A nonstationary parameter model for the sandstone creep tests

Abstract

The rock masses of hydro-fluctuation belt experience seepage pressure following impoundment in the Three Gorges Reservoir; its creep behaviors are significant for reservoir bank slopes. To study the creep behaviors under seepage pressure (0, 1.45, and 1.75 MPa), we performed creep tests using representative landslide sandstone in the Three Gorges Reservoir and investigated the sandstone creep behaviors under the coupling effects of seepage pressure and stress. Previous researches on rocks have usually regarded the creep constitutive parameter as a constant; however, in this study, a nonlinear, nonstationary, plastic-viscous (NNPV) creep model which can describe the curve of sandstone creep tests is proposed. The rock-creep parameters under three levels of seepage pressure were identified, and theoretical curves using the NNPV model agreed well with the experimental data, indicating that the new model cannot only describe the primary creep and secondary creep stages under varying seepage pressures but also, in particular, perfectly describes the tertiary creep stage. Finally, the sensitivity of the NNPV model parameters is analyzed, and the result shows that the nonstationary coefficient α and the nonlinear coefficient m are main parameters affecting the tertiary creep stage.

Appendix 1

One-dimensional creep constitutive equations of the NNPV

• (1) When σ₀ ≤ σ_s, the NNPV elements satisfy the following conditions

$$ \left\{\begin{array}{l}\varepsilon ={\varepsilon}_1+{\varepsilon}_2+{\varepsilon}_3\\ {}\sigma ={E}_1{\varepsilon}_1\\ {}\sigma ={E}_2{\varepsilon}_2+{\eta}_1{\dot{\varepsilon}}_2\\ {}\sigma ={E}_3{\varepsilon}_3+{\eta}_2{\dot{\varepsilon}}_3\end{array}\right. $$

(10)

where σ is the total stress and ε is the total strain.

• (2) When σ₀ ≥ σ_s and t ≤ t_P, the NNPV elements satisfy the following conditions:

$$ \left\{\begin{array}{l}\varepsilon ={\varepsilon}_1+{\varepsilon}_2+{\varepsilon}_3+{\varepsilon}_4\\ {}\sigma ={E}_1{\varepsilon}_1\\ {}\sigma ={E}_2{\varepsilon}_2+{\eta}_1{\dot{\varepsilon}}_2\\ {}\sigma ={\eta}_2{\dot{\varepsilon}}_3\\ {}\sigma ={\sigma}_S+{\eta}_3{\dot{\varepsilon}}_4/\left(\left({\left(\frac{t}{t_0}\right)}^m-1\right)\times m\times {\left(\frac{t}{t_0}\right)}^{\left(m-1\right)}\right)\end{array}\right. $$

(11)

• (3) When σ₀ ≥ σ_s and t ≥ t_P, let ${\left(\frac{t-{t}_P}{t_0}\right)}^m=A$, and the NNPV elements satisfy the following conditions:

$$ \left\{\begin{array}{l}\varepsilon ={\varepsilon}_1+{\varepsilon}_2+{\varepsilon}_3+{\varepsilon}_4\\ {}\sigma ={E}_1{\varepsilon}_1\\ {}\sigma ={E}_2{\varepsilon}_2+{\eta}_1{\dot{\varepsilon}}_2\\ {}\sigma ={\eta}_2{\dot{\varepsilon}}_3\\ {}\sigma ={\sigma}_S+{\eta}_3{\dot{\varepsilon}}_4{e}^{-\alpha t}/\left(\left(A-1\right)\times m\times {\left(\frac{t-{t}_p}{t_0}\right)}^{\left(m-1\right)}-\left(A-1\right)\times \alpha \right)\end{array}\right. $$

(12)

In the tests, σ = σ₀ = constant and the initial conditions of stress strain are:

$$ \left\{\begin{array}{l}{\dot{\varepsilon}}_i(0)={\ddot{\varepsilon}}_i(0)=0\kern3.25em i=1,2,3\\ {}{\dot{\sigma}}_j(0)={\ddot{\sigma}}_j(0)=0\kern2.25em j=1,2,3,4\end{array}\right. $$

(13)

Based on Eqs. (10)–(13), and with Laplace and Laplace inverse transformation, then:

• (a) When σ₀ ≤ σ_s, the NNPV creep equation is Eq. (4),

• (b) When σ₀ ≥ σ_s and t ≤ t_P, the NNPV creep equation is Eq. (5),

• (c) When σ₀ ≥ σ_s and t ≥ t_P, the NNPV creep equation is Eq. (6).

Three-dimensional creep constitutive equations of the NNPV

In the 3D stress state, the rock stress tensor σ_ij can be decomposed into spherical tensor σ_m and partial tensor S_ij; the strain tensor can be decomposed into spherical strain tensor ε_m and partial strain tensor e_ij, then:

$$ \left\{\begin{array}{l}{\sigma}_m=\frac{1}{3}\left({\sigma}_1+{\sigma}_2+{\sigma}_3\right)=\frac{1}{3}{\sigma}_{kk}\kern0.3em \\ {}{S}_{ij}={\sigma}_{ij}-{\delta}_{ij}{\sigma}_m={\sigma}_{ij}-\frac{1}{3}{\delta}_{ij}{\sigma}_{kk}\\ {}{\sigma}_{ij}={S}_{ij}+\frac{1}{3}{\delta}_{ij}{\sigma}_{kk}\kern0.2em \end{array}\right.\kern0.1em $$

(14)

δ_ij is Kronecker’s sign, σ_m only changes the volume of an object, but not shape, and S_ijonly only changes the shape, but not volume. Thus, strain tensor can also be separated into spherical strain tensor ε_m and partial strain tensor e_ij, then:

$$ \left\{\begin{array}{c}{\varepsilon}_{ij}={e}_{ij}+{\delta}_{ij}{\varepsilon}_m\kern0.72em \\ {}{\varepsilon}_m=\frac{1}{3}{\varepsilon}_{kk}\kern1.8em \\ {}{e}_{ij}={\varepsilon}_{ij}-\frac{1}{3}{\delta}_{ij}{\varepsilon}_{kk}\;\end{array}\right. $$

(15)

Let rock shear module be G, volume module be K, elastic module be E, and Poisson’s ratio be μ, then:

$$ \left\{\begin{array}{c}K=\frac{E}{3\left(1-2\mu \right)}\kern0.75em \\ {}G=\frac{E}{2\left(1+\mu \right)}\end{array}\right. $$

(16)

In the 3D stress state, based on Hooke’s law (Wang et al. 2016), then:

$$ \left\{\begin{array}{c}{\sigma}_m=3K{\varepsilon}_m\\ {}{S}_{ij=}2G{e}_{ij}\kern1em \end{array}\right. $$

(17)

Based on Eqs. (4), (15), and (16), when the constant deviator stress (S_ij)₀ is smaller than the rock yield strength σ_s, the 3D creep equation is expressed as:

$$ {\varepsilon}_{ij}=\frac{{\left({S}_{ij}\right)}_0}{2{G}_1}+\frac{{\left({S}_{ij}\right)}_0}{2{G}_2}\left(1-{e}^{-\frac{G_2}{\eta_1}t}\right)+\frac{{\left({S}_{ij}\right)}_0}{2{G}_3}\left(1-{e}^{-\frac{G_3}{\eta_2}t}\right)+\frac{\sigma_m{\delta}_{ij}}{3K}\kern0.5em {\left({S}_{ij}\right)}_0\le {\sigma}_s $$

(18)

where G₁ , G₂, and G₃ are the shear elastic module corresponding to E₁ , E₂, and E₃ in NNPV.

In the 3D stress state, when (S_ij)₀ ≥ σ_s and t ≤ t_P, rock plastic deformation occurs; rock yield surface F and plastic potential function Q should be introduced (Yang et al. 2013b), then: viscoplastic deformation rate as the fourth part in NNPV model is:

$$ {\dot{\varepsilon}}_{ij}^4=\left(\frac{<F>}{2{\eta}_3}\right)\frac{\partial Q}{\partial {\sigma}_{ij}} $$

(19)

$$ <F>=\left\{\begin{array}{l}0\kern1.5em \left(f\le 0\right)\\ {}f\kern1.5em \left(f\ge 0\right)\end{array}\right. $$

(20)

where f is yield function; f=Q when relevant flow rules are used, then the 3D constitutive equation in the fourth part of NNPV model is:

$$ {\varepsilon}_{ij}^4=\left(\frac{f}{2{\eta}_3}\right)\frac{\partial f}{\partial {\sigma}_{ij}}t\kern1.5em \left(f\ge 0\right) $$

(21)

Yield function f can be expressed as (Jiang et al. 2013):

$$ f=\sqrt{J_2}-{\sigma}_s/\sqrt{3} $$

(22)

where J₂ is the second stress deviator invariant.

Based on Eqs. (18) and (22), when (S_ij)₀ ≥ σ_s and t ≤ t_P, the constitutive equation of the NNPV model under 3D stress is:

$$ {\displaystyle \begin{array}{l}{\varepsilon}_{ij}(t)=\frac{{\left({S}_{ij}\right)}_0}{2{G}_1}+\frac{\sigma_m{\delta}_{ij}}{3K}+\frac{{\left({S}_{ij}\right)}_0}{2{G}_2}\left(1-{e}^{-\frac{G_2}{\eta_1}t}\right)+\frac{{\left({S}_{ij}\right)}_0}{2{G}_3}\left(1-{e}^{-\frac{G_3}{\eta_2}t}\right)+\\ {}\kern2.75em \left(\frac{f}{2{\eta}_3}\right)\frac{\partial f}{\partial {\sigma}_{ij}}\times \left({\left(\frac{t}{t_0}\right)}^{\mathrm{m}}-1\right)\kern1em \left(f\ge 0\right)\end{array}} $$

(23)

Similarly, in the 3D stress state, when (S_ij)₀ ≥ σ_s and t ≥ t_P, the constitutive equation of NNPV model under 3D stress is expressed as:

$$ {\displaystyle \begin{array}{l}{\varepsilon}_{ij}(t)=\frac{{\left({S}_{ij}\right)}_0}{2{G}_1}+\frac{\sigma_m{\delta}_{ij}}{3K}+\frac{{\left({S}_{ij}\right)}_0}{2{G}_2}\left(1-{e}^{-\frac{G_2}{\eta_1}t}\right)+\frac{{\left({S}_{ij}\right)}_0}{2{G}_3}\left(1-{e}^{-\frac{G_3}{\eta_2}t}\right)+\\ {}\kern2.5em \left(\frac{f}{2{\eta}_3\times {\mathrm{e}}^{-\alpha t}}\right)\frac{\partial f}{\partial {\sigma}_{ij}}\times \left({\left(\frac{t\hbox{-} {t}_P}{t_0}\right)}^{\mathrm{m}}-1\right)\kern0.75em \left(f\ge 0\right)\end{array}} $$

(24)

The triaxial compression creep stress state is σ₂ = σ₃ and constant; all levels of loads σ₁ after loading are constant; then from Eqs. (18), (23), and (24):

• (a) When σ₁ − σ₃ ≤ σ_S, the 3D creep equation of the NNPV model is expressed as Eq. (7),

• (b) When σ₁ − σ₃ ≥ σ_S and t ≤ t_P, the 3D creep equation of the NNPV model is expressed as Eq. (8),

• (c) When σ₁ − σ₃ ≥ σ_S and t ≥ t_P, the 3D creep equation of the NNPV model is expressed as Eq. (9).