Particle-Scale Insight into Deformation Noncoaxiality of Granular Materials

Li, X.; Yu, H.-S.

doi:10.1061/(ASCE)GM.1943-5622.0000338

Open access

Technical Papers

Jul 4, 2013

Particle-Scale Insight into Deformation Noncoaxiality of Granular Materials

Authors: X. Li [email protected] and H.-S. YuAuthor Affiliations

Publication: International Journal of Geomechanics

Volume 15, Issue 4

https://doi.org/10.1061/(ASCE)GM.1943-5622.0000338

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Abstract

This paper explores the mechanism of deformation noncoaxiality from the particle scale. A multiscale investigation has been carried out with the particle-scale information obtained from discrete element simulation on granular materials. The specimens were prepared anisotropically and sheared in various loading directions. The deformation noncoaxiality, i.e., noncoincidence between the principal stress direction and the principal strain increment direction, was observed. The directional statistical theory has been used to study the anisotropies in material fabric and particle interactions, and to characterize them in terms of direction tensors. Based on the stress-force-fabric relationship, the stress direction was determined from these direction tensors. In monotonic loading, it was observed that the force anisotropy is always coaxial with the loading direction, i.e., the strain increment direction, while the fabric anisotropy only gradually approaches the loading direction. This noncoincidence between the fabric anisotropy and the strain increment direction is found to be the main cause of deformation noncoaxiality. An expression to calculate the degree of noncoaxiality from the ratio and the direction deviation between the fabric and force anisotropies has been proposed.

Introduction

The theory of plasticity took experimental observations of solids, especially metal, as its starting point. Materials are assumed to be isotropic. With the principle of material objectivity, the plasticity theories, including the flow rules, were developed on the magnitude of the three principal stresses but not on their directions. With great physical insight, in 1870 Saint-Venant proposed that the principal axes of the plastic strain increment coincided with the axes of the principal stress. Hill (1950) subsequently pointed out that material may become more anisotropic as the deformation continues and the theoretical framework needs to be broadened to include anisotropy. This is particularly true for granular materials such as sand.

Experimental and numerical observations have demonstrated that anisotropy is an important feature of granular material. As a consequence, the principal plastic strain increment direction may not be coincident with the principal stress direction. Such a phenomenon is referred to as noncoaxial behavior, and the deviation between the two directions is referred to as the degree of noncoaxiality (Yu 2006). One of the earliest experimental studies on the noncoaxial behavior of granular materials concerning sand behavior subjected to simple shearing was reported by Roscoe et al. (1967). Arthur and Menzies (1972) and Oda (1972) studied the influence of the initial fabric on the strength and deformation of granular materials by preparing samples in a tilting mold to give different directions of sample deposition with respect to the applied principal stress directions. Arthur et al. (1977) applied controlled changes of principal stress directions on dense sand samples in plane strain condition with the directional shear cell (DSC). The samples experienced an initial loading to a high stress ratio followed by unloading to an isotropic stress state. After rotation of the principal stress direction, the sample was loaded incrementally to failure with the principal stress direction held constant. The observed differences in the soil behavior resulted from the pre-shearing confirming that the influence of induced anisotropy is at least as important as that of the initial anisotropy. The tests revealed a large divergence between the stress and strain increment directions at low stress ratios. The divergence diminished as shearing continued.

The collective evidence of deformation noncoaxiality provoked the development of conventional plastic potential theory to better predict intrinsic soil behaviors (Li and Dafalias 2004; Tsutsumi and Hashiguchi 2005; Yang and Yu 2006). Most developments in plasticity theory rely on empirical observation. Deviation from model predictions to real material behavior can be significant when the in situ loading path or working conditions are different from those encountered in model developments. A fundamental understanding of deformation noncoaxiality is yet to be found.

Multiscale investigations and homogenization theory have become fast developing subjects of research over the past few decades and are now important branches of study in relation to granular materials. Oda et. al (1985) discussed stress-induced anisotropy on the basis of biaxial compression tests with two-dimensional (2D) assemblies of oval cross-sectional rods and presented detailed fabric information during loading. Numerical simulation using the discrete element method (DEM) has gained popularity in multiscale investigations [Cundall and Strack 1979; Itasca Consulting Group (ICG) 1999]. Its unique advantage is that detailed particle-scale information, including particle displacements and interactions, can be easily retrieved at any stage of loading (Cundall and Strack 1983). Thornton and Zhang (2006) presented DEM simulation results of simple shear on spherical particles. Li and Yu (2009) prepared and tested two anisotropic specimens with varying loading directions. Both have reported significant noncoaxiality between the principal directions of stress and strain increments.

This paper addresses the particle-scale mechanism of deformation noncoaxiality. DEM simulation results have been used to facilitate the multiscale investigation (Li and Yu 2009). Directional statistical theory has been used to study the anisotropies in material fabric and particle interactions, and to characterize them in terms of direction tensors. The principal stress direction is determined by these direction tensors from the stress-force-fabric (SFF) relationship (Li and Yu 2013). Because the shearing is applied with a fixed strain increment direction, the deviation between the stress and loading directions is known as the degree of noncoaxiality.

For completeness, the paper is organized as follows. First, information on the DEM simulations and observed deformation noncoaxiality, as given in Li and Yu (2009), is briefly presented. Next, directional statistical theory is introduced and the SFF relationship is presented. Then, an elaboration of the particle-scale mechanism of deformation noncoaxiality in light of the SFF relationship is given. Finally, the main conclusions are summarized.

DEM Simulation Results of Deformation Noncoaxiality in Monotonic Loading

In an effort to study the dependence of granular material behavior on initial fabric and loading paths, Li and Yu (2009) prepared and tested two anisotropic specimens with fixed loading directions using the 2D commercial discrete element package, PFC2D 3.1 (ICG 1999).

Material Properties and Test Procedures

Each particle was formed by clumping two equal-sized disks together. The distance between the centers of the two disks was equal to 1.5 times the disk radius,

r

. The particle size was uniformly distributed within the range of 0.2–0.6 mm in terms of equivalent diameter, and the disk thickness was

t = 0.2 mm

. The contact law included two stiffness models (normal and tangential) and a slip model. In the simulations, the elastic models were linear, and the particle stiffnesses were set as

k_{n} = k_{s} = 10^{5} N / m

. The coefficient of friction was 0.5.

Two anisotropic specimens were used in the simulations. One was the initially anisotropic sample prepared using the deposition method. The void ratio of the prepared sample was 0.204 at

p_{c} = 1,000 kPa

. The other was the preloaded sample, prepared by shearing the initially anisotropic sample vertically up to 25% deviatoric strain and then unloaded to the isotropic stress state. The void ratio of the preloaded sample was 0.222 at the mean normal stress

p_{c} = 1,000 kPa

. The two specimens were consolidated, and sheared up to 25% deviatoric strain at varying loading directions. The loading direction was denoted by angle

α

, which is the deviation to the horizontal direction, i.e., the

x_{1} - axis

direction. Loading was applied in strain-controlled mode with the principal strain direction fixed. The strain increment direction coincides with the major principal strain direction.

Observation of Deformation Noncoaxiality in Anisotropic Specimens

The major principal directions of the stress and strain increment for the two specimens are plotted versus the stress ratio in Fig. 1, where the strain increment direction is indicated by the vertical lines while the stress direction is shown as hollow symbols. Laboratory test results similar to the behavior of initially anisotropic specimens have been previously reported (Arthur et al. 1977; Gutierrez et al. 1991; Miura et al. 1986). The principal stress direction is observed to slightly deviate from the strain increment direction, although the difference in the two directions was found to be very limited at the greatest 5°. The maximum deviation took place around

α = 45 °

, where the deviation was toward the deposition direction. The observation of the preloaded specimen has been found to be similar to that in laboratory experiments with a fixed principal stress direction (Symes et al. 1988; Wong and Arthur 1985). The major principal stress direction always deviates from the major principal strain direction toward the preloading direction. Significant noncoincidence between the stress and strain increment directions has been observed for the preloaded sample. This is dramatically different from that of the initially anisotropic sample. The differences diminished gradually as shearing progressed to a higher shear strain and larger stress ratio.

Fig. 1. Deformation noncoaxiality of anisotropic specimens (Reprinted from *International Journal of Engineering Science,* X. Li and H.-S. Yu, “Influence of Loading Direction on the Behavior of Anisotropic Granular Materials,” pp. 1284–1296, © 2009, with permission from Elsevier): (a) initially anisotropic specimen; (b) preloaded specimen

Directional Statistics and SFF Relationship

The continuum-scale stress-strain relationship of granular material is governed by its constitutive particles in terms of the material fabric and the particle interactions. Directional statistics theory has been developed (Kanatani 1984; Li and Yu 2011). It is a powerful tool in processing the particle-scale information and characterizing anisotropy in material fabrics and interactions in terms of direction tensors.

Integral Form of the Microstructural Stress Tensor

Let

Ω

represent the unit circle in 2D spaces (

D = 2

) and the unit sphere in three-dimensional (3D) spaces (

D = 3

). Denote the total number of contacts in a granular assembly as

M

, and let

Δ M (n)

represent the number of contacts whose normal directions fall into the stereoangle element

Δ Ω

centered at direction

n

. The discrete spectra of function

e^{c} (n) = Δ M (n) / Δ Ω

gives the probability density of the contact normal in direction

n

. Viewing a granular material as an assembly of granular particles with point contacts, the macroscale tensor over volume

V

can be evaluated from the tensor product of contact forces

f_{i}^{c}

and contact vectors

v_{i}^{c}

as follows (Li and Yu 2013; Rothenburg and Bathurst 1989):

σ_{ij} = \frac{1}{V} \sum_{c \in V} v_{i}^{c} f_{j}^{c} = \frac{M}{V} \sum_{Ω} e^{c} (n) {〈 v_{i} f_{j} 〉 |}_{n} Δ Ω

(1)

where

{* |}_{n}

= value of variable

*

in direction

n

;

{〈 * 〉 |}_{n}

= average value of all terms of

*

sharing the same contact normal direction

n

; and

Δ Ω

= elementary solid angle segment. The coordination number, i.e., the average number of contacts per particle, is

ω = M / N

, where

N

is the total number of particles. When

Δ Ω \to 0

, the transition leads to an expression of the stress tensor in terms of integration over all stereoangles as follows:

σ_{ij} = \frac{ω N}{V} \oint_{Ω} e^{c} (n) {〈 v_{i} f_{j} 〉 |}_{n} d Ω

(2)

Eq. (2) is a directional integration over the multiplication of the contact normal probability density

e^{c} (n)

and the joint product

{〈 v_{i} f_{j} 〉 |}_{n}

. Directional statistics theory is used to study the directional distributions of contact normal density

e^{c} (n)

, mean contact vector

{〈 v 〉 |}_{n}

, and mean contact force

{〈 f 〉 |}_{n} m

, as well as the statistical dependence between the contact vectors and contact forces. In this paper, the particle-scale data obtained by shearing the initially anisotropic specimen vertically, i.e.,

α = 90 °

, to 2% deviatoric strain is used to exemplify the analyses.

Observations of Statistical Dependence

In general,

{〈 v_{i} f_{j} 〉 |}_{n} \neq {〈 v_{i} 〉 |}_{n} {〈 f_{j} 〉 |}_{n}

, where

{〈 v_{i} 〉 |}_{n}

and

{〈 f_{j} 〉 |}_{n}

denote the mean contact vector and the mean contact force along direction

n

, respectively. Li and Yu (2013) compared the directional distribution of

{〈 v_{i} f_{j} 〉 |}_{n}

and

{〈 v_{i} 〉 |}_{n} {〈 f_{j} 〉 |}_{n}

. The results suggested that the effect of the statistical dependence between contact vectors and contact forces can be taken into account by approximating

{〈 v_{i} f_{j} 〉 |}_{n} = ς {〈 v_{i} 〉 |}_{n} {〈 f_{j} 〉 |}_{n}

(3)

where

ς

= direction-independent variable. Directional statistical analyses were carried out on two anisotropic specimens undergoing shearing in different directions. The results suggest that Eq. (3) remains a reasonable approximation.

Contact Normal Density $e^{c} (n)$

Applying directional statistical theory, the direction distributions can be approximated with polynomial functions of unit directional vector

n

. The directional distribution of contact normal density

e^{c} (n)

can be approximated as

E^{c} (n) = \frac{1}{E_{0}} (1 + D_{i_{1} i_{2}}^{c} n_{i_{1}} n_{i_{2}} + \dots + D_{i_{1} i_{2} \dots i_{n}}^{c} n_{i_{1}} n_{i_{2}} \dots n_{i_{n}} + \dots)

(4)

where

D_{i_{1} i_{2} \dots i_{n}}^{c}

= deviatoric direction tensor for the contact normal density. In view of its symmetry,

D_{i_{1} i_{2} \dots i_{n}}^{c}

should be symmetric with respect to subscripts

i_{1}, i_{2}, \dots, i_{n}

, i.e.,

D_{i_{1} i_{2} \dots i_{n}}^{c} = D_{(i_{1} i_{2} \dots i_{n})}^{c}

, ( ) over the subscripts designates the symmetrisation of the indices. Because Eq. (4) is an orthogonal decomposition,

D_{i_{1} i_{2} \dots i_{n}}^{c}

is deviatoric, i.e.,

D_{i_{1} \dots i_{k} \dots i_{l} \dots i_{n}}^{c} δ_{i_{k} i_{l}} = 0

. Here,

D_{i_{1} i_{2} \dots i_{n}}^{c}

can be calculated based on the particle-scale information following the method introduced in Li and Yu (2013), where the deviatoric tensor of the second-order power term was found dominant and those of higher-order power terms were negligible. Hence, the directional distributions of the contact normal density can be sufficiently approximated with up to second-rank power terms of unit directional vector

n

as

E^{c} (n) = \frac{1}{2 π} [1 + d_{2}^{c} \cos (2 θ - ϕ_{2}^{c})]

(5)

where

d_{2}^{c}

denotes the magnitude of directional variation and

{f i}_{2}^{c}

indicates the preferred principal direction. (Please see the attached file for symbols). The deviatoric direction tensor of the second-order power term can be expressed as

D_{i_{1} i_{2}}^{c} = d_{2}^{c} (\begin{matrix} \cos ϕ_{2}^{c} & \sin ϕ_{2}^{c} \\ \sin ϕ_{2}^{c} & - \cos ϕ_{2}^{c} \end{matrix})

(6)

For the initially anisotropic specimen sheared vertically up to 2% deviatoric strain, it can be determined that

d_{2}^{c} = 0.46

and

ϕ_{2}^{c} = 178 °

. The approximation using Eq. (5) is plotted in Fig. 2 together with the discrete data spectra from the DEM simulations. Fig. 2 suggests that Eq. (5) gives a good approximation of the directional distribution of the contact normal density.

Fig. 2. Comparison with DEM data [contact normal density $e^{c} (n)$ ]

Mean Contact Force ${〈 f 〉 |}_{n}$

The mean contact force can be approximated by an antisymmetric function with respect to direction

n

, i.e.,

{〈 f 〉 |}_{n} = - {〈 f 〉 |}_{- n}

. The contact forces averaged over contacts sharing the same normal direction

{〈 f 〉 |}_{n}

can be approximated in the form of orthogonal decomposition as

{〈 f_{j} 〉 |}_{n} = f_{0} (n_{j} + G_{j i_{1}}^{f} n_{i_{1}} + \dots + G_{j i_{1}, \dots, i_{n}}^{f} n_{i_{1}}, \dots, n_{i_{n}} + \dots)

(7)

where

f_{0}

= directional average of mean normal contact force

{〈 f^{n} 〉 |}_{n} = {〈 f 〉 |}_{n} \cdot n

, i.e.,

f_{0} = \oint_{Ω} {〈 f 〉 |}_{n} \cdot n d Ω / E_{0}

; and

G_{j i_{1} \dots i_{n}}^{f}

= deviatoric direction tensor for the mean contact force (Li and Yu 2013).

Here,

G_{j i_{1} \dots i_{n}}^{f}

is deviatoric and symmetric with respect to subscripts

i_{1} \dots i_{n}

, i.e.,

G_{j i_{1} i_{2} \dots i_{n}}^{f} = G_{j (i_{1} i_{2} \dots i_{n})}^{f}

and

G_{j i_{1} \dots i_{k} \dots i_{l} \dots i_{n}}^{f} δ_{i_{k} i_{l}} = 0

. Direction tensors

G_{j i_{1} \dots i_{n}}^{f}

of different ranks are determined independently based on the obtained particle-scale information. The magnitudes of the direction tensors are calculated and found to decrease quickly as the order of the power terms increases. In 2D spaces, the mean contact force can be sufficiently approximated with up to third-power terms as

{〈 f 〉 |}_{n} = f_{0} {(\begin{matrix} \cos θ \\ \sin θ \end{matrix}) + B_{1}^{f} [\begin{matrix} \cos (θ - β_{1}^{f}) \\ - \sin (θ - β_{1}^{f}) \end{matrix}] + A_{3}^{f} [\begin{matrix} \cos (3 θ - α_{3}^{f}) \\ \sin (3 θ - α_{3}^{f}) \end{matrix}]}

(8)

with the corresponding expressions of direction tensors

G_{j i_{1}}^{f} = B_{1}^{f} (\begin{matrix} \cos β_{1}^{f} & \sin β_{1}^{f} \\ \sin β_{1}^{f} & - \cos β_{1}^{f} \end{matrix}), G_{j i_{1} 11}^{f} = A_{3}^{f} (\begin{matrix} \cos α_{3}^{f} & \sin α_{3}^{f} \\ - \sin α_{3}^{f} & \cos α_{3}^{f} \end{matrix})

(9)

The remaining component of

G_{j i_{1} i_{2} i_{3}}^{f}

can be determined because it is symmetric and deviatoric with respect to

i_{1}

,

i_{2}

,

i_{3}

. Hence, the normal and tangential components of the mean contact forces can be determined as

{〈 f^{n} 〉 |}_{θ} = f_{0} [1 + B_{1}^{f} \cos (2 θ - β_{1}^{f}) + A_{3}^{f} \cos (2 θ - α_{3}^{f})] = f_{0} [1 + C_{n}^{f} \cos (2 θ - ϕ_{n}^{f})]

(10)

{〈 f^{t} 〉 |}_{θ} = f_{0} [- B_{1}^{f} \sin (2 θ - β_{1}^{f}) + A_{3}^{f} \sin (2 θ - α_{3}^{f})] = - f_{0} c_{t}^{f} \sin (2 θ - ϕ_{t}^{f})

(11)

where

C_{n}^{f} = \sqrt{{(B_{1}^{f})}^{2} + {(A_{3}^{f})}^{2} + 2 B_{1}^{f} A_{3}^{f} \cos (β_{1}^{f} - α_{3}^{f})}, \tan ϕ_{n}^{f} = \frac{B_{1}^{f} \sin β_{1}^{f} + A_{3}^{f} \sin α_{3}^{f}}{B_{1}^{f} \cos β_{1}^{f} + A_{3}^{f} \cos α_{3}^{f}}

and

c_{t}^{f} = \sqrt{{(B_{1}^{f})}^{2} + {(A_{3}^{f})}^{2} - 2 B_{1}^{f} A_{3}^{f} \cos (β_{1}^{f} - α_{3}^{f})}, \tan ϕ_{t}^{f} = \frac{B_{1}^{f} \sin β_{1}^{f} - A_{3}^{f} \sin α_{3}^{f}}{B_{1}^{f} \cos β_{1}^{f} - A_{3}^{f} \cos α_{3}^{f}}

For the initially anisotropic specimen sheared vertically up to 2% deviatoric strain, it is determined that

f_{0} = 0.076 N

,

c_{n}^{f} = 0.48

,

ϕ_{n}^{f} = 180 °

,

c_{t}^{f} = 0.26

, and

ϕ_{t}^{f} = 179 °

. Fig. 3 plots the particle-scale information collected from the DEM simulation and the approximations using Eqs. (10) and (11); approximations using up to the third-rank terms were found to be sufficient.

Fig. 3. Comparison with DEM data (mean contact force ${〈 f 〉 |}_{n}$ ): (a) mean normal contact force; (b) mean tangential contact force

Mean Contact Vector ${〈 v 〉 |}_{n}$

For contact vectors, it is assumed that on average particles have center-point symmetric geometries, i.e.,

{〈 v 〉 |}_{n} = - {〈 v 〉 |}_{- n}

. The directional statistical analyses and data processing are similar to those for contact forces and hence not repeated here. The anisotropy in the mean contact vector is found to be very small despite the noncircular particle shape used in simulations. In 2D spaces, the directional distributions of mean contact vectors are expressed up to the first-power terms of directional vector

n

as

{〈 v 〉 |}_{θ} = v_{0} {(\begin{matrix} \cos θ \\ \sin θ \end{matrix}) + B_{1}^{v} [\begin{matrix} \cos (θ - β_{1}^{v}) \\ - \sin (θ - β_{1}^{v}) \end{matrix}]}

(12)

The corresponding direction tensor is expressed as

G_{j i_{1}}^{v} = B_{1}^{v} (\begin{matrix} \cos β_{1}^{v} & \sin β_{1}^{v} \\ \sin β_{1}^{v} & - \cos β_{1}^{v} \end{matrix})

(13)

And the mean normal and tangential contact vectors are

{〈 v^{n} 〉 |}_{θ} = v_{0} [1 + B_{1}^{v} \cos (2 θ - β_{1}^{v})]

(14)

{〈 v^{t} 〉 |}_{θ} = v_{0} [- B_{1}^{v} \sin (2 θ - β_{1}^{v})]

(15)

For the initially anisotropic specimen sheared vertically up to 2% deviatoric strain, it has been calculated that

v_{0} = 0.193 mm

,

B_{1}^{v} = 0.048

, and

β_{1}^{f} = - 8 °

. The approximations are plotted in Fig. 4 together with the collected information from the DEM simulation. The first-rank approximation is found to be sufficient for the mean normal contact vector. A disparity in the mean tangential contact vector is observed; however, this is considered to be insignificant because its magnitude is found to be extremely small.

Fig. 4. Comparison with DEM data (mean contact vector ${〈 v 〉 |}_{n}$ ): (a) mean normal contact vector; (b) mean tangential contact vector

SFF Relationship in Granular Materials

Substituting Eqs. (3), (5), (8), and (12) into Eq. (2) leads to an expression of the SFF relationship truncated by selected ranks for approximations. Because the anisotropic magnitude of the contact vector is small, the joint product terms involving

G_{ij}^{v}

are expected to be extremely small; therefore, they are neglected. Invoking symmetry in the Cauchy stress tensor, i.e.,

σ_{12} = σ_{21}

, a simplified expression of the SFF relationship is obtained as follows:

σ_{ij} = \frac{ω N}{2 V} ς v_{0} f_{0} [(1 + C) δ_{ij} + G_{ji}^{f} + G_{ij}^{v} + \frac{1}{2} D_{ij}^{c}]

(16)

where

C = (1 / 2) G_{i l_{1}}^{f} G_{i l_{1}}^{v} + (1 / 4) (D_{i k_{1}}^{c} G_{i k_{1}}^{f} + D_{i k_{1}}^{c} G_{i k_{1}}^{v}) + (1 / 8) D_{k_{1} k_{2}}^{c} G_{ii k_{1} k_{2}}^{f}

. The expressions for direction tensors

D_{ij}^{c}

,

G_{ji}^{f}

,

G_{j i_{1} i_{2} i_{3}}^{f}

, and

G_{ji}^{v}

are given in Eqs. (6), (9), and (13).

With the direction tensors determined from the particle-scale information obtained from the DEM data, the stress tensor can be calculated from Eq. (16). The result is plotted in Fig. 5 in terms of the stress invariants

p = (σ_{11} + σ_{22}) / 2

and

η = q / p = \sqrt{{(σ_{11} - σ_{22})}^{2} + 4 σ_{12} σ_{21}} / p

, and the principal stress direction

θ_{a}

. In Fig. 5, the solid symbols indicate data from the DEM specimen boundary and the open symbols indicate the prediction from Eq. (15). The accuracy of the derived SFF relationship is checked by comparing the two curves. In general, the SFF relationship as predicted from Eq. (16) gives an excellent interpretation of the specimen stress state in terms of both the principal stresses and the principal stress direction.

Fig. 5. Accuracy of the SFF relationship: (a) initially anisotropic specimen; (b) preloaded specimen

Micromechanics of Deformation Noncoaxiality

The SFF relationship from Eq. (16) explicitly expresses the stress tensor in terms of the direction tensors (Li and Yu 2013). In the DEM simulations presented in Li and Yu (2009), shearing is applied with the strain increment direction fixed. The deformation noncoaxiality behavior can be investigated as the deviation of the principal stress directions determined from Eq. (16) to the loading direction.

Principal Stress Direction

According to Eq. (16), the principal stress direction is coaxial with

[G_{ji}^{f} + G_{ij}^{v} + (1 / 2) D_{ij}^{c}]

. Thus, the principal stress direction is determined by the anisotropic magnitudes and the principal directions of the deviatoric direction tensors for the contact normal density, mean contact force, and mean contact vector. The contact normal density and the mean contact vector are associated with the particle geometries and their spatial arrangements. Because

G_{ij}^{v}

and

D_{ij}^{c}

are deviatoric symmetric tensors,

[G_{ij}^{v} + (1 / 2) D_{ij}^{c}]

is also deviatoric and symmetric. Hence, an anisotropic fabric tensor

C_{ij}

that reflects the combined influence of the fabric anisotropy from both the contact normal density and the mean contact vector can be defined as

C_{ij} = G_{ij}^{v} + \frac{1}{2} D_{ij}^{c} = Δ (\begin{matrix} \cos ψ & \sin ψ \\ \sin ψ & - \cos ψ \end{matrix})

(17)

where

Δ

= information for the magnitude of fabric anisotropic; and

ψ / 2

= principal fabric direction.

With Eqs. (9) and (17), by denoting

\tan θ = \frac{(Δ / B_{1}^{f}) \sin (ψ - β_{1}^{f})}{(Δ / B_{1}^{f}) \cos (ψ - β_{1}^{f}) + 1}

(18)

then

G_{ji}^{f} + G_{ij}^{v} + \frac{1}{2} D_{ij}^{c} = {[Δ^{2} + 2 Δ B_{1}^{f} \cos (ψ - β_{1}^{f}) + {(B_{1}^{f})}^{2}]}^{1 / 2} [\begin{matrix} \cos (θ + β_{1}^{f}) & \sin (θ + β_{1}^{f}) \\ \sin (θ + β_{1}^{f}) & - \cos (θ + β_{1}^{f}) \end{matrix}]

(19)

Thus, the deviation of the principal stress direction to the principal direction of force anisotropy is

θ / 2

. From Eq. (18), it is known that angle

θ / 2

is only dependent on the ratio of the fabric and force anisotropies in terms of

Δ / B_{1}^{f}

and the deviation between their principal directions

(ψ - β_{1}^{f}) / 2

.

Fig. 6 gives information on the principal stress direction in terms of

θ / 2

at different values of

Δ / B_{1}^{f}

and

(ψ - β_{1}^{f}) / 2

. From Fig. 6 it is observed that when the fabric anisotropy is much smaller than the force anisotropy the deviation of the principal stress direction to the principal force direction is small and the maximal deviation occurs around

(ψ - β_{1}^{f}) / 2 = 45 °

. When the fabric anisotropy becomes comparable with the force anisotropy as

Δ / B_{1}^{f}

increases, the deviation of the principal stress direction to the principal force direction is larger and the maximal value skews toward higher

(ψ - β_{1}^{f}) / 2

. When

Δ / B_{1}^{f} = 1

, the principal stress is always in the middle between the principal force direction and the principal fabric direction, i.e.,

θ = (ψ + β_{1}^{f}) / 2

. When the ratio further increases to

Δ / B_{1}^{f} > 1

, the principal stress direction becomes closer to the fabric anisotropy direction than to the force anisotropy direction.

Particle-Scale Statistics

Directional analyses have been carried out to study the particle-scale information obtained from the previous two series of simulations on anisotropic specimens and the direction tensors have been calculated. The results from the two anisotropic specimens sheared at

α = 45 °

are given in Fig. 7.

Fig. 7. Direction tensor evolution when subjected to shearing at $α = 45 °$ : (a) initially anisotropic specimen; (b) preloaded specimen

The two specimens were initially anisotropic, as evidenced by the nonzero values of the anisotropic magnitudes before shearing. When sheared, the anisotropy in the mean contact force is the most significant of the three sources of anisotropies. Its principal direction almost spontaneously becomes coaxial with the loading direction. The anisotropic magnitude of the contact normal density is also observed to be important. However, its principal direction gradually approaches the loading direction in the first few percentages of shear strain and becomes coaxial with the loading direction at large strain levels. The anisotropic magnitude of the mean contact vector is limited.

The information for the proposed anisotropic fabric tensor

C_{ij}

is also included in Fig. 7 in terms of its anisotropic magnitude

Δ

and phase angle

ψ / 2

. Before shearing, the stress is isotropic, corresponding to the observations that the magnitudes in the force and fabric anisotropies are of equal magnitudes and their principal directions are normal to each other. For both specimens at their initial states, the principal direction for the contact normal density is vertical while the principal direction for the contact vector is horizontal. When sheared, the force anisotropy is found to develop more quickly and to remain at a higher value than the fabric anisotropy.

Particle-Scale Mechanism of Deformation Noncoaxiality

For the two series of simulations used here, the strain increment direction is fixed and is referred to as the loading direction. Fig. 7 suggests that the force anisotropy almost spontaneously becomes coaxial with the loading direction when shearing is applied, while the fabric anisotropy does not. Given the assumption that the force anisotropy is coaxial with the strain increment, the degree of noncoaxiality is hence equal to the deviation of the principal stress direction to that of force anisotropy, i.e.,

θ / 2

. From Eq. (18), the degree of noncoaxiality can be calculated based on the ratio

Δ / B_{1}^{f}

and the direction deviation between the contact force and material fabric

(ψ - β_{1}^{f}) / 2

. The dependence on the degree of noncoaxiality of

Δ / B_{1}^{f}

and

(ψ - β_{1}^{f}) / 2

is shown in Fig. 6.

For the isotropic specimen with

Δ = 0

, the degree of noncoaxiality is equal to zero. The material behavior is always coaxial. The deformation noncoaxiality is manifested only when the material is anisotropic and the force and fabric anisotropies are not coaxial. Therefore, the deviation of the principal fabric direction from the loading direction, which is also the principal direction of force anisotropy, is the main cause of the deformation noncoaxiality.

Fig. 7 shows that the initially anisotropic sample and the preloaded sample are of similar anisotropic degrees before shearing. However, upon shearing, the evolutions of the directional distributions are different in terms of both their anisotropic magnitudes and phase angles. Comparing the fabric information of the initially anisotropic specimen and the preloaded specimen, as plotted in Fig. 7, the evolutions of their anisotropic magnitudes are similar although the phase angle of the initially anisotropic specimen approaches the loading direction at a slightly faster pace. Noticeable difference lies in the evolution of the contact force anisotropy. The initially anisotropic specimen quickly develops a high anisotropic magnitude, while the increase in the preloaded specimen is much slower. As a result, the ratio of

Δ / B_{1}^{f}

in the preloaded specimen is much higher than that in the initially anisotropic specimen. This explains why the deformation noncoaxiality of the preloaded specimen is much more significant than that in the initially anisotropic specimen.

The ratio of

Δ / B_{1}^{f}

and the deviation of the phase angles

(ψ - β_{1}^{f}) / 2

are the two governing parameters affecting deformation noncoaxiality. They are plotted in Fig. 8. In general, for the initially anisotropic specimen the ratio of

Δ / B_{1}^{f}

is small. Hence, the degree of noncoaxiality is very limited. For the preloaded specimen, the ratio of

Δ / B_{1}^{f}

starts with a high value. Hence, the degree of noncoaxiality is more remarkable, and the maximal degree of noncoaxiality occurs when the loading direction shifts away from the initial fabric direction.

Fig. 8. Influential factors of deformation noncoaxiality: (a) initially anisotropic specimen; (b) preloaded specimen

Material Anisotropy and Deformation Noncoaxiality

For isotropic specimens, the material responses remain the same irrespective of the loading directions. For an anisotropic specimen, the material exhibits a different response when the loading is applied in different directions. As seen from Eq. (18), besides the ratio of anisotropic magnitudes

Δ / B_{1}^{f}

, the degree of noncoaxiality depends on the relative direction between the fabric and force directions, i.e., the loading direction,

(ψ - β_{1}^{f}) / 2

. When the loading direction is different, the deviation between the phase angles

(ψ - β_{1}^{f}) / 2

varies. This leads to a different observation of deformation noncoaxiality.

As seen from Fig. 8, when the loading direction varies from

α = 0 to 90 °

, the ratio of

Δ / B_{1}^{f}

becomes larger, while the deviation in phase angle

(ψ - β_{1}^{f}) / 2

becomes smaller. For the initially anisotropic specimen, because the contact force anisotropy increases rapidly upon shearing, the ratio of

Δ / B_{1}^{f}

is small. Referring to Fig. 6, when the ratio of

Δ / B_{1}^{f}

is small, generally the degree of noncoaxiality is small and the maximal degree of noncoaxiality appears around

α = 0 °

. This explains the small degrees of noncoaxiality observed for the initially anisotropic specimen.

For the preloaded sample, the ratio of

Δ / B_{1}^{f}

starts at a high value. When the loading direction varies from

α = 0 to 90 °

, the deviation in phase angle

(ψ - β_{1}^{f}) / 2

becomes smaller, while the ratio of

Δ / B_{1}^{f}

sees a sharper decrease, and then regains its value. From Eq. (18) and Fig. 6, a much higher degree of noncoaxiality is expected and the maximal degree of noncoaxiality skews toward the high value of

(ψ - β_{1}^{f}) / 2

, i.e., when the loading direction is closer to the horizontal direction. This is exactly the observation for the preloaded specimen given in Fig. 1. For both specimens, at large strain levels, the fabric and force anisotropies are coaxial with the loading direction, while the ratio of

Δ / B_{1}^{f}

stabilized around 0.6. The material behavior is coaxial.

Discussion on Material Anisotropy

The anisotropic fabric tensor defined in Eq. (17) has two components, where one is a result of the contact normal anisotropy and the other is a result of the mean contact vector anisotropy. Anisotropy in the mean contact vector was observed as a result of the nonspherical particles used in the simulations, although its magnitude remained limited. For the two anisotropic specimens, the anisotropic magnitudes of the mean contact vectors were about 0.05 throughout the loading. The principal direction gradually rotates from being horizontal—i.e., normal to the deposition direction at the initial states—to being normal to the loading direction.

The directional dependence in the mean contact vector is expected to have a strong correlation with the particle shape and orientations. The internal structure formed during particle deposition has a preferred particle orientation in the horizontal plane. When subjected to preloading, this concentration of particle orientation is further enhanced, while its principal direction of the contact vector remains horizontal. As loading is applied, the directional distribution of particle orientations tends to be normal to the loading direction with the magnitude of the particle orientation density slightly decreasing. As evidenced in Fig. 9, where the information on particle orientation and mean contact vector is drawn together for comparison, the anisotropy of the mean contact vector and that of particle orientation density remains closely correlated throughout the loading process.

Fig. 9. Contact vector anisotropy and particle orientation when sheared at $α = 45 °$ : (a) anisotropy magnitude; (b) phase angle

Historically, inherent, initial, and induced anisotropies are terminologies frequently used in the literature to categorize the anisotropy of granular materials. Casagrande and Carrillo (1944) distinguished two different kinds of anisotropy as being induced and inherent anisotropies. Oda et. al (1985) proposed three sources of material anisotropy: (1) preferred orientation of contact normals; (2) preferred alignment of nonspherical particles; and (3) elongated shape of voids. Among these sources, the experimental data on 2D oval cross-sectional rods suggest that the contact normal anisotropy and anisotropy as a result of the void shape are sensitive to shearing and are considered to be the dominant sources of stress-induced anisotropy. The anisotropy as a result of the preferred alignment of nonspherical particles varies slightly, is more likely to be preserved until failure, and is considered to be inherent anisotropy.

It is observed from Fig. 9 that the particle orientation also changes as a result of shearing, and tends to be normal to the loading direction at large strain levels. As seen from the expression of the anisotropic fabric tensor defined in Eq. (17), the effect of particle orientation anisotropy on material strength is reflected in terms of

G_{ij}^{v}

, i.e., the term quantifying the anisotropy in mean contact vectors. Furthermore, Li et al. (2009) demonstrated a strong correlation between the contact normal and void space anisotropies based on 2D DEM simulation results, which may slightly depend on the particle orientation anisotropy.

It can be argued that the particle orientation, contact normal, and void shape are different aspects of describing fabric anisotropy. Rather than categorizing them into either inherent or stress-induced anisotropy and studying them separately, the usage of the combined anisotropic fabric tensor

C_{ij}

is proposed to quantify the fabric anisotropy, whose effect on material strength can well be accounted for through the analytical SFF expression, Eq. (16). The material may initially possess a different anisotropic structure because of previous geological and loading histories. Upon shearing, the material anisotropy gradually evolves. The DEM simulation results suggest that specimens approach the same fabric anisotropy when shear is continually applied. Therefore, material anisotropy evolution can be considered as a kinetic process during which the material fabric gradually develops from its initial state to an ultimate state determined by the applied loading.

Conclusions

This paper reports a multiscale investigation of the deformation noncoaxiality of granular materials in monotonic shearing. Direction statistical analyses have been carried out on the particle-scale information obtained from 2D discrete element simulations. Two anisotropic specimens are used, i.e., initially anisotropic and preloaded. Despite the anisotropy at the initial states being similar, the noncoaxiality between the principal directions of the stress and strain increment is significant for the preloaded sample but negligible for the initially anisotropic sample, as shown in Fig. 1.

Starting from the microstructural expression of the stress tensor, the statistical feature of the contact normal directions, contact vectors, and contact forces have been studied and characterized in terms of the direction tensors, which are used as macroscale variables characterizing the material internal state. The SFF relationship has been established and used to explain the deformation noncoaxiality. In monotonic loading, it was observed that the force anisotropy is always coaxial with the loading direction, i.e., the strain increment direction, while the fabric anisotropy only gradually approaches the loading direction. This noncoincidence between the fabric anisotropy and the strain increment direction is the main cause of deformation noncoaxiality.

An anisotropic fabric tensor

C_{ij} = G_{ij}^{v} + (1 / 2) D_{ij}^{c}

is proposed to take into account both the contact normal density and contact vector anisotropies. Instead of distinguishing between inherent and stress-induced anisotropies,

C_{ij}

is used as a state variable quantifying the material fabric impact on the stress state. Material anisotropy evolution is considered as a kinetic process during which the material fabric gradually develops from its initial state to an ultimate state determined by the applied loading.

The principal stress direction can be determined by from the deviatoric direction tensors characterizing material fabric and contact force anisotropies. From Eq. (18), the degree of noncoaxiality can be determined given the information of the ratio between the fabric and force anisotropy

Δ / B_{1}^{f}

and the direction deviation between them,

(ψ - β_{1}^{f}) / 2

. For the initially anisotropic specimen, the ratio between the magnitude of the fabric and force anisotropies is small. Hence, the degree of noncoaxiality is small, and the maximal value appears around

α = 45 °

, as given by the observed behavior of the initially anisotropic specimen plotted in Fig. 1(a). When the ratio between the magnitude of the fabric and force anisotropies is higher, the degree of noncoaxiality is larger, and the maximal value appears when the deviation between the phase angles skews toward

α = 90 °

. This explains the higher deformation noncoaxiality observed in the preloaded specimen, as shown in Fig. 1(b).

Acknowledgments

The work reported in this paper is financially supported by the Engineering and Physical Sciences Research Council (EPSRC), United Kingdom, and a Nottingham Advance Research Fellowship.

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Information & Authors

Information

Published In

International Journal of Geomechanics

Volume 15 • Issue 4 • August 2015

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.

History

Received: Jan 16, 2013

Accepted: Jul 1, 2013

Published online: Jul 4, 2013

Published in print: Aug 1, 2015

Authors

Affiliations

X. Li [email protected]

Lecturer, Materials, Mechanics and Structures Research Division & Process and Environmental Research Division, Faculty of Engineering, Univ. of Nottingham, University Park, Nottingham NG7 2RD, U.K. (corresponding author). E-mail: [email protected]

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H.-S. Yu

Professor, Materials, Mechanics and Structures Research Division, Faculty of Engineering, Univ. of Nottingham, University Park, Nottingham NG7 2RD, U.K.

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Abstract

Introduction

DEM Simulation Results of Deformation Noncoaxiality in Monotonic Loading

Material Properties and Test Procedures

Observation of Deformation Noncoaxiality in Anisotropic Specimens

Directional Statistics and SFF Relationship

Integral Form of the Microstructural Stress Tensor

Observations of Statistical Dependence

Contact Normal Density ec(n)

Mean Contact Force 〈f〉|n

Mean Contact Vector 〈v〉|n

SFF Relationship in Granular Materials

Micromechanics of Deformation Noncoaxiality

Principal Stress Direction

Particle-Scale Statistics

Particle-Scale Mechanism of Deformation Noncoaxiality

Material Anisotropy and Deformation Noncoaxiality

Discussion on Material Anisotropy

Conclusions

Acknowledgments

References

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Contact Normal Density $e^{c} (n)$

Mean Contact Force ${〈 f 〉 |}_{n}$

Mean Contact Vector ${〈 v 〉 |}_{n}$