Constitutive model for soil-rock mixtures in the light of an updated skeleton void ratio concept

Abstract

As a type of special geological body, soil-rock mixtures (SRMs) are widely found in nature and used in civil engineering. Many structures, such as rockfill dams, highways and tunnels, have used SRMs as building materials. Proper modeling of SRMs is of great importance to capture the complex behavior of this heterogeneous material. In this manuscript, a simple constitutive model incorporating the skeleton void ratio concept is proposed for SRMs with varying soil contents (sc). A prominent feature of the model is a unified description of the behavior of SRMs with varying sc such that only model parameters of pure rock and of pure soil are required. After calibration, the model shows a good capacity to predict the stress–strain response of SRMs under a wide range of sc, void ratios, and confining pressures. In particular, it captures well the non-associated behavior of rock-dominated SRMs with different sc. Furthermore, the sc-value is shown to modify the plastic flow direction of the material without influencing its yield surface.

Appendix: Calibration of model parameters

• (1) Elastic parameters

  The initial elastic shear modulus, G, can be obtained from the experimental data of deviatoric stress, q, versus the deviatoric strain, ${\varepsilon }_{s}$ when the axial strain is lower than 0.2%. Rearrangement of Eq. (10) gives:

  $${G}_{0}=G\frac{\left(1+e\right)}{{\left(2.97-e\right)}^{2}\sqrt{{P}^{{^{\prime}}}{P}_{a}}}$$

  (24)

  The values of the elastic constant, G₀, at various confining pressures, can be determined from Eq. (24). The average value of G₀ under different confining pressures is adopted.

  Based on Eqs. (7)–(11), the Poisson’s ratio, $\nu$ can be obtained by:

  $$v=\frac{9\mathrm{d}{\varepsilon }_{s}^{e}-2\mathrm{d}{\varepsilon }_{v}^{e}}{18\mathrm{d}{\varepsilon }_{s}^{e}+2d{\varepsilon }_{v}^{e}}\approx \frac{9{\varepsilon }_{s}-2{\varepsilon }_{v}}{18{\varepsilon }_{s}+2{\varepsilon }_{s}}$$

  (25)

• (2) Critical state parameters

  ${e}_{\Gamma }$, $\lambda$ and $\xi$ can be determined by directly fitting the experimental data for the critical state line. The critical state stress ratio M can be obtained by fitting critical state test data in p’-q plane with a function of q = Mp’.

• (3) Dilatancy parameters

  The parameter m is determined from Eq. (17) at a phase transformation state, at which D = 0, and thus,

  $$m=\frac{1}{{\psi }^{d}}ln\frac{{M}^{d}}{M}$$

  (26)

  where ${\psi }^{d}$ and ${M}^{d}$ are the values of $\psi$ and $\eta$ at the phase transformation state.

  Ignoring the small elastic strain, we have:

  $$\frac{\mathrm{d}{\varepsilon }_{v}}{\mathrm{d}{\varepsilon }_{q}}\approx \frac{\mathrm{d}{\varepsilon }_{v}^{p}}{\mathrm{d}{\varepsilon }_{q}^{p}}=D={d}_{0}\left(\mathrm{exp}(m\psi )-\frac{\eta }{M}\right)$$

  (27)

  The parameter d₀ is determined based on the $d{\varepsilon }_{v}$-$d{\varepsilon }_{q}$ curve.

• (4) Hardening parameters

The parameter n is determined by Eq. (14) at a peak stress state, at which K_p = 0:

$$n=\frac{1}{{\psi }^{b}}ln\frac{M}{{M}^{b}}$$

(28)

where ${\psi }^{b}$ and ${M}^{b}$ are the values of $\psi$ and $\eta$ at the peak stress state.

Combining Eqs. (10) (13) and (14) for conventional drained tests (dp’ = dq/3) yields:

$$\frac{\mathrm{d}q}{\mathrm{d}{\varepsilon }_{q}}\approx \frac{\mathrm{d}q}{\mathrm{d}{\varepsilon }_{q}^{p}}=\frac{{K}_{p}}{1-\eta /3}=h\left\{\frac{{{G}_{0}\left(2.97-e\right)}^{2}\sqrt{{p}^{{^{\prime}}}{p}_{a}}\left(\frac{M}{\eta }-\mathrm{exp}(n\psi )\right)}{\left(1+e\right)(1-\eta /3)}\right\}$$

(29)

As all the model parameters in the brackets are known, h is determined based on $\mathrm{d}q-\mathrm{d}{\varepsilon }_{q}$ curves along drained triaxial loading paths. Then parameters h₁ and h₂ can be obtained by equation $h={h}_{1}-{h}_{2}{e}_{0}$.