretized according to the material zoning and construction design (see Figure 3). Then, the 2D mesh was extended to 3D mesh in accordance with contour line of the river valley. Figure 10 shows the 3D mesh of the Nuozhadu dam with 8095 brick and degenerated brick elements and 8340 nodes. The numerical simulations contain two stages, filling and impounding. During the filling stage, the dam body mainly subjects to body weight. en, Th at the end of construction, upstream water level goes up to the normal storage water Figure 10: 3D FEM mesh of Nuozhadu dam. level. The interaction between pore water and soil skeleton was considered through the whole numerical computation. 5. Three-Dimensional Finite Element Analyses 5.2. Results and Analyses 5.1. Computation Model. The numerical analyses were per- 5.2.1. Numerical Results Analyses. Figures 11 and 12 show the formed to simulate the performance of the dam during numerical results of finite element analyses with Duncan construction and impounding periods with eecti ff ve stress and Chang’s EB model and the modified PZ-III model, finite element analysis. respectively. −𝜎 (kPa) −𝜎 (kPa) 1 3 1 3 (%) (%) 0.5 0.4 0.2 0.3 0.1 2.5 0.2 3.5 2.5 1.5 0.3 1.5 0.5 1.5 0.5 0.6 0.7 0.1 0.5 Journal of Applied Mathematics 9 0.9 0.7 −2 0.1 (a) (b) −0.2 1.3 0.3 (c) (d) Figure 11: Displacement and stress contour of the maximum section for Duncan and Chang’s EB model: (a) displacement along river (m), (b) vertical displacement (m), (c) major principle stress (MPa), and (d) minor principle stress (MPa). −0.5 −2.9 −2.5 (a) (b) 0.1 3.5 (c) (d) Figure 12: Displacement and stress contour of the maximum section for the modified PZ-III model: (a) displacement along river (m), (b) vertical displacement (m), (c) major principle stress (MPa), and (d) minor principle stress (MPa). Through the comparison and analysis of the numerical (1) Aeft r the reservoir is impounded, upward displace- results (Figures 11 and 12), we can nd fi some similarities and ment as large as 0.7 m (see Figure 11(b))develops differences for these two models. on the upstream shell near dam crest for EB model On one hand, we can see many similar places in the and nearly 0 m for modified PZ-III model (see distributions of displacements and stresses. Figure 12(b)). In fact, monitoring data of practical engineering projects shows that no large upward (1) Aeft rthereservoirimpounding,duetothehugewater displacement happened aer ft impounding. This is pressure on upstream dam, horizontal displacement due to its weakness of EB model to distinguish the develops toward the downstream, and the largest loading and unloading condition during the water displacement is about 1.05 m for EB model and 0.74 m impounding. for modified PZ-III model. (2) In the distribution of minor principle stress (Figures (2) eTh maximum settlement occurs in the middle of core 11(d) and 12(d)), negative stress (i.e., tensile stress) wall due to lower modulus of clayey soil. occurs in theupstreamshell forEBmodel,whereas very little tensile stress exists for modified PZ-III (3) Because of the tremendous differences of modulus model. As we know, rockfill material is a typical kind between rockfill material and clayey soil, there exists of cohesionless coarse-grained soil, which means that obvious arching eeff ct in the core wall. it has no tensile strength. er Th efore, the existence of (4) Eeff ctive stress in upstream shell is less than the large area of tensile stress in the upstream shell is downstream shell due to the water pressure in the unreasonable. upstream shell. On the other hand, some differences also exist, which 5.2.2. Comparison between Numerical and In Situ Monitoring illustrate the advantages of modified PZ-III model. Data. Settlement is a key indicator to assess the safety of an 0.6 0.5 0.5 −0.5 1.5 0.5 −1 −1 −1.5 −2 1.5 2.5 −1.5 −2.4 −0.2 0.5 3 10 Journal of Applied Mathematics Elevation 655 m Time (a) −1000 0 1000 2000 3000 4000 Elevation 701 m Settlement (mm) In-situ EB Modified PZ Time Figure 13: Comparison between in situ monitoring settlement and (b) FEM results. Table 2: Material parameters of the modified PZ-III model. Material Rockfill I Rockfill II Mixed gravel clay 𝐾 500 1000 300 𝐺 1500 3000 900 Elevation 751 m 𝑚 0.50 0.50 0.50 𝑛 0.50 0.50 0.50 𝛼 0.45 0.45 0.45 𝛼 0.45 0.45 0.45 𝑀 1.05 0.90 0.60 Time 𝑀 1.60 1.35 1.10 𝑔𝑐 In-situ 𝛽 0.00 0.00 0.00 EB 𝛽 0.00 0.00 0.00 Modified PZ Γ 0.34 0.31 0.34 (c) 𝜆 0.10 0.09 0.03 𝑚 0.35 0.40 0.0 Figure 14: Comparison between in situ monitoring settlement and FEM results. 𝐻 800 1200 900 𝛾 55 5 𝛾 55 5 of rockfill materials was not considered, the FEM result of 𝐻 /MPa 9 9 10 𝑢0 settlement was below than the in situ monitoring data. As an elastoplastic model, the PZ-III model is capable of representing the mechanical behavior of soils better than earth dam. Figures 13 and 14 show the in situ monitoring data nonlinear elastic model such as Duncan and Chang’s EB and FEM results of settlement in the maximum cross-section. model. And the above ni fi te element analyses also proved it. The in situ data were obtained from electromagnetism type settlement gauges which were embedded during construction in the dam (as shown in Figure 3(a)). Through the compar- 6. Conclusions isons of in situ monitoring and numerical results, we can see that the modified PZ-III model gave a better prediction than This paper presents a modified PZ-III model based on the the EB model. However, as deformation induced by wetting generalized theory and original Pastor-Zienkiewicz-Chan Elevation (m) Settlement (mm) Settlement (mm) Settlement (mm) 2010/9/1 2010/1/1 2011/10/1 2010/6/10 2011/2/28 2011/12/20 2010/11/17 2012/3/9 2011/8/27 2011/4/26 2011/10/3 2012/5/28 2012/2/23 2012/3/11 2012/8/16 2012/8/18 2012/8/21 𝑓𝑐 Journal of Applied Mathematics 11 modeltosimulatethe stress-strainrelationshipofrockfill [11] P. V. Lade and J. M. Duncan, “Elastoplastic stress-strain theory for cohesionless soil,” Journal of the Geotechnical Engineering materials. Division,vol.101,no. 10,pp. 1037–1053, 1975. Triaxial test results of the filling materials of Nuozhadu [12] I. S. Sandler, F. L. DiMaggio, and G. Y. Baladi, “Generalized dam were used to validate the proposed model and determine cap model for geological materials,” Journal of the Geotechnical the model parameters of Duncan and Chang’s EB model and Engineering Division,vol.102,no. 7, pp.683–699,1976. the modified PZ-III model, respectively. eTh simulations of [13] X.-S. Li, Y. F. Dafalias, and Z.-L. Wang, “State-dependent dila- triaxial stress-strain response show that the modified PZ- tancy in critical-state constitutive modelling of sand,” Canadian III model is capable of representing the key features of Geotechnical Journal,vol.36, no.4,pp. 599–611, 1999. cohesionless soil, such as nonlinearity, dilatancy, and pressure [14] Y.-P. Yao and D. Sun, “Application of Lade’s criterion to Cam- dependency. clay model,” Journal of Engineering Mechanics,vol.126,no. 1, The proposed model has been incorporated into a ni fi te pp. 112–119, 2000. element code to simulate the static response of a high earth- [15] G.Y.Baladiand B. Rohani,“Elastic-plasticmodel forsaturated rockfill dam in China. eTh results were compared with those sand,” Journal of the Geotechnical Engineering Division,vol.105, of Duncan and Chang’s EB model. The two set of results no. 4, pp. 465–480, 1979. have both similarities and differences and the differences [16] O. Zienkiewicz and Z. Mroz, “Generalized plasticity formu- illustrate the advantages of the modified PZ-III model. The lation and applications to geomechanics,” in Mechanics of comparisons of FEM results, and in situ monitoring data Engineering Materials,C.S.Desai andR.H.Gallagher,Eds., pp. showed that the modified PZ-III model can give a better 655–679, John Wiley & Sons, New York, NY, USA, 1984. description of deformation of the earth-rockfill dam than [17] C. S. Desai and M. O. Faruque, “Constitutive model for Duncan and Chang’s EB model. geological materials,” Journal of Engineering Mechanics,vol.110, no.9,pp. 1391–1408, 1984. Acknowledgments [18] S. B. R. Murthy, A. Vatsala, and T. S. Nagaraj, “Revised Cam- clay model,” Journal of Geotechnical Engineering,vol.117,no. 6, This work was supported by the National Nature Science pp.851–871,1991. Foundation of China (51179092) and the State Key Laboratory [19] M. Pastor,O.C.Zienkiewicz, andA.H.C.Chan, “Generalized of Hydroscience and Engineering Project (2012-KY-02 and plasticity and the modelling of soil behaviour,” International 2013-KY-4). Journal for Numerical & Analytical Methods in Geomechanics, vol. 14,no. 3, pp.151–190,1990. [20] Z. Mroz and O. Zienkiewicz, “Uniform formulation of constitu- References tive equations for clays and sands,” in Mechanics of Engineering Materials,C.S.Desai andR.H.Gallangher, Eds.,pp. 415–449, [1] J. M. Duncan, “State of the art: limit equilibrium and finite- John Wiley & Sons, New York, NY, USA, 1984. element analysis of slopes,” Journal of Geotechnical and Geoen- vironmental Engineering,vol.122,no. 7, pp.577–596,1996. [21] G. Wang and J.-M. Zhang, “Dynamic consolidation finite element analysis of a sediment-protecting dyke under ocean [2] M. A. Biot, “General theory of three-dimensional consolida- wave loading,” Rock and Soil Mechanics,vol.27, no.4,pp. 555– tion,” Journal of Applied Physics,vol.12, no.2,pp. 155–164, 1941. 560, 2006. [3] R. S. Sandhu and E. L. Wilson, “Finite element analysis of [22] M. Alyami, M. Rouainia, and S. M. Wilkinson, “Numerical anal- seepage in elastic media,” Journal of the Engineering Mechanics ysis of deformation behaviour of quay walls under earthquake Division,vol.95, no.3,pp. 641–652, 1969. loading,” Soil Dynamics and Earthquake Engineering,vol.29,no. [4] J.T.Christian andJ.W.Boehmer,“Planestrainconsolidation 3, pp. 525–536, 2009. by finite elements,” JournalofSoilMechanics &Foundations [23] H. Li,P.Manuel, andT.Li, “Application of an generalized Division,vol.96, no.4,pp. 1435–1457, 1970. plasticity model to ultra-high rockfill dam,” in Proceedings [5] J.M.Duncanand C.-Y.Chang,“Nonlinearanalysisofstressand of the 12th International Conference on Engineering, Science, strain in soils,” Journal of the Soil Mechanics and Foundations Construction, and Operations in Challenging Environments— Division,vol.96, no.5,pp. 1629–1653, 1970. Earth and Space, pp. 385–398, Honolulu, Hawaii, USA, March [6] J. M. Duncan, P. M. Byrne, K. S. Wong, and P. Mabry, “Strength, stress-strain and bulk modulus parameters for finite element [24] T. Li and H. Zhang, “Dynamic parameter verification of P- analyses of stresses and movements in soil masses,” Tech. Rep. Z model and its application of dynamic analysis on rockfill UCB/GT/80-01, University of California, Berkeley, Calif, USA, dam,” in Proceedings of the 12th International Conference on Engineering, Science, Construction, and Operations in Challeng- [7] D.C.Drucker,R.E.Gibson, andD.J.Henkel, “Soilmechanics ing Environments—Earth and Space, pp. 2706–2713, Honolulu, and work-hardening theories of plasticity,” Transactions of the Hawaii, USA, March 2010. American Society of Civil Engineers,vol.122,pp. 338–346, 1957. [25] M. Pastor, “A generalized plasticity model for anisotropic [8] K. Roscoe, A. Schofield, and C. Wroth, “On the yielding of soils,” behaviour of sand,” Computer Methods and Advances in Geome- Geotechnique,vol.8,no. 1, pp.22–53,1958. chanics,vol.1,pp. 661–668, 1991. [9] K. Roscoe, A. Schofield, and A. u Th rairajah, “Yielding of clays [26] G. Bolzon,B.A.Schrefler, andO.C.Zienkiewicz, “Elastoplastic in states wetter than critical,” Geotechnique,vol.13, no.3,pp. soil constitutive laws generalized to partially saturated states,” 211–240, 1963. Geotechnique,vol.46, no.2,pp. 279–289, 1996. [10] J. Burland, “Correspondence on ‘eTh yielding and dilation of [27] H. I. Ling and H. Liu, “Pressure-level dependency and densifi- clay’,” Geotechnique,vol.15, pp.211–214,1965. cation behavior of sand through generalized plasticity model,” 12 Journal of Applied Mathematics Journal of Engineering Mechanics,vol.129,no. 8, pp.851–860, [28] H. I. Ling and S. Yang, “Unified sand model based on the critical state and generalized plasticity,” Journal of Engineering Mechanics,vol.132,no. 12,pp. 1380–1391, 2006. [29] N. D. Marschi, C. K. Chan, and H. B. Seed, “Evaluation of properties of rockfill materials,” Journal of the Soil Mechanics and Foundations Division,vol.98, no.1,pp. 95–114,1972. [30] R. J. Marsal, “Large scale testing of rockfill materials,” Journal of the Soil Mechanics and Foundations Division,vol.93, no.2,pp. 27–43, 1967. [31] R. J. Marsal, “Mechanical properties of rockfill,” in Embankment Dam Engineering, pp. 109–200, John Wiley & Sons, New York, NY, USA, 1973. [32] P. V. Lade, J. A. Yamamuro, and P. A. Bopp, “Significance of particle crushing in granular materials,” Journal of Geotechnical and Geoenvironmental Engineering,vol.122,no. 4, pp.309–316, [33] B. O. Hardin, “Crushing of soil particles,” Journal of Geotechnical Engineering, vol. 111, no. 10, pp. 1177–1192, 1985. [34] R. L. Kondner, “Hyperbolic stress-strain response: cohesive soils,” Journal of the Soil Mechanics and Foundations Division, vol. 89, no. 1, pp. 115–143, 1963. [35] Z.-L. Wang, Y. F. Dafalias, X.-S. Li, and F. I. Makdisi, “State pres- sure index for modeling sand behavior,” Journal of Geotechnical and Geoenvironmental Engineering,vol.128,no. 6, pp.511–519, 2002. 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Comparison between Duncan and Chang’s EB Model and the Generalized Plasticity Model in the Analysis of a High Earth-Rockfill Dam

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Hindawi Publishing Corporation
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Copyright © 2013 Weixin Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1687-0042
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10.1155/2013/709430
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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 709430, 12 pages http://dx.doi.org/10.1155/2013/709430 Research Article Comparison between Duncan and Chang’s EB Model and the Generalized Plasticity Model in the Analysis of a High Earth-Rockfill Dam Weixin Dong, Liming Hu, Yu Zhen Yu, and He Lv State Key Laboratory of Hydro-Science and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China Correspondence should be addressed to Yu Zhen Yu; yuyuzhen@tsinghua.edu.cn Received 4 June 2013; Revised 19 August 2013; Accepted 20 August 2013 Academic Editor: Fayun Liang Copyright © 2013 Weixin Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Nonlinear elastic model and elastoplastic model are two main kinds of constitutive models of soil, which are widely used in the numerical analyses of soil structure. In this study, Duncan and Chang’s EB model and the generalized plasticity model proposed by Pastor, Zienkiewicz, and Chan was discussed and applied to describe the stress-strain relationship of rockfill materials. The two models were validated using the results of triaxial shear tests under different confining pressures. The comparisons between the tfi tings of models and test data showed that the modified generalized plasticity model is capable of simulating the mechanical behaviours of rockfill materials. The modified generalized plastici ty model was implemented into a finite element code to carry out static analyses of a high earth-rockfill dam in China. Nonlinear elastic analyses were also performed with Duncan and Chang’s EB model in the same program framework. eTh comparisons of FEM results and in situ monitoring data showed that the modified PZ-III model can give a better description of deformation of the earth-rockfill dam than Duncan and Chang’s EB model. 1. Introduction mainly grouped in two categories: nonlinear elastic models and elastoplastic models. The constitutive model of soil is the keystone in the ni fi te For nonlinear elastic model, the nonlinear characteristic element analyses of geotechnical structures. A suitable con- of soil stress-strain relationship is considered by sectionalized stitutive model can simulate the stress-strain relationships of linearization. A typical nonlinear elastic model is Duncan soils under static or dynamic conditions. Numerical analysis, and Chang’s Model [5, 6], which has been widely used in especially for ni fi te element method incorporated with soil the numerical analyses of earth-rockfill dams, as the model constitutive models, has played a very important role in parameters are quite easy to be determined from conven- geotechnical analyses which always include complex bound- tional triaxial tests. And, a lot of experience of application has ary conditions, nonlinearity of material, and geometry [1]. been accumulated for this model. However, nonlinear elastic Biot presented the famous three-dimensional consolida- models also have some inherent limitations to represent the tion theory based on the eecti ff ve stress theory, equilibrium stress-strain characteristics of soils, such as shear-induced equation, and continuity condition [2]. However, it is quite dilatancy and stress path dependency. dicffi ulttogivethe theoreticalsolutionofBiot’sconsolidation Elastoplastic models would be very adequate in describ- theory except for few simple problems. Up to the 1960s, ing many key features of soils. Classical elastoplastic models with the rapid development of electronic computer and are based on the plastic incremental theory composed of yield constitutive models of soils, Biot’s consolidation theory was condition, flow rule, and hardening law. In the 1950s, Drucker successfully implemented in finite element codes to study the et al. (1957) [7] suggested a cap yield surface controlled by behavior of geotechnical structures [3, 4]. So far, thousands volumetric strain. Roscoe et al. [8, 9]proposedthe concepts of constitutive models have been proposed, which can be of critical statelineand stateboundarysurface,and then 2 Journal of Applied Mathematics they built the Original Cam Clay Model based on triaxial Consider tests. Burland [10]suggested adieff rentenergyequation andthenestablished theModiefi d CamClayModel.Since 𝜎 −𝜎 = , 1 3 (1) 𝑎+𝑏𝜀 the establishment of Cam Clay Model, some other types of elastoplastic constitutive models have also achieved great in which𝑎 and𝑏 are model constants. development [11–18]. Among these models, the generalized In this constitutive model, the tangential Young’s modu- plasticity model [16, 19, 20] can simulate the static and lus𝐸 and tangential bulk modulus𝐵 areusedtosimulatethe 𝑡 𝑡 dynamicmechanicalbehaviorsofclaysandsands.Thismodel nonlinear elastic response of soils, which are assumed to be is very flexible and convenient to extend, as the complicated yield or plastic potential surfaces need not to be specified 𝑛 𝜎 2 explicitly. And the model has been used successfully in the 𝐸 =𝐾𝑃 ( ) (1−𝑅 𝑆 ), 𝑡 𝑎 𝑓 𝑙 static or dynamic analyses of some geotechnical structures (2) [21–24]. Furthermore, based on the framework of generalized plasticity theory [16], some limitations of the original model 𝐵 =𝐾 𝑃 ( ) , 𝑡 𝑏 𝑎 have been solved [25–28], such as pressure dependency, den- sification under cyclic loading. The details of the generalized where𝑃 is the atmospheric pressure,𝐾 and𝐾 are modulus 𝑎 𝑏 plasticity theory and the original and proposed modified numbers, 𝑛 and 𝑚 are exponents determining the rate of Pastor-Zienkiewicz-Chan’s models will be introduced in the variation of moduli with confining pressure, and 𝑅 is the sections below. failure ratio with a invariable value less than 1. However, little experience has as yet been accumulated in The Mohr-Coulomb failure criterion is adopted in the applying the generalized plasticity model to the simulation model, and 𝑆 is a factor defined as shear stress level given of rockfill materials. And we know that rockfill material is by quite different from sands in mechanical properties [ 29–31]. eTh rockfillmaterialhas largeparticlesizeand sharpedges (1−sin𝜙)(𝜎 −𝜎 ) 1 3 𝑆 = . (3) and corners, which can result in remarkable particle breakage 2𝑐⋅ cos𝜙+2𝜎 ⋅ sin𝜙 andchangethe shear-induceddilation[32, 33]. On the other hand, though the generalized plasticity model has gained In theunloading andreloading stage, thetangential great success in the modeling of soils, the application of this Young’s modulus is defined as model in the large-scale ni fi te element analyses of earth dams was less reported. 𝐸 =𝐾 𝑃 ( ) . (4) 𝑢𝑟 𝑢𝑟 𝑎 In this study, the original generalized plasticity model was modified to consider the stress-strain relationships of rockfill materials, as most of previous studies focused on sands So far, the model has 8 parameters, 𝑐 , 𝜑 , 𝐾 , 𝐾 , 𝑛 , 𝑅 , 𝑢𝑟 𝑓 and clays. en, Th based on conventional triaxial test data, 𝐾 , 𝑚 . es Th e parameters can be determined with a set of conventional triaxial tests. the model parameters for dam materials of the Nuozhadu high earth-rockfill dam in Southwest China are determined. In general, a curved Mohr-Coulomb failure envelop is Finally, the static simulation of this dam is carried out by adopted by setting𝑐=0 and letting 𝜑 vary with confining using a finite element code incorporating with Duncan and pressure according to Chang’s EB model and the modified generalize plasticity model. The comparison of numerical results and in situ mon- 3 𝜑=𝜑 −Δ𝜑 log( ). (5) itoring data illustrates the advantages of modified generalized plasticity model in the simulation of earth-rockfill dams. eTh n parameters 𝑐 and𝜑 are replaced by𝜑 andΔ𝜑 . Although Duncan and Chang’s EB constitutive model is 2. Constitutive Model Descriptions quite simple, it has gained significant success in geotechnical engineering. On one hand, it is easy to obtain the model Two constitutive models of soils were used in the n fi ite parameters; on the other hand, much experience has been element analyses. One is the Duncan and Chang’s EB model accumulated. Nevertheless, it cannot incorporate dilatancy belonging to nonlinear elastic model, the other one is the which has an important inu fl ence in the mechanical behavior generalized plasticity model. of soils. And furthermore, it can only consider unloading process in a crude way. 2.1. Duncan and Chang’s Model. Duncan and Chang’s model [5] is a nonlinear elastic model, which has been widely used 2.2. Generalized Plasticity eTh ory and Its Original in the geotechnical engineering, especially in the numerical Constitutive Model analyses of earth dams. It is attributed to Kondner [34] who proposed the hyperbolic stress-strain function below to 2.2.1. Basic eTh ory. The generalized plasticity theory was describe the deviatoric stress-axial strain curve obtained from proposed by Zienkiewicz and Mroz (1984) [16]tomodel the triaxial tests. behaviors of sand under monotonic and cyclic loading. The Journal of Applied Mathematics 3 key futures of this theory are that neither yield surface nor In order to determine the plastic stiffness tensor, variables plastic potential surface needs to be defined explicitly, and n , n,and 𝐻 need to be den fi ed. n and n are /𝑈 𝐿/𝑈 /𝑈 consistency law is not required to determine plastic modulus. expressed as follows: In the theory, the total strain increment is divided into elastic and plastic components. 𝑔 1 Consider n =( , ) , 𝑔𝐿 2 2 1+𝑑 1+𝑑 √ √ 𝑔 𝑔 𝑒 𝑝 = 𝜀𝑑 +𝑑𝜀 , (6) (12) 𝑒 𝑝 where and = elastic and plastic strain increments, 𝑓 1 n=( , ) . respectively. 2 2 1+𝑑 1+𝑑 √ √ 𝑓 𝑓 eTh relationship between strain and stress increments is expressed as The dilatancy 𝑑 and stress ratio𝜂=𝑞/𝑝 are related as follows: = D :𝑑𝜀, (7) where D is the elastoplastic stiffness tensor given as 𝑑 = = (1+𝛼 )(𝑀 −𝜂 ). (13) 𝑔 𝑝 𝑔 𝑔 𝑒 𝑇 𝑒 D : n : n : D /𝑈 D = D − , (8) And𝑑 has a similar expression as 𝑇 𝑒 𝐻 + n : D : n 𝐿/𝑈 /𝑈 𝑑 =(1+𝛼 )(𝑀 −𝜂), (14) 𝑓 𝑓 𝑓 where D , n , n,and 𝐻 are elastic stiffness tensor, /𝑈 𝐿/𝑈 plastic flow direction vector, loading direction vector, and where 𝛼 , 𝛼 are model parameters and 𝑀 /𝑀 is equal plastic modulus under loading or unloading conditions, 𝑓 𝑔 𝑔 𝑓 to relative density. If 𝑑 = 𝑑 , associated flow rule is used, respectively. 𝑓 𝑔 otherwisenonassociated flowruleisused. The loading direction vector n is used to judge the loading In the case of unloading, the unloading plastic flow and unloading conditions: direction vector n is defined as 𝑔𝑈 ⋅ n>0 loading, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑑 󵄨 󵄨 󵄨 1 ⋅ n=0 neutral loading, (9) 󵄨 󵄨 󵄨 󵄨 (15) n =(− , ) . 𝑔𝑈 󵄨 󵄨 󵄨 󵄨 󵄨 2󵄨 2 󵄨 1+𝑑 󵄨 1+𝑑 √ √ 󵄨 𝑔 󵄨 𝑔 󵄨 󵄨 ⋅ n<0 unloading. The loading plastic modulus 𝐻 is proposed as Then, the elastoplastic stiffness tensor D can be obtained corresponding to the loading and unloading con- 𝐻 =𝐻 𝑝 𝐻 (𝐻 +𝐻 )𝐻 , (16) ditions. 𝐿 0 𝑓 V 𝑠 In the framework of generalized plasticity theory, all the components of the elastoplastic constitutive matrix are deter- where𝐻 =(1−𝜂/𝜂 ) limits the possible state and𝜂 =(1+ 𝑓 𝑓 𝑓 mined by the current state of stress and loading/unloading 1/𝛼 )𝑀 ,𝐻 =1−𝜂/𝑀 accounts for phase transformation; 𝑓 𝑓 V 𝑔 condition. 𝐻 =𝛽 𝛽 exp(−𝛽 𝜉) considers soil degradation and 𝜉 is the 𝑠 0 1 0 accumulated plastic shear strain;𝐻 =(𝜍 /𝜍) accounts MAX (−1/𝛼) 2.2.2. Pastor-Zienkiewicz-Chan Model. This model was pre- for past history and𝜍=𝑝[1−𝛼 𝜂/(1+𝛼 )/𝑀 ] which 𝑓 𝑓 𝑓 sented by Pastor et al. [19]. The relationships between elastic is the mobilized stress function; and 𝐻 , 𝛽 , 𝛽 , 𝛾 are model 0 0 1 volumetric and shear strain increments and stress increments parameters. are defined as Under unloading condition, the plastic modulus is defined as 󸀠 𝑒 𝑒 =𝐾 ,𝑑𝑞=3𝐺 , (10) 𝑒 V 𝑒𝑠 V 𝑠 𝑀 𝑀 𝑔 𝑔 𝐻 =𝐻 ( ) , >1, where𝐾 , 𝐺 are tangential bulk and shear moduli, respec- 𝑈 𝑢0 𝑒 V 𝑒𝑠 𝜂 𝜂 tively, and they are assumed to be (17) 󸀠 󸀠 𝐻 =𝐻 , ≤1, 𝑈 𝑢0 𝑝 𝑝 (11) 𝐾 =𝐾 ,𝐺 =𝐺 , 𝑒 V 𝑒𝑠𝑜 𝑒𝑠 𝑒𝑠𝑜 𝑝 𝑝 𝑜 𝑜 respectively, where𝐻 ,𝛾 are model parameters and𝜂 is the 𝑢0 𝑢 𝑢 where𝐾 ,𝐺 ,and𝑝 are model parameters. stress ratio from which unloading takes place. 𝑒𝑠𝑜 𝑒𝑠𝑜 𝑜 𝑑𝜀 𝑑𝜀 𝑑𝑝 𝐷𝑀 𝐷𝑀 𝑒𝑝 𝑑𝜎 𝑑𝜎 𝑑𝜎 𝑔𝐿 𝑔𝐿 𝑒𝑝 𝑔𝐿 𝑑𝜀 𝑑𝜀 𝑒𝑝 𝑑𝜎 𝑒𝑝 𝑑𝜀 𝑑𝜀 𝑑𝜀 𝑔𝐿 𝑔𝐿 4 Journal of Applied Mathematics 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =300 kPa 𝜎 =300 kPa 3 3 𝜎 =700 kPa 𝜎 =700 kPa 3 3 𝜎 = 1200 kPa 𝜎 = 1200 kPa 3 3 (a) (b) Figure 1: Simulation of stress-strain relationships for Original PZ-III model. 2.2.3. Modified Model. The Pastor-Zienkiewicz-Chan model where 𝑚 is a model parameter and 𝐼 =𝑝/𝑝 in which 𝑝 𝑝 𝑝 𝑐 𝑐 (PZ-III for short) has gained considerable success in describ- is the mean pressure at critical state. The critical state line is ing the behavior of sands and clays under monotonic and given by cyclic loadings. But it still has some shortcomings to predict the static or dynamic responds of sands, especially for rockfill 𝑒 =Γ−𝜆 log(𝑝 ). (20) 𝑐 𝑐 materials which are widely used in earth-rockfill dams. eTh Original PZ-III model has serious limitation in reflecting pressure dependency of soils. 3. Nuozhadu Hydropower Project Figure 1 shows the stress-strain relationships of a rockfill material under drained conventional triaxial tests using a Nuozhadu hydropower project is located in the Lancang set of parameters under different confining pressures, but River which is also named Mekong River in the down- PZ-III model gives the same 𝜀 -𝜀 curve, where 𝜀 , 𝜀 are stream in Yunnan Province, Southwest China, as shown in 1 V 1 V axial strain and volumetric strain, respectively. As confining Figure 2(a). eTh installed capacity of the powerstation is pressure ranges from 0 kPa to several MPa for a rockfill dam 5850 MW.ThemostimportantpartofNuozhaduhydropower with height of 200–300 m, the original PZ-III model cannot project is the high earth-rockfill dam with a maximum height be used to describe the mechanical behavior of rockfill dams. of 261.5 m, which is the highest one with the same type in Some relations of the original model are modified to take China and the fourth highest in the world. eTh reservoir has 8 3 into account the influence of confining pressure as astorage capacity of 237.0 × 10 m ,withthe normal storage water level of 812.5 m and dead water level of 765 m. Figure 3 shows the material zoning and construction 𝑚 𝑛 󸀠 󸀠 stages of the maximum cross-section. The elevation of the 𝑝 𝑝 𝐾 =𝐾 𝑝 ( ),𝐺 =𝐺 𝑝 ( ) , earthcorebottomand thecrest of thedam are562.6mand 𝑒 V 𝑒0 𝑎 𝑒𝑠 𝑒0 𝑎 𝑝 𝑝 𝑎 𝑎 824.1 m, respectively. The dam crest has a longitudinal length (18) 󸀠 of 630 m with a width of 18 m. eTh upstream and downstream 𝐻 =𝐻 𝑝 ( ) 𝐻 (𝐻 +𝐻 )𝐻 , slopes are at 1.9 : 1 and 1.8 : 1, respectively. eTh dam body is 𝐿 0 𝑎 𝑓 V 𝑠 composed of several different types of materials. The shells of upstream and downstream are composed of decomposed rock materials. Anti-seepage material in the earth core is clay where 𝐾 and 𝐺 are elastic constants, 𝑚 and 𝑛 are model 𝑒0 𝑒0 mixed with gravel. Adding gravel to the clay can improve the parameters to consider the eeff ct of pressure dependency. strength of clay and reduce the arching effect between shells As sand behavior is dependent on densities or void ratio, and earth core. eTh gravel material consists of fresh crushed a state pressure index, 𝐼 ,proposedbyWangetal. [35]was 𝑝 stone of breccia and granite with a maximum diameter of introduced in the PZ-III model and (13)was modiefi das 150 mm. In addition to these, the ne fi rockfill and filter materials are filled against the earth core to prevent the fine particle from being washed away. 𝑚 The dam construction was started in 2008 and was 𝑑 = = (1+𝛼 )(𝑀 𝐼 −𝜂 ), (19) 𝑔 𝑝 𝑔 𝑔 𝑝 completedatthe endof2012. Figure 2(c) shows the dam −𝜎 (kPa) 1 3 (%) 𝑑𝜀 𝑑𝜀 𝐷𝑀 Journal of Applied Mathematics 5 China Burma Laos Thailand Vietnam (a) (b) (c) (d) Figure 2: Nuozhadu dam. (a) Nuozhadu dam location, (b) project blueprint, (c) Nuozhadu dam under construction, and (d) dam site geomorphology. Table 1: Material parameters of Duncan and Chang’s EB model. under construction. Figure 3(b) demonstrates the practical construction process. Material Rockfill I Rockfill II Mixed gravel clay 𝜑 / 55.82 54.33 39.30 Δ𝜑 / 12.29 12.07 9.80 4. Experimental Validation of 𝑅 0.73 0.74 0.77 Model Parameters 𝑓 𝐾 1450 1360 520 The modified PZ-III model was implemented in a finite 𝐾 550 600 250 element code which has been successfully used to analyze 𝐾 2800 2500 900 𝑢𝑟 earth dams with Duncan and Chang’s EB model and some 𝑛 0.30 0.43 0.42 other constitutive models. A set of triaxial test data was used 𝑚 0.13 0.08 0.25 to make sure that the model has been incorporated into the FEM code accurately. eTh proposed generalized plasticity model totally needs 17 parameters. eTh model parameters used in the computa- tion of the earth-rockfill dam were obtained by tt fi ing the behavior of rockfill materials and mixed gravel clay, especially triaxial test results. Drained triaxial tests under different con- for dilatancy. With the reduction of conn fi ing pressure, n fi ing pressures were conducted to test the rockfill materials the rockfill materials tend to dilate as the experimental and mixed gravel clay, which are the main parts of the dam volumetric strain curve shows. Especially for the rockfill body. materials under low confining pressure, negative volumetric Duncan and Chang’s EB model parameters are shown in strain rapidly develops aeft r a short stage of volumetric Table 1 and the modified PZ-III model parameters in Table 2. contraction. Due to the intrinsic limitation, Duncan and As shown in Figures 4, 5, 6, 7, 8,and 9, the modified PZ- Chang’s EB model cannot simulate the dilatancy which is a III model presents a better ability to simulate the mechanics crucial feature of rockfill materials. 6 Journal of Applied Mathematics Upstream Downstream 824.1 RU1 RD3 RU3 F2 F2 F1 RD2 F1 RU2 Cofferdam 600 ED RD1 F1/F2: filter material zone I/II RU1/RD1: upstream/downstream rockfill zone I RU2/RD2: upstream/downstream rockfill zone II ED: clay mixed gravel RU3/RD3: upstream/downstream fine rockfill : electromagnetism type settlement gauges (a) 2012.12.31 812.5 2011.05.31∼2012.05.31 2010.05.31∼2011.05.31 2009.05.31∼2010.05.31 2008.02.15∼2008.05.31 2008.05.31∼2009.05.31 (b) Figure 3: eTh maximum cross-section. (a) Material zoning and (b) construction stage. −4 −3 −2 6000 −1 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 1 300 kPa 900 kPa 300 kPa 900 kPa 2500 kPa 2500 kPa 1500 kPa 1500 kPa Duncan-Chang EB Duncan-Chang EB (a) (b) Figure 4: Comparison between tfi tings of Duncan and Chang’s EB model and experimental triaxial tests results for rockfill material I. −𝜎 (kPa) 1 3 (%) Journal of Applied Mathematics 7 10000 −4 −3 −2 −1 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 1 1 300 kPa 900 kPa 300 kPa 900 kPa 2500 kPa 2500 kPa 1500 kPa 1500 kPa Modified PZ Modified PZ (a) (b) Figure 5: Comparison between tfi tings of the modified PZ-III model and experimental triaxial tests results for rockfill material I. −3 −2 −1 0 3 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =900 kPa 𝜎 =900 kPa 𝜎 =300 kPa 𝜎 =300 kPa 3 3 3 𝜎 = 1500 kPa 𝜎 = 2500 kPa 𝜎 = 1500 kPa 𝜎 = 2500 kPa 3 3 3 3 Duncan-Chang EB Duncan-Chang EB (a) (b) Figure 6: Comparison between tfi tings of Duncan and Chang’s EB model and experimental triaxial tests results for rockfill material II. −3 −2 −1 0 3 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =900 kPa 𝜎 =900 kPa 𝜎 =300 kPa 𝜎 =300 kPa 3 3 3 𝜎 = 1500 kPa 𝜎 = 1500 kPa 𝜎 = 2500 kPa 𝜎 = 2500 kPa 3 3 3 Modified PZ Modified PZ (a) (b) Figure 7: Comparison between tfi tings of the modified PZ-III model and experimental triaxial tests results for rockfill material II. −𝜎 (kPa) −𝜎 (kPa) −𝜎 (kPa) 1 3 1 3 1 3 (%) (%) (%) 8 Journal of Applied Mathematics 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =300 kPa 𝜎 =900 kPa 𝜎 =300 kPa 𝜎 =900 kPa 3 3 3 3 𝜎 = 1500 kPa 𝜎 = 2500 kPa 𝜎 = 1500 kPa 𝜎 = 2500 kPa 3 3 3 3 Duncan-Chang EB Duncan-Chang EB (a) (b) Figure 8: Comparison between tfi tings of Duncan and Chang’s EB model and experimental triaxial tests results for clay. 6000 0 4000 1 0 5 10 15 0 5 10 15 𝜀 (%) 𝜀 (%) 𝜎 =900 kPa 𝜎 =300 kPa 𝜎 =900 kPa 𝜎 =300 kPa 3 3 3 3 𝜎 = 1500 kPa 𝜎 = 1500 kPa 𝜎 = 2500 kPa 𝜎 = 2500 kPa 3 3 3 Modified PZ Modified PZ (a) (b) Figure 9: Comparison between tfi tings of the modified PZ-III model and experimental triaxial tests results for clay. First, the 2D ni fi te element mesh of the maximum cross- section of the dam was discretized according to the material zoning and construction design (see Figure 3). Then, the 2D mesh was extended to 3D mesh in accordance with contour line of the river valley. Figure 10 shows the 3D mesh of the Nuozhadu dam with 8095 brick and degenerated brick elements and 8340 nodes. The numerical simulations contain two stages, filling and impounding. During the filling stage, the dam body mainly subjects to body weight. en, Th at the end of construction, upstream water level goes up to the normal storage water Figure 10: 3D FEM mesh of Nuozhadu dam. level. The interaction between pore water and soil skeleton was considered through the whole numerical computation. 5. Three-Dimensional Finite Element Analyses 5.2. Results and Analyses 5.1. Computation Model. The numerical analyses were per- 5.2.1. Numerical Results Analyses. Figures 11 and 12 show the formed to simulate the performance of the dam during numerical results of finite element analyses with Duncan construction and impounding periods with eecti ff ve stress and Chang’s EB model and the modified PZ-III model, finite element analysis. respectively. −𝜎 (kPa) −𝜎 (kPa) 1 3 1 3 (%) (%) 0.5 0.4 0.2 0.3 0.1 2.5 0.2 3.5 2.5 1.5 0.3 1.5 0.5 1.5 0.5 0.6 0.7 0.1 0.5 Journal of Applied Mathematics 9 0.9 0.7 −2 0.1 (a) (b) −0.2 1.3 0.3 (c) (d) Figure 11: Displacement and stress contour of the maximum section for Duncan and Chang’s EB model: (a) displacement along river (m), (b) vertical displacement (m), (c) major principle stress (MPa), and (d) minor principle stress (MPa). −0.5 −2.9 −2.5 (a) (b) 0.1 3.5 (c) (d) Figure 12: Displacement and stress contour of the maximum section for the modified PZ-III model: (a) displacement along river (m), (b) vertical displacement (m), (c) major principle stress (MPa), and (d) minor principle stress (MPa). Through the comparison and analysis of the numerical (1) Aeft r the reservoir is impounded, upward displace- results (Figures 11 and 12), we can nd fi some similarities and ment as large as 0.7 m (see Figure 11(b))develops differences for these two models. on the upstream shell near dam crest for EB model On one hand, we can see many similar places in the and nearly 0 m for modified PZ-III model (see distributions of displacements and stresses. Figure 12(b)). In fact, monitoring data of practical engineering projects shows that no large upward (1) Aeft rthereservoirimpounding,duetothehugewater displacement happened aer ft impounding. This is pressure on upstream dam, horizontal displacement due to its weakness of EB model to distinguish the develops toward the downstream, and the largest loading and unloading condition during the water displacement is about 1.05 m for EB model and 0.74 m impounding. for modified PZ-III model. (2) In the distribution of minor principle stress (Figures (2) eTh maximum settlement occurs in the middle of core 11(d) and 12(d)), negative stress (i.e., tensile stress) wall due to lower modulus of clayey soil. occurs in theupstreamshell forEBmodel,whereas very little tensile stress exists for modified PZ-III (3) Because of the tremendous differences of modulus model. As we know, rockfill material is a typical kind between rockfill material and clayey soil, there exists of cohesionless coarse-grained soil, which means that obvious arching eeff ct in the core wall. it has no tensile strength. er Th efore, the existence of (4) Eeff ctive stress in upstream shell is less than the large area of tensile stress in the upstream shell is downstream shell due to the water pressure in the unreasonable. upstream shell. On the other hand, some differences also exist, which 5.2.2. Comparison between Numerical and In Situ Monitoring illustrate the advantages of modified PZ-III model. Data. Settlement is a key indicator to assess the safety of an 0.6 0.5 0.5 −0.5 1.5 0.5 −1 −1 −1.5 −2 1.5 2.5 −1.5 −2.4 −0.2 0.5 3 10 Journal of Applied Mathematics Elevation 655 m Time (a) −1000 0 1000 2000 3000 4000 Elevation 701 m Settlement (mm) In-situ EB Modified PZ Time Figure 13: Comparison between in situ monitoring settlement and (b) FEM results. Table 2: Material parameters of the modified PZ-III model. Material Rockfill I Rockfill II Mixed gravel clay 𝐾 500 1000 300 𝐺 1500 3000 900 Elevation 751 m 𝑚 0.50 0.50 0.50 𝑛 0.50 0.50 0.50 𝛼 0.45 0.45 0.45 𝛼 0.45 0.45 0.45 𝑀 1.05 0.90 0.60 Time 𝑀 1.60 1.35 1.10 𝑔𝑐 In-situ 𝛽 0.00 0.00 0.00 EB 𝛽 0.00 0.00 0.00 Modified PZ Γ 0.34 0.31 0.34 (c) 𝜆 0.10 0.09 0.03 𝑚 0.35 0.40 0.0 Figure 14: Comparison between in situ monitoring settlement and FEM results. 𝐻 800 1200 900 𝛾 55 5 𝛾 55 5 of rockfill materials was not considered, the FEM result of 𝐻 /MPa 9 9 10 𝑢0 settlement was below than the in situ monitoring data. As an elastoplastic model, the PZ-III model is capable of representing the mechanical behavior of soils better than earth dam. Figures 13 and 14 show the in situ monitoring data nonlinear elastic model such as Duncan and Chang’s EB and FEM results of settlement in the maximum cross-section. model. And the above ni fi te element analyses also proved it. The in situ data were obtained from electromagnetism type settlement gauges which were embedded during construction in the dam (as shown in Figure 3(a)). Through the compar- 6. Conclusions isons of in situ monitoring and numerical results, we can see that the modified PZ-III model gave a better prediction than This paper presents a modified PZ-III model based on the the EB model. However, as deformation induced by wetting generalized theory and original Pastor-Zienkiewicz-Chan Elevation (m) Settlement (mm) Settlement (mm) Settlement (mm) 2010/9/1 2010/1/1 2011/10/1 2010/6/10 2011/2/28 2011/12/20 2010/11/17 2012/3/9 2011/8/27 2011/4/26 2011/10/3 2012/5/28 2012/2/23 2012/3/11 2012/8/16 2012/8/18 2012/8/21 𝑓𝑐 Journal of Applied Mathematics 11 modeltosimulatethe stress-strainrelationshipofrockfill [11] P. V. Lade and J. M. Duncan, “Elastoplastic stress-strain theory for cohesionless soil,” Journal of the Geotechnical Engineering materials. Division,vol.101,no. 10,pp. 1037–1053, 1975. Triaxial test results of the filling materials of Nuozhadu [12] I. S. Sandler, F. L. DiMaggio, and G. Y. Baladi, “Generalized dam were used to validate the proposed model and determine cap model for geological materials,” Journal of the Geotechnical the model parameters of Duncan and Chang’s EB model and Engineering Division,vol.102,no. 7, pp.683–699,1976. the modified PZ-III model, respectively. eTh simulations of [13] X.-S. Li, Y. F. Dafalias, and Z.-L. Wang, “State-dependent dila- triaxial stress-strain response show that the modified PZ- tancy in critical-state constitutive modelling of sand,” Canadian III model is capable of representing the key features of Geotechnical Journal,vol.36, no.4,pp. 599–611, 1999. cohesionless soil, such as nonlinearity, dilatancy, and pressure [14] Y.-P. Yao and D. Sun, “Application of Lade’s criterion to Cam- dependency. clay model,” Journal of Engineering Mechanics,vol.126,no. 1, The proposed model has been incorporated into a ni fi te pp. 112–119, 2000. element code to simulate the static response of a high earth- [15] G.Y.Baladiand B. Rohani,“Elastic-plasticmodel forsaturated rockfill dam in China. eTh results were compared with those sand,” Journal of the Geotechnical Engineering Division,vol.105, of Duncan and Chang’s EB model. The two set of results no. 4, pp. 465–480, 1979. have both similarities and differences and the differences [16] O. Zienkiewicz and Z. Mroz, “Generalized plasticity formu- illustrate the advantages of the modified PZ-III model. The lation and applications to geomechanics,” in Mechanics of comparisons of FEM results, and in situ monitoring data Engineering Materials,C.S.Desai andR.H.Gallagher,Eds., pp. showed that the modified PZ-III model can give a better 655–679, John Wiley & Sons, New York, NY, USA, 1984. description of deformation of the earth-rockfill dam than [17] C. S. Desai and M. O. Faruque, “Constitutive model for Duncan and Chang’s EB model. geological materials,” Journal of Engineering Mechanics,vol.110, no.9,pp. 1391–1408, 1984. Acknowledgments [18] S. B. R. Murthy, A. Vatsala, and T. S. Nagaraj, “Revised Cam- clay model,” Journal of Geotechnical Engineering,vol.117,no. 6, This work was supported by the National Nature Science pp.851–871,1991. Foundation of China (51179092) and the State Key Laboratory [19] M. Pastor,O.C.Zienkiewicz, andA.H.C.Chan, “Generalized of Hydroscience and Engineering Project (2012-KY-02 and plasticity and the modelling of soil behaviour,” International 2013-KY-4). Journal for Numerical & Analytical Methods in Geomechanics, vol. 14,no. 3, pp.151–190,1990. [20] Z. Mroz and O. Zienkiewicz, “Uniform formulation of constitu- References tive equations for clays and sands,” in Mechanics of Engineering Materials,C.S.Desai andR.H.Gallangher, Eds.,pp. 415–449, [1] J. M. Duncan, “State of the art: limit equilibrium and finite- John Wiley & Sons, New York, NY, USA, 1984. element analysis of slopes,” Journal of Geotechnical and Geoen- vironmental Engineering,vol.122,no. 7, pp.577–596,1996. [21] G. Wang and J.-M. Zhang, “Dynamic consolidation finite element analysis of a sediment-protecting dyke under ocean [2] M. A. Biot, “General theory of three-dimensional consolida- wave loading,” Rock and Soil Mechanics,vol.27, no.4,pp. 555– tion,” Journal of Applied Physics,vol.12, no.2,pp. 155–164, 1941. 560, 2006. [3] R. S. Sandhu and E. L. Wilson, “Finite element analysis of [22] M. Alyami, M. Rouainia, and S. M. Wilkinson, “Numerical anal- seepage in elastic media,” Journal of the Engineering Mechanics ysis of deformation behaviour of quay walls under earthquake Division,vol.95, no.3,pp. 641–652, 1969. loading,” Soil Dynamics and Earthquake Engineering,vol.29,no. [4] J.T.Christian andJ.W.Boehmer,“Planestrainconsolidation 3, pp. 525–536, 2009. by finite elements,” JournalofSoilMechanics &Foundations [23] H. Li,P.Manuel, andT.Li, “Application of an generalized Division,vol.96, no.4,pp. 1435–1457, 1970. plasticity model to ultra-high rockfill dam,” in Proceedings [5] J.M.Duncanand C.-Y.Chang,“Nonlinearanalysisofstressand of the 12th International Conference on Engineering, Science, strain in soils,” Journal of the Soil Mechanics and Foundations Construction, and Operations in Challenging Environments— Division,vol.96, no.5,pp. 1629–1653, 1970. Earth and Space, pp. 385–398, Honolulu, Hawaii, USA, March [6] J. M. Duncan, P. M. Byrne, K. S. Wong, and P. Mabry, “Strength, stress-strain and bulk modulus parameters for finite element [24] T. Li and H. Zhang, “Dynamic parameter verification of P- analyses of stresses and movements in soil masses,” Tech. Rep. Z model and its application of dynamic analysis on rockfill UCB/GT/80-01, University of California, Berkeley, Calif, USA, dam,” in Proceedings of the 12th International Conference on Engineering, Science, Construction, and Operations in Challeng- [7] D.C.Drucker,R.E.Gibson, andD.J.Henkel, “Soilmechanics ing Environments—Earth and Space, pp. 2706–2713, Honolulu, and work-hardening theories of plasticity,” Transactions of the Hawaii, USA, March 2010. American Society of Civil Engineers,vol.122,pp. 338–346, 1957. [25] M. Pastor, “A generalized plasticity model for anisotropic [8] K. Roscoe, A. Schofield, and C. Wroth, “On the yielding of soils,” behaviour of sand,” Computer Methods and Advances in Geome- Geotechnique,vol.8,no. 1, pp.22–53,1958. chanics,vol.1,pp. 661–668, 1991. [9] K. Roscoe, A. Schofield, and A. u Th rairajah, “Yielding of clays [26] G. Bolzon,B.A.Schrefler, andO.C.Zienkiewicz, “Elastoplastic in states wetter than critical,” Geotechnique,vol.13, no.3,pp. soil constitutive laws generalized to partially saturated states,” 211–240, 1963. Geotechnique,vol.46, no.2,pp. 279–289, 1996. [10] J. Burland, “Correspondence on ‘eTh yielding and dilation of [27] H. I. Ling and H. Liu, “Pressure-level dependency and densifi- clay’,” Geotechnique,vol.15, pp.211–214,1965. cation behavior of sand through generalized plasticity model,” 12 Journal of Applied Mathematics Journal of Engineering Mechanics,vol.129,no. 8, pp.851–860, [28] H. I. Ling and S. Yang, “Unified sand model based on the critical state and generalized plasticity,” Journal of Engineering Mechanics,vol.132,no. 12,pp. 1380–1391, 2006. [29] N. D. Marschi, C. K. Chan, and H. B. Seed, “Evaluation of properties of rockfill materials,” Journal of the Soil Mechanics and Foundations Division,vol.98, no.1,pp. 95–114,1972. [30] R. J. Marsal, “Large scale testing of rockfill materials,” Journal of the Soil Mechanics and Foundations Division,vol.93, no.2,pp. 27–43, 1967. [31] R. J. Marsal, “Mechanical properties of rockfill,” in Embankment Dam Engineering, pp. 109–200, John Wiley & Sons, New York, NY, USA, 1973. [32] P. V. Lade, J. A. Yamamuro, and P. A. Bopp, “Significance of particle crushing in granular materials,” Journal of Geotechnical and Geoenvironmental Engineering,vol.122,no. 4, pp.309–316, [33] B. O. Hardin, “Crushing of soil particles,” Journal of Geotechnical Engineering, vol. 111, no. 10, pp. 1177–1192, 1985. [34] R. L. Kondner, “Hyperbolic stress-strain response: cohesive soils,” Journal of the Soil Mechanics and Foundations Division, vol. 89, no. 1, pp. 115–143, 1963. [35] Z.-L. Wang, Y. F. Dafalias, X.-S. Li, and F. I. Makdisi, “State pres- sure index for modeling sand behavior,” Journal of Geotechnical and Geoenvironmental Engineering,vol.128,no. 6, pp.511–519, 2002. 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