Damage Constitutive Model of Mudstone Creep Based on the Theory of Fractional Calculus

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1 Advaces i Peroleum Exploraio ad Developme Vol. 1, No. 2, 215, pp DOI:1.3968/773 ISSN X [Pri] ISSN [Olie] Damage Cosiuive Model of Mudsoe Creep Based o he Theory of Fracioal Calculus ZENG Jia [a],* ; ZHANG Ju [a] [a] Norheas Peroleum Uiversiy Peroleum Egieerig Isiue, Daqig, Chia. *Correspodig auhor. Received 18 Ocober 215; acceped 2 December 215 Published olie 31 December 215 Absrac Whe swellig mudsoe, is various mechaical parameers will chage sigificaly. Esablish mudsoe s fracioal calculus creep damage model srucure diagram, which describe he exe of damage i rock damage variable. Assumed ha he sress level of rock exceeded is he log-erm sregh ad uder he codiio of mudsoe acceleraed srai rae loadig, oliear damage model based o fracioal calculus heory was esablished. Therefore mudsoe creep cosiuive model cosiderig he acceleraed srai rae loadig codiios was obaied. The resuls of cosiuive model showed ha here appears o be a expoeial fucio bewee srai ad ime i he acceleraed creep sage. The creep es daa were used o verify he proposed model ad he resuls suggesed ha he fracioal order b was he irisic parameers of rock mass which could reflec is hardess. Durig he acceleraed creep sage, fracioal creep damage cosiuive model could describe he sress-srai relaioship well. Key words: Fracioal calculus; Sress-srai relaioship; Damage mechaics; Mudsoe acceleraed creep Zeg, J., & Zhag, J. (215). Damage cosiuive model of mudsoe creep based o he heory of fracioal calculus. Advaces i Peroleum Exploraio ad Developme, 1(2), Available from: URL: hp:// DOI: hp://dx.doi.org/1.3968/773 INTRODUCTION Rock creep is oe of he impora mechaical properies of rock ad also is oe of he impora reasos for egieerig wall rock deformaio isabiliy. Be esablished o fully reflec he acceleraed creep characerisics of mudsoe rock creep model is a impora subjec of curre research of rock creep properies. Rock creep characerisics showed hree disic phases i he log erm loads. Aeuaio creep sage, he creep rae decreases, showig a sigifica o-liear pheomea; whe eerig he seady-sae phase, creep rae remaied a a cosa value, showig a early liear feaures; whe he creep io he acceleraio phase, he deformaio rae bega o icrease, showig a edecy o icrease he oliear acceleraio, showig a more sigifica oliear characerisics. I rece years, may scholars have doe a lo of research o mudsoe creep. Bu durig he sudy, he mai cosideraio is he firs creep sage ad secod creep sage. I fac, i he process of rock creep, he hird sage of creep properies has more sigificace [1]. Krag e al. (1977) hrough es resuls, he volume of rock was ielasic srai reaches a criical value, acceleraig creep phase bega ad led o he fial desrucio of he specime, failure ime associaed wih sress. Geevois e al. (1979) hrough he experime daa of iformaio, ge a graph which is reflecig he relaioship bewee sress raio ad acceleraig creep ime; Drago e al. (1979) proposed a viscoplasic model of rock, ad assumed crack desiy parameer reaches a criical value, creep rae icreases rapidly lead o rock failure; Okubo e al. (1911) poied ou a acceleraig creep sage, he relaioship bewee srai rae ad rock is iversely proporioal o he icrease [2]. Deg e al [3]. iroduced a oliear viscous dampers, ad used viscous damper o esablished comprehesive rheological mechaics model, his model ca describe hree kids of creep deformaio a he same ime. Che e al. [4] pu forward wo kids of oliear eleme o se up composie rheological mechaics model, ad his model ca well describe he characerisics of sof rock acceleraig creep sage. Cao e al. [5] chaged viscous coefficie o oliear, i viscosiy model, ad improved 83 Copyrigh Caadia Research & Developme Ceer of Scieces ad Culures

2 Damage Cosiuive Model of Mudsoe Creep Based o he Theory of Fracioal Calculus Nishihara Masao model. This model ca reflec he oaeuaio creep properies of rocks. Wag e al. [6] based o he improved Nishihara model esablished parameric oliear creep model, which could reflec he rock specimes of he hree sages of creep process, especially i oliear acceleraig creep deformaio sage. Yi e al. [7] used he heory of fracioal order calculus, pu forward a kid of sofware compoe, which is used o simulae he geoechical maerials bewee ideal solid ad fluid. Alhough he combied model which is made up of sofware compoe ad classical liear mechaical compoes, ca describe he oliear behavior of he aeuaio of he rock creep ad seady creep of he rock creep, ca depic he acceleraio of rock creep properies. Based o he above research, esablish mudsoe s fracioal calculus creep damage model srucure diagram, which describe he exe of damage i rock damage variable. Assumed ha he sress level of rock exceeded is he log-erm sregh ad uder he codiio of mudsoe acceleraed srai rae loadig, oliear damage model based o fracioal calculus heory was esablished. Therefore mudsoe creep cosiuive model cosiderig he acceleraed srai rae loadig codiios was obaied. The resuls of cosiuive model showed ha here appears o be a expoeial fucio bewee srai ad ime i he acceleraed creep sage. The creep es daa were used o verify he proposed model. The Laplace rasform formula of fracioal order calculus is: L f (),p = p f ( p), L f (),p = p f ( p). (4) ( f () i he viciiy of = ca be iegral, 1) I Equaio - f (p) is a Laplace rasform of f (). Accordig o he classical heory of solid mechaics ad fluid mechaics, he cosiuive relaios of ideal solid should mee he Hooke s law () - ε(), ideal fluid should saisfy Newo law of viscosiy () - d 1 ε()/ 1. Fracioal order calculus is a mahemaical problem, which is sudy ay order differeial, iegral operaor feaures ad applicaios. If chage he () - ε() o () - d ε()/, he, he geoechical maerials which is bewee ideal solid ad fluid, is fracioal order differeial form of sress-srai relaio is d ε ( ) ( ) = ξ. (5) Whe < < 1, Equaio (5) describe he sae of maer ha is bewee ideal solid ad fluid; whe > 1, Equaio (5) describe he acceleraig rheological sae of maer, his paper maily sudies accelerae rheological maerial saus. Accordig o his arraive, we call i he sofware compoes whe > 1 (as show i Figure 1), ξ is viscoelasic coefficie, similar o he elasic modulus of Hooke s law, he ξ dimesio is [sress ime ]. 1. NONLINEAR MODELING 1.1 The Basic Compoes of Fracioal Calculus Fracioal order calculus is expaded he order of calculus io areas of fracio or eve egaive umber. The sofware compoes coais fracioal order calculus is regarded as a compoe model. The model is bewee he ideals of solid ad fluid ad ca well reflec he viscoelasic characerisics of geoechical maerials. This aricle uses he Riema-Liouville fracioal calculus operaor heory, for order iegral of fucio f(), is defied as: d f ( ) d ( ) = f () = f (). (1) Fracioal differeial is defied as: d f ( ) d ( ) = D f () = D f (). (2) I Equaios (1) & (2): >, moreover - 1 < ( is a posiive ieger). Γ() is he Gamma fucio, is defied as: Γ 1 ( ) = e ( Re( ) ) >. (3) Figure 1 Fracioal Order Calculus of he Viscoelasic Eleme Whe () = cos, ha is sress is a cosa value, o Equaio (5) o he basis of Riema-Liouville fracioal order operaor heory o carry o he iegral operaio is ε ( ) =. (6) ξ Γ ( 1+ ) I he case of cosa sress whe ake differe values, aalysis of he relaioship bewee srai ad ime ad heir chage red. Creep curve as show whe > 1, based o he fracioal order creep model, he curves of srai wih ime as show i Figure 2. Combied wih he characerisics of ypical rock creep deformaio. Accordig o derivae ad chage he fracioal heory model, we could use o simulae creep curve. > 1 (Sofware eleme) describe he viscoplasic deformaio ha is acceleraig rheological maerial saus. Copyrigh Caadia Research & Developme Ceer of Scieces ad Culures 84

3 ZENG Jia; ZHANG Ju (215). Advaces i Peroleum Exploraio ad Developme, 1(2), ε Figure 2 > 1 Curve Diagram Whe ε() = cos, hrough he fracioal order calculus deduce he relaxaio equaio is ( ) = ξ ε. (7) Γ( 1 ) ε is he iiial sress of he iiial srai. For differe maerials, adjus he parameers ad ξ of compoes o chage he creep curve or relaxaio curve, hus accuraely fiig maerial es resuls. 1.2 I he Process of Rock Creep Damage Rock creep damage sress hreshold is he log-erm sregh of rock, whe he load sress is greaer ha he log-erm sregh of rock mass iself, wih he icrease of ime rock begis o creep damage. Based o damage mechaics heory, o accou of he ieral micro cracks of rock mass expadig, ecrypio, icrease uder he acio of exeral loadig, he abiliy o resis damage ad disorio of rock mass decreases causig damage, ad led o he decrease of he mechaical parameers of rock mass, so he rock damage degree relaed o he size of he load sress ad ime. Accordig o he defiiio of damage variable mehod, his paper discussed damage variable of rock mass damage is defied by adopig he mehod of elasic modulus (, ) (, ) 1 E D =. (8) E I Equaio (8), E is he iiial elasic modulus of rock mass; D (,) is he damage variable of rock mass i ime; E (,) is he elasic modulus of he rock mass i ime, maily relaed o Ouside he load sress level a his mome. Afer damage due o creep, rock mass has o be capable of bearig, accordig o Liu Baoguo mudsoe creep damage experime research [8], defiiio of E(,) is defie E (,) = E exp[- - /b]. (9) I Equaio (9), is he log-erm sregh of rock maerial which ca be deermied by ess; b is rock maerial cosa; - is sep fucio, is defied as: ( ) ( ) =. (1) > Take he Equaio (9) io he Equaio (8) D (,) = 1 - exp[- - /b]. (11) Whe he exeral load sress is greaer ha log-erm sregh of rock mass by (11), ha is >,, D = 1, rock mass maerials compleely damage. Wih he icrease of he ime ad sress, maerial elasic modulus decay gradually, damage variable was gradually icreased. Accordig o he Equaio (11) damage variable D defie effecive sress 8 ad omial sress mee relaios 8 = 1 D (,. (12) ) I Equaio (12), 8 is effecive sress, is omial sress. Take he Equaio (7) io he Equaio (8) 8 = exp [ - /b]. (13) I Equaio (13), is he log-erm sregh of rock maerial, could be deermied by ess; b is rock maerial cosa; - is sep fucio. 1.3 Mudsoe Fracioal Order Damage Creep Model The iiial load, rock mass maerials exiss elasic deformaio. Acceleraed creep of rock mass characerisics is based o ime ad exeral load sress chage gradually, here is damage hreshold, so by referece fracioal order damage compoes which could reflec he creep damage process o describe he viscoplasic acceleraig deformaio process of rock mass, model srucure as show i Figure 3. Figure 3 Fracioal Order Damage Creep Model Whe he exeral load sress is greaer ha log-erm sregh of rock mass, ha is mudsoe acceleraed creep, accordig o he model diagram available = 1 = 2 = 3, ε = ε 1 + ε 2 + ε 3. (14) Amog hem 1 = Eε1 d ε 2 2 = ξ1. (15) d ε3 3 = + ξ2 I Equaio (15), whe he load sress is less ha log-erm sregh of rock mass codiios, ha is Creep 85 Copyrigh Caadia Research & Developme Ceer of Scieces ad Culures

4 Damage Cosiuive Model of Mudsoe Creep Based o he Theory of Fracioal Calculus sage Ⅰ ad Creep sage Ⅱ mudsoe viscoelasic deformaio, < < 1. Laplace rasform ad Laplace iverse rasform of he Equaio (15) ad ake io he Equaio (14), for he resulig oliear creep cosiuive equaio whe he load sress greaer ha log-erm sregh of rock mass is ha: ε = + + E ξ1 ξ2. (16) Because of he load sress greaer ha log-erm sregh of rock mass, ha is, rock mass will eer he sage of creep damage. The Equaio (16) ca be rewrie as % ε = + +. (17) E ξ1 ξ2 Takig he iiial codiios = ad = io Equaio (17) ad accordig o he Riema-Liouville fracioal order calculus, acquire he oliear creep cosiuive equaio %. (18) E ξ Γ 1+ ξ Γ 1+ ( ) ( ) 1 2 Takig he Equaio (13) io (18) acquire rock creep damage cosiuive model which is based o he heory of fracioal order calculus, ha is exp b. E ξ1 Γ ( 1+ ) ξ2 Γ ( 1+ ) (19) 2. ACCELERATE THE STRAIN RATE LOAD THE MUDSTONE CONSTITUTIVE MODEL Kow o accelerae creep sage based o he heory of fracioal calculus of Rock creep damage cosiuive model is exp b. E ξ1 Γ ( 1+ ) ξ2 Γ ( 1+ ) Accordig o he acual siuaio, assumig ha ε = ae ad akig io Equaio (19), ca be calculaed uder he codiio of accelerae he srai rae load. The sresssrai relaioship of mudsoe creep is (lε l a) exp (lε l a) b (lε l a) E ξ Γ + ξ Γ + ( 1 ) ( 1 ) 1 2. (2) I Equaio (2), a is cosa. By Equaio (2), ca be deduced accelerae he srai rae load fracioal order he mudsoe creep damage cosiuive model is 1 ( ) 2 ( ) 1 ( ) ( ) ( ) ( ) ( ) E ξ Γ 1+ ξ Γ 1+ ε + E ξ Γ 1 + (lε l a) =. (21) ξ 1 Γ 1 + ξ 2 Γ E ξ 2 Γ 1 + (l ε l a) + E ξ 1 Γ 1 + (l ε l a) exp (l ε l a) b Whe he ouer load sress is greaer ha log-erm sregh of rock mass, ha is, rock mass will eer E 1Γ ( 1+ ) 2Γ ( 1+ ) + E 1Γ ( 1 + )(l l a) ( ) ( ) ( ) ( ) he sage of creep damage, ha is exp[- - /b], he Equaio (21) could be wrie as ξ ξ ε ξ ε =. (22) ξ Γ 1 + ξ Γ E ξ Γ 1 + (l ε l a) + E ξ Γ 1 + (l ε l a) Figure 4 Rock Creep Sress-Srai Curve Diagram Accordig o he Equaio (22), fiig he sresssrai curve diagram uder he codiio of rock creep, as show i Figure 4. From he sress-srai curve diagram aalysis wih srai icrease gradually, sress icreases gradually, he srai rae also will icrease. Namely ha i he process of mudsoe creep, he greaer he srai, he greaer he sress also. Whe he value is o a he same ime, accordig o he Equaio (22). Describe he sress-srai curve of he maerial as show i Figure 5. I ca be see from he Figure 5, wih he icrease of value, sress icreasig ampliude decreases. The smaller he value of, he greaer he depede variable, he chage of sress more obvious. Copyrigh Caadia Research & Developme Ceer of Scieces ad Culures 86

5 ZENG Jia; ZHANG Ju (215). Advaces i Peroleum Exploraio ad Developme, 1(2), Figure 5 The Rock Creep Sress-Srai Curve Diagram Uder Differe Values 3. THE FRACTIONAL ORDER MUDSTONE CREEP DAMAGE CONSTITUTIVE MODEL VALIDATION Because he coveioal riaxial es is srai rae loadig es ha Srai could be corolled, hrough he es verify he fracioal order mudsoe creep damage cosiuive model. To verify he correcess ad he raioaliy of he model i his paper, he auhor carries o he riaxial compressio experime, sample ake from he souh area of Daqig oil field s oil ad waer wells casig damage zoe. Through he experime obaied rock mass mechaics parameers ad sress-srai curve, compared o he experimeal curves ad fiig curve from Equaio (18), as show i Figure 6. Figure 6 Sress-Srai Curves Accordig o he Figure 6, accelerae srai rae loadig fracioal order mudsoe creep damage cosiuive model s resul agree well wih experimeal daa fiig curve, resuls idicae ha he correcess of he cosiuive model is esablished i his paper. Wih he icrease of srai rae, all he value of is 1.13, so he fracioal order of he same kid of mudsoe do chage wih cofiig pressure, ad be able o reflec he hardess of mudsoe. CONCLUSION (a) Accordig o classical heory of solid ad fluid mechaics, based o he heory of fracioal order calculus, iroducig damage variable, esablish acceleraed srai rae loadig fracioal order mudsoe creep damage cosiuive model. The model ca beer reflec he mudsoe of sress-srai relaioship i he acceleraed srai rae loadig codiios. (b) By acceleraed srai rae loadig fracioal order mudsoe creep damage cosiuive model, he relaioship bewee srai ad ime is expoeial fucio i he acceleraed creep sage, srai icreases, he sress icreases ad he srai rae also icrease. Wih he icrease of value, sress icreasig ampliude decreases. The smaller he value of, he greaer he depede variable, he chage of sress more obvious. (c) Coveioal riaxial es of mudsoe (acceleraed srai rae loadig) is able o verify he accelerae srai rae loadig fracioal order mudsoe creep damage cosiuive model. The model s resul agrees well wih experimeal daa fiig curve. The fracioal orders of he same kid of mudsoe do chage wih cofiig pressure, ad be able o reflec he hardess of mudsoe. I his paper he iovaio lies i he acceleraed srai rae loadig sage, he resuls of cosiuive model showed ha here appears o be a expoeial fucio bewee srai ad ime. The mudsoe creep cosiuive model cosiderig he acceleraed srai rae loadig codiios was esablished based o he heory of fracioal order ca describe he relaio bewee sress ad srai. REFERENCES [1] Sog, Y. J., & Lei, S. Y. (213). Mechaical model of rock oliear creep damage based o fracioal calculus. Chiese Joural of Udergroud Space ad Egieerig, (9), [2] Zhag, Z. T., Wag, H., & Tao, Z. Y. (1996). The sudy of rock creep propery. Joural of Yagze River Scieific Research Isiue, S1, 2-6. [3] Deg, R. G., Zhou, D. P., & Zhag, Z. Y. (21). A ew rheological model for rocks. Chiese Joural of Rock Mechaics ad Egieerig, 2(6), [4] Che, Y. J., Pa, C. L., & Cao, P. (23). A ew mechaical model for sof rock rheology. Rock ad Soil Mechaics, 24(2), [5] Cao, S. G., Bia, J., & Li, P. (22). Rheologic cosiuive relaioship of rocks ad a modifical model. Chiese Joural of Rock Mechaics ad Egieerig, 21(5), [6] Wag, L. G., He, F., & Liu, X. F. (24). No-liear creep model ad sabiliy aalysis of rock. Chiese Joural of Rock Mechaics ad Egieerig, 23(1), [7] Yi, D. S., Re, J. J., & He, C. L. (29). Sress-sai relaio of sof soil based o fracioal calculus operaor s heory. Chiese Joural of Rock Mechaics ad Egieerig, 28, [8] Liu, B. G., & Cui, S. D. (21). Experimeal sudy of creep damage of mudsoe. Chiese Joural of Rock Mechaics ad Egieerig, 1(29), Copyrigh Caadia Research & Developme Ceer of Scieces ad Culures