Coupled thermo-mechanical constitutive damage model for sandstone

Savani Vidana Pathiranagei,Ivan Gratchev

School of Engineering and Built Environment,Gold Coast Campus,Griffith University,Southport,QLD 4222,Australia

Keywords:High temperature Confining pressure Thermo-mechanical (TM) damage Thermal damage Mechanical damage

ABSTRACT Underground rock dynamic disasters are becoming more severe due to the increasing depth of human operations underground.Underground temperature and pressure conditions contribute significantly to these disasters.Therefore,it is important to understand the coupled thermo-mechanical(TM)behaviour of rocks for the long-term safety and maintenance of underground tunnelling and mining.Moreover,investigation of the damage,strength and failure characteristics of rocks under triaxial stress conditions is important to avoid underground rock disasters.In this study,based on Weibull distribution and Lemaitre’s strain equivalent principle,a statistical coupled TM constitutive model for sandstone was established under high temperature and pressure conditions.The triaxial test results of sandstone under different temperature and pressure conditions were used to validate the model.The proposed model was in good agreement with the experimental results up to 600 °C.The total TM damage was decreased with increasing temperature,while it was increased with increasing confining pressure.The model’s parameters can be calculated using conventional laboratory test results.

1.Introduction

Rock burst and slabbing have recently become more dangerous due to the increasing depth and intensity of deep underground applications (Li et al.,2018).These disasters can threaten both the workers′safety and the underground operations’ efficiency.The main reasons for such disasters are the rock failure that occurs due to high underground temperature and pressure conditions(Xu and Karakus,2018).Therefore,research into the effect of high temperature and pressure on the engineering properties of rocks is significant for the long-term safety and maintenance of underground applications (Yin et al.,2018).

In recent decades,several authors have been studying the physical and mechanical properties of rocks under high temperature or pressure conditions (Chen et al.,2012; Wasantha et al.,2015; Yin et al.,2015,2016; Zhang et al.,2016; Yang et al.,2017;Gratchev et al.,2019).A good deal of research has been conducted to predict and model thermo-mechanical(TM)damage for different rocks such as granite (Hu et al.,2018; Xu and Karakus,2018; Zhu et al.,2021),claystone (Takarli et al.,2008; Morales-Monsalve et al.,2018),marble (Jiang et al.,2018; Zhu et al.,2021),sandstone(Hassanzadegan et al.,2014;Zhu et al.,2021),shale(Karrech et al.,2014) and limestone (Rong et al.,2018).As a result,a few thermal damage models have been introduced to estimate the changes of elastic modulus,unconfined compressive strength,and porosity with increasing temperatures(Su et al.,2011;Poulet et al.,2012; Tian,2013; Ersoy et al.,2017; Karrech et al.,2017; Shi et al.,2021).Xiao et al.(2020) conducted their research about the microscopic fracture morphology and damage constitutive model of red sandstone under seepage pressure.In addition to that,a damaged visco-plasticity model for pressure and temperature sensitive geomaterials was proposed by Karrech et al.(2011).However,only limited investigation into the coupling effect of temperature and pressure with the focus on heated rock properties under three-dimensional (3D) stress conditions has been conducted to date(Xu and Karakus,2018).

This study seeks to investigate the engineering properties of sandstone,which is a common sedimentary rock representing main components of the roof and floor strata of coal seams(Brook et al.,2020).It is also a dominant rock type(Douglas et al.,2009)in Southeast Queensland,Australia,where this study takes place.Only limited research on the coupled TM damage model for sandstone(Hassanzadegan et al.,2014; Yin et al.,2018; Zhu et al.,2021) has been conducted so far.Hassanzadegan et al.(2014)investigated the porosity of sandstone heated to a temperature of 140°C only,while Zhu et al.(2021) validated the proposed thermal damage constitutive model for sandstone using the published data thus ignoring the effect of local geology and lithological characteristics of rock.It is evident that there is need for high-quality data on the properties of heated sandstone to better understand the effect of higher temperatures on rock strength under triaxial stress conditions (Xu and Karakus,2018).In addition,a recent study by Pathiranagei et al.(2021) and Pathiranagei and Gratchev (2021)indicates that for sandstone,the temperature threshold affecting rock properties can vary significantly depending on mineral composition.

This study aims to bridge the research gap and provide highquality data on the engineering properties of sandstone heated to a higher temperature range (from 25°C to 800°C).A series of traixial tests was conducted to estimate the effect of confining pressure on rock strength and thus to clarify the rock behavior under the application of 3D stresses.Finally,the TM damage model using Weibull distribution and Mohr-Coulomb criterion was modified to incorporate a wider temperature range,and it was validated against the reported laboratory tests.This paper presents and discusses the obtained results.

2.Theoretical background

In this study,the Weibull damage method has been implemented to analyse the mechanical characteristics of sandstone.The thermal damage factor has been used to identify the influence of temperature on rock mechanical properties.The generalised damage variable has been utilised to define the total damage due to thermal stress and load.Then,Mohr-Coulomb failure criterion has been used to derive the coupled TM damage model of sandstone.

2.1.Thermal damage (DT)

The thermal damage factor was first introduced by Dougill et al.(1976) and represents the influence of temperatures on the rock mechanical properties.The thermal damage factor can be defined using the elastic modulus based on linear damage mechanics as follows (Wang et al.,2007):

where Eois the elastic modulus of rock at room temperature,and ETis the elastic modulus of rock at the temperature T.

2.2.Mechanical damage (DM)

The main factors for rock damage can be identified as the rock’s internal defects,voids and microcracks (Bandara et al.,2018;Kumari et al.,2018;Yin et al.,2018).In rock materials,the Weibull damage method is generally used,which can be stated as

where ω(F)stands for the distribution function of micro-intensity,F is an elemental strength parameter,Fois the mean value of the elemental strength parameter,and m is the shape parameter of material(Wang et al.,2007).The elemental strength parameter can be explained as follows.The occurrence of microcracks in the rock sample leads to variation in the strength and shape of each element.Therefore,the failure of different elements in the rock under loading is irregular.Generally,the strength of microscopic element is supposed to follow the Weibull distribution function.In Weibull distribution function,the parameter that represents the elemental strength is named as‘elemental strength parameter’(Xie et al.,2020).

Since the damage of rock is a gradual accumulation of failures and is continuously happening,the relationship between ω(F)and DMcan be defined as follows:

where DMdenotes the Lemaitre’s damage variable or mechanical damage.

To describe the mechanical damage,the concept of effective stress (σ*ij) is used.Effective stress is the stress acting on the effective area of the damaged material.

The component of nominal stress (σ11,σ22,and σ33(σ22=σ33))can be obtained directly from a conventional triaxial test.When the effective stress components denote σ*11,σ*22and σ*33(σ*22=σ*33),the relationship between effective stress and apparent stress can be expressed using Lemaitre’s strain equivalent principle as follows(Lemaitre,1992):

where σ*denotes the effective stress,and σ is the normal stress.The strain can be obtained by combining Hooke’s law for linear elasticity as follows:

where Ceijklis the material compliance matrix.

Then we have

where Deijkldenotes the elastic stiffness matrix.Substituting Eq.(3)into Eq.(6),the stress-strain relationship is expressed as follows:

Two Weibull distribution parameters (Foand m) in Eq.(7) are not determined.

In the stress-strain curve,there is a peak value for shear stress.The peak point is named as (εpeak,σpeak).Eq.(8) has been used to explain it.The above mentioned peak point can be used to find out the model parameter m and Fo.At the peak point(εpeak,σpeak),the derivative of the shear stress with respect to the shear displacement should be equal to zero and that is shown in Eq.(9).This method is mentioned as the ‘extremum method’ (Shi et al.,2021).

The following geometrical conditions are satisfied by the peak stress σpeakand the corresponding strain εpeak:

Substituting Eqs.(8) and (9) into Eq.(7),m can be obtained as

The relationship between Foand m is expressed as

2.3.Coupled TM damage model

Concerning the thermal damage,Eq.(1) can be incorporated with elastic stiffness tensor as given in the following equation:

Substituting Eq.(12) into Eq.(7),the coupled TM damage constitutive equation can be given as

Rocks indicate various damage characteristics due to thermal stress and load.The total damage(D)of the rock loaded under high temperature can be evaluated by the generalised damage variables as follows (Yin et al.,2018):

Substituting Eqs.(3) and (1) into Eq.(12),the total damage (D)can be written as

Eq.(15) can be used to calculate the coupled TM damage using the various failure criteria.

2.4.The Mohr-Coulomb criterion in the damage model

The Mohr-Coulomb criterion can be stated in the principal stress space as follows (Xu and Karakus,2018):

where I′1defines the first effective stress invariant,J′2defines the second effective deviator stress invariant,θσis the Lode angle,φ is the internal friction angle,c is the cohesion of the rock,and s*ijis the effective stress deviator tensor.

Substituting Eq.(16) into Eq.(3),then DMcan be expressed as

Substituting Eq.(16) into Eq.(15),the total TM damage(D) can be stated as

The equation of the damage evolution rate of the rock can be expressed by deriving Eq.(19) as follows:

However,in the triaxial test,σ1＞σ2=σ3,Eqs.(19)and(20)can be simplified as

with

where v denotes the Poisson′s ratio.

Using the above formulas and test results,the different damage characteristics of the rocks can be determined with different pressure and temperature conditions.

3.Experimental programme

3.1.Rock used

Sandstone from Gold Coast,Australia was used for this study.Xray diffraction analysis indicated that the sandstone was mainly composed of 45% quartz,39% feldspar,10% kaolinite and 6% illite.The X-ray diffraction spectroscopy of sandstone is shown in Fig.1.Twenty-four cubic (50 mm3) sandstone samples were prepared to perform the triaxial test and sample surfaces were polished.

3.2.Testing programme

The samples were first placed in a furnace at room temperature and then heated with a 5°C/min increase until a pre-determined temperature (400°C,600°C and 800°C) was reached.This slow heating rate was used to prevent possible thermal shock.Once the pre-determined temperature was reached,the sample was kept in the furnace for another 2 h to allow the temperature to be distributed uniformly(Glover et al.,1995;Chaki et al.,2008;Ersoy et al.,2017; Xiong et al.,2018).On switching off the heat,the samples remained in the furnance until room temperature was reached.

Triaxial tests were performed using a true triaxial testing system located at the Geotechnical Engineering Centre within the School of Civil Engineering at the University of Queensland,Australia.At each temperature point (25°C,400°C,600°C and 800°C),six samples were used.The triaxial tests were conducted under different confining pressures of 3 MPa,6 MPa and 9 MPa on the preheated samples.A loading rate of 0.2 MPa/s was applied for all testing.

Fig.1.The X-ray diffraction spectroscopy of sandstone.

4.Experimental results and analysis

In this section,the stress-strain relations,elastic modulus,peak stress,macroscopic failure mode,and failure criteria of sandstone with different temperatures and confining pressures are examined.These results are used to build up the coupled TM damage model.

4.1.Stress-strain relations

The stress-strain behaviour of sandstone at three different pressure conditions and four different temperatures is similar(Fig.2).The peak strength of the sandstone increases with increasing the confining pressure and temperature.However,the axial and lateral strains of the sandstone have a different behaviour.This indicates that increasing the confining pressure decreases the ability of the rock to resist deformation in the axial and lateral directions.With increasing temperature and confining pressure,the ability of the sandstone to fracture decreases.The stress-strain curves of sandstone illustrate remarkable ductile features at three different pressure conditions and four different temperatures.The summary of test data is shown in Table 1.

Table 1 Summary of laboratory test data.

4.2.Strength characteristics

Fig.2.Stress-strain curves of sandstone at different temperatures and confining pressures: (a) 25 °C,(b) 400 °C,(c) 600 °C,and (d) 800 °C.

The elastic modulus of sandstone was measured in this study using an approximately straight line on the stress-strain curve of the rock at different temperatures and confining pressures.As can be seen from Figs.3 and 4,the changes of elastic modulus of sandstone with temperature under the three different confining pressures are almost identical to each other.At the confining pressure of 3 MPa,the average elastic modulus increases up to 400°C and decreases with increasing temperature.However,the average elastic modulus at 6 MPa decreases up to 400°C and then increases up to 600°C and again decreases with temperature.The average elastic modulus at 9 MPa slightly changes with temperature.Generally,the elastic modulus becomes greater with increasing confining pressure.

Ranjith et al.(2012)observed the increase in elastic modulus at 400°C for sandstone.Zhang et al.(2009)also experienced the same behaviour for sandstone below 600°C.The reason behind the initial increase of the elastic modulus has been explained by Zhang et al.(2014) as follows.The evaporation of water inside the rock structure at a temperature up to 400°C causes increase in porosity,thus contributing to a rise in the density after reaching a state of stress in the elastic phase.This causes a well resistance of rock to deformation,and therefore increase in elastic modulus up to 400°C.

Fig.3.Average elastic modulus versus temperature under different confining pressures.

Fig.4.Average elastic modulus versus confining pressure under different temperatures.

The relationship between average peak stress versus temperature at different confining pressures and the average peak stress versus confining pressure at different temperatures are shown in Figs.5 and 6,respectively.It can be identified that the peak stress rises with increasing temperatures at 3 MPa,6 MPa and 9 MPa confining pressures.Moreover,the peak stress grows with increasing confining pressures at 25°C,400°C,600°C and 800°C.This peak stress increase is typically due to the crack healing caused by the extension of heated minerals when rock samples are subjected to higher temperatures(600°C and 800°C)(Sirdesai,2017).As explained by Wu et al.(2019),the strength increment of sandstone at higher temperatures(＞600°C)is mainly due to the higher porosity and weaker heterogeneity of mineral thermal expansion.Increasing temperature can enhance the rock strength by generating the plastic expansion of minerals and enhancing the frictions between the minerals (Ferrero and Marini,2001; Ranjith et al.,2012).Moreover,the air cooling treatment used in this study also helps to keep the strength higher.As mentioned by Wu et al.(2019),the strength weakening in the air cooling stage is not enough to compensate the strength enhancement created in the heating stage.Zhang et al.(2009) and Wei et al.(2019) also experienced strength improvement of sandstone with temperature.However,Zhang et al.(2009) explained that the reason behind the strength increment with temperature is evaporation of water and discharge of gases contained in the inner structure.

Fig.5.Average peak stress versus temperature under different confining pressures.

The frictional characteristics of sandstone were increased with greater confining pressures;thus,the cracks were closed under the confining pressures and this increased the rock stress.According to Hajpál and T?r?k(2004)and Zhang et al.(2019),thermally-induced cracks in sandstone are typically related to the decomposition of calcite and/or shrinkage of clay minerals.However,the sandstone in the current study has a low percentage of clay minerals and no calcite.Therefore,this sandstone showed less thermal damage.This can be further explained as follows.The mineral thermal expansion coefficient has a huge influence on the thermal loading process as explained by Wu et al.(2019).If the rock contains minerals,which have larger discrepancy in the thermal expansion coefficient,then it causes severe thermal damage.The quartz has a higher thermal expansion coefficient when compared to other minerals in the rock.Therefore,quartz content in the rock sample has a significant impact on the heterogeneity of the thermal expansion coefficient.Other than the thermal expansion coefficient,the pore size distribution of the rock sample also has a great influence on the thermal damage.Sandstone has a large pore space,which allows accommodating the thermal deformation in the thermal loading process(Wu et al.,2019).Moreover,the mineral dissolution process of sandstone (feldspar dissolves into cement materials) can increase the porosity and decrease the mineral heterogeneity,which ultimately enhance the thermal resistivity of sandstone even after 600°C.

Fig.6.Average peak stress versus confining pressure under different temperatures.

4.3.Failure mode

The typical failure modes after triaxial testing are shown in Figs.7-10.Figs.7-10 illustrate images of the failure modes at 3 MPa,6 MPa and 9 MPa confining pressures and 25°C,400°C,600°C and 800°C temperatures.It is understandable that the high temperature and pressure have a small impact on the failure behaviour of sandstone.Ductile flow is recognised in the deformation of the sandstone samples.Ductile flow is the dominant failure mode of these rock samples and noticeable lateral bulging is also evident.With increasing both temperature and confining pressure,the number of macro-cracks inside the samples gradually increases.During this period of rising temperature and confining pressure,the samples failed due to their decreasing carrying capacity.

4.4.Cohesion and friction angle

The strength of sandstone changed with higher temperature.Therefore,the failure criterion calculated by these tests also needs to change with temperature.In this study,the linear Mohr-Coulomb criterion is used based on the test data.The calculated cohesion and friction angle of sandstone after thermal treatment are summarised in Table 2.It can be identified that the R2values are close to 1,therefore,linear regression is acceptable and reasonable.This means that the thermally treated sandstone agrees with the Mohr-Coulomb criterion.

Table 2 Total cohesion and friction angle after heat treatment.

Fig.7.Failure behaviour of sandstone under temperature of 25 °C and confining pressures of (a) 3 MPa,(b) 6 MPa,and (c) 9 MPa.

Fig.8.Failure behaviour of sandstone under temperature of 400 °C and confining pressures of (a) 3 MPa,(b) 6 MPa,and (c) 9 MPa.

Fig.11 indicates the slight rise in friction angle with increasing temperature,while under the same condition,cohesion remains almost the same.The internal friction angle signified the frictional properties of sandstone.The increase in friction angle with temperature is due to the different thermal expansion coefficients of mineral particles.This creates uneven deformation in the rock sample and increases the mutual friction between the rock particles (Meng et al.,2020).

Fig.9.Failure behaviour of sandstone under temperature of 600 °C and confining pressures of (a) 3 MPa,(b) 6 MPa,and (c) 9 MPa.

Fig.10.Failure behaviour of sandstone under temperature of 800 °C and confining pressures of (a) 3 MPa,(b) 6 MPa,and (c) 9 MPa.

The cohesion value indicates the degree of cementation,the cementing composition and the particle size of the mineral particles(Meng et al.,2020).Even though the cohesion does not change significantly with the temperature,the friction angle has a positive relationship with temperature,which denotes that the peak stress of the sandstone is determined by the friction angle.

Fig.11.The changes in internal friction angle and cohesion with increasing temperature.

5.The analysis of damage characteristics

5.1.Damage evolution characteristics

The damage parameters of sandstone at different heating temperatures in triaxial tests are given in Table 3.Fig.12 shows the curves of stress-strain (σ-ε),thermal damage (DT),mechanical damage (DM) and total TM damage at 400°C.The stress-strain response of sandstone can be divided into three stages,as given in Fig.12.At first,the rock sample is in a linear elastic stage and a stage of compaction in which the total coupled TM damage curve is horizontal due to the temperature.The slight increase of stress in Stage 2 (strain-hardening stage) will cause the generation and propagation of new microcracks and the rock approaches a critical TM damage state where macro-cracks begin.During Stage 3,the failure of the rock is occurring.The total damage to the rock mainly depends on loading during this stage.There is no noticeable peak strain or residual strength in the sample.A brittle ductile failure transition phase is evident in the upwardly convex critical phase of the stress-strain curve.

Table 3 The damage parameters at different heating temperatures in triaxial tests.

The total coupled TM damage curves (D) at different temperatures using Eq.(21) are shown in Fig.13a.This indicates that the total coupled TM damage of sandstone follows the same trend at different temperatures.The total damage curve is horizontal at the early stage of deformation,and it is equal to the thermal damage during this stage.After that,the curve gradually increases with the increase of axial strain until the critical damage state.Using Eq.(22),the total damage evolution rate (dD/dε) at different temperatures is illustrated in Fig.13b.Until the temperature reaches 400°C,the damage accumulation increases and afterwards damage accumulation reduces with increasing temperature (Fig.14).This indicates that with the strain increment,the temperature can influence the ductility of the sandstone.

The different damage variables at various temperatures are shown in Fig.15.Polynomial growth of thermal damage occurs with rising temperature.From 25°C to 400°C,thermal damage slightly decreases and after that,the rise with increasing temperature may be due to the phase transition of quartz mineral.The mechanical damage curve linearly decreases with increasing temperature.However,the total TM damage curve shows a logistic decline with temperature,which corresponds to the peak strain.Hence,we can use critical damage (Dcr) as a parameter to characterise rock failure behaviour.

Fig.12.Stress-strain and damage curves at 400 °C.

Fig.13.(a) Coupled TM damage and (b) damage evolution rate at various temperatures.

Fig.14.The maximum value of dD/d? at various temperatures.

Fig.15.Different damage variables with temperature.

The various parameters of sandstone samples at different confining pressures in triaxial tests after 400°C are given in Table 4.The total coupled TM damage (D) is calculated using Eq.(21) and the total damage evolution rate(dD/dε)is calculated using Eq.(22)at three different confining pressures after 400°C,and these are depicted in Fig.16.The total damage grows when the confining pressure increases under the same strain level,which shows that the rising confining pressure weakens the stress state and speeds up the development of damage (Fig.16a).The damage evolution rate versus strain curves at different confining pressures show thesame type of behaviour as that of temperature (Fig.16b).The increase of confining pressure improves the strain recoverability and reduces the rock ductility.

Table 4 Parameters at different confining pressures in triaxial tests after 400 °C.

Fig.16.(a) Coupled TM damage and (b) damage evolution rate at various confining pressures.

The damage variables at different confining pressures are illustrated in Fig.17.Thermal damage decreases linearly with pressure.The mechanical damage curve logistically increases with greater confining pressure and the total TM damage curve follows the same trend.

Fig.17.Different damage variables with confining pressures.

The coupled TM damage value of sandstone decreases with increasing temperature and is consistent with the behaviour of average peak stress.The peak stress increases as the confining pressure rises.However,the coupled TM damage value also increases with greater confining pressure.As hinted above,the main reason for these variations is sandstone’s mineral composition.Sandstone is composed of different minerals such as quartz,feldspar and clay minerals.There is a huge variation in the thermal expansion coefficients of quartz and feldspar minerals.As an example,the thermal expansion of quartz is three times higher than that of feldspar(Wang et al.,2020;Pathiranagei and Gratchev,2021).Another reason for the increase in peak stress of sandstone with temperature is the ceramic like behaviour of clay minerals(Pathiranagei and Gratchev,2021).Moreover,the deformation characteristics during triaxial compression loading of heated sandstone induce pore closure and frictional slippage between the minerals.This causes an increase in peak stress.

5.2.Model parameters

Figs.18 and 19 show the behaviour of the two distribution parameters (m and Fo) in the damage model with different temperatures and confining pressures.The exponential growth of parameter m with the increasing temperature can be seen,while the polynomial trend of m is evident with confining pressure(Fig.18a and b)).

Foindicates linear growth with the increase of temperature(Fig.19a).It first shows a decreasing trend until 6 MPa and then an increasing trend with the rising confining pressure (Fig.19b)).The behaviour of Foand m with temperature is in accordance with the change of peak stress with temperature.Hence,Foand m can be used to signify the peak strength of the sandstone at different temperatures.

6.Validation of the model

To test the applicability and accuracy of the TM damage constitutive model for the Mohr-Coulomb criterion,the model was tested against standard test cases.The four stress-strain curves measured under temperatures of 25°C,400°C,600°C and 800°C and confining pressure of 3 MPa are used to validate the model using Eq.(13).As can be identified from Fig.20,the theoretical damage model is generally consistent with laboratory test curves for temperatures less than 600°C.This model shows linear elastic behaviour before reaching a yield point and then exhibits elasticplastic behaviour.However,there are some variations between the theoretical and experimental curves.

Test curves do not indicate the rock’s compaction stage while the theoretical curve reflects the rock’s initial compaction stage,resulting in the value of theoretical stress being lower than the test stress at the same strain before yield stress.This gap increases with the rise in temperature.Moreover,during the plastic deformation,the theoretical curve shows elastic-plastic behaviour while the test curve shows strain-hardening behaviour,which indicates that the theoretical curve represents a mechanical response similar to that of ideal plastic material during the post-yield phase.The current model is not applicable to high temperatures such as 600°C and 800°C due to the significant discrepancy between the test and theoretical curves.This discrepancy may be due to the phase transition of quartz mineral at around 573°C.

To expand its applicability,the model can be used with plasticity theory and the residual strength can be used when establishing the damage model.

7.Conclusions

Triaxial tests under four different temperatures and three different confining pressures were conducted on sandstone.The coupled TM damage model is in good agreement with the experimental results(triaxial test)at temperatures less than 600°C.This predicts the deformation process and identifies the failure mechanism of deep rocks.Conclusions can be drawn from the findings as follows:

Fig.18.Model parameter m with (a) temperature and (b) confining pressure.

Fig.19.Model parameter Fo with (a) temperature and (b) confining pressure.

Fig.20.Theoretical and test curves at 3 MPa confining pressure and various temperatures: (a) 25 °C,(b) 400 °C,(c) 600 °C,and (d) 800 °C.

(1) According to the triaxial test data,the ability of sandstone to fracture decreases with increasing temperature.The peak stress rose with greater heating temperature and confining pressures.A high temperature has a negative impact on the average elastic modulus while confining pressure has a positive impact.Through regression analysis of the triaxial test results,the Mohr-Coulomb criterion was found to be acceptable and reasonable to explain the geomechanical behaviour of the tested sandstone after thermal treatment.Moreover,the material cohesion does not change with the temperature while the friction angle becomes greater with rising temperature.

(2) The critical thermal damage curve indicates polynomial growth with the increase of temperature.However,because of a gradual decline of the critical mechanical damage curve,the total critical TM damage is reduced with increasing temperature.Due to the high temperature treatment,different minerals in the sandstone sample tend to expand under different thermal expansion coefficients,causing the decrease in total damage.

(3) The total TM damage increases as the confining pressure rises,showing that the confining pressure diminishes the rock resistance and plastic deformation.

(4) To enhance the applicability and accuracy of the current TM damage model,the model can be used in conjunction with plasticity models.

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.

Acknowledgments

The authors would like to acknowledge the Bowen Basin Underground Geotechnical Society for funding this project.