Application of granular solid hydrodynamics to a well-graded unbound granular material undergoing triaxial tests

Abstract

Unbound granular materials (UGMs) are widely used as a base or a subbase in pavement construction. They are generally well graded and exhibit a higher peak strength than that of conventional cohesionless granular materials. By using a simplified version of granular solid hydrodynamics (GSH), a set of GSH material constants is determined for a UGM material. The deviatoric stress and volumetric strain caused by triaxial compression are calculated and then compared with experimental data. The results indicate that the GSH theory is able to describe such a special type of granular materials.

Keywords

granular solid hydrodynamics / unbound granular material (UGM) / triaxial tests

Introduction

Unbound granular materials (UGMs) are generally used in road pavements, which are often the main load-bearing layer and are designed to evenly spread the load of the pavement to the subgrade below. The quality of the UGM is very important in determining the useful life of the road, and it is necessary to develop effective methods of testing and evaluating the strength of UGM materials under given loading conditions. As a conglomeration of crushed stones, UGM exhibits complex mechanical properties (roughly classified into a hardening stage, a softening stage and a critical state) that can be described with primarily elastoplastic or hypoplastic models. For example, Rondon et al. [1] conducted monotonic and cyclic triaxial loading on a well-graded UGM. A hypoplasticity model provided by von Wolffersdorff [2] was used as the constitutive model, and a set of constants was determined. However, hypoplastic models are phenomenological models, and the complicated mechanical behavior of UGMs is still poorly understood [3]. Because thermodynamics is universal, a framework that is developed based on its principles may be used to properly study the physics of granular mechanics and to predict macroscopic stress-strain relationships. Granular solid hydrodynamics (GSH) has been proposed based on the laws of thermodynamics; GSH is a complete continuum mechanical theory for granular materials, including explicit expressions for the energy current and entropy production [4,5]. GSH has been supported and quantified with arguments from physics and soil mechanics. Meanwhile, for steady-state deformations, it is equivalent to hypoplasticity.

However, it should be noted that there are up to 12 transport coefficients in GSH; even the granular medium is assumed to be intrinsically isotropic. In reality, due to the complex dissipation mechanisms in granular media, we have to consider a sufficient number of transport coefficients. As a result, it is not a simple matter to ascertain the models because they change with the state. Additional quantifications of GSH are required both from physics and engineering practices. We require 1) repeated estimates and calculations, 2) a large amount of experimental data for comparison and calibration and 3) study of the mesoscale characteristics.

Recently, by using GSH, the elastic energy and its relaxation (denoted by the granular temperature) were both calculated and explained [6]. For a dense assembly, it was found that the elastic energy and energy dissipation rate reach peak values simultaneously as the peak strength was reached. To observe the mesoscale characteristics, a two-dimension biaxial test was simulated with a discrete element method. The motion of particles and the evolution of force networks were observed at different strain values. The GSH was simplified for analysis of the mechanical properties under the triaxial compression test<FootNote>

Song S X, Sun Q C, Jin F, et al. Analysis of parameters in granular solid hydrodynamics for triaxial compression test. Submitted to Powder Technology

</FootNote> , and then it was applied to analyze the experimental data obtained by Cheng [8]. The influences of major GSH material constants, especially their cross-coupling influences, were analyzed, and their physical meanings were further clarified.

In this work, the results of triaxial tests of a well-graded UGM obtained by Rondon et al. [1] were used, and the simplified GSH theory was employed to study the stress-strain relations and to determine the GSH constants of the UGM material. The calculated results of GSH are compared with both experimental data and the calculated results from a hypoplasticity model [1,2].

Triaxial test setup

A well-graded UGM material was prepared. It was mixed from different gradations of quartz sands with subangular grain shapes. The maximum grain size of d_max = 16 mm was used so as not to not fall below a ratio a/d_max of 5, with a × b being the dimensions of the specimen cross section in the triaxial tests. The mean grain diameter was d₅₀ = 6.3 mm, and the coefficient of uniformity is C_u = d₆₀/d₁₀ = 100. For the fine particles, a quartz meal was used. The maximum density according to German Standard Code DIN 18126 is ρ_max = 2.163 g/cm³ (determined with a shaking table), and the minimum density is ρ_min = 1.835 g/cm³. These values correspond to e_min = 0.225 and e_max = 0.444 being the minimum and maximum void ratios, according to DIN 18126. For a Proctor test with modified energy, the maximum dry density can reach ρ_Pr = 2.30 g/cm³. The optimum water content is w_opt = 5.2%. Three monotonic triaxial tests on specimens with large initial relative densities (I_D0 = 1.06~1.13, dry density>95% of ρ_dr) were performed, where relative density is expressed by the index of I_D = (e_max—e)/(e_max—e_min), and I_D0 is the initial density. The limit void ratios at zero pressure were estimated from the relations e_d0 ≈ e_min, e_c0 ≈ e_max and e_i0 ≈ 1.15e_max (for well-graded soils). The effective lateral stresses were σ₃ = 50, 100 and 200 kPa, respectively. The shape of specimens used in tests were prismatic with a × b × h = 8.7 × 8.7 × 18.0 cm (h/a = 2.1), by means of a steel mold consisting of four plates. Specimens were prepared by tamping in n = 6 layers each with a thickness of 3 cm. After tamping the specimen was placed into the triaxial cell and the steel mold was removed. The test device is presented in Fig. 1. The deviatoric stress q, defined as q = σ₁—σ₃, is recorded. Both q and the volumetric strain ϵ_v are plotted against the axial strain ϵ₁ at various confining pressures: σ₃ = 50, 100 and 200 kPa (see Figs. 2 and 4).

Fundamentals of GSH for triaxial tests

GSH is a complete continuum mechanical theory for granular materials. Derivations and explanations can be found in Ref. [3-7] GSH consists of 1) five conservation laws for the energy w, mass ρ, and momentum ρv_i; 2) an evolution equation for the elastic strain u_ij; and 3) balance equations for two temperatures. Both the true and the granular temperature, T and T_g, respectively, are necessary. In particular, T_g is a new concept introduced in GSH to treat a granular material as a transient elastic object due to particles jiggling and sliding. T_g is similar to the heat-like seismicity.

The elastic strain u_ij is defined as the portion of the strain ϵ_ij that deforms the particles and leads to reversible storage of elastic energy. The plastic rest, ϵ_ij-u_ij, accounts for rolling and sliding. Because the energy depends on u_ij alone, this is a crucial step to retain many useful features of elasticity, particularly in an explicit expression of the stress. The evolution equation for u_ij and T_g and the expression for the Cauchy stress σ_ij are briefly introduced (see Refs. [3,4]. for a complete derivation). Considering granular materials as transiently elastic systems, the evolution equation of u_ij is {\partial }_t{u}_{ij}=v_{ij}+X_{ij}, where v_{ij}\equiv ({\nabla }_j{v}_i+{\nabla }_i{v}_j)/\mathrm{2} is the space derivative of velocity v_i and X_ij represents a relaxing, dissipative contribution. The Cauchy stress is given as {\sigma }_{ij}={\pi }_{ij}-{\sigma }_{ij}^D, with{\pi }_{ij}=-\partial E_e/\partial u_{ij}. Dividing u_ij into \Delta \equiv -u_{ii},u_{ij}^{\mathrm{0}}=u_{ij}-u_{ii}{\delta }_{ij}/\mathrm{3}, we define the elastic energy asE_e=B\sqrt{\Delta }(\mathrm{2}{\Delta }^{\mathrm{2}}/\mathrm{5}+u_{\mathbf{s}}^{\mathrm{2}}/\xi ), with u_s^{\mathrm{2}}=u_{ij}^{\mathrm{0}}{u}_{ij}^{\mathrm{0}}. The coefficient B accounts for overall rigidity and it increases with the density; x relates to the Mohr-Coulomb yield. The elastic energy E_e is convex only for u_s/\Delta \le \sqrt{\mathrm{2}\xi } (or equivalently q/P\le \sqrt{\mathrm{3}/\xi }), coinciding with the Drucker-Prager condition.

The energy conservation and entropy production R={\sigma }_{ij}^D{v}_{ij}+X_{ij}{\pi }_{ij}+\cdots, implying that the variables X_ij and {\sigma }_{ij}^Dare written as follows (the viscous terms, large only for high shear rate dense flows, are neglected):

(1)\begin{array}{l}{X}_{ij}={\lambda }_{ijkl}{\pi }_{kl}-{\alpha }_{ijkl}{v}_{kl}={\lambda }_s{\pi }_{ij}^{\mathrm{0}}+{\lambda }_v{\delta }_{ij}{\pi }_{ll}/\mathrm{3}-{\alpha }_{ijkl}{v}_{kl},\hfill \\ {\delta }_{ij}=\{\begin{array}{l}0,i\mathrm{}\ne \mathrm{}j;\hfill \\ 1,i=j.\hfill \end{array}\hfill \\ {\sigma }_{ij}^D={\alpha }_{ijkl}{\pi }_{kl}.\hfill \end{array}

The fourth-order transport coefficients in Eq. (1) are

(2)\begin{array}{l}{\alpha }_{ijkl}={\alpha }_s{\delta }_{ik}{\delta }_{jl}+({\alpha }_v-{\alpha }_s){\delta }_{ij}{\delta }_{kl}/\mathrm{3}+c_{\mathrm{1}}(u_{ij}^{\mathrm{0}}{\delta }_{kl}+u_{kl}^{\mathrm{0}}{\delta }_{ij}),\hfill \\ {\lambda }_{ijkl}={\lambda }_s{\delta }_{ik}{\delta }_{jl}+({\lambda }_v-{\lambda }_s){\delta }_{ij}{\delta }_{kl}/\mathrm{3}+{\lambda }_{\mathrm{1}}(u_{ij}^{\mathrm{0}}{\delta }_{kl}+u_{kl}^{\mathrm{0}}{\delta }_{ij}).\hfill \end{array}

Following the same rules for a triaxial test in which axial symmetry exists (i.e., σ₂ = σ₃), u_{ij},{\sigma }_{ij} and v_{ij}can be rewritten as

(3)\begin{array}{l}\Delta =-u_{ii},u_q=u_{\mathrm{11}}-u_{\mathrm{33}},\hfill \\ P={\sigma }_{ii}/\mathrm{3},q={\sigma }_{\mathrm{11}}-{\sigma }_{\mathrm{33}},\hfill \\ v=-v_{ii},v=v_{\mathrm{11}}-v_{\mathrm{33}}.\hfill \end{array}

The evolution equation for u_{ij} and {\sigma }_{ij} takes the following final form:

(4)\begin{array}{l}{\partial }_t\Delta =(\mathrm{1}-{\alpha }_v)v+\mathrm{2}{\alpha }_{\mathrm{1}}{u}_q{v}_q-{\lambda }_{\mathrm{0}}{T}_g\Delta ,\hfill \\ {\partial }_t{u}_q=(\mathrm{1}-{\alpha }_s)v_q+{\alpha }_{\mathrm{1}}{u}_{q}v-\lambda T_g{u}_q.\hfill \end{array}

(5)\begin{array}{l}P=(\mathrm{1}-{\alpha }_v)B{\Delta }^{\mathrm{1.5}}(\mathrm{1}+u_q^{\mathrm{2}}/\mathrm{3}\xi {\Delta }^{\mathrm{2}})+\mathrm{4}{\alpha }_{\mathrm{1}}B{u}_q^{\mathrm{2}}{\Delta }^{\mathrm{0.5}}/\mathrm{3}\xi ,\hfill \\ q=-\mathrm{2}(\mathrm{1}-{\alpha }_s)B{\Delta }^{\mathrm{0.5}}{u}_q/\xi -\mathrm{3}{\alpha }_{\mathrm{1}}B{\Delta }^{\mathrm{1.5}}(\mathrm{1}+u_q^{\mathrm{2}}/\mathrm{3}\xi {\Delta }^{\mathrm{2}})u_q.\hfill \end{array}

The mass conversation equation and entropy increase equations are

(6){\partial }_t\rho -\rho v=\mathrm{0},

(7)\rho {\partial }_t\vartheta =v^{\mathrm{2}}/\mathrm{3}+\mathrm{2}{v}_q^{\mathrm{2}}/\mathrm{3}-T_g^{\mathrm{2}},

(8)\vartheta =b{T}_g.

where \vartheta is the granular entropy per unit mass.

The relaxation of u_ij can be described by a simple relaxation time model, where u_ij relaxes with relaxation rate 1/τ. When T_g is finite and particles vibrate, they briefly lose contact with one another, during which time their elastic form will be partially lost. Macroscopically, the relaxation rate increases with T_g and disappears when T_g= 0. Therefore, for X_ij in Eq. (1), we assume {\lambda }_s{\pi }_{ij}^{\mathrm{0}}=-u_{ij}^{\mathrm{0}}/{\tau }_s=-\lambda T_{\mathrm{g}}{u}_{ij}^{\mathrm{0}} and {\lambda }_v{\pi }_{ll}=\Delta /{\tau }_v={\lambda }_{\mathrm{0}}{T}_{\mathrm{g}}\Delta.

With the conditions of σ₃ = const, and v₁₁ = const, triaxial tests could be studied by solving Eqs. (4-8).

Results and discussion

Figures 2 and 3 show the variations of volumetric strain ϵ_v and deviatoric stress q with axial strain ϵ₁. From the experimental results, we can see that the densities for the three dense specimens increases until the densest state is achieved at approximately ϵ₁ ≈ 1.0%, assuming compaction is positive. Then, the system begins to dilate. Meanwhile, there is a steady increase in q, which reaches peak values of approximately 490, 890 and 1610 kPa for σ₃ = 50, 100 and 200 kPa, respectively, i.e., the stress ratio q/σ₃ ≈ 9.8, 8.9 and 8.1 for the three loadings. Thereafter, it steadily decreases and reaches constant values of approximately 350 and 1190 kPa for σ₃ = 50 and 200 kPa, respectively. Hypoplastic constitutive models (e.g., Ref. [2,8]) were developed to describe the mechanical behavior of “grain skeletons” phenomenologically. Hypoplastic models (e.g., Ref [8]) are incrementally nonlinear, rate-independent, path-dependent and dissipative. The hypoplastic material constants were determined for the UGM material (see Table 1 in Ref. [1].). The calculated data with a hypoplasticity model are shown as dashed curves in Figs. 2 and 3.

In the simplified version of GSH, three major material constants have to be determined. Their physical meanings and values used in this work are provided below and are listed in Table 1. The calculated volumetric strain ϵ_v and deviatoric stress q are shown as solid curves in Figs. 2 and 3, and the values are acceptable compared with experiment data.

1) In Ref. [1], the maximum density according to German Standard Code DIN 18126 is ρ_max = 2.163 g/cm³ (determined with a shaking table), and the relative density is I_D>1 in the tests. However, for a Proctor test with modified energy, the maximum dry density can reach ρ_Pr = 2.30 g/cm³. In our study, we define the dimensionless density ρ, which is normalized with ρ_Pr, i.e., for random close-packed density ρ_Pr, the dimensionless density ρ_cp = 1. Similarly, for the random loose-packed density, ρ_lp = ρ_min/ρ_Pr.

2) When particles vibrate, they briefly lose contact with one another, and the deformation is slowly lost. The relaxation rate of the elastic strain will decrease with the increase in the density and vanish at the density ρ_cp, the point at which the system becomes elastic. This behavior is described by the parameter λ. A smaller λ implies a slower relaxation; therefore, a higher peak stress and greater residual stress would appear. In this work, we choose {\lambda }_{\mathrm{0}}=\lambda /\mathrm{5} and \lambda =\mathrm{300}{f}_{\rho }^{\mathrm{0.5}}, where f_{\rho }=(\mathrm{1}-\rho )/(\mathrm{1}-{\rho }_{lp}).

3) The slip between contact particles when the system is sheared is taken into account by the parameter α. As a result, only the (1–α) portion of v_ij will deform the particles. The variable α₁ implies the cross-effect of elastic volume strain and shear strain. Therefore, the shear flow v_q leads to a compression rate {\partial }_t\Delta and causes dilatancy. The dilatancy obviously increases with the increase in α₁. In this work, we choose{\alpha }_s=\mathrm{0.76}{f}_{\rho }^{\mathrm{0.15}}, {\alpha }_v=\mathrm{0.85}{f}_{\rho }^{\mathrm{0.15}} and {\alpha }_{\mathrm{1}}=\mathrm{160}{f}_{\rho }.

4) The constant x is related to the stability condition. The elastic energy E_e is convex only for q/P\le \sqrt{\mathrm{3}/\xi }. The peak friction angle {\phi }_{\mathrm{P}} is expressed as \sin {\phi }_{\mathrm{P}}=\mathrm{3}q/(6p+q), i.e., \sin {\phi }_P=\mathrm{3}/(\mathrm{2}\sqrt{\mathrm{3}\xi }+\mathrm{1}). For the friction angle of approximately 54.4° as e_p = 0.210, a smaller x should be used. We choose x= 1 for the UGM material in this study.

5) The value of constant B₀ is relatively simple to determine. For quartz sand, it is usually taken as B₀ = 7 GPa (see Ref. [4-7]).

From Fig. 3, we notice that the GSH results may be better than the hypoplasticity results. It has been proved that GSH could be reduced to hypoplasticity for steady-state deformations. Further comparisons will be conducted. The peak stresses P_p and q_p are extracted from Fig. 3 and are shown in the P-q plane in Fig. 4. The calculated GSH results match well with the experimental data. A slight decrease in the peak stress ratio ŋ_p= P_p/q_p with increasing lateral stress σ₃ is also observed, which coincides with the conventional trend in triaxial tests.

Outlook

The simplified granular solid hydrodynamics theory has been supported and qualified in triaxial tests with river sand in the Netherlands by Cheng [7]. As a well-graded granular material, the UGM material used in this work exhibits much higher peak strength than that of conventional granular materials. Thus, this observation raises the question of whether the simplified GSH theory is still applicable for such special UGM materials. The results obtained in this work obviously show that GSH is still applicable if appropriate GSH constants are carefully chosen for such a UGM, which enhances our confidence in GSH theory. Determination of the values of material constants in GSH would depend on analyses of the structure and motions at a given particle scale (e.g., mesoscale). At this stage, we are systematically conducting careful measurements using CT, MRI and large-scale DEM simulations.

References

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[4] Jiang Y, Liu M. Granular solid hydrodynamics. Granular Matter, 2009, 11(3): 139–156

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[7] Cheng X H. Localization in Dutch Dune Sand and Organic Clay. Delft: Delft University Press, 2004

[8] Kolymbas D. Introduction to Hypoplasticity. Rotterdam: Balkema, 2000

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