Loading history effect on time-dependent deformations after unloading – reversible creep of soft rock (marl)

Abstract

Practical problem in geotechnical design of support systems for tunnels, slopes, and excavation pits requires numerical modeling of different phases of excavation and support installation. The applied constitutive model should therefore include aspects of the influence of the loading history on the deformation response of the rock material during both loading and unloading. Notwithstanding the practical significance, researches into this subject-matter with a view to soft rocks are rare. Some aspects of deformational behavior of soft rock after unloading given in this paper are based on results of time-cycling loading and stepwise unloading long-term creep experiments conducted by the authors on the marl specimens. The results of the experiment indicate that the time of unloading $t_{\mathrm{c}}$ measured from the beginning of a specific load history is a basic parameter, which can describe the influence of history on the deformation response of marl because it implicitly reflects the process of preconsolidation (hardening) of the material. The deformation response after unloading is described by determining the dependence of the parameters of the modified Wallner rheological model of $t_{\mathrm{c}}$. This approach requires a larger volume of experimental data, so a semiempirical approach has been applied as an alternative, which can include the influence of magnitude of creep stress, the time of maintaining such stress before unloading, and the duration of the last period in which rock material was completely unloaded during load history.

Appendices

Appendix 1: Modified Wallner’s model

Modified Wallner’s model consists of five different rheological bodies, each of which corresponds to an appropriate deformation component (Fig. 34).

Fig. 34

Structure of modified Wallner’s rheological model

When a stress level is below the plastic limit, the creep strain is limited to primary and secondary creep, and hence the total strain is a sum of elastic strain and strains of primary and secondary creep:

$$ \varepsilon = \varepsilon ^{el} + \varepsilon ^{p} + \varepsilon ^{s}. $$

(19)

Table 4 Review of constitutive equations for rock (stress level below the plastic limit)

Elastic component

In case of the uniaxial stress state, equation (20), provided that the stress and strain states are time-independent, becomes

$$ \varepsilon _{1}^{el} = \frac{\Delta \sigma _{1}}{E}, $$

(23)

where $E$ refers to the Young modulus of elasticity, $\Delta \sigma _{1}$ denotes the main normal stress difference, and $\varepsilon _{1}$ stands for the main strain (in the direction of axis of the main normal stress).

Primary creep

With the introduction of $\sigma _{2} = \sigma _{3} = 0$, the following applies:

$$ \Delta S_{1} = \frac{2}{3}\Delta \sigma _{1},\qquad \Delta S_{2} = - \frac{1}{3}\Delta \sigma _{1},\qquad \Delta S_{3} = - \frac{1}{3}\Delta \sigma _{1}, $$

(24)

$$ \Delta \sigma _{eff} = \Delta \sigma _{1},\qquad \Delta \varepsilon ^{p}_{eff} = \frac{2}{3}\varepsilon _{1},\qquad \left \langle G \right \rangle = E_{p}\left [ \left ( \frac{\Delta \sigma _{1}}{E_{p}} \right )^{m} - \varepsilon _{1}^{p} \right ]. $$

(25)

When equations (24) and (25) are included into equation (21), for the case of uniaxial stress state, equation (21) has the following form:

$$ \frac{d\varepsilon _{1}^{p}}{dt} = \frac{1}{\mu _{p}}E_{p}\left [ \left ( \frac{\Delta \sigma _{1}}{E_{p}} \right )^{m} - \frac{2}{3}\varepsilon _{1}^{p} \right ]. $$

(26)

After regrouping, we obtain:

$$ \frac{\partial \varepsilon _{1}^{p}}{\partial t} + \frac{2}{3}\frac{E_{p}}{\mu _{p}}\varepsilon _{1}^{p} = \frac{E_{p}}{\mu _{p}}\left ( \frac{\Delta \sigma _{1}}{E_{p}} \right )^{m},\qquad \alpha = \frac{2}{3}\frac{E_{p}}{\mu _{p}},\qquad \beta = \frac{E_{p}}{\mu _{p}}\left ( \frac{\Delta \sigma _{1}}{E_{p}} \right )^{m}. $$

(27)

A linear nonhomogenous differential equation has the following form:

$$ \frac{\partial \varepsilon _{1}^{p}}{\partial t} + \alpha \varepsilon _{1}^{\mathrm{p}} = \beta . $$

(28)

After separation of variables and integration, we obtain

$$ \varepsilon _{1}^{p} = \frac{\beta }{\alpha } + Ce^{ - \alpha t}, $$

(29)

where the integration constant C is determined based on the initial condition $\varepsilon ( t = 0 ) = 0$:

$$ C = - \frac{\beta }{\alpha }. $$

(30)

Hence, after the constant $C$ is included in equation (29), a solution of the equation is

$$ \varepsilon _{1}^{p} = \frac{\beta }{\alpha } \left ( 1 - e^{ - \alpha t} \right ). $$

(31)

After replacement of $\alpha$ and $\beta$ and regrouping, we obtain the following final form of equation for primary creep at uniaxial loading:

$$ \varepsilon _{1}^{p} = \frac{3}{2}\left ( \frac{\Delta \sigma _{1}}{E_{p}} \right )^{m}\left ( 1 - e^{ - \frac{2}{3}\frac{E_{p}}{\eta _{p}}t} \right ). $$

(32)

Secondary creep

For the uniaxial compression, with introduction of $\sigma _{2} = \sigma _{3} = 0$ and $\sigma _{eff} = \sigma _{1}$, equation (22) is reduced to

$$ \frac{\partial \varepsilon _{1}^{s}}{\partial t} = \frac{1}{\mu _{p}}\sigma _{1}^{n} = a\sigma _{1}^{n}, $$

(33)

that is, after integration of (33), we obtain:

$$ \varepsilon _{1}^{s} = a\sigma _{1}^{n}t + C, $$

(34)

where the integration constant $C$ is defined based on the initial condition $\varepsilon _{1\,\left ( t = 0 \right )}^{p} = 0\ \Rightarrow\ C=0$. Hence the final form of the equation for secondary creep at uniaxial compression is

$$ \varepsilon _{1}^{s} = a\sigma _{1}^{n}t. $$

(35)

Final constitutive equation of modified Wallner’s model at uniaxial stress state below the yield condition

By substituting equation (23) for elastic component of deformation and equation (32) for primary creep and equation (35) for secondary creep into equation (19) we obtain the following constitutive equation for rock creep at uniaxial stress state below the plastic limit:

$$ \varepsilon = \frac{\Delta \sigma _{1}}{E} + \frac{3}{2}\left ( \frac{\Delta \sigma _{1}}{E_{p}} \right )^{m}\left ( 1 - e^{ - \frac{2}{3}\frac{E_{p}}{\eta _{p}}t} \right ) + a\left ( \sigma _{1} \right )^{n}t. $$

(36)

Appendix 2

Table 5 Measured values of the reversible creep strains accompanying diagram illustrated in Fig. 11

Table 6 Measured average values of the reversible creep strains accompanying diagram illustrated in Fig. 12

Table 7 Measured average values of the reversible creep strains accompanying diagram illustrated in Fig. 13

Table 8 Measured average values of the reversible creep strains accompanying diagram illustrated in Fig. 14

Table 9 Measured average values of the reversible creep strains accompanying diagram illustrated in Fig. 15