Jin-feng Zou and Jia-min Du. 1. Introduction

Transcription

1 Mathematical Poblems in Engineeing Volume 206, Aticle ID , 0 pages Reseach Aticle A Numeical Appoach fo the Quasi-Plane Stain-Softening Poblem of Cylindical Cavity Expansion Based on the Hoek-Bown Failue Citeion Jin-feng Zou and Jia-min Du School of Civil Engineeing, Cental South Univesity, No. 22, Shaoshan South Road, Cental South Univesity Railway Campus, Changsha, Hunan Povince 40075, China Coespondence should be addessed to Jin-feng Zou; zoujinfeng csu@63.com Received 2 Apil 206; Revised 26 June 206; Accepted 20 July 206 Academic Edito: Sebastian Heideneich Copyight 206 J.-f. Zou and J.-m. Du. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. This pape focuses on a novel appoach fo the quasi-plane stain-softening poblem of the cylindical cavity expansion based on genealized Hoek-Bown failue citeion. Because the intemediate pincipal stess is defomation-dependent, the quasi-plane stain poblem is defined to implement the numeical solution of the intemediate pincipal stess. This appoach assumes that the initial total stain in axial diection is a nonzeo constant (ε 0 ) and the plastic stain in axial diection is not zeo. Based on 3D failue citeion, the numeical solution of plastic stain is given. Solution of the intemediate pincipal stess can be deived by Hooke s law. The adial and cicumfeential stess and stain consideing the intemediate pincipal stess ae obtained by the poposed appoach of the intemediate pincipal stess, stess equilibium equation, and genealized H-B failue citeion. The numeical esults can be used fo the solution of stain-softening suounding ock. In additional, the validity and accuacy of the poposed appoach ae veified with the published esults. At last, paametic studies ae caied out using MATLAB pogamming to highlight the influences of the out-of-plane stess on the stess and displacement of suounding ock.. Intoduction Cavity expansion theoy has been widely used in Geotechnical Engineeing such as pessuemete test and othe geotechnical poblems. Based on the Moh-Coulomb (M- C) and Hoek-Bown (H-B) failue citeia, many eseaches have solved many engineeing poblems using the analytical solution and semianalytical solution. Vesic [] poposed an appoximate solution based on the Moh-Coulomb failue citeion fo both spheical and cylindical cavity expansion poblems. Cate et al. [2] pesented an explicit pessueexpansion elation and deived an analytical solution fo cavity expansion in nonassociated Moh-Coulomb media. Duban and Papanastasiou [3], enhanced the solutions of cylindical cavity expansion and contaction in pessue sensitive geomateials fo the Tesca and Mises models, by incopoating pessue sensitivity in the plastic potential and effective stess. Duban [4 6] poposed the lage stain and geneal solutions fo pessuized elastoplastic tubes, and the finite staining of pessuized compessible elastoplastic tubes, espectively. Papanastasiou and Duban pesented the elastoplastic analysis method of cylindical cavity poblems in geomateials. Papanastasiou discussed the influence of plasticity in hydaulic factuing. Collins and Yu [7] poposed the solutions fo lage stain undained cavity expansion. Pan and Bown [8] consideed the effects of out-of-plane stess and dilation on the convegence and stability of the suounding ock. Cao et al. [9] expanded the solutions on modified Cam Clay model. Alonso et al. [0] obtained the self-simila solution fo the cicula tunnel in stain-softening ock masses. Lee and Pietuszczak [] and Pak et al. [2] pesented the solutions fo the cavity expansion with unloading by finite diffeence method. Yang and Zou [3] pesented a numeical solution of cavity expansion in the genealized H-B media. Chen and Abousleiman [4, 5] pesented an analytical solution of cavity expansion based on the modified Cam-Clay model. Wang et al. [6] studied the influence of out-of-plane stess on the distibution of stess, stain, and

2 2 Mathematical Poblems in Engineeing displacement based on plane stain assumption. Zhou et al. [7 9] poposed an analytical solution consideing the influence of the shea stess on cylindical cavity expansion in an undained elastic-pefect soil. Although some liteatues [8, 6 22] focused on the effect of out-of-plane stess have been published, those studies assume that the plastic stain in axial diection is zeo and the plastic potential function is simple. Fo example, Pan and Bown [8] poposed an appoach in which the axial in situ stess of the plastic zone is defomation-dependent and the fomula fo the calculation of intemediate pincipal stess was deived, only numeical solution though finite element method was pesented. In the pesented solutions, a numeical stepwise pocedue that consides the quasi-plane stain-softening behavio is adopted and impoved, whee the deteioation of stength, defomation, and dilation angle in the plastic egion ae consideed. The impoved constitutive model consides the stain-softening behavio and the defomation dependence of intemediate pincipal stess compatible with genealized H-B failue citeion. 2. Objective and Scope The main objective of the pesent study is to intoduce a novelappoachfothemechanicalanalysisofcylindical cavity expansion consideing the influence of the axial stess based on the assumptions of the quasi-plane stain-softening poblem, and the coesponding theoetical solutions fo the defomation-dependent intemediate pincipal stess and axial stain ae poposed. 3. Methodology 3.. Definition of the Poblem and Assumptions. As shown in Figue, a cylindical cavity expansion with an initial adius (a 0 ) is subjected to a hydostatic pessue (p 0 )inockmass which is consideed as continuous, homogeneous, isotopic, and initially elastic. The out-of-plane stess (σ z )alongthe axis diection of the cylindical cavity is also consideed. The cylindical cavity expands to a adius of a as the intenal cavity pessue inceases fom p to p i. Coespondingly, an element initially located at a distance ( 0 ) fom the cente of thecavity wall movesto a newadial position (). The egion of ock mass aound the cylindical cavity is in the elastic state when the cylindical cavity expansion pessue p is small. As the cylindical cavity expansion pessue p inceases to the citical value, the plastic egion of ock mass aound the cavity would appea with plastic defomation. If the cylindical cavity expansionpessuep inceases continually, the plastic flow zone is fomed, and the stength of suounding ock mass aound cavity is educed to the esidual stength. The cylindical cavity expands to a adius of a as the intenal cavity pessue inceases fom p to p i.theplasticegion aound the cylindical cavity is divided into two aeas (i.e., plastic stain-softening egion ( p ) and plastic flowing egion ( s )). The ock mass beyond the plastic egion would emain elastic. q z y p 0 p in a 0 a i th annulus Plastic egion Elastic egion Figue : Model of the axisymmetic and quasi-plane stainsoftening poblem fo cylindical cavity expansion Hoek-Bown Failue Citeion. The genealized Hoek- Bown failue citeion is adopted and expessed by [22 24] σ =σ 3 +σ c ( mσ 3 σ c p x p 0 a +s), () whee σ and σ 3 ae the majo and mino pincipal stesses, espectively. σ c is the uniaxial compessive stength of the ock mass. a, m, and s ae the stength paametes of the genealized H-B failue citeion. These vaiables ae expessed as m = m i exp[(gsi 00)/(28 4D)], s = exp[(gsi 00)/(9 3D)], andn=/2+[exp( GSI/5) exp( 20/3)]/6. D is a facto that depends on the degee of distubance to which the ock has been subjected in tems of blast damage and stess elaxation. Its value vaies between 0 and.gsiisthegeologicalstengthindexoftheockmass, and its value anges between 0 and Plastic Potential Function. In the pape, the plastic potential function based on 3D M-C failue citeion is adopted to obtain the solution of stain. The plastic potential function poposedbypanandbown[8]canbeexpessedby Q (σ) = n 3 I + 3 σ c J n J 2, (2) whee I = σ +σ 2 +σ 3, J 2 = (/6)[(σ σ 2 ) 2 +(σ 2 σ 3 ) 2 +(σ 3 σ ) 2 ]. σ, σ 2,andσ 3 ae the majo, intemediate, and mino pincipal stess, espectively. n is the dilation paamete which is poposed by Pan and Bown [8]. Based on the plastic flow ule, the plastic stain incement is given by dε p =λ f σ. (3)

3 Mathematical Poblems in Engineeing 3 The incements of majo, intemediate, and mino plastic stains ae pesented by dε p =[( 3(2σ σ 2 σ 3 ) 2 J 2 3 )n + σ c (2σ σ 2 σ 3 )] dλ dε p 2 =[( 3(2σ 2 σ 3 σ ) 2 J 2 + σ c (2σ 2 σ 3 σ )] dλ dε p 3 =[( 3(2σ 3 σ 2 σ ) 2 J 2 + σ c (2σ 3 σ 2 σ )] dλ, 3 )n 3 )n whee ε p, εp 2,andεp 3 ae the majo, intemediate, and mino pincipal stains, espectively. dλ is the plastic constant Deteioation of Stength and Defomation Paametes of Suounding Rock. BasedontheeseachesultsofAlonso et al. [0], the stength and defomation paametes of the stain-softening ock mass ae evaluated based on plastic defomation and ae contolled by the deviatoic stain γ p =ε p εp 3, (5) whee ε p and εp 3 ae the majo and mino plastic stains, espectively. The physical paametes of the suounding ock mass ae descibed accoding to the bilinea function of plastic shea stain as follows: ω(γ p ω )= p (ω p ω ) γp (6) ω, γ p γ p, whee ω epesents a stength paamete, such as m, s, and a; γ p is the citical deviatoic plastic stain fom which the esidual behavio is fist obseved and should be identified though expeimentation. The subscipts p and epesent the peak and esidual values, espectively. When the axial foce is consideed as the middle pincipal stess, the elastic modulus and Poisson s atio in the plastic zone of the defomation and stess evolution can be epesented by a piecewise linea function as follows: (4) E(γ p E )= p (E p E ) γp E, γ p γ p, V (γ p V )= p (V p V ) γp V, γ p γ p. (7) To take into account the effect of vaiable dilation in the plastic egion, dilation angle φ(γ p ) pesumably deceases linealy with stain fom its peak value φ p at γ p =0to the esidual value φ at γ p =γ p, as in the following equation: ξ(γ p ξ )= p (ξ p ξ ) γp (8) ξ, γ p γ p, whee ξ p and ξ ae the peak and esidual values of the dilation angle of the ock, espectively; γ p is the softening paamete; γ p is the value of the softening paamete that contols the tansition between the softening and esidual stages. In (5) (8), γ p is obtained by (5). γ p is the citical deviatoic plastic stain and should be detemined though expeimentation. The coesponding of paametes (m, s, anda) ae as follows: m(γ p m )= p (m p m ) γp m, γ p γ p, s(γ p s )= p (s p s ) γp s, γ p γ p, a(γ p a )= p (a p a ) γp a, γ p γ p. ξ p and ξ ae the peak and esidual values of the dilation angle of the ock and detemined by expeiments, espectively. Then, the deteioation paametes of stength and defomation of suounding ock ae detemined [25]. 4. Solutions of Stess and Displacement in Elastic Zone 4.. Equilibium Equations and Stess Bounday Conditions. Unde the assumption of small defomation, the ock mass satisfiesthegenealizedhooke slawintheelasticegionand obeys genealized H-B failue citeion in the plastic egion. The stess equilibium equation of an element nea an cavity wallcanbeepesentedby dσ d + σ σ θ =0, (0) whee σ is the adial stess; σ θ is the tangential stess. Stess bounday conditions can be given by σ =0 lim σ =p =σ 0. (9) () The elationship of the adial and the tangential stesses at the inteface between the elastic and plastic zones is pesented by σ θ =2σ 0 σ R. (2)

4 4 Mathematical Poblems in Engineeing The adial and the tangential stesses at the elastoplastic inteface ae satisfied with the yield citeion and can be given by y Elastic egion σ R =σ θ +σ c ( mσ θ σ c Combination of (2) and (3) leads to a +s). (3) i P(0) σ 2σ 0 σ θ } } = σ } θ. (4) } } q } σ θ σ z P i a 0 E R x 4.2.StessandStainSolutions. The solutions of stess and displacement in the elastic zone ae expessed as follows [2, 22, 26]: P(i) P(i ) i th annulus σ =p 0 (p 0 σ R )( R ) 2 Softening egion σ θ =p 0 +(p 0 σ R )( R ) 2 σ z =υ(σ θ +σ ) 2υp 0 +q u= 2G (σ R σ 0 ) R2, (5) whee R is the plastic adius of suounding ock, υ is Poisson s atio, and G = E/2( + υ) is the Shea modulus. 5. Stess and Stain in Plastic Region Analytical solutions of stess and displacement ae difficult to obtain in the stain-softening ock mass, especially consideing the axial stess. In this pape, the quasi-plane stain and displacement of stain-softening ock mass ae solved by iteation method which is illustated by Figue 2. The total plastic egion is divided into n connect annuli as shown in Figue 2. The ith annulus is delimitated by the intenal adius with (j ) = (j ) /R and the oute adius with (j) = (j) /R.Theadiusofthefistingis (0) =which is at the inteface between the elastic egion and the plastic zone; the suounding ock soil mass emains the citical state of the plastic. The incement of adial stess esults in the following: Δσ = p in σ R. (6) n So the adial stess can be expessed by σ (i) =σ (i ) +Δσ. (7) The coesponding tangential stess is given by σ (i) =σ θ(i) +σ c ( m a (i )σ (i ) θ(i) +s σ (i ) ). (8) c The axial stess σ z(i) poposed by Pan and Bown [8] can be epesented by σ z(i) = V (σ (i) +σ θ(i) )+( 2V) σ 0 Eε p z(i ). (9) Figue 2: Nomalized plastic egion with finite numbe of annuli. Combination of (7) and (8) leads to Δσ θ(i) =σ θ(i) σ θ(i ) Δσ z(i) =σ z(i) σ z(i ). The elastic stain can be expessed by ε e = E [σ V (σ θ +σ z ) (σ 0 Vσ 0 Vq)] ε e θ = E [σ θ V (σ +σ z ) (σ 0 Vσ 0 Vq)] ε e z = E [σ z V (σ θ +σ ) (q 2Vσ 0 )]. The stess equilibium equation is given by (20) (2) dσ d + σ σ θ =0. (22) The stess equilibium diffeential equation fo the ith annulus is expessed by σ (i) σ (i ) (i) (i ) + H(σ θ(i)) (i) =0, (23) whee H(σ θ(i) )=σ c (m (i ) σ θ(i) /σ c +s (i ) ) a (i ), (i) =( (i) + (i ) )/2,andσ (i) =(σ (i) +σ (i ) )/2. The nomalized inne adius (i) = (i) /R can be expessed as (i) = (i ) 2H (σ θ(i) ) Δσ 2H (σ θ(i) )+Δσ. (24) If the annuli ae sufficiently thin in the case of axial symmety, then the stain-displacement elationships can be

5 Mathematical Poblems in Engineeing 5 descibed accoding to the model pesented by Bown et al. [27] as j j = 2ε θ(j ) ε (j ) ε (j) 2ε θ(j) ε (j ) ε (j). (25) The nomalized inne adius is defined as follows: (j) = (j) R. (26) Then, (25) is simplified to (j) (j ) = 2ε θ(j ) ε (j ) ε (j) 2ε θ(j) ε (j ) ε (j). (27) The stain-displacement elationships can be given by ε = du d ε θ = u (28). The compatibility equation can be witten in the geneal fom as follows: dε θ d + ε θ ε =0. (29) Combining (26) and (29), the nomalized compatibility equation can be expessed as follows: dε θ d + ε θ ε =0. (30) Thetotalstaininplasticzoneisthesumoftheelasticand plastic stains as follows: ε θ =ε p θ +εe θ ε =ε p +εe ε z =ε p z +εe z. The following equations can be obtained by (30): dε p θ d + ε dε p θ d + ε p θ εp = dεe θ d εe θ εe ε e θ εe = +V E (σ θ σ ) p θ εp = dεe θ d + +V E Combination of (32) and (33) leads to Δε p θ(i) ( Δ (i) + (k 3 k ) (i) k 3 ) (3) (32) H(σ θ ). (33) = +V H(σ θ(i) ) Δεe θ(i) (ε p θ(i ) E (i) Δ (i) εp (i ) ), (i) (34) whee Δ (i) = (i) (i ), k =( 3(2σ σ 2 σ 3 )/2 J 2 /3)n + (/σ c )(2σ σ 2 σ 3 ),andk 3 = ( 3(2σ 3 σ 2 σ )/2 J 2 /3)n + (/σ c )(2σ 3 σ 2 σ ). The incements of the cicumfeential, adial, and axial stains can be, espectively, expessed by Δε p θ(i) = ((+V) /E) (H (σ θ(i))/ (i) ) Δε e θ(i) /Δ (i) (/ (i) )(ε p θ(i ) εp (i ) ) (/Δ (i) +(k 3 k )/ (i) k 3 ) Δε p k (i) =Δεp θ(i) k 3 (35) Δε p k 2 z(i) =Δεp θ(i). k 3 Then, the total stain can be given by ε θ(i) =ε θ(i ) +Δε e θ(i) +Δεp θ(i) ε (i) =ε (i ) +Δε e (i) +Δεp (i) ε z(i) =ε z(i ) +Δε e z(i) +Δεp z(i). Theefoe, we can obtain the plastic adius as follows: (36) R= 0 (i) (37) as The displacement at each ing can be obtained by u=ε θ(i) R. (38) The displacement of the annulus at adius j is expessed u (j) = ε θ(j) (j). (39) The displacements of the annulus at adius ( j )andplastic adius (R) ae obtained though the numeical stepwise pocedue in combination with MATLAB.

6 6 Mathematical Poblems in Engineeing Stess (MPa) σ σ z σ θ 0.00 γ p =0 γ p = γ p = 0.02 γ p = 00 γ p =0 γ p = γ p = 0.02 γ p = 00 (a) Stesses (b) Displacements Figue 3: Displacement and stess with the diffeent citical values of stain-softening paametes. Table:ResultsbythepoposedappoachandVesic ssolution[]. p /a u Vesic p (MPa) H-B Diffeences 4.% 2.5%.9%.% 0.3% 0.7% 6. Validations To confim the validity and accuacy of the poposed appoachbasedonthegenealizedh-bfailueciteion,the esults of the poposed appoach ae compaed with those of Vesic s solution [] fo ock mass with the following data: a u =0.25m,σ 0 =0MPa,E =5500MPa,V =0.25,a =0.55,s = , m =.7,σ c =0MPa,andψ =0 [28].Howeve,Vesic s esult [] fo ock mass is based on the M-C failue citeia. In ode to compae the esult of the poposed solution, the technique of the equivalent M-C and genealized H-B stength paametes is adopted [29]. The stength paametes fo the M-C failue citeion ae as follows: c =.3659 MPa, φ =8.8549MPa. As seen fom Table, the expansion pessues of the poposed appoach based on the genealized H-B failue citeion agee well with those of Vesic s solution []. In the compaison, the maxim diffeences of expansion pessue p do not exceed 5% fo the cylindical expansion cavity. The validations show that the numeical stepwise method is effective in analyzing the cavity expansion poblem. 7. Numeical Analysis and Discussions The stess and displacement of cylindical cavity with outof-plane stess consideed ae calculated to emphasize the influences of the out-of-plane stess. In ode to study the effect of stain-softening, dilation paamete, stength paamete, elastic modulus, and Poisson s atio with the outof-plane stess consideed, seveal examples ae pefomed in the poposed solution. The input data of the poposed solution based on the genealized H-B failue citeion pesented by Shaan [28] ae as follows: σ 0 =0MPa, E =5500MPa,V =0.25,p in = 30 MPa, 0 =0.25m,σ c =0MPa, q=0mpa, m p =.7,s p = , a p =0.55,m =0.8,s =0.009,a = Effects of the Stain-Softening Paametes. Softening paametes ae impotant chaacteistic of stain-softening suounding ock. Its value can be acquied fom two methods. One is detemined by the plastic shea stain γ p =ε p εp 3, andtheotheistousethepincipalstaintodetemineε p.in ode to analyze the influence of softening paametes, fou cases ae analyzed using the poposed appoach (i.e., Case : γ p =0;Case2:γ p =0.004;Case3:γ p =0.02;Case4:γ p = 00).TheesultsaeshowninFigue3. Figue 3 illustates that the values of stess and displacement decease with the softening paamete (γ p )inceasing. It can be seen fom Figue 3 that the displacements of the poposed appoach would be educed significantly if γ p inceases fom 0 to 00. Fo example, displacement is 0.04 m, m, and 0.03 m when γ p equals 0, 0.004, and 00, espectively. The eduction of the displacement is 28% if γ p inceases fom 0 to Effect of Dilation Paametes. In ode to examine the effects of dilation paametes that conside out-of-plane stess, fou cases ae pefomed using the poposed appoach (i.e., Case : n =0.;Case2:n=;Case3:n=2;Case4: n=4). The esults ae shown in Figues 4 and 5.

7 Mathematical Poblems in Engineeing Stess (MPa) n = 0. n= n=2 n=4 n = 0. n= n=2 n=4 (a) Stesses (b) Displacements Figue 4: Displacement and stess with the diffeent dilation paametes Dilation paamete (n) Figue 5: Displacements with the diffeent dilation paametes. The effects of dilation paametes on the displacement and stess consideing out-of-plane stess ae significant. As shown in Figue 4, displacements of this study deceases fom m to 0.05 m with dilation paametes inceasing fom 0. to 4 and it educes by Δ = ( )/ % = 54.5%. Theefoe, the effects of dilation paametes should be takenintoaccountcaefully. Figue 5 demonstates the elationship between the displacements of cavity and the dilation paamete. The displacements decease continuously with the inceasing of the dilation paamete. Howeve, when the dilation paamete is geate than 3.0, the displacements decease slightly. But the displacements decease significantly when the dilation paamete anges fom 0 to 3.0. Theefoe, the effect of dilation paamete changes with diffeent dilation paamete Effects of Stength Paametes. In ode to detect the effect of stength paamete (m) on stess and displacement, thee diffeent cases ae analyzed using the poposed appoach. They ae Case : m p = m =0.8;Case2:m p =.7,m =0.8; Case 3: m p = m =.7. The esults ae shown in Figue 6. It is shown fom Figue 6 that the effect of stength paamete on stess and displacement is significant, and all of them decease with stength paamete inceasing. Moeove, the plastic adius as shown in Figue 6(a) is.423 m and m when m p and m equal 0.8 and.7, espectively. It canbeseenfomfigue6(b)thatthedisplacementdeceases fom m to 0.04 m with stength paametes inceasing fom 0.8 to.7. Thus, the effects of stength paamete on stess and displacement ae significant Effects of Elasticity Modulus. To identify the effect of modulus of elasticity on the stess and displacement, thee diffeent cases ae analyzed using the poposed method. They aecase:e p = E = 3500; Case 2: E p =5500,E =3500;Case 3: E p = E = The esults ae depicted in Figue 7. The effect of the elastic modulus on the displacement that consides out-of-plane stess is significant, while the influence of the elastic modulus on stess is insignificant. As showninfigue7,thedisplacementdeceasesfom0.056m to 0.04 m as the elastic modulus inceases fom 3500 MPa to 5500 MPa. Howeve, the stesses fo thee diffeent cases ae elatively steady. Theefoe, the effects of elastic modulus on displacement should be taken into consideation Effects of Poisson Ratio. To detemine the effects of Poisson s atio on stess and displacement, the esults ae shown in Figue 8, fo thee diffeent cases of Poisson s atio using the poposed appoach: Case : V p = V =0.25;Case2: V p =0.35,V = 0.25; Case 3: V p = V =0.35.

8 8 Mathematical Poblems in Engineeing Stess (MPa) m p =m = 0.8 m p =.7, m = 0.8 m p =m =.7 m p =m = 0.8 m p =.7, m = 0.8 m p =m =.7 (a) Stesses (b) Displacements Figue 6: Displacement and stess with the diffeent stength paametes m Stess (MPa) E p =E = 3500 E p = 5500, E = 3500 E p =E = 5500 E p =E = 3500 E p = 5500, E = 3500 E p =E = 5500 (a) Stesses (b) Displacements Figue 7: Displacement and stess with the diffeent elastic modulus. Appaently, it can be seen fom Figue 8 that vaiations of diffeent cuves ae vey small. Hence, Poisson s atio has little effect on the stess and displacement. Fo example, displacements vay fom 0.033m to 0.03m, and plastic adius vaies fom.22 m to.8 m as V incease fom 0.25 to Conclusions Solutions of stess, displacement, and plastic adius fo cylindical cavity expansion wee poposed by consideing out-of-plane stess and the quasi-plane stain-softening poblem based on the genealized Hoek-Bown failue citeion. The intemediate pincipal stess is obtained by 3D plastic potential function and Hooke s law, which is defomationdependent. The validity and accuacy of the poposed solution ae confimed by Vesic s solution []. Futhemoe, the effects of stain-softening, dilation paamete, stength paamete, elastic modulus, and Poisson s atio on stesses and displacement of cavity expansion ae studied with the new appoach.

9 Mathematical Poblems in Engineeing 9 Stess (MPa) p = = 0.25 p = 0.35, = 0.25 p = = 0.35 p = = 0.25 p = 0.35, = 0.25 p = = 0.35 (a) Stesses (b) Displacements Figue 8: Displacement and stess with the diffeent Poisson s atio. Competing Inteests The authos declae that they have no competing inteests. Refeences [] A. S. Vesic, Expansion of cavities in infinite soil mass, Jounal of Geotechnical Engineeing, vol. 98, no. 3, pp , 972. [2] J.P.Cate,J.R.Booke,andS.K.Yeung, Cavityexpansionin cohesive fictional soils, Géotechnique, vol. 36, no. 3, pp , 986. [3] D. Duban and P. Papanastasiou, Cylindical cavity expansion and contaction in pessue sensitive geomateials, Acta Mechanica,vol.22,no. 4,pp.99 22,997. [4] D. Duban, Lage stain solution fo pessuized elasto/plastic tubes, Jounal of Applied Mechanics, vol. 46, no., pp , 979. [5] D. Duban and M. Kubi, Geneal solution fo the pessuized elastoplastic tube, Jounal of Applied Mechanics, vol. 59, no., pp.20 26,992. [6] D. Duban, Finite staining of pessuized compessible elastoplastic tubes, Intenational Jounal of Engineeing Science, vol. 26, no. 9, pp , 988. [7]I.F.CollinsandH.S.Yu, Undainedcavityexpansionsin citical state soils, Intenational Jounal fo Numeical and Analytical Methods in Geomechanics, vol. 20, no. 7, pp , 996. [8] X.-D. Pan and E. T. Bown, Influence of axial stess and dilatancy on ock tunnel stabblity, Jounal of Geotechnical Engineeing,vol.22,no.2,pp.39 46,996. [9] L.F.Cao,C.I.Teh,andM.F.Chang, Undainedcavityexpansion in modified Cam clay I: theoetical analysis, Géotechnique, vol. 5, no. 4, pp , 200. [0]E.Alonso,L.R.Alejano,F.Vaas,G.Fdez-Mańin, and C. Caanza-Toes, Gound esponse cuves fo ock masses exhibiting stain-softening behaviou, Intenational Jounal fo Numeical and Analytical Methods in Geomechanics,vol.27,no. 3, pp , [] Y.-K. Lee and S. Pietuszczak, A new numeical pocedue fo elasto-plastic analysis of a cicula opening excavated in a stain-softening ock mass, Tunnelling and Undegound Space Technology,vol.23,no.5,pp ,2008. [2] K.-H. Pak, B. Tontavanich, and J.-G. Lee, A simple pocedue fo gound esponse cuve of cicula tunnel in elastic-stain softening ock masses, Tunnelling and Undegound Space Technology,vol.23,no.2,pp.5 59,2008. [3] X. L. Yang and J. F. Zou, Cavity expansion analysis with nonlinea failue citeion, Geotechnical Engineeing,vol.64,no., pp.4 49,20. [4] S. L. Chen and Y. N. Abousleiman, Exact dained solution fo cylindical cavity expansion in modified cam clay soil, Géotechnique,vol.63,no.6,pp.50 57,203. [5] S. L. Chen and Y. N. Abousleiman, Exact undained elastoplastic solution fo cylindical cavity expansion in modified cam clay soil, Géotechnique, vol. 62, no. 5, pp , 202. [6] S. L. Wang, S. D. Yin, and Z. J. Wu, Stain-softening analysis of a spheical cavity, Intenational Jounal fo Numeical and Analytical Methods in Geomechanics,vol.36,no.2,pp , 202. [7] H. Zhou, H. Liu, and G. Kong, Influence of shea stess on cylindical cavity expansion in undained elastic-pefectly plastic soil, Geotechnique Lettes,vol.4,pp ,204. [8]H.Zhou,H.Liu,G.Kong,andZ.Cao, Analyticalsolution fo pessue-contolled elliptical cavity expansion in elasticpefectly plastic soil, Géotechnique Lettes,vol.4,no.2,pp.72 78, 204. [9]H.Zhou,H.L.Liu,G.Q.Kong,andX.Huang, Analytical solution of undained cylindical cavity expansion in satuated soil unde anisotopic initial stess, Computes and Geotechnics, vol. 55, pp , 204. [20] A.-Z. Lu, G.-S. Xu, F. Sun, and W.-Q. Sun, Elasto-plastic analysis of a cicula tunnel including the effect of the axial in

10 0 Mathematical Poblems in Engineeing situ stess, Intenational Jounal of Rock Mechanics and Mining Sciences,vol.47,no.,pp.50 59,200. [2] J. F. Zou and Z. He, Numeical appoach fo stain-softening ock with axial stess, Poceedings of the Institution of Civil Enginees Geotechnical Engineeing, vol.69,no.3,pp , 206. [22] Z. Jin-Feng and S. Yu, Theoetical solutions of a cicula tunnel with the influence of the out-of-plane stess based on the genealized Hoek-Bown failue citeion, Intenational Jounal of Geomechanics, vol.6,no.3,aticleid , 206. [23] J. F. Zou and S. Q. Zuo, An appoximate solution fo the cylindical cavity expansion unde the non-axisymmetic displacement bounday condition on hypotenuse, Intenational Jounal of Geotechnical Engineeing,206. [24] J. F. Zou and Z. Q. Xia, Solutions fo displacement and stess in stain-softening suounding ock incopoating the effects of hydaulic-mechanical coupling and ockbolt effectiveness, Geotechnical and Geological Engineeing,206. [25] J. F. Zou and S. S. Li, Theoetical solution fo displacement and stess in stain-softening suounding ock unde hydaulicmechanical coupling, Science China Technological Sciences,vol. 58,no.8,pp.40 43,205. [26] J. F. Zou, W. Q. Tong, and J. Zhao, Enegy dissipation of cavity expansion based on genealized non-linea failue citeion unde high stesses, JounalofCentalSouthUnivesity, vol. 9, no. 5, pp , 202. [27] E. T. Bown, J. W. Bay, B. Ladanyi, and E. Hoek, Gound esponse cuves fo ock tunnels, Jounal of Geotechnical Engineeing,vol.09,no.,pp.5 39,983. [28] S. K. Shaan, Exact and appoximate solutions fo displacements aound cicula openings in elastic-bittle-plastic Hoek- Bown ock, Intenational Jounal of Rock Mechanics & Mining Sciences,vol.42,no.4,pp ,2005. [29] X.-L. Yang and J.-H. Yin, Slope equivalent Moh-Coulomb stength paametes fo ock masses satisfying the Hoek-Bown citeion, Rock Mechanics and Rock Engineeing,vol.43,no.4, pp , 200.

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