Deformation-softening behaviors of high-strength and high-toughness steels used for rock bolts

Ding Wang,Manhao He,Zhigang Tao,Aipeng Guo,Xuhun Wang

a State Key Laboratory for Geomechanics and Deep Underground Engineering,China University of Mining & Technology,Beijing,100083,China

b Research Institute for Deep Underground Science and Engineering,China University of Mining & Technology,Beijing,100083,China

c School of Civil Engineering,Qingdao University of Technology,Qingdao,266033,China

Keywords:Rock bolt High-strength and high-toughness steels Loading rate Pretension Deformation-softening Crystal plasticity

ABSTRACT In deep ground engineering,the use of high-strength and high-toughness steels for rock bolt can significantly improve its energy absorption capacity.However,the mechanisms and effects of rock loading conditions on this kind of high energy-absorbing steel for rock bolt remain immature.In this study,taking Muzhailing highway tunnel as the background,physically based crystal plasticity simulations were performed to understand the effect of rock loading rate and pretension on the deformation behaviors of twinning induced plasticity(TWIP)steel used for rock bolt.The material physical connecting to the underlying microscopic mechanisms of dislocation glide and deformation twinning were incorporated in numerical modeling.The rock loading conditions were mimicked by the real-time field monitoring data of the NPR bolt/cable equipment installed on the tunnel surrounding rock surface.The results indicate that the bolt rod exhibits pronounced deformation-softening behavior with decrease of the loading rate.There is also a sound deformation-relaxation phenomenon induced by the dramatic decrease of loading rate after pre-tensioning.The high pretension (＞600 MPa or 224 kN) can help bolt rod steel resist deformation-softening behavior,especially at low loading rate (＜10-1 MPa/s or 10-2 kN/s).The loading rate was found to be a significant factor affecting deformation-softening behavior while the pretension was found to be the major parameter accounting for the deformation-relaxation scenario.The results provide the theoretical basis and technical support for practical applications.

1.Introduction

High in situ stresses and large deformations of surrounding rocks,or dynamic failures such as rock burst and rock explosions,highlight the importance of improving the energy absorption capacity of rock bolts in deep ground engineering (He et al.,2014a).The deformation characteristics of bolt rod play key roles on energy-absorption capacity in rock bolts.Therefore,it is crucial to understand the mechanisms and effects of different rock loading conditions on the deformation behaviors of bolt rod,especially the high-strength and high-toughness steels.

Owing to relatively lower tensile strength (＜400 MPa) and smaller elongation(＜10%)after plastic yielding,the deformation of bolt rod made of conventional steels,such as plain carbon steels,is strictly limited due to small deformation range(Kang et al.,2015).In contrast,a rock bolt made of high-strength and high-toughness steels,such as twinning induced plasticity (TWIP) steels,exhibits tensile strength up to 1 GPa and good ductility with more than 50%uniform strain(Bouaziz et al.,2011;De Cooman et al.,2018),which can significantly improve its energy-absorption performances.These materials are complex,with a carefully balanced microstructure,where metastable austenite,precipitation state,complex dislocation substructures,twinning effects and deformation-driven athermal transformation phenomena are formed and activated(Raabe et al.,2020).The microstructure evolution and deformation mechanisms,such as interactions of dislocation glide and deformation twinning,determine the sustained high performance behaviors of these materials (Martin et al.,2016; Kim et al.,2017).Deformation twinning increases the hardening rate mainly by acting as interface obstacles to dislocation motion (Pierce et al.,2015).To date,many studies have been conducted to understand the deformation behaviors of high-strength and high-toughness steels at different temperatures (Martin et al.,2016; Wong et al.,2016; Madivala et al.,2018),strain rates (Benzing et al.,2018; Lee et al.,2018; Li et al.,2019),loading paths (Lapovok et al.,2018;Pereira et al.,2019).However,most of the studies are carried out in the loading conditions of material forming processes which are quite different from the deep ground engineering conditions.For instance,the time scales of static rock deformation are generally in the range of several hours or days,while those of material forming processes are generally in the range of seconds or minutes.In addition,materials are seldom pretensioned during forming processes,while the influence of pretensioning (Jalalifar et al.,2006;Hosseinitoudeshki and Fazeli,2014; Yu et al.,2019; Kang et al.,2020; Gao et al.,2021) on rock bolt is significant and should be considered.

For the complex geological conditions and unstable loading process from surrounding rocks,numerical simulation is a feasible method to overcome the high-cost and time-consuming processes of experimental and field tests.Many numerical studies were conducted to understand the deformation and failure mechanisms of rock bolt (Maghous et al.,2012; Cao et al.,2013; Li,2014; Wei et al.,2017; Li et al.,2018a,2021a; Wang et al.,2020).However,the numerical models for material deformation in these studies were of macroscopic yield criterion models,which are suitable for conventional steels with simple microstructure.For high-strength and high-toughness steels with complex microstructures,the numerical model should consider the detailed physical evolution process of microscopic deformation mechanisms,such as dislocation slip (Wang et al.,2018a; Zhang et al.,2021) and deformation twinning (Steinmetz et al.,2013).In computational material science,crystal plasticity model is a powerful tool to investigate the elastoplastic deformation behavior and the corresponding microstructure-property relations in crystalline materials (Roters et al.,2019).Thereinto,the physically based crystal plasticity model (as opposed to phenomenological models or yield criterion models)is established to include microscopic physical quantities as internal variables (i.e.dislocation densities) and rate-dependent equations based on the active deformation mechanisms(Segurado et al.,2018; Lu et al.,2020).The crystal plasticity constitutive model utilized in this study incorporates the dislocation and twinning mechanisms as given in Wong et al.(2016).The model was developed in the framework of the Düsseldorf Advanced Material Simulation Kit (DAMASK) (Roters et al.,2019),which is a unified multi-physics crystal plasticity simulation tool for investigating the response of metallic microstructures to applied loads(Wang et al.,2018a,b; Diehl et al.,2020).

The mechanical boundary conditions of numerical model are effective to be established by matching the monitoring data from deep ground sites.However,the surrounding rocks in deep ground are feathered with high instability and large nonlinear deformation,which makes it difficult to figure out deformation boundary conditions and loading path on bolt rod in numerical simulation.The rock instability can be improved by embedding support equipment to alter the stress state of the surrounding rocks.Many support techniques and products have been developed to accommodate rock deformation in deep ground.Typical tensile bolts are the cone bolt,D bolt,Roofex bolt,and negative Poisson’s ratio (NPR) bolt/cable (Jager,1992; Charette and Plouffe,2007; Li,2012; Li and Doucet,2012; He et al.,2014a).Among them,the NPR bolt/cable(He et al.,2014a; Tao et al.,2016) has the advantages of high resistance friction and can resist large deformation of the surrounding rocks.This novel support equipment coupled with rock mass will form a stable load-bearing ring,which can make the excavated roadway be stabilized (He et al.,2014a,b).

In this study,the field monitoring data of anchor rod axial stress obtained from Muzhailing highway tunnel were utilized as mechanical boundary conditions.Based on field monitoring data,crystal plasticity simulations were performed to understand the effects of different loading rates and pretension levels on deformation behaviors of TWIP steel for bolt rod in deep ground engineering.

2.Methods

2.1.Field monitoring techniques

In this study,the NPR bolt/cable equipment was utilized and multiple anchor rod dynamometers were set on the surrounding rocks of the Muzhailing highway tunnel,where the geological conditions of Muzhailing highway tunnel on Weiwu Expressway are rather complex.The surrounding rocks are mainly of thin carbonaceous slate,and the joint fissures are well developed.The maximum depth of the tunnel is about 630 m,which is subjected to north-south compressive stress.The maximum horizontal stress is 25 MPa,which is of extremely high in situ stress area.The rocks in the whole area are mainly of V-class surrounding rocks.In this study,K1780 section in the#2 inclined shaft of Muzhailing highway tunnel was selected as the field monitoring section.As large nonlinear deformation(2.3 m)of surrounding rocks in Muzhailing highway tunnel was observed,the conventional bolt support scheme could not be used to resist larger deformations of surrounding rocks.After the NPR bolt/cable (as shown in Fig.1) was used to improve the support of #2 inclined shaft,the deformation of surrounding rocks was successfully controlled within 300 mm.In order to verify the supports of the NPR bolt/cable and variations of anchor rod axial forces during the supporting process,multiple monitoring points were set on the surrounding rocks.Fig.2a shows the schematic diagram of the sectional arrangement of monitoring points.

The NPR bolt/cable(He et al.,2014a,b)has five major parts,i.e.rod body,constant resistance body,constant resistance sleeve,tray,and lockset.It has the capacities of constant resistance,energy absorption and large elongation and other perfect mechanical properties (Tao et al.,2016,2021; Sun et al.,2019).The schematic diagram and photo of NPR bolt/cable are represented in Fig.1.

Fig.1.(a) Schematic diagram of NPR bolt/cable structure; and (b) Photo of the NPR bolt/cable (Sun et al.,2019).

Fig.2.(a) Schematic diagram of the sectional arrangement of monitoring points; (b)The structure of monitoring device on the NPR bolt/cable; and (c) Photo of installed NPR bolt/cable with dynamometer (Wang et al.,2021).

The axial force resulted from the deformation energy of surrounding rocks was applied to the bolt through the bolt tray and the internal anchoring section.The axial force increases gradually with increase of the deformation energy of surrounding rocks.When the axial force applied to the rod body is greater than or equal to the design constant resistance value of the NPR bolt/cable,the constant resistance body in the constant resistance sleeve will rub and slide along the cylinder wall to realize the unique capabilities of NPR bolt/cable.

The anchor rod dynamometer was used to record the variation of anchor rod axial forces.Fig.2b and c shows the structure and photo of monitoring device on the NPR bolt/cable.The monitoring data of anchor rod axial forces were utilized as the target tensile stress used for crystal plasticity simulations.

2.2.Crystal plasticity model

2.2.1.Modeling and numerical solution

The crystal plasticity model used in this context is a physically based constitutive model developed within the framework of the DAMASK(Roters et al.,2019)that incorporates the dislocation glide,twinning and ε-martensite phase transformations.The constitutive model for strain hardening and twinning evolution has been developed over the years,and has been verified by the experiments(Steinmetz et al.,2013; Wong et al.,2016; Madivala et al.,2018;Roters et al.,2019).A detailed theoretical description of the constitutive model used in this study is given in Wong et al.(2016).The conventional rock bolt models (Li et al.,2020b,2021b; Chen et al.,2021) consider the load-displacement behaviors of rock bolt based on the criterion of relative movement between rock bolt,grout and rock interface.The crystal plasticity model utilized in this context is to address the nonlinear responses of plastic deformation of bolt rod,incorporating the dislocation and twinning mechanisms.

In this study,a spectral solver was employed to solve the mechanical boundary problem of crystal plasticity simulations(Roters et al.,2019; Diehl et al.,2020).Moulinec and Suquet (1994,1998)firstly proposed the use of a spectral method for solving the mechanical boundary problem.Lebensohn (2001) further developed the fast Fourier transform (FFT) approach of spectral method for prediction of micromechanical properties in polycrystalline materials.Eisenlohr et al.(2013) extended the DAMASK framework using the spectral solver in combination with multiple constitutive models.Shanthraj et al.(2015) developed the spectral method to predict the micromechanical fields of plastically deforming heterogeneous materials.The computational efficiency of spectral method over finite element method has been demonstrated(Eisenlohr et al.,2013),particularly for heterogeneous material with large local stiffness contrast (Wang et al.,2018a,b).

2.2.2.Simulation setup

The representative volume element(RVE)was first constructed as a spectral grid with dimensions 16 ×16 ×16,which comprises 20 grains.The grain shapes were generated using the Voronoi tessellation method.The crystallographic orientations of the grains were randomly assigned from a uniform orientation distribution.The uniaxial tension boundary conditions are given by

The stress P11and deformation gradient F11along tensile direction at each incremental step obtained from numerical simulations were plotted to reveal the mechanical and deformation behaviors of the material.The optimized parameters of crystal plasticity constitutive model used for TWIP steel strengthened by dislocation glide and twinning mechanisms were adopted from previous work (Madivala et al.,2018),which have already been calibrated by experimental results (Steinmetz et al.,2013; Wong et al.,2016; Madivala et al.,2018).

3.Results

3.1.Crystal plasticity simulations

In this section,the crystal plasticity simulations with mechanical boundary conditions were showed based on field monitoring data obtained from Muzhailing highway tunnel.Fig.3a shows the variation of monitored anchor rod axial forces with time,where the data points of right arch shoulder are shown in red color and those of left arch shoulder are in black.The rod axial forces on the right and left arch shoulders are not the same but with a similar variation trend.The pre-tensioning process of tightening anchor bolt into the rock mass generated initial tensile force f0on rod body.After pretensioning,the monitoring tensile force ftincreases monotonously with increasing loading time t until the onset of constant resistance slip of the NPR bolt/cable(6 d for left arch shoulder,and 7 d for right arch shoulder).The applied tensile force can induce elastic and plastic deformations of the bolt rod.When the tensile force reached a constant resistance,the deformation of the NPR bolt/cable was achieved by frictional sliding of the constant resistance body through the internal surface of the casting tube.The monitoring tensile force ftthen oscillated around a constant resistance value.In this study,the crystal plasticity simulations were performed in the range prior to the onset of constant resistance slip.The tensile stress(σt) can be calculated by

where d0is the rod diameter,and d0=21.8 mm.In the simulation,the mechanical boundary conditions were set as P11=σtat monitoring time t.The loading time of pre-tensioning was set as 30 s and the post-loading time was set as 86,400 s for a single day.The detailed monitoring data and the corresponding mechanical boundary conditions are given in Table 1.

Based on the mechanical boundary conditions,crystal plasticity simulations were performed to predict the stress and deformation of steel strengthened by dislocation glide and twinning mechanisms.The stress path of numerical simulations in Fig.3b was plotted by the average stress values P11at each incremental time step.The corresponding monitoring forces were marked as hollow points.The three-dimensional (3D) patterns of deformation gradient F11and stress P11distributions of the deformed RVE (see Fig.4) reveal the non-uniform deformation characteristics at microscopic scale.The curves of stress P11versus deformation gradient F11are plotted in Fig.5a.There are pronounced deformation-relaxation plateaus illustrated in Fig.5b.The deformation-relaxation is an elongation improvement phenomenon induced by the dramatic decrease of the loading rate(Hariharan et al.,2013).It shows that the deformation-relaxation occurs immediately after pre-tensioning.The relaxation plateau contains about 2.5%elongation strain under the applied pretension level.

Fig.3.Field monitoring force and stress path of numerical simulations: (a) Field monitoring data of the axial tensile forces obtained from K1780 section in Muzhailing highway tunnel,and (b) The stress paths of numerical simulations (solid lines) were set in accordance with the field monitoring force (hollow points) before the onset of constant resistance slip.

The variations of average dislocation density ρd(Fig.6a) and twin volume fraction ftw(Fig.6b)with deformation gradient F11are plotted in Fig.6.The positions of corresponding day points were marked on the curves.Compared with Fig.5,the activity intensities of the underlying deformation mechanisms increase continuously with increase of deformation gradient,even during the relaxation process.In addition,the variation curves of the non-zero stress components with deformation gradient F11(see Fig.7) show that P12,P13,P21,P23,P31,P32have an obvious variation during deformation-relaxation process after pre-tensioning.

3.2.Crystal plasticity simulations under different loading rates and pretension levels

According to the crystal plasticity simulations based on field monitoring data in Section 3.1,the loading stress rates obtained from Muzhailing highway tunnel are in the magnitudes of 10-4MPa/s (Table 1).This real-time loading rates are very slow compared to the normal experimental loading conditions,e.g.the test under stress-controlled rate of 10-2MPa/s to 102MPa/s(Gong et al.,2019) and the test based on force-controlled rate of 1 kN/s(Komurlu,2018).In this section,in order to study the effects of loading rates and pretension,a wide range of loading rates was achieved by setting up pretension stress σ0,loading time t and final tensile stress for three groups of simulations(Table 2):Group 1(G1)with pretension,Group 2 (G2) without pretension,and Group 3(G3)with different pretensions.The pretension stress of G1 was set as 867 MPa and the loading durations were 30 s,1 min,1 h,6 h,24 h,6 d,and 10 d.The loading durations of G2 were set the same as G1,but without pretension.The loading process of G1 and G2 are presented in Fig.8.The pretension stress list for G3 was 300 MPa,400 MPa,500 MPa,600 MPa,700 MPa,800 MPa,900 MPa with 6 d of loading time.The final tensile stress of all three groups was set as 919 MPa.

Table 1 The monitoring data of anchor rod axial force at K1780 section of Muzhailing highway tunnel and the corresponding mechanical boundary conditions (tensile stress,σt) at respective monitoring time.

For G1,the curves of stress P11versus deformation gradient F11along tensile direction are shown in Fig.9a.It shows the deformation behavior of bolt materials loaded with different loading times varies from each other after pre-tensioning.The extent of deformation-relaxation increases with increase of the loading time.The enlarged view in Fig.9b presents the maximum difference of deformation gradient F11is between 1 min and 1 h.The difference between 6 d and 10 d is nearly negligible.For G2,there are no deformation-relaxation plateau as shown in Fig.9c.However,there is a pronounced deformation-softening effect with increase of the loading time as shown in the enlarged view of Fig.9d.

Fig.4.The 3D patterns of (a) Deformation gradient F11,and (b) Stress P11 distribution of simulation results.

Fig.5.Numerical simulation results of (a) Stress P11 versus deformation gradient F11 curves with (b) Its enlarged view based on field monitoring data.

The comparison of the deformations between simulations with pretension and without pretension is shown in Fig.10a.During the pre-tensioning process,the stress of material with pretension is obviously higher than that without pretension.However,the two stress curves overlap eventually with each other during postloading stage.The deformation-relaxation plateau links the two curves.Similar variation trend is observed in G3,as shown in Fig.10b.

Fig.6.Numerical simulation results of (a) dislocation density ρd and (b) twin volume fraction ftw (%) versus deformation gradient F11 curves based on field monitoring data.

The curves of deformation gradient F11versus loading time for material with and without pretension are plotted in Fig.11.For G1(Fig.11a),F11changes in a proportional way with loading time,which is similar to the stress P11in Fig.8a.For G2 (Fig.11b),F11changes in a nonlinear way with loading time.For G2 of 6 d and 10 d,there are almost no deformation change during the initial 2 d and 4 d before the onset of plastic deformation.

4.Discussion

4.1.Effects of loading rate and pretension on deformation-softening behavior

The simulations are summarized in Table 2 and visualized in Fig.12,where variations of characteristic values,i.e.the final deformation gradient F11and the extent of deformation-relaxation ΔF11,are plotted with the applied loading rates.The observed effects (deformation-softening and deformation-relaxation behaviors) are analyzed as illustrated in Figs.13 and 14.

Table 2 The boundary conditions,loading rates and simulation responses for G1,G2,and G3.The final target stresses of all three groups were set as 919 MPa(equivalent to 343 kN for rock bolt).

As illustrated in Figs.9 and 12a,it shows that the bolt rod exhibits pronounced deformation-softening behaviors with decrease of the static loading rate.This matches with the experimental studies of Benzing et al.(2018)and Xu et al.(2019)that the tensile strength of high-strength and high-toughness steels generally decrease with decrease of the static loading rate.It is also reported that the tensile strength (Gong et al.,2019) and brittleness(Komurlu,2018) of rock materials were found to notably decrease as a result of decrease in loading rate.For rock bolt,most studies were conducted under dynamic load test that the yield stress and tensile strength of rock bolt steel decrease considerably with the decreasing loading rate (Ansell,2006; Wu et al.,2019).However,the studies of effect of static loading rate on rock bolt steel material are still limited.

As shown in Fig.13,the variation tendencies of dislocation density ρdand twin volume fraction ftwwith stress rate match that of final deformation gradient F11(see Fig.12a).It shows that the improvement of elongation is related to the increase of dislocation density ρd(1.45×1015-1.75×1015m-2)and twin volume fraction ftw(5.8-6.3 vol%),which is also revealed via nanoscale deformation experiment by Li et al.(2019).

As shown in Fig.12a,the final deformation gradient F11of G1 is lower than that of G2 under the same loading rate level.The deformation difference between G1 and G2 becomes larger with decrease of the loading rate.This signifies that the effect of high pretension can help bolt rod material resist the deformationsoftening behavior,especially at low loading rate (＜10-1MPa/s or 10-2kN/s).In addition,the curve of G3 links the curves of G1 and G2.For the G3 cases with low pretension levels (＜400 MPa or 149 kN) that is close to the material yield stress (398 MPa),the positions of F11are just located above the curve of G2.For G3 cases with intermediate pretension levels (500-600 MPa or 187-224 kN),the positions of F11locate between the curves of G1 and G2.For G3 cases with pretension stress higher than 600 MPa or 224 kN,the positions of F11just overlap on the curve of G1.The variation of the G3 curves indicates that the effect of pretension becomes significant with increase of the pretension stresses higher than the material yield stress.

4.2.Effects of loading rate and pretension on deformationrelaxation phenomenon

Our simulations (Figs.5,9 and 10) reveal that the loading rate and pretension have a pronounced effect on deformationrelaxation phenomenon of bolt rod material.As shown in Fig.12b,the extent of deformation-relaxation increases from 0 to 3% with decrease of the loading rate (100-10-5MPa/s or 10-1-10-5kN/s)and/or the increase of pretension levels(300-900 MPa or 112-336 kN).

As shown in Fig.7,the variations of the non-zero stress components (P12,P13,P21,P23,P31,P32) after pre-tensioning indicates that the stress state of material adjusts itself in a way to be favorable for the dramatic change of the stress rate.A similar phenomenon was observed by Hariharan et al.(2013) that there is a pronounced improvement of the uniform elongation by 0.1-3.5%in the interrupted tensile test under the relaxation time of 60 s.They suggested that the redistribution of dislocations allows further increase in dislocation density which could be the reason for the relaxation phenomenon.This is also revealed in our study as illustrated in Fig.14.The variations of dislocation density Δρd(0.05 × 1015-0.225 × 1015m-2) and twin volume fraction Δftw(0.1-0.5 vol%) show the similar variation patterns as that of the deformation-relaxation ΔF11in Fig.12b.The extent of deformationrelaxation follows the activity intensities of the underlying deformation mechanisms.

Fig.7.The variation curves of non-zero stress components of(a)Stress P12,(b)Stress P13,(c)Stress P21,(d)Stress P23,(e)Stress P31 and(f)Stress P32 with deformation gradient after pre-tensioning based on the monitoring data from the left arch shoulder of K1780 section.The red circle and black square markers represent the start and end points of deformation relaxation,respectively.

Furthermore,it is found that the effect of pretension has a larger effect on the deformation-relaxation phenomenon compared with that of loading rates.As shown in Fig.12b,besides the case with loading time of 30 s,the extent of deformation-relaxation ΔF11of G1 increases linearly with the decrease of the logarithm of stress rate.However,for G3 with different pretension levels,ΔF11decreases rapidly to zero as the pretension level drops to 300 MPa which is below the material yield stress(398 MPa),even though the magnitudes of loading rate are very slow (10-4-10-3MPa/s or 10-5-10-4kN/s).Therefore,the deformation-relaxation caused by pretensioning should also be taken carefully for the problems of debonding slip(Yu et al.,2019)and rock/soil creep(Gao et al.,2021)for pretensioned rock bolt.

Fig.8.Curves of stress P11 versus loading times for (a) G1,and (b) G2.

The effects of loading rate and pretension on deformationsoftening (final deformation gradient F11) and deformationrelaxation ΔF11were weighted based on analysis of variance(ANOVA) (Yang and Tarng,1998; Wasantha and Ranjith,2014; Li et al.,2018b,2020a),as tabulated in Tables 3 and 4.The ANOVA is an effective post-data processing method to quantify the contribution of each independent parameter on the dependent ones.The associated calculation equations for the sum of square,degree of freedom,mean square and F value are given in the works by Li et al.(2018a,2020a).The loading rate was found to be a significant factor affecting deformation softening with 59% weighting contribution(Table 3),while the pretension held 66% weighting contribution to the deformation-relaxation(Table 4).The results from ANOVA are in agreement with those observed in Figs.9 and 12.

Table 3 Summary of ANOVA on final deformation gradient F11.

Table 4 Summary of ANOVA on deformation-relaxation ΔF11.

Fig.9.Curves of stress P11 and deformation gradient F11 along tensile direction for (a) G1 and (b) The enlarged view of G1,(c) G2 and (d) The enlarged view of G2.

Fig.10.Curves of stress P11 versus deformation gradient F11 along tensile direction for(a)Material deformation loaded with pretension and without pretension,and(b)G3 loaded with different pretensions.

5.Conclusions

Fig.11.Curves of deformation gradient F11 versus loading times for (a) G1 and (b) G2.

Fig.12.Curves of(a)Final deformation gradient F11 and(b)Deformation-relaxation ΔF11 versus stress rates(and corresponding force rates)for results of G1,G2 and G3.The details of boundary conditions setup are described in Table 2.

Based on field monitoring data obtained from Muzhailing highway tunnel,crystal plasticity simulations incorporating dislocation glide and deformation twinning mechanisms were conducted to study the effects of rock loading rate and pretension on deformation behavior of high-strength and high-toughness TWIP steel used for rock bolt rod.The main conclusions are as follows:

(1) The bolt rod exhibits a pronounced deformation-softening behavior with decrease of the static loading rate.The high pretension level (＞600 MPa or 224 kN) can help bolt rod steel resist the deformation-softening behavior,especially at low loading rate(＜10-1MPa/s or 10-2kN/s).

(2) There is a sound deformation-relaxation behavior induced by dramatic decrease of loading rate after pretensioning.The extent of deformation-relaxation increases with increase of the pretension levels (larger than material yield stress) and the decrease of the loading rate.

(3) The loading rate was found to be relatively more significant factor affecting deformation-softening behavior while the pretension was found to be the most influence parameter accounting for the deformation-relaxation behavior.Nevertheless,studies of experimental and field tests involving manufacture and application of highperformance and high-toughness steels for rock bolt are needed.

Fig.13.Curves of(a)Dislocation density ρd,and(b)Twin volume fraction ftw (%)versus stress rates(and corresponding force rates)for results of final deformation states from G1,G2 and G3.The details of boundary conditions setup are described in Table 2.

Fig.14.Curves of (a) Dislocation relaxation Δρd and (b) Twin relaxation Δftw (%) versus stress rates (and corresponding force rates) for results of G1,G2 and G3.The details of boundary conditions setup are described in Table 2.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No.41941018),the Science and Technology Major Project of Gansu Province (Grant No.19ZD2GA005),and the Research Institute for Deep Underground Science and Engineering Foundation (Grant No.XD2021023).We thank two anonymous reviewers for their constructive comments and suggestions.