Analysis and modelling of longitudinal deformation profiles of tunnels excavated in strain-softening time-dependent rock masses

Abstract

Rock mass behaviour model selection, in particular, to represent the post-failure behaviour and time-dependent behaviour of rock masses, are critical issues in the correct application of tunnelling design techniques such as the convergence-confinement method or numerical modelling. This study provides a general numerical approach for predicting longitudinal deformation profiles using a coupled ViscoElastic-ViscoPlastic Strain-Softening (VEVP-SS) model. A viscous dashpot and the strain-softening model are coupled to simulate the progressive damage process and creep failure behaviour of rock masses. Different failure criteria are considered to simulate the post-failure behaviour. As a verification step, numerical creep tests are carried out to analyse the coupled behaviour, and the basic viscoelastic and strain-softening results of the VEVP-SS model are compared with analytical solutions and numerical results. The proposed method is able to consider the coupling between post-failure behaviour and time-dependent behaviour, thus providing a new alternative method for preliminary tunnel design. Parametric analyses are then carried out to investigate the influence of different aspects on the longitudinal deformation profiles. The tunnel deformation based on the VEVP-SS model is larger than the corresponding elastic–plastic results due to the contribution of the creep behaviour, and the excavation rate becomes relevant when considering time-dependent behaviour.

Introduction

In a geomechanics framework, accidents are frequently related to fractures. The complexity of different geomaterials makes the study of these fractures a critical topic for the understanding of their behaviour; such a study constitutes a very important step in the mitigation of accidents that may occur in mining or civil engineering works such as tunnel excavation (Zhao et al., 2015, Zhou et al., 2019). Rock masses undergo progressive damage and long-term viscous behaviour throughout excavation and construction. Some underground structures show large delayed displacements that could lead to failure (Fabre and Pellet, 2006). The effect of time on rock mass deformability and strength is a topic of considerable interest in rock mechanics (Eberhardt et al., 1999, Damjanac and Fairhurst, 2010). Therefore, a proper selection of rock mass behaviour models and a proper simulation of the entire process of excavation and construction are crucial to obtain a reliable tool to achieve the optimal design of tunnels.

Most tunnel designs are currently based on empirical, analytical or numerical methods (Zhao et al., 2015, Fabre and Pellet, 2006, Barla and Borgna, 2000, Paraskevopoulou, 2016, Wang and Nie, 2010, Song et al., 2018b, Wang et al., 2018, Conte et al., 2013, Deng and Liu, 2020, Doležalová, 2001, Galli et al., 2004, Sainoki et al., 2017, Wang et al., 2020, Wang et al., 2017, Zhang et al., 2019, Zhu et al., 2013, Zou et al., 2017, Song et al., 2018a, Manica, 2018, Paraskevopoulou et al., 2018, Alonso et al., 2003, Barla et al., 2011, Alejano et al., 2010, Carranza-Torres and Fairhurst, 1999, Cui et al., 2019, Cui et al., 2015, Sulem et al., 1987, de la Fuente et al., 2019, Vlachopoulos and Diederichs, 2009, Wang et al., 2013, Wang et al., 2015, Alejano et al., 2009, Alejano et al., 2012, Carranza-Torres and Fairhurst, 2000, Yi et al., 2019, Debernardi and Barla, 2009, Kitagawa et al., 1991, Paraskevopoulou and Diederichs, 2018, Sterpi and Gioda, 2009, Wang et al., 2014, Wang et al., 2017). Among all these methods, the convergence-confinement method (CCM) is an analytical method that was developed in the 1930s (Fenner, 1938) and later refined by other researchers (Paraskevopoulou, 2016, Alejano et al., 2010, Cui et al., 2015, Vlachopoulos and Diederichs, 2009, Alejano et al., 2012, Carranza-Torres and Fairhurst, 2000, Corbetta et al., 1991). It provides an efficient way to determine supporting forces by considering the rock-support interactions (Alejano et al., 2010, Alejano et al., 2012). The CCM consists of three basic components in the form of graphs:

• The longitudinal deformation profile (LDP) relates the radial displacements of an unsupported tunnel section with its longitudinal distance to the tunnel face.

• The ground reaction curve (GRC) establishes the relationship between the decreasing inner pressure and the increasing radial displacements of the tunnel wall.

• The support characteristic curve (SCC) represents the stress–strain relationship of the support system (Alejano et al., 2010, Cui et al., 2015, Alejano et al., 2012). Then, an adequate design of the required support system can be achieved by taking into account the distance from the tunnel face at which the support will be installed and the supporting forces to which the support will be subjected, which can be obtained by the intersection of the GRC and the SCC, as shown in Fig. 1.

The GRC has been studied by various researchers (Zhang et al., 2019, Zou et al., 2017, Alonso et al., 2003, Cui et al., 2019, Alejano et al., 2009, Kabwe et al., 2020, Wang et al., 2010, Zhang et al., 2012). Hoek and Brown initially presented the SCC for different types of support structures (Hoek and Brown, 1980), and then further research on the topic was carried out by other researchers (Carranza-Torres and Fairhurst, 2000, Oreste, 2008, Oreste, 2003).

However, the main focus of this article also includes the longitudinal deformation profile (LDP). Many researchers have derived solutions for the LDP, most of which are based on the elastic (Corbetta et al., 1991, Panet, 1993, Panet, 1995, Unlu and Gercek, 2003) and elastic-perfectly plastic (EPP) behaviour of rock masses (Carranza-Torres and Fairhurst, 1999, Vlachopoulos and Diederichs, 2009, Carranza-Torres and Fairhurst, 2000). These models, nonetheless, do not seem to properly model the behaviour for average-quality rock masses (Alejano et al., 2012, Hoek and Brown, 1997). The response of rock masses will differ depending on the selected model. Considering post-failure behaviour, Alejano et al. (Alejano et al., 2012) extended the Vlachopoulos and Diederichs (Vlachopoulos and Diederichs, 2009) approach to the case of strain-softening rock masses, representing a wider range of rock masses, which can be used to obtain a more realistic approach to calculate the LDP. In fact, the LDP and the GRC heavily depend on the behaviour model chosen for the rock mass (Alejano et al., 2012).

In Fig. 1, the support design of a tunnel is estimated using CCM. If the support system is installed at a distance of 1.5 times the tunnel radius from the tunnel face, the support strength is enough to withstand the load when considering an EPP approach for the calculation of both the LDP and the GRC. However, at the same distance from the tunnel face, the support will collapse if the GRC and the LDP are calculated according to a strain-softening approach. Therefore, if the rock mass model cannot reproduce the actual behaviour of the rock mass, the resulting design may be unsafe. In the current research, a strain-softening model will be adopted to simulate the post-failure behaviour of rock masses.

Nevertheless, the solutions mentioned above do not consider the ductile properties of rock masses. Most types of rock masses exhibit significant ductile characteristics (Sainoki et al., 2017, de la Fuente et al., 2019, Malan, 2002), which are known to induce gradual deformations over time that occur even after the completion of underground excavations. Some researchers presented solutions for tunnels excavated in viscoelastic geomaterials (Wang and Nie, 2010, Song et al., 2018b, Song et al., 2018a, Wang et al., 2015), but plastic behaviour cannot be accounted for in viscoelastic models. On the other hand, many researchers proposed elastic-viscoplastic models (Conte et al., 2013, Zhu et al., 2013, Debernardi and Barla, 2009, Sterpi and Gioda, 2009, Desai and Zhang, 1987, Karim et al., 2013, Kutter and Sathialingam, 1992, Pellet et al., 2005), three-stage creep (3SC) model (Barla et al., 2011, Sterpi and Gioda, 2009) or Stress Hardening ELastic VIscous Plastic (SHELVIP) model (Barla et al., 2011, Debernardi and Barla, 2009) to simulate the creep and damage behaviour of geomaterials, but few of these models were used for the estimation of the LDP.

Moreover, few of these models consider a Mohr-Coulomb/Hoek-Brown strain-softening model, which may be important to model tunnel behaviour, especially in average-quality rock masses (Alejano et al., 2009, Alejano et al., 2012). Based on the Burgers-creep viscoplastic (CVISC) model introduced by Itasca (2011)), Paraskevopoulou (2016) presented LDP simulations for viscoelastic rock masses. However, in the CVISC model, the plastic slider is not coupled with viscous dashpot plastic yielding, which means that the model behaves similarly to a viscoelastic body if the stress states are below the yielding threshold (Paraskevopoulou et al., 2018). However, in many engineering cases, it is essential to consider the coupling between the plastic behaviour and the creep behaviour of the rock mass.

In summary, research on this topic has been mostly concerned with elastic, plastic, viscoelastic, or viscoplastic problems. To overcome these limitations, in this paper, we present a new coupled ViscoElastic-ViscoPlastic Strain-Softening (VEVP-SS) model, which considers the following:

• Time-dependent creep deformation.

• Mohr-Coulomb and Hoek-Brown strain-softening models.

• Progressive damage coupled with creep behaviour to simulate failure induced by creep and the subsequent progressive damage. This is the most significant part of the proposed model.

• The existence of the ‘limited stress level’, which will be explained in section 2. Our approach intends to be a general numerical approach for obtaining the longitudinal deformation profile (LDP) of tunnels excavated in time-dependent strain-softening rock masses.

The coupled ViscoElastic-ViscoPlastic Strain-Softening (VEVP-SS) model is first introduced and implemented into the finite element software CODE_BRIGHT (Olivella et al., 2020). Numerical tests are carried out to calibrate the numerical implementation and to verify the coupled behaviour of viscous dashpot and strain-softening models. Then, the CODE_BRIGHT results are compared with analytical solutions and FLAC (Itasca, 2011, FLAC3D, 2007) results to verify the viscoelastic and strain-softening behaviour in the VEVP-SS model, respectively. Finally, a comprehensive parameter analysis is provided to illustrate the sensitivity of the model to the excavation rate and rock mass behaviour model selection. It should be noted that the VEVP-SS model is currently under development, and further improvements are in progress. For example, a primary creep stage may be introduced in the VEVP-SS model in the near future.