How contact stiffness and density determine stress-dependent elastic moduli: a micromechanics approach

Abstract

To investigate the origin of the stress-level dependency of soil elasticity, a series of stress-path experiments were simulated for loose and dense soil specimens with three different contact surfaces. In the discrete element analyses, an assumption was introduced in which the contact body had the geometry of an elastic sphere with local, axi-symmetric irregularity. To evaluate the cross-anisotropic elastic shear moduli, small-strain cyclic shear tests were simulated under stress conditions along four stress-probing paths. For dense specimens with high coordination numbers, the internal structure was represented by the degree of fabric anisotropy and the coordination number remained unchanged during shearing, thus leading to the coincidence of the sum of the exponents in the contact stiffness model. For the loose specimens with low coordination numbers, the fabric structure evolved continuously during shearing, which resulted in the increase of the exponents in the power function of the elastic modulus. The rearrangement of particles and the transition of contact-force chains, along with the evolution of the fabric, manifested as increasing dependency of the elastic moduli on the stresses in such loose specimens.

1 Introduction

Stress-level-dependent elasticity is a major feature of soil behavior that needs to be investigated independently within the framework of micromechanics theory. The nonlinear anisotropic elastic response, which has commonly been observed in a large amount of experimental data, can be represented by a well-known empirical equation of three-dimensional elastic moduli [1] given by the following equation:

$$\begin{aligned} G_{ij} =C_{ij} p_a f(e)(\sigma ^{\prime }_i /p_a )^{m}(\sigma ^{\prime }_j /p_a )^{n} \end{aligned}$$

(1)

where $G_{ij}$ is the elastic shear modulus in the $i$–$j$ plane, $C_{ij}$ is a dimensionless constant relating to the directional properties of the elastic shear moduli, $p_a$ is the atmospheric pressure, used as a normalizing constant, $f(e)$ is the void ratio function, $\sigma _i^{\prime }$ and $\sigma _j^{\prime }$ are the principal stresses in the $i$ and $j$ directions respectively, and the exponents $m$ and $n$ are material constants. The sum of exponents $m$ and $n$ (hereafter referred to as $s$) determines the magnitude of the stress-level dependency of the elastic stiffness of soils. While the stress-level dependency of soil elasticity has long been known, the physical origin of this relationship has not yet been clarified.

The particular form of the power function in Eq. (1) has drawn researchers to investigate an explicit relationship between the elastic response of soils and theoretical contact equations. For instance, Hardin and Blandford [1] explained that in Eq. (1), $s$ is analog to the power of 1/3 in the classical solution of the elastic moduli for a regular assembly of elastic spheres [2]. Cascante and Santamarina [3] argued that under conditions of isotropic stress, $s$ is determined by the contact-force–displacement relationship, regardless of what mechanisms caused the contact deformation. Jung and Chung [4] investigated the stress-level dependency of cross-anisotropic elastic moduli in granular soils via the analytical micromechanics theory. They showed that under conditions of isotropic stress, the power function of normal contact stiffness manifests itself in the sum of exponents, $s$, whereas under conditions of anisotropic stress, the evolution of the fabric also has a significant influence on $s$ in the power function.

In experiments, a recent survey of data from resonant column testing [5] for both sands and clays showed that $s$ has been found to vary between approximately 0.4 and 0.6, with a value of 0.5 having been observed by many researchers. By comparing wave velocities, Lee and Huang [6] showed that $s$ becomes larger as the grains of sand became more angular. Kohata et al. [7] reported that $s$ was nearly 0.5 for various sands with minor angles on their contact surfaces, zero for soft mudstone and tuff featuring flat contact surfaces, and 0.8 for weathered granite, which likely had an angular contact surface.

Clayton [8] recently proposed three factors that might influence the shear modulus of a granular material at very small strain levels: the void ratio, inter-particle contact stiffness, and deformation and flexing within individual particles. When considering minor changes in the void ratio and limited particle deformation under service loading in granular soils, it is obvious that the shear modulus the value of $s$ is mostly affected by the inter-particle contact stiffness, which itself depends on particle mineralogy, angularity and roughness.

While the contact stiffness likely determines the power equation of soil elastic stiffness, some other factors may also influence the elastic modulus. Houlsby and Schnaid [9] showed that the value of $s$ increased as the density increased despite significant scatter in the loose sands testing data. AhnDan and Koseki [10] showed that there was no apparent pattern in the magnitude of $s$ in relation to variations in sample density. Makse et al. [11] pointed out the importance of the increasing number of contacts under pressure to explain the pressure dependence of elastic shear moduli. Unfortunately, the combined effect of the major factor (i.e., contact stiffness) and minor factor (i.e., density of the specimen) on soil elasticity has not yet been scrutinized.

To investigate the origin of the stress-level dependency of soil elasticity, we focused on the significance of microscopic parameters such as contact geometry, the evolution of the contact distribution, and changes in contact densities. We conducted a systematic investigation via the discrete element method to simulate small-strain, cyclic-loading tests. Two synthetic specimens, one loose and the other dense, were used for the simulations. The stress path experiment was simulated under various contact conditions, from which the patterns of stress-dependent elastic stiffness were analyzed.

2 Microscopic contact model

Let our hypothesis be that the power law of the contact model at the microscopic scale determines the power law of the elastic modulus at the macroscopic scale. To investigate this argument, it is necessary to employ a versatile model which is able to express various contact conditions. Thus we introduce a model proposed by Jäger [12] which assumes the geometry of the contact body to be an elastic sphere with local, axi-symmetric irregularity. Figure 1 illustrates contacting particles with small irregularities and the geometry of the contact surface with locally elastic half-space. The irregularity is expressed as $A_\alpha r^{\alpha }$ where $r$ is the radius from the center of the rigid irregularity, $A_\alpha$ is the geometric constant, and $\alpha$ is the exponent determining the shape of the irregularity. In this study, three contact geometries are chosen: a conical irregularity ($\alpha =1$), a spherical irregularity ($\alpha =2$), and a flat surface ($\alpha =\infty$). Figure 2 shows the three different contact geometries.

Fig. 1

Geometry of contact surface. a Contacting particles with local irregularities, b geometry of contact surface (after Jäger [12])

Fig. 2

Three types of contact geometry. a Conical contact surface, b spherical contact surface, c flat contact surface

Consequently, the contact normal stiffness for the conical contact surface with $\upalpha =1$ can be expressed as

$$\begin{aligned} k_n =\left( {\frac{10G_g }{\pi (1-\nu _g )}} \right) ^{1/2}\left( {f_n } \right) ^{1/2} \end{aligned}$$

(5)

Using Eq. (4), the contact normal stiffness for the spherical contact surface with $\upalpha = 2$ becomes

$$\begin{aligned} k_n =\left( {\frac{2G_g \sqrt{2R}}{3(1-\nu _g )}} \right) ^{2/3}\left( {f_n } \right) ^{1/3} \end{aligned}$$

(6)

Note that Eq. (6) is identical to the Hertzian contact model. The contact normal stiffness for the flat surface is independent of the contact force because $\upalpha =\infty$ in Eq. (4), thus removing the term including $f_n$, so that

$$\begin{aligned} k_n =\frac{4G_g }{1-\nu _g } \end{aligned}$$

(7)

Discrete element modeling (DEM) codes were modified to accommodate the three contact models described above. Regarding the tangential contact stiffness, we employed the expression proposed by Walton [13] which assumes that the tangential contact stiffness $k_t$ has a linear relationship with $k_n$ which can be stated as

$$\begin{aligned} k_t /k_n =2(1-\nu _g )/(2-\nu _g) \end{aligned}$$

(8)

3 Discrete element simulations

Axi-symmetric stress loadings on an assembly of spherical particles were simulated using a non-commercial DEM code called OVAL developed by Kuhn [14, 15]. The simulations performed in the present study involved a three-dimensional assembly of 10,000 uniform spheres with a particle diameter of 0.5 mm.

To prepare an initially isotropic granular assembly within a specified volume ($20\times 20\times 20 \hbox { mm}$), 10,000 seed particles with an artificially smaller diameter (0.1 mm) than the desired value were evenly distributed. Subsequently, the particles were numerically expanded until their diameters reached the desired value of 0.5 mm. The void ratio of the assembly at the completion of particle expansion was 0.56. This generation method involves multiple collisions between particles such that the particles push each other into an evenly-dense arrangement [16]. During expansion, the friction between the particles was purposely ignored for better stabilization. The external boundaries remained fixed during the expansion of the particles, and thus the particle configuration was locally adjusted to attain equilibrium. The model parameters for the simulation are summarized in Table 1.

Table 1 Model parameters used in the DEM simulations

The increment of time needed for stable computation is proportional to the square root of the particle mass divided by the sum of the stiffnesses acting on the particles. This suggests that the simulation of particles of high stiffness demands extremely small time increments and thus significant computational costs. To complete the simulations within a reasonable timeframe, the mass scaling technique [15, 17], which assumes a fictitious mass density for the particles, was employed to ensure stable and expedient simulations. Figure 3 shows the initial assembly of particles prior to stress path probing.

Fig. 3

Assembly of particles after preparation

Both loose and dense specimens were prepared by adjusting the frictional coefficient between the particles. For the dense specimen, the particles began in a frictionless condition and a hydrostatic pressure of 10 kPa was applied to all the faces of the isotropic specimen. To obtain a loose specimen, friction was activated prior to applying the hydrostatic pressure such that the movement of the particles was more restrained than it was for the dense specimen. As mentioned previously, three contact stiffness models were selectively employed so that a total of six specimens with different combinations of contact stiffness and density were prepared for the simulations. Table 2 summarizes the initial conditions of the six specimens. Note that small differences in the void ratio between the loose and dense specimens is common in discrete element methods [18].

Table 2 Configuration of the initial specimens after isotropic compression

Prior to stress probing, the specimen was loaded by increasing the hydrostatic pressure from 10 to 100 kPa, and this isotropic loading is called ISO. Four different stress paths—TXC, RTXC, RTXE, and TXE—were chosen for stress probing. Figure 4 shows the stress paths where the mean normal effective stress $p^{\prime }$ and deviator stress $q$ are defined as

$$\begin{aligned} p^{\prime }=( {\sigma ^{\prime }_x +\sigma ^{\prime }_y +\sigma ^{\prime }_z } )/3 \quad \hbox {and}\quad q=\sigma ^{\prime }_z -\sigma ^{\prime }_h \end{aligned}$$

(9)

where $\sigma _z^{\prime }$ is the vertical (axial) effective stress, $\sigma _x^{\prime }$ and $\sigma _y^{\prime }$ are the two horizontal stresses respectively, and $\sigma _h^{\prime }$ is the average value of the two horizontal stresses. Table 3 summarizes the notation used to describe the stress paths in Fig. 4. Note that instead of the minor principal stress, $\sigma _h^{\prime }$ was used to define the deviator stress and $q$ to account for the minor differences in the two horizontal stresses from the simulations. The difference between the two horizontal stresses was consistently less than 1 kPa. From the same initial stress point, the axial stress increased along the TXC path, the radial stress decreased along the RTXC path, the axial stress decreased along the RTXE path, and the radial stress increased along the TXE path. The stresses were servo-controlled by moving the rigid boundary walls in accordance with continuous calculation of the wall stress defined as the average reaction force acting on the rigid wall divided by the area of the wall.

Fig. 4

Applied stress paths and the stress points for cyclic loadings

Table 3 Triaxial experiments performed for stress probe tests

To measure the macroscopic elastic moduli along the stress path, a number of small-strain, cyclic-loading tests which are frequently performed in soil experiments [19] were simulated. We took advantage of the numerical simulation in that a specimen under a specific stress condition was “frozen” and later used for the small-strain, cyclic-loading tests. Every detail of information on the status of the assembly (e.g., the location of every particle and the contact force distribution) was stored at a specific stress point indicated by a small solid circle in Fig. 4. The stresses at which the cyclic loading tests were conducted were chosen at intervals of 10 kPa along the ISO path of increasing $p^{\prime }$ and at every increment of the stress ratio ($\eta =q{/}p^{\prime })$ of 0.05 along the other stress-probing paths.

The small-strain, cyclic-loading test was conducted by tilting a pair of opposite rigid walls towards a given direction while holding the normal component of the wall stresses constant via a servo-control algorithm. Assuming that the “frozen” specimen preserved its cross-anisotropic nature likely induced from the initial depositional preparation, three tilting directions were chosen as shown in Fig. 5. The elastic shear modulus $G_{hh}$ can be measured by slightly distorting the specimen in the horizontal plane as shown in Fig. 5a. As illustrated in Fig. 5b, c, the small-strain cyclic loadings in the vertical plane were used to measure the elastic moduli $G_{vh}$ and $G_{hv}$ respectively, which should be identical according to the theory of cross-anisotropic elasticity.

Fig. 5

Three different shearing modes for small cyclic tests

4 Results and discussion: macroscopic responses

Figure 6 shows the variation of the stress ratio during shearing. Stress–strain curves typical of frictional material were observed. One response to the axial compression was that the curves for TXC and RTXC overlapped; another was the axial extension of RTXE and TXE. The stress–strain curve for the dense specimens exhibits sharp stress peaks, while the curve for the loose specimens shows a mild change of slopes at the yield stress. The mobilized friction angles at the maximum stress ratios in Fig. 6 are summarized in Table 4. The friction angles at the failure range from 26$^{\circ }$ to 31$^{\circ }$, close to the inter-particle friction angle of $26.5^{\circ }$ given by the frictional coefficient between particles. Particular variations caused by different contact geometries and densities were not observed.

Fig. 6

Macroscopic stress ratio–axial strain relationships. a Conical contact in dense specimen, b conical contact in loose specimen, c spherical contact in dense specimen, d spherical contact in loose specimen, e flat contact in dense specimen, f flat contact in loose specimen

The general expression of the contact normal force–displacement relationship is given by

$$\begin{aligned} k_n =df_n /d\delta _n \end{aligned}$$

(2)

where $k_n$ is the contact normal stiffness, $f_n$ is the contact normal force, and $\delta _n$ is a contact normal displacement. In the case of the contact conditions shown in Fig. 1, the relationship between the contact force and displacement is defined as

$$\begin{aligned} f_n&= c_n (\delta _n )^{1+1/\alpha } \nonumber \\&\quad \hbox {where }c_n =\frac{4G_g \alpha }{(1-\nu _g )(\alpha +1)}\left( {\frac{\Gamma \left( {\frac{\alpha +1}{2}} \right) }{\sqrt{\pi }A_\alpha \Gamma \left( {\frac{\alpha +2}{2}} \right) }} \right) ^{1/\alpha }\nonumber \\ \end{aligned}$$

(3)

where $G_g$ and $\nu _g$ are the elastic shear modulus and Poisson’s ratio of elastic particle grains respectively, and $\Gamma ( x)$ is the gamma function. For example, the contact force–displacement relationship is $f_n =c_n (\delta _n )^{2}$ for the conical asperity $(\upalpha =1), f_n =c_n (\delta _n )^{3/2}$ for the spherical surface ($\upalpha =2$), and $f_n =c_n \delta _n$ for the flat contact surface ($\upalpha =\infty$).

Differentiating and rearranging Eq. (3) with respect to $f_n$ allows the tangent value of the contact normal stiffness $k_n$ to be stated as

$$\begin{aligned} k_n =\frac{df_n }{d\delta _n }=\left( {1+\frac{1}{\alpha }} \right) c_n^{\frac{\alpha }{1+\alpha }} (f_n )^{\frac{1}{1+\alpha }} \end{aligned}$$

(4)

Table 4 Macroscopic friction angle at failure

Unlike the friction angle, the macroscopic stiffness represented by the initial slope of the stress–strain ratio curves is highly dependent on the initial density of the specimens. Figure 6 shows that the stress–strain curves of the dense specimens had a sharper slope than did the stress–strain curves of the loose specimens. As shown in Fig. 6, the stress–strain curves for the dense specimens consistently maintain linearity before a clear peak in stress, whereas the loose specimens show mild nonlinearity in the stress–strain curves before failure. Figure 7 shows that for both the loose and dense specimens, the volume expands similarly during shearing. This dilatancy might be expected if the void ratio values are relatively low (e.g., 0.56–0.57) in both the dense and loose specimens, as summarized in Table 2.

Fig. 7

Macroscopic volumetric strain–axial strain relationships. a Conical contact in dense specimen, b conical contact in loose specimen, c spherical contact in dense specimen, d spherical contact in loose specimen, e flat contact in dense specimen, f flat contact in loose specimen

Figure 8 shows three typical shear stress–strain curves obtained from the small cyclic loading tests on the dense specimen with a conical contact surface along the TXC path. An apparently linear and elastic stress–strain response was observed at the beginning of the shearing when $\upeta =0$ as shown in Fig. 8a. When = 0.5 in Fig. 8b, the stress–strain response exhibits nonlinear hysteresis, indicating energy dissipation due to friction. At $\upeta =1.1$ (close to the value at failure), the hysteretic stress–strain curve is not enclosed as shown in Fig. 8c. Except for a few cases near the point of failure, the initial linear portion of the stress–strain curve was used to define the elastic moduli $G_\mathrm{hh} ,G_\mathrm{vh}$ and $G_\mathrm{hv}$.

Fig. 8

Stress–strain curve obtained from the small strain cyclic shear tests for $\hbox {G}_\mathrm{hh }$. a Linear response at $\upeta =0$, b non-linear response at $\upeta =0.5$, c unstable response at $\upeta = 1.1$

Figures 9, 10 and 11 present variations among the three elastic shear moduli during stress probing on dense specimens. The effect of the packing density will be discussed after providing data for the loosely packed specimens in Figs. 12, 13 and 14. The results show that $G_{vh}$ is practically identical to $G_{hv}$, which confirms that the specimens preserve the cross-anisotropic nature induced by the deposition process during preparation. As described in Eq. (1), both $G_{vh}$ and $G_{hv}$ increase as $p^{\prime }$ increases. However, such increases do not continue as the stress ratio approaches a critical value corresponding to frictional failure. For example, the value of $G_{hv}$ in the TXC path increased until $\upeta = 0.9$ and then fell sharply as shown in Fig. 9. For the RTXC and RTXE paths in which the axial or radial stress decreased, the elastic shear moduli $G_{vh}$ and $G_{hv}$ decreased as predicted by Eq. (1) and subsequently fell sharply as $\upeta$ neared the failure values of 0.9 and $-$0.7 respectively. According to Eq. (1), it is expected that $G_{hh}$ relates only to horizontal stress, as can be seen in Figs. 9, 10, 11, 12, 13 and 14. Along the TXC and RTXE paths in which the horizontal stress remained constant, the value of $G_{hh}$ remained approximately constant until $\upeta =\pm 0.6.$ A significant drop in the value of $G_{hh}$ near the point of failure also appears as it does for $G_{vh}$ and $G_{hv}$.

Fig. 9

Elastic shear moduli for conical contact in dense specimens. a $\upeta \hbox {--} \hbox {G}_\mathrm{hh}$, b $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hh }$, c $\upeta \hbox {--} \hbox {G}_\mathrm{hv }$, d $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hv }$, e $\upeta \hbox {--} \hbox {G}_\mathrm{vh}$, f $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{vh}$

Fig. 10

Elastic shear moduli for spherical contact in dense specimens. a $\upeta \hbox {--} \hbox {G}_\mathrm{{hh}}$, b $\hbox {p}' \hbox {--} \hbox {G}_{\mathrm{{hh}}}$, c $\upeta \hbox {--} \hbox {G}_\mathrm{hv}$, d $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hv}$, e $\upeta \hbox {--} \hbox {G}_\mathrm{vh}$, f $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{vh}$

Fig. 11

Elastic shear moduli for flat contact in dense specimens. a $\upeta \hbox {--} \hbox {G}_\mathrm{hh}$, b $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hh}$, c $\upeta \hbox {--} \hbox {G}_\mathrm{hv}$, d $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hv}$, e $\upeta \hbox {--} \hbox {G}_\mathrm{vh}$, f $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{vh}$

Fig. 12

Elastic shear moduli for conical contact in loose specimens. a $\upeta \hbox {--} \hbox {G}_\mathrm{hh}$, b $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hh}$, c $\upeta \hbox {--} \hbox {G}_\mathrm{hv}$, d $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hv}$, e $\upeta \hbox {--} \hbox {G}_\mathrm{vh}$, f $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{vh}$

Fig. 13

Elastic shear moduli for spherical contact in loose specimens. a $\upeta \hbox {--} \hbox {G}_\mathrm{hh}$, b $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hh}$, c $\upeta \hbox {--} \hbox {G}_\mathrm{hv}$, d $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hv}$, e $\upeta \hbox {--} \hbox {G}_\mathrm{vh}$, f $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{vh}$

Fig. 14

Elastic shear moduli for flat contact in loose specimens. a $\upeta \hbox {--} \hbox {G}_\mathrm{hh}$, b $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hh}$, c $\upeta \hbox {--} \hbox {G}_\mathrm{hv}$, d $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{hv}$, e $\upeta \hbox {--} \hbox {G}_\mathrm{vh}$, f $\hbox {p}'\hbox {--} \hbox {G}_\mathrm{vh}$

It should be noted that the elastic shear moduli for flat contact surfaces are insensitive to variations in stress. As shown in Fig. 11, all of the elastic moduli remained approximately constant until $\upeta =\pm 0.6$. This also means that the exponents in the empirical expression of Eq. (1) are zero in that the elastic shear modulus is independent of these stresses. The exponent used in Eq. (1) is closely related to the value of $\alpha$ used in the microscopic contact law in Eq. (4). For flat contact, the value of $\alpha$ is infinite, so the value of $1/(1+\alpha )$ in Eq. (4) becomes zero and is independent of the contact normal force as in Eq. (7). Consequently, the macroscopic responses reflect the microscopic behavior of the particle contacts.

Unfortunately, the relationship observed between the macroscopic response and microscopic cause in dense specimens does not seem to be valid in loose specimens. Figures 12, 13 and 14 show the simulation results for the different elastic shear moduli of loose specimens. For the loose specimens with conical contact surfaces, $G_{vh}$ and $G_{hv}$ increase along the TXC and TXE paths, but not as quickly as they did with the dense specimens. In addition, a sharp reduction of the elastic moduli as $\upeta$ neared the failure value is not observed in the loose specimens. As shown in Fig. 14, it appears illogical to assume that the elastic moduli are independent of the variations in stress for specimens with flat contact surfaces.

The differences in the responses of dense and loose specimens can be investigated further by quantitatively evaluating the relationship between the elastic moduli and the stresses. Figure 15 summarizes the value of $s$, which is the sum of exponents $m$ and $n$, by fitting the data shown in Figs. 9, 10, 11, 12, 13 and 14 into Eq. (1). The Levenberg–Marquardt iterative algorithm [20] was employed for nonlinear regression analysis to estimate the two exponents. The void ratio function in Eq. (1) is assumed to be $f(e)=(2.17-e)^{2}/(1+e)$ as suggested by Hardin and Richart [21]. Note that interpolations of data were made for stress ratios ranging between $-$0.6 and +0.6 to exclude irrelevant data points corresponding to sharp drops in value when near failure. Note also that the reliability of the shear elastic moduli near failure, suggested in Figs. 9, 10, 11, 12, 13 and 14, is low due to severe nonlinearity in the stress–strain curves as illustrated in Fig. 8c. The parameters obtained from the regression analysis are summarized in Table 5.

Fig. 15

Sum of exponent, s = m + n, for different contact geometries

Table 5 Parameters of empirical fitting equation with $f(e)$

As can be seen in Fig. 15, for a given contact surface condition, there is no difference among the values of $s$ for the different elastic shear moduli. For the different contact surfaces, however, the values of $s$ are quite different. As shown in Fig. 15, the dense specimens have values of $s = 0.5$ for conical contact surfaces, $s=0.33$ for spherical surfaces, and $s=0.0$ for flat surfaces. Surprisingly, these values are exactly the same as the exponents for the expressions of microscopic contact stiffness in Eqs. (5)–(7). These results also confirm the conclusion reached by Jung and Chung [4] that the power function of normal contact stiffness manifests itself in the stress-level-dependent elastic moduli.

However, the values of $s$ for the loose specimens (0.24–0.67) are much higher than they are for the dense specimens (0.02–0.51), with an average difference between the loose and dense specimens of approximately 0.2 for each contact surface condition. To explain the higher value of $s$ in the loose specimens, we need to further investigate the differences in the microscopic responses of the loose and dense specimens.

5 Results and discussion: microscopic responses

The microscopic status of an assembly is statistically described by two quantities. The first is the coordination number $C_n$, which is the average number of contacts per particle and is a measure of the packing density. Table 2 summarizes the initial values of $C_n$ for the different contact models. Second is the contact distribution function $E(\gamma ,\beta )$, which measures the spatial distribution of contacts with respect to angular orientations $\gamma$ and $\beta$ in the spherical coordinate system. Figure 16 shows the spherical coordinate system used in this study. The contact distribution function for a symmetric assembly to the vertical axis [4, 22, 23] can be expressed as

$$\begin{aligned} E(\gamma ,\beta )=\frac{3(1+a\cos 2\gamma )}{4\pi (3-a)} \end{aligned}$$

(10)

which has a symmetry such that $E(\gamma +\pi )=E(\gamma )$ and is also independent of $\beta$. Using Eq. (10), a single parameter $a$, denoted as the degree of fabric anisotropy, can be used to describe the status of the contact normal distribution of the axi-symmetric assembly. The value of $a$ can fall between $-$1.0 and 1.0. The internal fabric structure becomes isotropic when $a = 0.$ The majority of the contact normals are distributed vertically for positive $a$, whereas the contact normals are distributed horizontally for negative $a$. Figure 16b illustrates different probability distributions of the contact normal expressed by Eq. (10) with positive, negative and zero values for $a$. A detailed summary of the expressions of the contact distribution is given in Jang et al. [23].

Fig. 16

Contact normal distribution function of axi-symmetric condition in spherical coordination system. a Spherical coordinate system, b contact normal distribution function

Microscopic parameters $C_n$ and $a$ were evaluated using the data obtained from a “frozen” assembly, which was also used for the small-strain, cyclic-loading tests. Figures 17, 18 and 19 show the variations of $C_n$ and $a$ in loose and dense specimens with different contact models. For the dense specimens, the values of $C_n$ and $a$ remain constant within the range of stress ratios measuring between $-$0.6 and 0.6. Beyond this range of stress ratios, the value of $C_n$ dramatically decreases, which is surprisingly similar to the macroscopic response—a sudden drop in the value of the elastic shear moduli in dense specimens. So long as the value of $C_n$ remains constant, the value of $a$ also remains constant at zero, implicating isotropy in the fabric structure. Beyond this range of constant value, the value of $a$ significantly increases for stress ratios larger than 1.0 and decreases for stress ratios of less than $-$0.6.

Fig. 17

Evolution of microscopic parameters for conical contact. a $\upeta \hbox {--}\hbox {C}_\mathrm{n}$ in dense specimens, b $\upeta \hbox {--}\hbox {C}_\mathrm{n}$ in loose specimens, c $\upeta \hbox {--} \hbox {a}$ in dense specimens, d $\upeta \hbox {--} \hbox {a}$ in loose specimens

Fig. 18

Evolution of microscopic parameters for spherical contact. a $\upeta \hbox {--} \hbox {C}_\mathrm{n}$ in dense specimens, b $\upeta \hbox {--} \hbox {C}_\mathrm{n}$ in loose specimens, c $\upeta \hbox {--} \hbox {a}$ in dense specimens, d $\upeta \hbox {--} \hbox {a}$ in loose specimens

Fig. 19

Evolution of microscopic parameters for flat contact. a $\upeta \hbox {--} \hbox {C}_\mathrm{n}$ in dense specimens, b $\upeta \hbox {--} \hbox {C}_\mathrm{n}$ in loose specimens, c $\upeta \hbox {--} \hbox {a}$ in dense specimens, d $\upeta \hbox {--} \hbox {a}$ in loose specimens

The patterns observed in the loose specimens, however, are quite different from those observed in the dense specimens. For the conical contact model, the value of $C_n$ varies nonlinearly—initially increasing and then decreasing—on the TXC and TXE paths, whereas the value of $C_n$ gradually decreases without any constancy on the RTXC and RTXE paths. Unlike the dense specimens, the loose specimens do not exhibit a dramatic decrease of $C_n$ near the failure values of the stress ratio. For the spherical and flat contact conditions, the value of $C_n$ steadily increases. The variations of $a$ in the loose specimens are also considerably different from those in the dense specimens. Unlike the dense specimens, the loose specimens do not display a constant value of $a$ for the stress ratios between $-$0.6 and 0.6. Instead the value of $a$ nonlinearly varies with the stress ratio. This implies that the change of fabric structure manifested by the variation of $a$ is much easier for the loose specimens than it is for the dense specimens. As shown in Table 2, a lower value of $C_n$ for the loose specimens provides a higher degree of freedom in particle movement than for the dense specimens such that the contact points are continuously reallocated in favorable directions. In the dense specimens with high $C_n$ values, the movement of particles is highly restrained but these restraints are eventually released at high stress ratios accompanying an abrupt reduction of in the value of $C_n$. Similar responses have also been observed by Maeda et al. [24].

The macroscopic response shows large values for $s$ for the loose specimens, which implies that, with respect to elastic shear moduli, the loose specimens are more dependent on stresses than the dense specimens are. The gap between the values for the loose and dense specimens supposes that the loose specimens have a causal agent other than contact stiffness, which has already been reflected in the low values for $s$ for the dense specimens. It is estimated that microscopic variation in the particle contacts causes the additional dependency of the elastic shear modulus on the stresses in the loose specimens. Due to a lack of constraints corresponding to low $C_n$ values, the loose specimens experience the continuous rearrangement of their particles as well as the transition of contact force chains during shearing. This mobility in the contacts of the loose specimens eventually increases the dependency of the elastic shear moduli on stresses.

6 Conclusions

To investigate the origin of the stress-level dependency of soil elasticity, a series of stress-path experiments were conducted on loose and dense specimens with three different contact surface shapes (i.e., conical, spherical and flat). An analytical model assuming the geometry of the contact body as an elastic sphere with a local axi-symmetric irregularity was introduced in the discrete element simulations. To evaluate the cross-anisotropic elastic shear moduli, small-strain, cyclic-loading tests were simulated on assemblies under the stress conditions along four stress probing paths. To study the stress-level dependency of soil elasticity, the sums of the exponents in the expression relating the elastic shear moduli and stresses has been focused upon.

For the dense specimens, the contact geometry had a decisive influence on the sum of the exponents, expressed as $s$. The simulation results show that $s = 0.5$ for the conical contact surface, 0.33 for the spherical surface, and 0.0 for the flat surface. Surprisingly, these values are exactly identical to the value of the exponents for the expressions of the contact stiffness equation. For the loose specimens, however, the value of $s$ is much higher than it is for the dense specimens such that there is no correlation between the value of $s$ and the exponent in the contact stiffness model.

Microscopic data explains the reason why the loose and dense specimens exhibit different degrees of stress-level dependency with regard to the elastic shear moduli. For the dense specimens with high $C_n$ values, the internal structure represented by the degree of fabric anisotropy and the $C_n$ value remain unchanged during shearing, thus leading to the coincidence of the $s$ value and the exponent in the contact stiffness model. For the loose specimens with a low $C_n$ value, however, the fabric structure evolves continuously during shearing, which results in an increase in the exponents in the power function of the elastic modulus. The rearrangement of particles and the transition of contact force chains with the evolution of the fabric manifest themselves as increasing dependency with regard to the elastic moduli on the stresses in the loose specimens.