Manifestation of particle morphology on the mechanical behaviour of granular ensembles

Abstract

This paper presents the effect of particle morphology (grain shape) on the mechanical response of granular materials. Two model systems with extreme differences in morphology were selected (spherical glass ballotini and angular sand) for this experimental programme. A series of hollow cylinder torsion tests were conducted in this programme under monotonic drained conditions on specimens reconstituted to the same relative density. Tests were conducted under different intermediate principal stress ratio (b) on both the model materials. The glass ballotini shows increased dilation at the outset of the test, however, at large strains, the particle rearrangement in the sand and the increased interlocking leads to higher strength at the critical state. The effect of individual particle morphology is manifested in both the increased friction angle and a larger sized failure locus in stress space with increase in angularity. The stresses developed in these two model materials are also accompanied by intriguing volume change behaviour. The glass ballotini despite a lower strength presents a predominantly dilative response immaterial of the ‘b’ value, while the angular sand shows increased strength at large strains, while showing a contractive response. These results allow incorporation of particle morphology effects at the ensemble level in plasticity based constitutive models.

Appendix

• $\hbox {D}_{50}$—mean grain size/ median obtained from a particle size distribution (plot between cumulative percentage finer and particle size where the particle size corresponding to 50% finer).

• $\hbox {C}_{\mathrm{u}}$—uniformity coefficient which is nothing but the ratio between D60 (particle size corresponding to 60% finer obtained from the grain size distribution plot) to D10 (particle size corresponding to 10% finer obtained from the grain size distribution plot). $C_u=\frac{D_{60}}{D_{10}}$

• $\hbox {C}_{\mathrm{c}}$—coefficient of curvature. $C_u=\frac{D_{30}^{2}}{D_{60} D_{10}}$ (D30—particle size corresponding to 30% finer obtained from a grain size distribution plot).

• Void ratio (e) is the ratio of volume of voids to the volume of solids.

• Porosity (n) is the ratio of volume of voids to the total volume.

• Packing fraction $\chi =\frac{1}{1+e}$.

Sayao and vaid [50]—Criteria put forth to fix the specimen dimensions so as to minimize non-uniformities.

The specimen geometry is fixed such that the stress distribution across the wall is kept as uniform as possible i.e. the stress difference between real and calculated values should be minimized. In order to fix the dimensions of the hollow cylinder specimen, extensive numerical studies have been carried out using both elastic and plastic formulations to arrive at various sample dimensions.

The criterion proposed by Sayao and Vaid [50] in fixing the specimen geometry is given below:

• 1. Inner radius: $0.65\le \frac{r_{i}}{r_{o}}\le 0.82$

• 2. Height: $1.8\le \frac{H}{2r_{o}}\le 2.2$

• 3. Wall thickness: $r_{o}-r_{i}$ = 20 to 60 mm

where $\hbox {r}_{\mathrm{i}}$—internal radii of the specimen (mm), $\hbox {r}_{\mathrm{o}}$—external radii of the specimen (mm), H is the height of the specimen (mm). The above criteria can be used when the sand specimen volume is sufficiently large compared to the volume change during shearing, the density is uniform across the wall and the wall thickness should be large compared to the maximum grain size so that the failure mechanism would not be constrained.

Pressure ratio at the end of each test:

Equations:

$$\begin{aligned}&\overline{\sigma _z } =\frac{W}{\pi \left( {r_o^2 -r_i^2 } \right) }+\frac{\left( {p_o r_o^2 -p_i r_i^2 } \right) }{\left( {r_o^2 -r_i^2 } \right) } \end{aligned}$$

(11)

$$\begin{aligned}&\overline{\sigma _r } =\frac{\left( {p_o r_o +p_i r_i } \right) }{\left( {r_o +r_i } \right) } \end{aligned}$$

(12)

$$\begin{aligned}&\overline{\sigma _\theta } =\frac{\left( {p_o r_o -p_i r_i } \right) }{\left( {r_o -r_i } \right) }\end{aligned}$$

(13)

$$\begin{aligned}&\overline{\tau _{\theta z} } =\frac{3.M_T }{2.\pi \left( {r_o^3 -r_i^3 } \right) }\end{aligned}$$

(14)

$$\begin{aligned}&\overline{\varepsilon _z } =\frac{\Delta H}{H}\end{aligned}$$

(15)

$$\begin{aligned}&\overline{\varepsilon _r } =-\frac{\left( {u_0 -u_i } \right) }{\left( {r_0 -r_i } \right) }\end{aligned}$$

(16)

$$\begin{aligned}&\overline{\varepsilon _\theta } =-\frac{\left( {u_0 +u_i } \right) }{\left( {r_0 +r_i } \right) }\end{aligned}$$

(17)

$$\begin{aligned}&\overline{\gamma _{\theta z} } =-\frac{2\beta \left( {r_o^3 -r_i^3 } \right) }{3H\left( {r_o^2 -r_i^2 } \right) }\end{aligned}$$

(18)

$$\begin{aligned}&\sigma _1 = \frac{\sigma _z +\sigma _\theta }{2}+\sqrt{\left( {\left( {\frac{\sigma _z -\sigma _\theta }{2}} \right) ^{2}+\left( {\tau _{\theta z} } \right) ^{2}} \right) }\end{aligned}$$

(19)

$$\begin{aligned}&\sigma _3 =\frac{\sigma _z +\sigma _\theta }{2}-\sqrt{\left( {\left( {\frac{\sigma _z -\sigma _\theta }{2}} \right) ^{2}+\left( {\tau _{\theta z} } \right) ^{2}} \right) }\end{aligned}$$

(20)

$$\begin{aligned}&p^{{\prime }}=\frac{\sigma _1^{\prime } +\sigma _2^{\prime } +\sigma _3^{\prime } }{3}\end{aligned}$$

(21)

$$\begin{aligned}&q=\frac{1}{\sqrt{2}}\left[ {\left( {\sigma _1^{\prime } -\sigma _2^{\prime } } \right) ^{2}+\left( {\sigma _2^{\prime } -\sigma _3^{\prime } } \right) ^{2}+\left( {\sigma _3^{\prime } -\sigma _1^{\prime } } \right) ^{2}} \right] ^{1/2}\end{aligned}$$

(22)

$$\begin{aligned}&\tau _{oct} =\frac{1}{3}\left[ {\left( {\sigma _1^{\prime } -\sigma _2^{\prime } } \right) ^{2}+\left( {\sigma _2^{\prime } -\sigma _3^{\prime } } \right) ^{2}+\left( {\sigma _3^{\prime } -\sigma _1^{\prime } } \right) ^{2}} \right] ^{1/2}\end{aligned}$$

(23)

$$\begin{aligned}&J_2 =\frac{1}{6}\left[ {\left( {\sigma _1^{\prime } -\sigma _2^{\prime } } \right) ^{2}+\left( {\sigma _2^{\prime } -\sigma _3^{\prime } } \right) ^{2}+\left( {\sigma _3^{\prime } -\sigma _1^{\prime } } \right) ^{2}} \right] \end{aligned}$$

(24)

$$\begin{aligned}&J_3 =\frac{1}{27}\left[ \left( {\sigma _1^{\prime } -\sigma _2^{\prime } } \right) ^{2}\left( {\sigma _1^{\prime } +\sigma _2^{\prime } -2.\sigma _3^{\prime } } \right) \right. \nonumber \\&\quad \quad +\left( {\sigma _2^{\prime } -\sigma _3^{\prime } } \right) ^{2}\left( {\sigma _2^{\prime } +\sigma _3^{\prime } -2.\sigma _1^{\prime } } \right) \nonumber \\&\qquad \left. +\left( {\sigma _3^{\prime } -\sigma _1^{\prime } } \right) ^{2}\left( {\sigma _3^{\prime } +\sigma _1^{\prime } -2.\sigma _2^{\prime } } \right) \right] \end{aligned}$$

(25)

$$\begin{aligned}&q=\sqrt{3.J_2 } (\hbox { for TX-C and TX-E }q=\sigma _1-\sigma _3)\end{aligned}$$

(26)

$$\begin{aligned}&\gamma _{oct} =\frac{2}{3}\left[ {\left( {\varepsilon _1 -\varepsilon _2 } \right) ^{2}+\left( {\varepsilon _2 -\varepsilon _3 } \right) ^{2}+\left( {\varepsilon _3 -\varepsilon _1 } \right) ^{2}} \right] ^{1/2} \end{aligned}$$

(27)

$$\begin{aligned}&\hbox {Relative density, }\nonumber \\&R.D_r =\frac{e_{\max } -e_{nat} }{e_{\max } -e_{\min } }\end{aligned}$$

(28)

$$\begin{aligned}&\varepsilon _v = \varepsilon _1 + \varepsilon _2 +\varepsilon _3 \end{aligned}$$

(29)

$$\begin{aligned}&\varepsilon _q = \frac{\sqrt{2}}{3}\left[ \left( {\varepsilon _{1}-\varepsilon _{2}}\right) ^{2} +\left( {\varepsilon _{2}-\varepsilon _{3}}\right) ^{2}+\left( {\varepsilon _{3}-\varepsilon _{1}}\right) ^{2} \right] ^{1/2} \end{aligned}$$

(30)

$\hbox {u}_{\mathrm{o}}$ is the external radial displacement, $\hbox {u}_{\mathrm{i}}$ is the internal radial displacement, $\upvarepsilon _{\mathrm{z}}$ is the axial strain, $\upvarepsilon _{\mathrm{r}}$ is the radial strain, $\upvarepsilon _{\uptheta }$ is the circumferential strain, $\upgamma _{\uptheta \mathrm{z}}$ is the shear strain, $\upbeta$ is the rotation angle, H is the initial height of the specimen, $\Delta \hbox {H}$ is the change in axial displacement, p$^{\prime }$ is the mean effective stress, q is the three dimensional form of deviatoric stress, $\hbox {J}_{2}$ is the second deviatoric stress, invariant, $\hbox {J}_{3}$ is the third deviatoric stress invariant, $\uptau _{\mathrm{oct}}$ is the octahedral shear stress, $\upgamma _\mathrm{oct}$ is the octahedral shear strain, $\hbox {e}_{\max }$—maximum void ratio the material can achieve, $\hbox {e}_{\min }$—minimum void ratio the material can achieve, $\hbox {e}_{\mathrm{nat}}$—void ratio at which the specimen is prepared, $\varepsilon _v$—volumetric strain and $\varepsilon _q$—deviatoric strain.