Extension to barodesy to model void ratio and stress dependency of the K ₀ value

Abstract

Barodesy is a new framework for constitutive modelling of soils. The actual version uses a constant value of the coefficient of earth pressure at rest K ₀, which contradicts experimental findings. A straightforward modification is presented here to remove this shortcoming. The calibration of the material parameters remains as simple as in the original model.

1 Barodesy

Barodesy has been recently invented by Kolymbas [12–15] as a framework for constitutive modelling of soils. Barodesy is mainly based on explicit modelling the asymptotic behaviour of soil [8] which was firsty recognised for sand by Goldscheider [7] and later confirmed, for example, by Chu and Lo [4]. Asymptotic stress paths are reached when soil is deformed with proportional strain paths. ¹ This behaviour is modelled by means of a function

$$ {\bf R}({\bf D}) = {\bf I} \; \hbox{tr} \, {\bf D}^0 + c_1 \exp (c_2 {\bf D}^0), $$

(1)

that relates the stretching D to a tensor R pointing in direction of the corresponding asymptotic stress path [12]. The stretching D is the symmetric part of the velocity gradient, and c ₁, c ₂ are material constants. The superscript 0 denotes the normalisation of a tensor X, i.e. X ⁰ = X |X| with $|{\bf X}|=\sqrt{\hbox{tr}\,{\bf X}^2}.\,\,{\bf I}$ is the identity tensor. The stress direction R ⁰ acts as attractor for any stress path during constant stretching via the evolution equation involving the Jaumann objective stress rate [14]

$$ \mathop{{\bf T}}\limits^{\circ} = h \cdot \left(f {\bf R}^0 + g {\bf T}^0 \right) \cdot |{\bf D}|. $$

(2)

The functions h, f and g are set for sand to [13, 15]

$$ h = |{\bf T}|^{c_3}\;, $$

(3)

$$ f = ( c_4\,\hbox{tr}\,{\bf D}^0 - c_5 e_c) \exp \big( c_6 |{\bf R}^0-{\bf T}^0 | \big ) $$

(4)

$$ g = c_5 e, $$

(5)

with e denoting the void ratio, and e _c the pressure dependent critical void ratio

$$ e_c = (1+e_{c_{0}}) \exp \left(\frac{|{\bf T}|^{1-c_3}}{c_4 (1-c_3)}\right) -1. $$

(6)

$e_{c_{0}}$ is the critical void ratio at zero pressure, and c ₃, c ₄, c ₅ as well as c ₆ are additional material constants. The calibration of all material constants is described in [13, 15]. A set of parameters for Hostun RF sand [5] is given in Table 1.

Table 1 Material parameters of barodesy for Hostun RF sand

In the special case of isochoric stretching (tr D = 0), the stress directions R ⁰(D) belong to critical states and the corresponding asymptotic stress states T ⁰ = R ⁰ are steady states [20]. The critical stress states defined by the R-function (1) of barodesy are practically equal [6] to those modelled by the Matsuoka and Nakai stress criterion [18].

2 Prediction of K ₀

The stress path modelled by (2) for the constant principal stretching D ₁ and D ₂ = D ₃ = 0 will approach the proportional stress path T ⁰ = R ⁰, as obtained by the R-function (1)

$$ K_0 = \frac{\sigma_3}{\sigma_1} = \frac{R_3}{R_1} = \frac{c_1-1}{c_1\exp(-c_2)-1}, $$

(7)

with σ ₁ and σ ₃ denoting the major and minor principal stress, respectively. K ₀ is predicted to be constant. This contradicts experimental findings, which indicate that K ₀ depends on the void ratio or the mean stress or both, for example, [1–3, 9, 16, 17, 19, 22]. Most laboratory experiments indicate that K ₀ increases with void ratio. However, recent experiments by Talesnick [19] with stress gauges inside the specimen show the opposite trend. Dynamic compaction of the sample prior to testing may also lead to this opposite behaviour depending on grain-crushing phenomena [17]. It is an open question, which trend is more likely for soils. Both trends can be modelled with the here proposed model via a direct calibration from such experiments. For a more detailed review of other factors influencing K ₀ see [9].

3 Extension for void ratio and stress dependent K ₀ value

A straightforward approach to model the influence of void ratio and stress level on the K ₀ value is to replace the constant c ₁ in the R-function (1) by the expression β

$$ {\bf R}({\bf D}) = {\bf I} \; \hbox{tr} {\bf D}^0 + \beta \exp (c_2 {\bf D}^0), $$

(8)

with

$$ \beta = c_1 + c_7 (e-e_c). $$

(9)

Equation (9) includes the actual void ratio e and the stress level via the pressure dependent critical void ratio (6) and an additional constant c ₇.

4 Calibration

The constant c ₂ is a function of the critical friction angle solely and can therefore be calibrated as proposed in [14]. The calibration of the constants c ₃ and c ₄ via fitting the stress-strain response of a loose sample in a oedometric test is negligibly affected if reasonable values for K ₀ are chosen. Therefore, the values given in [13, 15] (Table 1) are used here.

The calibration of the material constants c ₁ and c ₇ follows directly the calibration of c ₁ for the original R-function (1) given in [14], which is based on evaluating the peak stress state of a drained triaxial experiment on a dense sample. Using the peak states of experiments with different consolidation pressures and initial void ratios, c ₁ and c ₇ can be calibrated. In particular, the experiments host-d-triax-cd-600(hfdw09) and host-l-triax-cd-100(hosfl13) from Desrues et al. [5] are used here, see Table 2.

Table 2 Peak stress states of drained triaxial experiments on Huston RF sand: consolidation stress σ _c , peak friction angle $\varphi_p$, angle of dilatancy at peak ψ _p and void ratio at peak e _p

From a peak state, one can evaluate $K_a = {\sigma_3}/{\sigma_1} = {(1-\hbox{sin} \,\varphi_p)}/{(1+\hbox{sin} \,\varphi_p)} \,\hbox{and} \,\hbox{tan} \,{\hat{\psi}}_p = -1 - {(\hbox{sin} \,\psi_p+1)}/{(\hbox{sin}\, \psi_p-1)}$ [14]. Note that $\hat{\psi} := \hbox{arctan}(-\hbox{tr} {\bf D} / D_1)$ introduced in [14] is just another measurement of the dilatancy. With an arbitrary chosen D ₁ =  − 1 follows from the dilatancy $D_2 = D_3 = (\hbox{tan} \, {\hat{\psi}}_p+1)/2,$ hence, the value of β at the peak is

$$ \beta_p = \frac{(1-K_a) \hbox{tr} {\bf D}^0 }{K_a \exp(c_2 D^0_1)-\exp(c_2 D^0_2)}. $$

(10)

From the stress state at peak σ ₃ = σ ₂ = σ _c and σ ₁ = σ ₃/K _a one can evaluate the critical void ratio at peak e _c,p using (6).

The β-function is a linear interpolation between the peak states I and II in Table 2 and therefore

$$ c_7 = \frac{\beta_p^I-\beta_p^{II}} {e_p^I-e_{c,p}^{I}-e_p^{II}+e_{c,p}^{II}} = 4.827 $$

(11)

$$ c_1 = \beta_p^I - c_7 (e_p^I-e_{c,p}^{I}) = -1.489. $$

(12)

The constant c ₅ was calibrated in [13, 15] by fitting the constitutive equation to several experimentally obtained results and c ₆ by trial and error. Using the original value of c ₅ from Table 2 with the extended form of barodesy yields too small peak friction angles in numerical triaxial test simulations. Decreasing the constant to c ₅ =  −5,040 gives good results for the peak friction angles, compare Table 4. The constant c ₆ can be left unchanged. The whole set of parameters for Hostun RF sand [5] used here is collected in Table 3.

Table 3 Material parameters of extended barodesy for Hostun RF sand

Table 4 Peak friction angles $\varphi_p$ of drained triaxial experiments on dense Hostun RF sand and numerical simulation with original R-function (1) (material parameters from Table 1) and here proposed extended version (8) (material parameters from Table 3)

5 Numerical simulations

Oedometric and triaxial compression test were investigated to highlight the differences between the behaviour of the original and the extended versions of barodesy.

Figure 1a shows the loading branch of the oedometric compression test host-l-oed(Hn2) [5] on loose Huston RF sand (initial void ratio e _ini = 0.8703) as well as the computed response for the original and the extended versions of barodesy. As this experiment was used for calibration in [13, 15], the computation with the original model fits quite well. The difference to the computation with the extended model is negligible.

Fig. 1

Oedometric compression on Huston RF sand, initial void ratios: e _ini = 0.8703,  e _ini = 0.6667

Figure 1b shows the same comparison for the loading branch of the oedometric compression on dense Huston RF sand: host-d-oed(dhn3) [5] (initial void ratio e _ini = 0.6667). The extended version of barodesy fits slightly better the experimental results.

The original version of barodesy predicts a constant value of K ₀ = σ ₃/σ ₁. For the parameters in Table 1 and with (7) follows K ₀ = 0.4672, which is plotted in Fig. 2. The extended version predicts a higher K ₀-value for loose samples and a lower one for dense samples, which is in line with experimental observations, for example, [1, 9]. K ₀ increases by 9 % for a decrease in e _ini of 23 %, which seems to be rather small. A numerical simulation using Hypoplasticity [11] in the version of von Wolffersdorff [21] predicts a similar increase of 8 %. Moreover, K ₀ increased in experiments on Karlsruhe sand [10] by 13 % for a decrease in e _ini of 29 %.

Fig. 2

Oedometric compression on Huston RF sand: K ₀ = σ ₃/σ ₁, with σ ₁ and $\varepsilon_1$ the vertical stress and strain, respectively, and σ ₃ the horizontal stress

The empirical equation of Jáky, $1-\hbox{sin} \,\varphi_c$, is widely accepted to predict the K ₀ value of normally consolidated clays. The applicability of this relation to sands is questionable since ``K ₀ of granular materials is not a unique function of neither the critical friction angle nor the peak friction angle [9]”. However, in our case $\varphi_c=33.8^\circ$ [14] and ${K_0^{{\rm J}{{\acute{\rm a}}}{\rm ky}} = 0.44}$, which is similar to that one predicted by barodesy.

The difference between the original and the extended version of barodesy is even more pronounced in oedometric unloading, Fig. 3. The original version predicts only slightly larger pre-loaded state ratios K = σ ₃/σ ₁ for dense sand than for loose sand. The extended version shows considerable larger differences, which seems to be qualitatively more realistic.

Fig. 3

Oedometric compression and extension on Huston RF sand, initial void ratios: e _ini = 0.8703, e _ini = 0.6667

The proposed version is tested on three drained triaxial experiments on dense Hostun RF sand with the consolidation stresses 100, 300 and 600 kPa, see Table 4. The deviation of the numerical peak friction angle to the experimental one is ±1 % in computations with the extended R-function and 2–5 % in computations with the original R-function. Using the extended version allows a slightly better fit of the experimental peak friction angles over the whole experimental stress range. The angle of dilatancy at peak is underestimated with the original version and overestimated with the extended one, see Table 5. The behaviour of both models is exemplified on the numerical simulation of a triaxial test with a consolidation stress of 600 kPa, see Fig. 4. The simulation with the extended version fits slightly better to the experimental data.

Table 5 Angle of dilatancy at peak ψ _p of drained triaxial experiments on dense Hostun RF sand and numerical simulation with original R-function (1) (material parameters from Table 1) and here proposed extended version (8) (material parameters from Table 3)

Fig. 4

Triaxial compression on dense Huston RF sand (host-d-triax-cd-600(hfdw09) [5], e _ini = 0.6707): σ ₁, $\varepsilon_1$ and $\varepsilon_v$ are the vertical stress, the vertical and the volumetric strain, respectively; σ ₃ = σ ₂ = 600 kPa as horizontal stress

6 Conclusion

The original version of barodesy assumes a constant value of K ₀, which contradicts experimental findings. A straightforward small modification allows for modelling pressure and void ratio dependency of K ₀. The additional material constant can be calibrated with triaxial tests on loose and dense samples with different consolidation pressures. Almost all other constants can be used from the original model, with the exception of c ₅ which has to be changed slightly. The extended version of barodesy predicts a higher K ₀-value for loose samples than for dense samples in keeping with experimental observations. The original version predicts an intermediate value for K ₀ and can therefore be seen as reasonable simplification. A slightly better fit of peak friction angles is possible with the extended version.

The extended version may be used in cases when (almost) confined conditions have to be modelled and the stresses evolving due to such a confinement are of interest.