Anisometric granular materials: compression and shear properties

Abstract

Random layers of dry coarse anisometric grains of eight different materials were studied experimentally. The static and dynamic parameters of compression and shear tests were evaluated. The relation between compaction and flow properties was determined based on compression and shear models parameters. The results on anisometric particles were compared with published data on common powders.

Graphic abstract

Appendices

Appendix 1: Mohr Circles—definition of used quantities

In this study, many parameters related to the shear test are used and their mutual connections are discussed. It is instructive to look at the sketch drawn in Fig. 14 [7, 29] to see their definitions, geometrical properties and the physical implications. The rod-like graphite particles are used for demonstration. The used terminology follows the terminology of Schulze [7].

Fig. 14

Mohr circles analysis (definition sketch). The applied normal stress σ (vertical direction, cause, stimulus) compresses the granular bed. The corresponding shear stress τ (horizontal direction, effect, response) is needed to overcome the internal friction produced by the compression and make the sample flow. The demonstration is made with the experimental results for rod-like graphite. Similar highly instructive diagram was originally suggested in User Guide Data Analyses by Freeman Technology [29]

The experiment starts in the pre-shear steady state point [σ_ss, τ_ss] (black circle), where the sample is compressed by the largest load, i.e. preconsolidation stress σ_p (here 9.08 kPa). The normal vertical stress σ_ss = σ_p and the corresponding shear stress τ_ss needed to shear the sample horizontally in steady state (stress to make the sample flow in steady state, here 5.28 kPa) is measured as the steady state point. Then the load is reduced to the first lower value σ₁ and the shear τ₁ (incipient yield stress to make the sample flow) is measured. It repeats several times, here for σ_1,2,3,4,5 = 7, 6, 5, 4, 3 kPa to obtain the values of τ_1,2,3,4,5. The pairs [σ_i, τ_i] are the experimental shear stress points and are called the individual yield limit values (small diamonds in Fig. 14). These measured data [σ_i, τ_i] for i = 1, 2, 3, 4, 5, form the experimental yield locus curve τ = τ (σ). These generally nonlinear data are fitted with a straight line, the yield locus line, as an approximation. Its intercept at σ = 0 is the cohesion strength τ_c and its slope is the angle of internal friction φ. The slope of line connecting the pre-shear steady state point [σ_ss, τ_ss] and the zero point [0, 0] (dotted line in Fig. 14) gives the steady state angle of internal friction φ_ss.

The larger Mohr Circle is tangential to the yield locus line and intersects the steady state point [σ_ss, τ_ss]. Its intercepts with the horizontal axis give the major and minor principal stresses σ₁ and σ₂. Their proportion, called the principal stresses ratio λ = σ₂/σ₁, quantifies the stress transfer from the static normal load (σ₁) to the lateral flow direction (σ₂). The line passing through the origin [0, 0] and being tangent to the large Mohr Circle is called effective yield locus (dashed line in Fig. 14) and its angle is the effective angle of internal friction φ_e. Note that for cohesive (τ_c > 0) materials it is φ_e> φ and for non-cohesive (τ_c = 0) materials φ_{e =} φ.

The smaller Mohr Circle is tangential to the yield locus line and intersects the origin [0, 0]. Its rightward point of crossing the horizontal axis is the unconfined yield strength σ_c. The ratio of the major principal stress σ₁ and the unconfined yield strength σ_c is called the flowability index ffc = σ₁/σ_c and gives a rough estimate of the material flowability: the larger ffc, the easier the material flows.

There are some mathematical-geometrical relations between the various stress parameters, that are the consequences of the Mohr Circle analysis and that are not the material properties of granular samples. For instance, the principal stresses ratio λ is a function of effective angle of internal friction φ_e,

$$\lambda = \frac{{1 - \sin \varphi_{e} }}{{1 + \sin \varphi_{e} }},$$

(5)

see Fig. 15a.

Fig. 15

Mathematical relations resulting from the Mohr analysis. a λ decreases with φ_e, b for σ_c = σ₂ it is ffc = 1/λ, dividing the plane into two regions; easy flowing materials have higher ffc which corresponds to σ_c < σ₂

Further, when σ_c = σ₂, then the flowability and stress ratio are in the inverse relation, ffc = 1/λ, see the line in Fig. 15b that divides the parameter plane ffc and φ_e into two regions. When σ_c < σ₂, the value of ffc lies above the line, in the region of higher flowability. Therefore, this inequality can be used as the condition indicating better flowability. On the other hand, when σ_c > σ₂, the flowability index is located below the line, in the region of lower values of ffc.

Another consequence are the implications: if τ_c ≥ 0 (which is physically plausible), then φ_e ≥ φ, which is directly seen from Fig. 14. Be specific (1) if τ_c = 0 then φ_e = φ and (2) if τ_c > 0 then φ_e > φ. For many materials, the angles typically follow this ordering: φ < φ_ss < φ_e. Nevertheless, there are samples with negative τ_c < 0, which can be the result of the specific particles rearrangement. It can be due to some instabilities. During shear test, certain degradation of soil microstructure can be observed both for dense and loose samples [42] and therefore, the slope of yield locus should be so steep, that cohesivity exhibits a negative value. For instance, some samples of rod-like pasta displayed negative τ_c occasionally, see Fig. 4c, d.

Appendix 2: Typical values of λ

The typical values of the principal stress ratio λ = σ₂/σ₁ for common powders reported in the literature [7, 44, 45] are about 0.3–0.6. They are slightly higher than our values for anisometric grains, 0.1–0.5. Here we show that this difference is not due to the material properties but the way the former estimate was obtained.

Kwade et al. [44, 45] made an attempt to obtain the shear-related quantities from the compression test. They developed a clever device called Lambdameter that measured the vertical σ_v and horizontal σ_h stresses. The known normal load (up to 35 kPa) was laterally transmitted by the granular sample toward the ring wall whose extension was detected by the strain gauges lining the perimeter. The values σ_h and σ_v were close to σ₂ and σ₁, and were used to calculate the compression angle φ_c by their Eq. 9. The corresponding value of λ₀ was found by their Eq. 11, i.e. our Eq. (5). They tested 41 granular samples with both the compression device (Lambdameter) and the shear device (Jenike ring shear tester). The latter device gives the quantities pertaining to Mohr’s analysis. A selection of experimental results is in their Table 5 where the only shear quantity reported is effective angle of internal friction φ_e. The corresponding value of λ can be obtained by Eq. (5). They presented theoretical consideration about the correspondence between the compression and shear experiments and their quantities. A new parameter was introduced, the compression angle φ_c, that is the analogue of the shear angle φ_e. Because in compression the internal friction is not fully mobilized, φ_c < φ_e.

We thus have two sets of parallel data pairs, one for compression [φ_c, λ₀] and one for shear [φ_e, λ]. Their different plots are shown in Fig. 16a where the experimental data from Table 5 in [45] were used. The full line is the theoretical Eq. (5). If we choose to plot the compression couple [φ_c, λ₀], we obtain the grey circles following Eq. (5) with lambda within 0.3–0.6. If we choose to plot the shear couple [φ_e, λ], we obtain the black circles following Eq. (5) with lambda within 0.1–0.5 because φ_e > φ_c.

Fig. 16

Different plots of ratio of principal stresses (λ) vs angle of friction (φ). a Data from Kwade et al. [45], grey circles—compression couple [φ_c, λ₀], black circles—shear couple [φ_e, λ], white circles—mixed shear/compression couple [φ_e, λ₀], full line—Eq. (5), dashed line—Eq. (6). b Our data—shear couple [φ_e, λ], full line—Eq. (5)

If we choose to plot the mixed shear-compression couple [φ_e, λ₀], we obtain the disorganized cloud of white circles located above the full line of Eq. (5). This option was taken by the authors in [45] in their Fig. 13, where the lambda values lie within 0.3–0.6, which result passed into [7]. In an attempt to model these hybrid data in [45], a modification of Eq. (5) by Kezdi [51] was employed where the denominator is neglected,

$$\lambda_{0} = 1{-}\sin \varphi_{e} ,$$

(6)

see the dashed line in Fig. 16a. Our shear data on anisometric particles are displayed in Fig. 16b where the natural shear plot [φ_e, λ] is used. The marks follow the Eq. (5) line and have lambda within 0.1–0.5, similarly like the black circles in Fig. 16a for general powders from [45].