Comparison of hyperelastic models for granular materials

Three recently proposed hyperelastic models for granular materials are compared with experiment data. Though all three are formulated to give elastic moduli that are power law functions of the mean stress, they have rather different dependencies on individual stresses, and generally differ from well established experimental forms. Predicted static stress distributions are in qualitative agreement with experiments, but do not differ greatly from isotropic linear elasticity, and similarly fail to account for variability in experiment data that presumably occurs due to a preparation dependence of granular materials.

Figure 1

The yield angle ${\varphi }_y$ for GE as a function of $\xi$. The maximum occurs at $\sim 26.5°$ for $a=1/2$ and $\sim 17°$ for $a=1$.

Figure 2

The stress ratio ${\sigma }_s/P$ as a function of pressure, for $\beta =1$ and $B={10}^{12}$ (arbitrary units).

Figure 3

The isotropic Poisson’s ratio as a function of dimensionless material constant $\beta$ in the EP model.

Figure 4

The isotropic Poisson’s ratio as a function of dimensionless material constant $\alpha$ in the HAR model.

Figure 5

The vertical Young’s modulus $E_v$ as a function of the horizontal stress ${\sigma }_h$ for the GE-C model; ${\sigma }_v/\mathcal{A}=1$ and $\xi =5/3$. $E_v$ varies substantially with ${\sigma }_h$, in contrast with experiment data.

Figure 6

The vertical Young’s modulus $E_v$ as a function of the horizontal stress ${\sigma }_h$ for the EP model, ${\sigma }_v/B=1$. $E_v$ varies only slightly with ${\sigma }_h$, in relative agreement with experiment data, where there is no dependence of $E_v$ on ${\sigma }_h$.

Figure 7

The vertical Young’s modulus $E_v$ as a function of the horizontal stress ${\sigma }_h$ for the HAR model, ${\sigma }_v/A=1$. $E_v$ varies substantially with ${\sigma }_h$, in contrast with experiment data. In particular, the variation increases for increasing values of $\alpha$.

Figure 8

$E_v$ vs ${\sigma }_v$ for the EP model, and experiment fit.

Figure 9

$E_v$ vs ${\sigma }_v$ for the HAR model, and experiment fit.

Figure 10

$E_v$ vs ${\sigma }_v$ for the GE-C model, and experiment fit.

Figure 11

Abaqus results for the stress at the bottom of a conical sand pile: hyperelastic models (top) and isotropic linear elasticity (bottom).

Figure 12

Abaqus plane strain results for the stress at the bottom of a sand wedge: hyperelastic models (top) and isotropic linear elasticity (bottom).

Figure 13

Abaqus results for silo stresses using the EP model, with $\beta =3/2$.

Figure 14

Janssen’s constant as a function of the material constant $\beta$ for the EP model (left) and $\alpha$ for the HAR model (right). Values are taken at $r/R=0.5$ and $z/H=0.5$.

Figure 15

Hyperelastic models for the “overshoot” in stress when a load equal to ${\sigma }_{sat}$ is applied to the surface of the silo. Magnitudes of the overshoot are similar to those predicted by ILE, and smaller than observed in experiments [45].

Figure 16

Abaqus calculations of the response function for GE (top) and ILE (bottom). The presence of boundaries increases the peak height, and a frictionless bottom surface results in a higher peak than a rough surface (glued grains).

Figure 17

Abaqus calculations of the response function for the HAR (top) and EP (bottom) models. The EP model predicts a much narrower response function than is observed, and is not particularly sensitive to the value of $\beta$.