A DEM study of oedometric compression of model granular materials Initial state influence , stress ratio , elasticity , irreversibility .

A DEM simulation study of spherical beads with elastic-frictional contacts in oedometric compression is carried out for a wide variety of initial states, differing in solid fraction Φ, coordination number z (independent of Φ in dense systems) and inherent anisotropy. Stress ratio K0 = σ2/σ1, along with z, Φ and force and fabric anisotropies are monitored in compressions in which axial stress σ1 varies by more than 3 orders of magnitude. K0 tends to remain constant if the material was already one-dimensionally compressed in the assembling stage. Otherwise, it decreases steadily over the investigated stress range. K0 relates to force and fabric anisotropy parameters by a simple formula. Elastic moduli may express the response to very small stress increments about the transversely isotropic equilibrated states, although oedometric compression proves an essentially anelastic process, mainly due to friction mobilization. Despite apparent nearly reversible increases of axial strain 1 (or density Φ), especially in dense samples, internal state evolutions are strongly irreversible, as evidenced by changes in z and K0. Fabric changes are reflected by anisotropic elastic moduli.


Introduction
Oedometric compression, an axially symmetric process in which one principal strain component (ε 1 ) is increased, the others being maintained at zero (ε 2 = ε 3 = 0), is one of the simplest anisotropic loading processes, representative of natural materials under gravity (e.g.sediments consolidating under their weight).Oedometric compression leads to transversely isotropic structures, with the symmetry of revolution about axis 1.It is classically characterized by the ratio K 0 (termed coefficient of earth pressure at rest) of lateral stress σ 2 = σ 3 to axial stress σ 1 [1,2], which is often regarded as a constant.Oedometric compression is usually described as irreversible [2,3], with a net plastic increase of solid fraction Φ (or a decrease of void ratio e = −1 + 1/Φ) as σ 1 grows, while density changes occurring below the maximum past value of σ 1 (the preconsolidation pressure) are reversible.
The present contribution reports on a systematic DEM study of oedometric compression in a simple model material for a wide variety of initial states, with the aim of exploring, for such a simple case of an anisotropic material state under varying load intensity, stress ratio K 0 , elasticity and irreversibility, with reference to microstructural and micromechanical aspects.After a brief description of the model and initial states (Sec.2), Sec. 3 discusses stress anisotropy and its connections to force networks, and Sec 4 gives a brief account of elastic properties.Sec. 5 discusses irreversiblity, before a few words of conclusion (Sec.6).
One may regard the beads as frictionless in this assembling procedure, and possibly implement a "mixing" or "vibration" stage, as described in Ref. [4].As a result, one obtains the 6 different initial states of Tab.

Compression and stress ratio
All six initial states are subjected to an oedometric compression with σ 1 growing to 31.2 MPa (κ 180).In this process, while Φ increases by 0.01 to 0.02, coordination number z varies as in Fig. 1.In general, a gradual increase  of z under compression is observed, except for an initial stage in systems prepared with large coordination (DHo, DHi), in which z first decreases to a minimum.This shows that oedometric compression is a complex process that differs from a homogeneous uniaxial shrinking of the contact structure.The contact network, beyond its density, should also be characterized by fabric anisotropy, which is encoded, to first order, in parameter c2 (c 2 = 0 in isotropic states): c2 , initially zero in "XYi" systems by construction, gradually grows with σ 1 , reaching typical values of 0.02-0.04under high presure.It stays constant, near 0.04, in LLo systems, and achieves its highest values in the DLo states under low stress, slightly decreasing above 1 MPa (down to 0.06 for the highest σ 1 value).Fig. 2 plots stress ratio K 0 versus σ 1 in the oedometric compression of all six initial states up to a large axial stress value.Initially equal to 1 for "XYi" systems, and close to 1 in DHo ones, K 0 decreases, fast in DHo, much more gradually in isotropically assembled samples.It only stays roughly constant (near 0.5 in DLo, 0.7 in LLo, and somewhat above 0.6 in DHo for σ 1 ≥ 200 kPa) in oedometrically assembled systems.K 0 , using the classical formula relating stresses to contact forces in static granular packs [4], readily related to anisotropy parameters.A force anisotropy parameter, f2 is defined from the angular distribution of average normal force, F (|n 1 |), normalized by the average normal force in all contacts.By axial symmetry F is a function of the sole coordinate n 1 of the unit contact normal vector n.One writes Using the standard formula for stresses [4], noting that the contributions of tangential forces to stresses sigma 1 sigma 2 are negligible, and dealing with anisotropies to leading order, one readily obtains an approximation for principal stress ratio K 0 = σ 2 /σ 1 : As apparent in Fig. 2, relation (3) provides a good approximation of K 0 , which is thus simply related to anisotropy parameters c2 and f2 (as remarked in [5,6]).f2 steadily increases with σ 1 , from zero in initially isotropic assemblies, the faster in highly coordinate systems, up to about 0.11 in the MPa range.f2 and c2 are of the same order in the most anisotropic systems, DLo and LLo.

Elastic properties
An elastic response is probed on subjecting wellequilibrated states along the compression curve to small stress increments that do not cause rearrangements and for which the effects of friction are negligible [7].Elastic moduli are evaluated on building the stiffness matrix of the contact network, involving normal and tangential stiffnesses associated with small displacement and rotation increments [7].Given the transversely isotropic symmetry in oedometric compression, there are 5 independent moduli [8,9].Defining a quasielastic range for incre- ments of axial strains, Δε 1 , by the requirement that axial strain increment Δσ 1 differs from the elastic response C 11 Δε 1 by less than 5%, its typical variation with σ 1 , for both signs of Δε 1 , is shown in Fig. 3.A fully elastic response for the whole compression curve would imply stress ratios Δσ 2 /Δσ 1 equal to the corresponding ratio of elastic constants, C 12 /C 11 .However, those two quantities strongly differ (typically one has C 12 /C 11 < 0.4, while Δσ 2 /Δσ 1 values are similar to K 0 ).The oedometric compression entails friction mobilization and sliding (most frequent, somewhat surprisingly, in contacts with normal direction parallel to the transverse plane) and the quasielastic range is only associated to arrested, stabilized configurations (obtained after tiny creep intervals) in which full friction mobilization is suppressed.A full study of elastic moduli [10] reveals, on the one hand, similar trends as in isotropic systems [7], as to their dependence   σ 1 (kPa) 0.9 on stresses and coordination number (which is far mode influential than solid fraction), and, on the other hand, that moduli reflect the geometric and mechanical anisotropy of oedometrically compressed systems.Thus, Figs.4-5 shows how the ratio of moduli C 11 and C 22 (associated to longitudinal waves propagating in the axial and transverse directions) achieves large values in the most anisotropic systems (DLo, and LLo), while lower ratios characterize the gradual gains of fabric (a moderate effect) and force anisotropy (a faster phenomenon) as initially isotropic packs are oedometrically loaded.Those figures also show the predictions of the Voigt ("effective medium theory") prediction, based on an assumption of homogeneous strains doan to the grain scale, for such ratios of moduli.While the approximation itself fails to correctly predict the values of the moduli [7,10] its estimate of relative difference C 11 /C 22 − 1 is correct to within 20% for small levels of anisotropy (as C 11 /C 22 < 1.2).The variety of numerical results on elastic moduli for the different assembling procedures compares well [10] to that of published laboratory results on spherical bead assemblies [8,9].

Unloading and irreversibilities
A compression cycle in which σ 1 , after its maximum, is reduced back to its initial value, results in the variations of void index e and coordination number z shown in Fig. 6 The amount of irreversible compression (decrease pf e) af- ter one cycle is remarkably small, compared to typical experimental results on sands [11], a difference we attribute to the absence of plasticity or damage at the contact leve in the numerical model.The irreversibility is quite conspicuous, however, as regards z, which, as observed with isotropic bead packs [12], decreases to low values after one cycle in initially highly coordinated systems (this effect exists, to a smaller extent, in smaller stress cycles).Stress ratio K 0 also exhibits strongly irreversible behavior, with values larger than 1 obtained after a cycle (Fig. 7), as transverse stress σ 2 decreases slower than controlled, axial stress σ 1 along the unloading path.

Conclusions
This communication reports on some salient aspects of a systematic numerical study of oedometric compression of spherical grain assemblies [10], which investigates a wide variety of different initial states, subjected to a large increase of axial stress, and substantially extends the domains explored in the previous DEM literature [5,6].
The coefficient of earth pressure, K 0 , can be regarded as constant in systems prepared under similar initial oedometric load as the studied test.Otherwise, it appears to vary, reflecting a growing load-induced anisotropy.K 0 relates to both fabric and force anisotropy in a simple way.Despite the smallness of irreversible density changes, oedometric compression is a complex process involving friction-induced irreversible internal evolutions in the contact network, with possible decreases of initially large coordination numbers and hysteretic stress states.Elastic properties reflect material anisotropies and, given encouraging comparisons with experimental results, could provide access to fabric variables.

Figure 1 .
Figure 1.Coordination number z versus axial stress σ 1 for different initial states (DLo+ = DLo state with larger sample size).

Table 1 .
Solid fraction Φ, coordination numbers z and z * , rattler fraction x 0 , and stress ratio K 0 for the different initial states.Values averaged over 3 configurations of N = 4000 grains.
by construction in "XYi" states, might exhibit significant levels of anisotropy in oedometrically prepared ones.