An approximate nonlinear modified Mohr-Coulomb shear strength criterion with critical state for intact rocks

Baotang Shen,Jingyu Shi,Nik Barton

aState Key Laboratory of Mining Disaster Prevention and Control,Shandong University of Science and Technology,Qingdao,China

bCSIRO Energy,QCAT,1 Technology Court,Pullenvale,QLD,4069,Australia

cNick Barton&Associates,Oslo,Norway

Keywords:Shear strength Modified Mohr-Coulomb criterion Critical state Intact rock

ABSTRACT In this paper,the Mohr-Coulomb shear strength criterion is modified by mobilising the cohesion and internal friction angle with normal stress,in order to capture the nonlinearity and critical state concept for intact rocks reported in the literature.The mathematical expression for the strength is the same as the classical form,but the terms of cohesion and internal friction angle depend on the normal stress now,leading to a nonlinear relationship between the strength and normal stress.It covers both the tension and compression regions with different expressions for cohesion and internal friction angle.The strengths from the two regions join continuously at the transition of zero normal stress.The part in the compression region approximately satisfies the conditions of critical state,where the maximum shear strength is reached.Due to the nonlinearity,the classical simple relationship between the parameters of cohesion,internal friction angle and uniaxial compressive strength from the linear Mohr-Coulomb criterion does not hold anymore.The equation for determining one of the three parameters in terms of the other two is supplied.This equation is nonlinear and thus a nonlinear equation solver is needed.For simplicity,the classical linear relationship is used as a local approximation.The approximate modified Mohr-Coulomb criterion has been implemented in a fracture mechanics based numerical code FRACOD,and an example case of deep tunnel failure is presented to demonstrate the difference between the original and modified Mohr-Coulomb criteria.It is shown that the nonlinear modified Mohr-Coulomb criterion predicts somewhat deeper and more intensive fracturing regions in the surrounding rock mass than the original linear Mohr-Coulomb criterion.A more comprehensive piecewise nonlinear shear strength criterion is also included in Appendix B for those readers who are interested.It covers the tensile,compressive,brittle-ductile behaviour transition and the critical state,and gives smooth transitions.

1.Introduction

The traditional Mohr-Coulomb shear strength criterion considers the strength to be linearly dependingon the normal stresson the shear plane.It has been widely reported that the shear strength of many rocks actually follows a nonlinear relationship with the normal compressive stress,especially at extremely high confining pressure(e.g.Barton,1976),and even at relatively low confining pressure(e.g.Mogi,1974),if the rock is weak.The shear strength envelopes in the τ-σnplane are concave towards the normal compressive stress axis,whereτis the shear strength andσnis the normal stress on the shear plane.Many nonlinear shear strength criteria exist in the literature,including the Barton criterion(Barton,1976,2013)and Hoek-Brown criterion(Hoek and Brown,1980a,b,Hoek and Brown 1988).Barton(2013)summarised the nonlinear shear strengths for intact rocks,fractured rocks,jointed rocks and rock fills.

Mogi(1966)compiled a large body of triaxial experimental data for rocks from a variety of sources.Fig.1(from Barton(1976))reproduces Mogi’s test data for dry carbonate rocks and shows the variation of shear strength with the confining pressure.It can be seen that for most rock samples,the increase in the shear strength reduces and becomes negligibly small beyond a certain confining pressure.Mogi(1966)’s results also showed transition of behaviour from brittle to ductile.

Fig.1.Triaxial data for dry carbonate rocks compiled by Mogi(1966)in his Fig.3.All tests were performed at room temperature.Brittle,brittle-ductile transitional and ductile behaviours are indicated by closed,half-open and open symbols,respectively.The numbers represent the type of rocks given in Mogi(1966).The two solid lines representσ1=3σ3,the proposed critical state line of Barton(1976),and σ1=2σ3.

Based on this and many other experiments in the literature,Barton(1976)proposed a critical state concept for rocks at which the tangent of the shear strength envelope approaches horizontal in the τ-σnplane(see Fig.2,from Barton(2006,2013)).Fig.2 illustrates a shear strength envelope for intact rocks(solid curve)with the Mohr circles of stress states at four strength limit stages:uniaxial tensile strength(UTS),uniaxial compressive strength(UCS),brittle-ductile transition and critical state as well as the critical state line.The critical state line is also shown in Fig.1.The dashed curve(denoted by J)in Fig.2 represents the shear strength envelope for fractured rocks.At high confining pressure,for a given rock type,the two strength envelopes coincide,indicating that after the intact rock is fractured(reaching the ultimate strength),the stress will not drop and thus the rock mass behaves as a ductile material.A complete set of Mohr circles for high-pressure tests on strong limestones(with UCS of 250 MPa)derived from Byerlee(1968)data is shown in Fig.3,reproduced from Barton(1976).From this,it can be seen that at the critical state,the major principal stress(σ1)is about three times the minor one(σ3).Furthermore,the minor principal stress(or critical confining pressure)is close to the UCS,and the value of peak shear strength is half of the normal compressive stress.These represent the quantitative aspects of the critical state concept for rock shear strength.

Fig.2.Brittle-ductile transition and critical state of intact rocks,with Mohr circles for four strength limit stages,from Barton(1976,2013).The critical state is atσ1=3σ3with critical state internal friction angle φc=26.6°.Curve “J”represents the shear strength of the rock when fractured,also for a given rock type.τmaxis the maximum shear strength at the critical state;σtand σcare the UTS and UCS,respectively.

Fig.3.Mohr circles for high-pressure tests on strong limestones.Data are derived from Byerlee(1968).The figure is reproduced from Barton(1976).The small numbers(1,3,6,10,12 and 15)refer to specific Byerlee triaxial test numbers.The critical state line was added by Barton.The conjugate-fracture angle 2β progresses towards 90°with increase ofσ.The small open circles nearer the axis represent the peak strength of fractures in the same limestone,as measured by Byerlee.The dotted curved envelope,extended into the inadmissible area for reasons of clarity,was used to demonstrate the maximum empirical strength of(rough)rock fractures in the same limestone.

Singh et al.(2011)incorporated the critical state concept of Barton into a modified Mohr-Coulomb triaxial strength criterion for intact rocks,which was of quadratic form.Their analysis of data from a huge body of reported triaxial tests showed that the critical confining pressure is approximately equal to the UCS.This agrees with that shown in Fig.3.Their data came from 158 sets of triaxial tests with over 1000 individual tests.With such a large set of data,this value for critical confining pressure can be treated as generally applicable,and the result is also verified in Ma et al.(2014)’s tests on rock salt.From the observations of the data of Mogi(1966)and Byerlee(1968)and the Mohr circles shown in Fig.3,Barton(1976)provided another critical state conclusion that the peak shear strength at critical state corresponded to one half of the normal stress.This indicates that the peak shear strength is numerically approximatelyequal to the UCS.Singh et al.(2011)also investigated the effect of the intermediate principal stress on strength,using polyaxial test data.They found that the critical state condition was satisfied by both the minimum and intermediate principal stresses.Singh and Singh(2012)later employed the same form of strength criterion for jointed rock masses.

Barton(1976)employed the classical form ofτ= σntanφfor shear strength for both intact rocks and joints by embedding different nonlinear stress-dependences into the internal friction angle(φ).There is no specific cohesion term.We propose in this paper to employ a piecewise nonlinear expression for the shear strength envelope of intact rocks,which will cover the tension,compression and critical state regimes.The mathematical expression for the strength is the same as the classical form,both in the compression and tension regions.But the expressions for the cohesion and internal friction angle are functions of the normal stress.Although the terms in the tension and compression regions are different,the strengths in the two regions join continuously at the transition of zero normal stress.The part in the compression region approximatelysatisfies conditions of critical state.Due tothe nonlinearity,the classical simple relationship between the parameters cohesion,internal friction angle and UCS from the linear Mohr-Coulomb criterion does not hold anymore.The equation for determining one of the three parameters in terms of the other two is supplied.For simplicity,the classical form of relationship is taken as an approximation.A complicated piecewise nonlinear shear strength criterion is also included in Appendix B.It covers tension,compressive brittle-ductile behaviour transition and critical state.Although it is piecewise joined,the linking is smooth at the transitions.Comparisons of the approximate modified Mohr-Coulomb shear strength criterion,the piecewise nonlinear criterion and the Hoek-Brown criterion are presented in Appendix B.Both the approximate modified Mohr-Coulomb criterion and the piecewise nonlinear criterion show lower shear strength than the Hoek-Brown criterion at high confining pressure,in other words greater nonlinearity.However,the new piecewise nonlinear criterion as developed in Appendix B is more complicated and needs four strength parameters.

The modified Mohr-Coulomb criterion has been implemented in the fracture mechanics based numerical code FRACOD,and an example case of tunnel failure has been presented in this paper to demonstrate the difference between the original Mohr-Coulomb criterion and the new suggestion for a modified nonlinear Mohr-Coulomb criterion.

2.An approximate modified Mohr-Coulomb shear strength criterion with critical state

As a starting point,we employ a criterion for shear strengthτ with the same form as the classical Mohr-Coulomb criterion:

where c is the cohesion.However,we consider that bothφand c are functions of the normal stressσn.This means that there is a nonlinear relationship between the shear strength magnitude and the normal(confining)stress.We consider the shear strength in both the normal compressive and tensile stress regimes,as follows.

(1)Compression region(σn＞ 0)

When the normal stress is in the compression state,we assume the following functions:

(2)Tension region(σn＜ 0)

We propose the functions below for the tension region:

The proposed shear strength envelope is simple and the terms have well-known meanings.We have introduced three parameters,UCS,σc,apparent internal friction angle,φ0and cohesion,c0,for the compressionregion andoneextraparameterσtinthetensionregion.Atσn=-σt,the shear strength τ=0,thusσtis clearly the UTS.

It can be seen that ifσn=0,then c=c0,φ= φ0andτ=c0from both the compression and tension sides.Hence the two parts are continuously joined atσn=0 and c0is the cohesion atσn=0.The parameterφ0could be taken as the internal friction angle atσn=0.However,it is noted that the real angle of the tangent slope of the strength envelope atσn=0 from the compression region calculated by differentiating Eq.(1)with Eqs.(2)and(3)with respect toσnand then evaluating withσn=0 is notφ0.The tangentof the envelope at σn=0 is

This would be greater than φ0if c0＜σcand smaller than φ0if c0＞σc.From the tensile region,the real angle of the tangent slope is indeed equal toφ0.Thus the two envelopes have different slope gradients atσn=0,although they join continuously at one point.

Barton(1976)showed that there is a confining stress state at which the shear strength reaches a maximum and that there cannot be further increase of the strength with increase of the confining stress(see Fig.2).This state is called the critical state and the corresponding confining stress is termed the critical confining stress.This indicates that the strength envelope becomes horizontal at the critical state.Barton(1976)also showed that the value of the maximum shear strength is half of the normal compressive stress at the critical state.From the Mohr circle at the critical state,it can be seen that the normal compressive stress at critical state is twice the critical confining pressure.

From re-analysis of 158 reported sets of test data incorporating over a thousand triaxial tests,and using their single quadratic expression for the strength,Singh et al.(2011)suggested that the critical confining pressure is approximately equal to the UCS.Thus the finding by Singh et al.(2011)indicates that at critical state,the normal compressive stress is indeed equal to twice the UCS.It should be noted that in their derivation,the UCS is related to the cohesion and internal friction angle from the linear Mohr-Coulomb strength criterion.

It can also be seen that atσn=2σc,the expressions in Eqs.(1)-(3)giveφ=0°andτ= σc,which are the apparent conditions for the critical state.However,atσn=0,the realangleof tangentslopeof the strength envelope is not exactly equal toφ,thus generally it will not be quite zero at σn=2σc,though it should be very small.The tangentof the strengthenvelopeatσn=2σcis(1-c0/σc-φ0)/2.Thus the real critical state conditions are not satisfied exactly atσn=2σc.This is acceptable,since the value of critical confining pressure found by Singh et al.(2011)is approximate,and they used a relation between the UCS,cohesion and internal friction angle that was derived from a linear shear strength envelope.Hence the critical state described here,as formulated by Eqs.(1)-(3),is an approximation,but it predicts an acceptably close-to-horizontal strength envelope.

If we enforce that the strength envelope given by Eqs.(1)-(3)touches tangentially the Mohr circle with zero confining pressure andσcas the major compressive stress,thenσcwill be the real UCS.This condition will yield a relation between UCS,σc,cohesion,c0,and the apparent internal friction angleφ0.However,due to the nonlinearity,the relation is not simple,and cannot be expressed in an explicit closed-form solution.It is solved iteratively with a small computer program.For simplicity,we employ the classical form of relation among the three parameters:

and leave the discussion for the exact equations and solutions for the relation in Appendix A.

Besides the not quite horizontal slope atσn=2σc,the joining of the compression and tension parts atσn=0 is not smooth in general.The tangent slope of the tension part atσn=-σtis not vertical,thus the strength envelope does not quite touch the Mohr circle for the stress state corresponding to the UTS,i.e.tangentially at σn=-σt.In addition,it has been shown in extensive early research by Mogi(1966)and Byerlee(1968)and others(reviewed in Barton(1976))that the behaviour of intact rocks changes from brittle to ductile if the confining pressure is suf ficiently high.The brittle-ductile transition means that the different strengths in the two regimes should meet at the transition smoothly(tangentially).This has not been addressed in the above expression.A three-part piecewise nonlinear shear strength envelope is given in Appendix B which can satisfy all these aspects.Because of its complexity,we leave it in the appendix for interested readers.

3.Example of approximate nonlinear shear strength criterion

To illustrate the approximate modified Mohr-Coulomb shear strength criterion,we consider three cases with parameters σt=8 MPa and φ0=35°:

(1)The first case is the original Mohr-Coulomb shear strength criterion with the extra parameters c0=10 MPa and σc=38.42 MPa satisfying Eq.(8)which is from the original Mohr-Coulomb criterion.

(2)The second case is the approximate nonlinear shear strength criterion using c0= 10 MPa and then calculating σc=38.42 MPa from Eq.(8)which is an approximation for the nonlinear criterion.

(3)The third case is the approximate nonlinear shear strength criterion usingσc=38.42 MPa and calculating c0=9.03 MPa from the equations given in Appendix A to make the strength envelope touching the Mohr circle for UCS tangentially.

The strength envelopes and corresponding Mohr circles are shown in Fig.4.As expected,the original Mohr-Coulomb criterion yields higher shear strength than the modified criterion.For the case(2),the angle of tangent atσn=0 from the compression side is about 47°,while from the tension side,it is equal to φ0=35°.The tangents are not the same at this point,as explained above.As for the previouslymentioned reasons,the strength envelope of case(2)is close to but not exactly touching tangentially the Mohr circles for the uniaxial compression case,while the strength envelope of case(3)is.At the critical state(σn=2σc),the strength envelope has an angle of tangent of about 3°,slightly off the horizontal direction.It can be seen that the two curves for cases(2)and(3)are close to each other and this indicates that the approximation with Eq.(8)is satisfactory.It can also be observed from Fig.4 that large difference between the classical and the modified criteria exists when the confining stress is more than 50%of the UCS.

4.Simulation of tunnel failure with the approximate shear strength criterion

The above shear strength criterion,as described by Eqs.(1)-(5),has been implemented into the numerical code FRACOD to model the failure process around underground excavations.FRACOD is a code that predicts the explicit fracturing process in rocks using fracture mechanics principles(Shen et al.,2014).Over the past three decades,significant progress has been made in developing this approach to a level that it can predict actual rock mass stability at an engineering scale.The code also includes complex coupling processes between the rock mechanical response,thermal process and hydraulic flow,making it possible to handle complex coupling problems frequently encountered in geothermal,hydraulic fracturing,nuclear waste disposal,and underground lique fied natural gas(LNG)storage.During this period,FRACOD has been applied in the modelling of numerous problems,which include borehole stability in deep geothermal reservoirs,pillar spalling under mechanical and thermal loading,prediction of tunnel and shaft stability under high stress,excavation disturbed zones(EDZ),etc.(Shen et al.,2014).

4.1.Basic principles in FRACOD

FRACOD is a two-dimensional(2D)code which is based on the displacement discontinuity method(DDM)principles for stress analysis.In the FRACOD model,fractures are represented by a number of displacement discontinuity elements.When fracture propagation is detected,new displacement discontinuity elements will be added to the existing fracture tips to simulate fracture growth.The criterion used in FRACOD to detect fracture propagation is the F-criterion developed by Shen and Stephansson(1993).This criterion is capable of predicting both tensile(mode I)and shear(mode II)fracture propagationwhich is particularly useful for rock fracture propagation as both tensile and shear failures are common in rock masses.According to the F-criterion,in an arbitrary direction(θ)at a fracture tip,there exists an F-value,which is calculated by

Fig.4.Variation of approximate modified shear strength envelopes with the normal stress.Mohr circles show the three special stress states.Comparison of the results of the original Mohr-Coulomb(M-C),the modified Mohr-Coulomb shear strength from the linear relation Eq.(8)and that from the nonlinear equation given in Appendix A is presented.

Fig.5.Modelling results of a tunnel failure under high stresses.(a,b)Initial fracture initiation pattern(exclusively shear failure-green);(c,d)Final fracturing pattern after fracture propagation(tensile failure mode-red);(e,f)Major principal stress distribution after failure;and(g,h)Displacement distribution after failure.Note that in this example,fracture initiation using only the original and modified Mohr-Coulomb criteria has been allowed,in order to demonstrate their difference.The fractures are all initiated in shear.They also propagated mainly in shear with some tensile segments during fracture coalescence.The extensional strain criterion discussed in Barton and Shen(2017)for predicting the initial tensile spalling in tunnels has been switched off in this example in order to better observe the effect of original and modified Mohr-Coulomb criteria on shear failure.

where GIcand GIIcare the critical strain energy release rates for mode I and mode II fracture propagation,respectively;GI(θ)and GII(θ)are the strain energy release rates due to the potential mode I and mode II fracture growth of a unit length,respectively.The direction of fracture propagation will be the direction where F reaches the maximum value.If the maximum value of F reaches 1,fracture propagation will occur.

FRACOD predicts fracture initiation from initially intact rock.Because FRACOD considers the intact rock as a flawless and homogeneous medium,any fracture initiation from such a medium represents a localised failure of the intact rock.The localised failure will be predicted by the intact rock failure criterion.

A fracture initiation can be formed due to tension or shear.For tensile fracture initiation,both the conventional tensile stress criterion and the extensional strain criterion(Barton and Shen,2017)can be used in FRACOD.The new rock fracture will be generated in the direction perpendicular to the maximum tensile stress or extensional strain.The length of the newly generated fracture can be specified bythe user,orcan be automatically defined bythe code based on the element length used for fractures and by the spacing of the grid points used in the intact rock.

For a shear fracture initiation,both the original linear Mohr-Coulomb failure criterion and the modified nonlinear Mohr-Coulomb failure criterion given by Eqs.(1)-(5)have been implemented in FRACOD.When the shear stress at a given point of the intact rock exceeds the shear strength of the intact rock,a new rock fracture will be generated in the direction of the potential shear failure plane.

4.2.Numerical model and results

To demonstrate the effect of the modified Mohr-Coulomb failure criterion using critical state driven nonlinearity,we simulated a case of tunnel failure under high stresses using FRACOD.By choosing both the original Mohr-Coulomb failure criterion and the modified nonlinear Mohr-Coulomb failure criterion in the model,we obtained two sets of results and compared them.

An 8 m-diameter tunnel in an elastic and massive rock mass was simulated.The in situ stress state in the 2D plane perpendicular to the tunnel cross-section was assumed to represent a stress ratio of σHmax/σv=2 at a simulated depth of 1000 m,where σHmaxis the maximum horizontal normal stress,andσvis the vertical normal stress due to the gravity.The numerical model has boundary stressesσHmax=50 MPa andσv=25 MPa.The strength and fracture toughness of the rock were assumed to be the same as discussed in theexamplecase ofSection3:UCS=38.4 MPa,cohesionc=10MPa,internal friction angleφ=35°at zero normal stress,tensile strength σt=8 MPa,mode I fracture toughness KIc=1.9 MPa m1/2,and mode II fracture toughness KIIc=2.35 MPa m1/2.Fracture initiation length is set to be 0.15 m.The boundary of the tunnel is simulated by 120 displacementdiscontinuityelementswithanequallengthof0.21m.The internal tunnel boundary is stress-free and no lining or other tunnel support is considered.

The modelling results,including the fracturing patterns,major principal stress distribution and displacement distribution,are shown in Fig.5.The figures on the left arethe results obtained using the modified nonlinear Mohr-Coulomb failure criterion,whereas the figures on the right are those with the original linear Mohr-Coulomb failure criterion.

Differences can be noticed in the predicted fracture initiation pattern between Fig.5a and b.The modified nonlinear Mohr-Coulomb failure criterion led to somewhat deeper failure regions in the surrounding rock mass than that of the original Mohr-Coulomb criterion.The depth of fracture initiation regions reaches respectively 1 m and 0.75 m into the tunnel roof/ floor for the modified and original Mohr-Coulomb criteria.The initial fractures propagate under the high stress in the roof and floor,and the final fracture patterns are shown in Fig.5c and d for both cases.Thus slightly deeper and more intensive fracturing regions in the roof and floor are predicted using the modified nonlinear Mohr-Coulomb failure criterion than that of the original linear Mohr-Coulomb criterion.Similar differences can also be observed in the stress plots shown in Fig.5e and f and displacement plots shown in Fig.5g and h.

The difference in the modelling results is expected,given that the modified Mohr-Coulomb criterion predicts lower shear strength than the original Mohr-Coulomb criterion when the confining stress is high.At the location of 1 m into the roof and floor of the tunnel,the initial confining stress is 17.5 MPa and the major principal stress is 89 MPa.Based on the strength curves shown in Fig.4,this stress combination produces a Mohr circle(red circle in Fig.4)which intersects the strength envelope of the modified Mohr-Coulomb criterion but not that of the original Mohr-Coulomb criterion.Consequently fracture initiation has been predicted at this location in the FRACOD model using modified Mohr-Coulomb criterion but not with the original Mohr-Coulomb criterion.

5.Conclusions

We have investigated the nonlinear shear strength envelopes using the critical state concept for intact rocks.First we employ the classical Mohr-Coulomb form and introduce the nonlinearity by modifying the cohesion and internal friction angle terms to change with normal stress.The expressions are simple and the terms have physical meaning.One of the critical state conditions is satisfied,but the other one is not exactly satisfied.Due to the nonlinearity,the classical simple relation among cohesion,internal friction angle and UCS can no longer be valid.The equations and solution scheme for some new relations are presented in Appendix A,but they are complicated and their solution needs a nonlinear equation solver program.For practical purposes,the classical simple relation between cohesion,internal friction angle and UCS is accepted as giving close approximations to the intended nonlinear critical state strength limits.

In Appendix B,we present a complicated piecewise nonlinear expression for the shear strength envelope which includes the brittle-ductile transition behaviour and critical state concepts for rocks reported in Barton(1976).The expression covers the tensile region too.The strength envelope tangentially links the parts at the tensile-compressive transition and at the brittle-ductile transition.In the tensile regime,for the envelope to have vertical tangent with the Mohr circle corresponding to the UTS,we employed a square root term together with a quadratic term.The square root term is similar to the Hoek-Brown shear strength.In the brittle and ductile regimes,weassume a quadratic form.The coefficients for the brittle quadratic formulation are determined by requiring the shear strength envelope tobe tangenttothe Mohrcircle corresponding to the UCS.The quadratic formulation in the ductile regime satisfies the Barton(1976)’s specification of critical state:the envelope reaching a horizontal tangent with the maximum shear strength equal to half of the compressive normal stress on the shear plane.

The modified nonlinear Mohr-Coulomb criterion has been implemented in a fracture mechanics based numerical code FRACOD,and an example case of tunnel failure has been presented.It is shown that the modified nonlinear Mohr-Coulomb criterion predicts somewhat deeper and more intensive fracturing regions in the surrounding rock mass than the original linear Mohr-Coulomb criterion.One might conclude that this seems to be more in line with the intensely fractured state of the debris both described and seen in photographs after(sometimes tragic)rockburst accidents in deep tunnels and mines.

The modified nonlinear Mohr-Coulomb criterion is relatively simple to apply in practical rock engineering as it only requires one to estimate the mobilised internal friction angle and cohesion using the ratio of normal stress to rock UCS.It is recommended that this criterion be used for rock engineering in high stress environments when the confining stress is more than 50%of the UCS of the intact rock.

Conflict of interest

We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have in fluenced its outcome.

Acknowledgements

We are very grateful for the financial support from(1)the International Collaboration Project on Coupled Fracture Mechanics Modelling(project team consistingof CSIRO,SDUST,Posiva,KIGAM,KICT,CAS-IRSM,DUT/Mechsoft,SNU,LBNL,ETH,Aalto Uni.,GFZ and TYUT);and(2)Taishan Scholar Talent Team Support Plan for Advantaged&Unique Discipline Areas,Shandong Province.

Appendices A and B.Supplementary data

Supplementary data related to this article can be found at https://doi.org/10.1016/j.jrmge.2018.04.002.