State-dependent Dilatancy Theory and Numerical Modelling of Rockfills

Abstract: The dilatancy behavior of rockfills is relate to the stress level, the initial state and particle breakage. In this paper, based on the critical state theory, the state-dependent dilatancy theory of rockfills is established, and it is introduced into the state-dependent constitutive model of coarse materials, so the state-dependent constitutive model of rockfills is formulated. According to the large-scale triaxial testing results, using the Fortran program modelling the experimental results, then, comparing the test results and simulation results, only one set parameters of state-dependent constitutive model of rockfills can reflect the strain softening and dilatancy properties of rockfills under the condition of different density, gradation and confining pressure. Therefore, the rationality of the state-dependent constitutive model of rockfills is verified.


Introduction
Dilatancy is an important engineering characteristic of granular materials.In general, volume expansion in shearing is called shear expansion, and volume shrinkage is called shear contraction [1].Classical stress dilatancy theory (Rowe,1962) is used for describing the relationship between the dilatancy and stress ratio.In 1962, Rulong Wei [2] discussing the dilatancy theory of soil in detail, he believed that the dilatancy of granular materials is result of the particle interlocking, the main reason is particle turning over adjacent particle or having tendency in the shear process.
Base on the classical stress dilatancy theory, many scholars at home and abroad have studied the shear dilatancy behaviors of coarse materials.The dilatancy behavior of sand has been studied [3][4][5][6][7][8], and the state dependent dilatancy theory of sand has been established.Both rockfills and sand are coarse granular materials, but rockfills has many different engineering properties as large particle size, high strength, small deformation and particle breakage.
Liu [9,10] studies the dilatancy behavior of rockfills by large-scale triaxial experiment, pointing out that stress path and consolidation stress are the main external factors, establishing new stress dilatancy function.Xu [11] establishes strain hardening constitutive model of rockfills, which can describe the dilatancy and contraction of rockfills.Chu [13] researches the dilatancy behavior of coarse materials, he believes that the dilatancy function of Modified Cambridge model can't reflect the dilatancy behavior of coarse materials, Row's dilatancy function underestimating its compressibility in the contraction stage, and overestimating its dilatancy behavior in the shear expansion stage, so he studies the relationship between the density, confining pressure and dilatancy by large scale triaxial experiment.
Correct understanding the dilatancy behavior is the key for establishing reasonable constitutive model.In this paper, referring to the current research results, based on the critical state theory, establishing the state dependent dilatancy theory of rockfills, which is introduced into the state-dependent constitutive model of coarse granular materials, so the state-dependent constitutive model of rockfills is formulated.A numerical simulation of largescale triaxial test is carried out by compiling Fortran program, comparing the test results and simulation results, only one set parameters of state-dependent constitutive model of rockfills can reflect the strain softening and dilatancy properties.

State-dependent dilatancy theory of rockfills
The dilatancy equation can directly reflect the relationship between the shearing deformation behavior and its influencing factors.When dilatancy ratio is greater than or equal to zero, it is shrinkage in the shearing process, and when dilatancy ratio is less than zero, it is dilatancy.Li [6] proposes dilatancy function of sand, which covering the void ratio and other intrinsic state variables, the dilatancy equation is as follows: (4) In which, d is dilatancy ratio, d1 and m are model parameters, ψ [6] is state variable, η is stress ratio, M is critical state stress ratio, e is the current void ratio, ec is the critical void ratio, eΓ is the void ratio when the effective mean normal stress is zero, λ is the slope of critical state line, ξ is material parameter, p is effective mean normal stress, pa is atmospheric pressure, q is deviatoric stress.

Journal of Civil Engineering and Construction
Rockfills and sand are all coarse granular materials, they have similar dilatancy behaviors, classical stress dilatancy function can't reflect their deformation characteristics.But, comparing to sand, rockfills has larger particle size, easier to breakage, they have different engineering behaviors.In this paper, referring to the state dependent dilatancy theory of sand, studying the dilatancy behavior of rockfills and establishing the dilatancy function, so the state dependent dilatancy theory of coarse granular material is improved.
The dilatancy of rockfills is relate to the factors of gradation, density, stress level and particle breakage, based on the critical state theory of rockfills, the general state dependent dilatancy function of rockfills is as follows.0 () In which, d is dilatancy ratio, D0 is the initial dimension of particle distribution, e is the current void ratio, η is stress ratio, p is effective mean normal stress, q is deviatoric stress, C is other internal state parameters.In the equation ( 5), when the sample reach critical state, the current void ratio is critical void ratio, the current stress ratio is critical stress ratio, the shear deformation tends to be stable, and the dilatancy ratio equal to zero.In the shearing process, in the begin, it is shear contraction then is dilatancy, when the shear shrinkage is transited to the dilatancy, the dilatancy ratio is equal to zero also, but the current void ratio is not critical void ratio, the current stress ratio is phase transitional stress ratio.
According the above analysis, the state dependent dilatancy function of rockfills is similar to sand's, which is proposed by Li [6], the state dependent dilatancy function of rockfills is as follows,   In which, d is dilatancy ratio, d0 and m are model parameters, ψ [6] is state variable, η is stress ratio, M is critical state stress ratio, e is the current void ratio, ec is the critical void ratio, eΓ is the void ratio when the effective mean normal stress is zero, λc is the slope of critical state line, ξ is material parameter, p is effective mean normal stress, pa is atmospheric pressure, D is the fractal dimension, e is the void ratio after consolidation, σ3 is confining pressure, D0 is the initial dimension of particle distribution, q is deviatoric stress, a, b, c, l, α and β are material constants.
It can be seen from equation (10) that the void ratio of sand is different from the rockfills' when the effective mean normal stress is zero, which is relate to the initial state, stress level and particle breakage.when the shearing shrinkage is transited to the dilatancy, the dilatancy ratio is equal to zero, the model parameter , in which, M d is phase transitional stress ratio, M is critical stress ratio, ψ d is phase transitional state variable.

State-dependent constitutive model of rockfills
In order to reflect the shear deformation behavior of rockfills rationally, the newly state dependent dilatancy function of rockfills is introduced into the state dependent constitutive model of coarse granular materials, which is proposed by Li and Dafalias [17], the constitutive model of rockfills is as follows.
In which, G is the elastic shear modulus, K is the elastic bulk modulus, L is plastic loading factor, h(L) is Heaviside equation, when L is greater than zero, h(L)=1, when L is less than or equal to zero, h(L)=0, Kp is plastic modulus, other symbols are the same as the previous text.
The elastic shear modulus can be calculated by equation ( 14), as In which, G0 is material constant, e is the void ratio after consolidation, υ is Poisson's ratio.The plastic modulus Kp can be calculated by state variable ψ, it can reflect the strain hardening and softening of rockfills, the expression is as follows.The 16 parameters of state dependent constitutive model of rockfills are calibrated by the large triaxial experimental results, which can be seen in Table 1.

Numerical modelling
The state dependent constitutive model of rockfills is compiled by Fortran program, using the parameters calibrated by triaxial experiments and the testing initial conditions, modeling the triaxial tests, the test results and numerical results are listed in Figure 1~4.
Comparing the results in Figure 1~4, it is demonstrated that modelling results match the testing results very well.The state dependent constitutive model of rockfills can reflect the stress and deformation characteristics under the condition of different density, gradation and confining pressure, it can also reflect the strain hardening and softening, volumetric dilatancy and shrinkage.The constitutive model of rockfills is formulated considering the influence of density, gradation, stress state and particle breakage, but the particle shape, composition, the particle recombination in the shearing process and the technical level of the experimenter are not considered, so the model need further exploration.

Conclusions
Based on the large triaxial experimental results, according the theory analysis and numerical modelling, the main conclusions are listed as follows.
(1) Based on the critical state theory of rockfills, the state dependent dilatancy theory is formulated, it is introduced into the state dependent constitutive model of coarse granular materials, so the state dependent constitutive model of rockfills is established.
(2) The large triaxial consolidation and drainage shear tests are simulated by Fortran program, comparing the testing results and numerical modelling results, it is found that the two results match very well.
(3) The state-dependent constitutive model of rockfills using only one set parameters can reflect the stress and deformation characteristics under the condition of different density, gradation and confining pressure.It also can reflect the strain hardening and softening, volumetric dilatancy and shrinkage.
which, hG is hardening parameter, h=h1-h2•e, h1, h2 and n are model parameters.When the strain hardening is transited to strain softening, the plastic modulus is equal to zero, so, and ψ b are the stress ratio and state variable in the strain transitional phase.
Testing results (The relative density is 0.60) (a2) Modelling results (The relative density is 0.60) Testing results (The relative density is 0.90) (c2) Modelling results (The relative density is 0.90) Figure1Deviatoric stress and axial strain of testing results and modelling results (gradation 1) Testing results (gradation 2) (b2) Modelling results (gradation 2) Figure2 Deviatoric stress and axial strain of testing results and modelling results (the relative density is 0.75) Testing results (The relative density is 0.60) (a2) Modelling results (The relative density is 0.60) Testing results (The relative density is 0.90) (b2) Modelling results (The relative density is 0.90) Figure3 Volumetric strain and deviatoric strain of testing results and modelling results (gradation 1) Testing results (gradation 2) (b2) Modelling results (gradation 2) Figure4 Volumetric strain and deviatoric strain of testing results and modelling results (the relative density is 0.75)

Table 1
Parameters of state dependent constitutive model of rockfills