Modelling Hydromechanical Dilation Geomaterial Cavitation and Localization

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1 Modelling Hydomechanical Dilaion Geomaeial Caviaion and Localizaion Y. Sieffe, O. Buzzi, F. Collin and R. Chambon Absac This pape pesens an exension of he local second gadien model o muliphasic maeials (solid paicles, ai, wae) and including he caviaion phenomenon. This new developmen was made in ode o model he esponse of sauaed dilaan maeials unde deviaoic sess and undained condiions and possibly, in fuue, he behaviou of unsauaed soils. 1 Inoducion A chaaceisic of geomaeials is o develop inelasic volume change. Clays, sands, ocks and concee ae dilaan maeials, i.e. he poosiy inceases in he plasic egime. In he case of sauaed sae, he poes ae sauaed wih fluid. Wih a maeial pemeabiliy dependence, he poe volume inceases moe apidly han he fluid can flow inside. Then he fluid is in ension which leads o a decease of he poe pessue unil negaive poe pessue could be achieved. This well-known poblem in numeical modelling leads o a dilaan hadening behaviou because he decease of poe wae pessue is coupled wih an incease of effecive sesses (compession). This is paiculaly poblemaic wih consiuive Y. Sieffe (*) R. Chambon Laboaoie 3S-R, Genoble-INP, CNRS, Genoble Univesié Joseph-Fouie, B.P. 53X, Genoble Cedex, Fance yannick.sieffe@3s-genoble.f O. Buzzi Pioiy Reseach Cene fo Geoechnical and Maeials Modelling, The Univesiy of Newcasle, Callaghan, NSW, Ausalia F. Collin Depamen of AGEnCo Insiu de Mécanique e Génie Civil, Univesié de Liège, 1 Chemin Des Cheveuils, 4000 Liège, Belgium Spinge Inenaional Publishing Swizeland 2015 K.-T. Chau and J. Zhao (eds.), Bifucaion and Degadaion of Geomaeials in he New Millennium, Spinge Seies in Geomechanics and Geoengineeing, DOI / _3 13

2 14 Y. Sieffe e al. equaions modelling he degadaion of he sengh of maeials as pessue of poe wae may have an influence on shea band fomaion (Loe and Pevos 1991). 2 Pesenaion of he Model Befoe saing, i is impoan o specify he main limiaions of his wok. The fis esicion of his sudy is ha we deal only wih quasi-saic poblems in unsauaed condiions, unde Richad s assumpions (vapou wae pessue is consan). Fuhemoe fo he sake of simpliciy, isohemal condiion, incompessible solid gains, incompessible vapou wae, no flow fo gas (no eny of ai) ae assumed. Howeve phase changes beween fluid and vapou wae ae consideed. In all he compuaion lage sains effecs ae aken ino accoun. As in Collin e al. (2006), he poe fluid and wae vapou ae assumed no influence a he micosucue level, mico kinemaic gadien is no geneaed by poe pessue and vapou wae vaiaions. The unknowns of he second gadien mechanical and he flow poblems ae especively he (maco) displacemen u i, he mico kinemaic v and he poe wae pessue p w (possibly negaive in unsauaed case). In ode o ge second gadien models, we add he assumpion ha he mico kinemaic gadien v is equal o he maco displacemen gadien F. This implies simila elaions fo viual quaniies. v = u i xj = F σ In he famewok of Schefle s sess, he effecive sess is: is he oal sess, σ is he effecive sess, pw, is he fluid pessue, p v is he vapou pessue, S w, is he wae elaive sauaion and δ is Konecke s dela. S v, σ = σ + S w, S w, is he vapou elaive sauaion. The mass densiy of he mixue is: mix, = s( 1 φ ) + S w, p w, δ + ( 1 S w, ) p v δ + S v, = 1 w, φ + ( 1 S w, ) vφ S is he solid gain densiy (assumed o be incompessible, i.e. S = ce), w, is he fluid densiy, φ is he poosiy defined as φ = p, / whee is he cuen volume of a given mass of skeleon and p, he coesponding poous volume. In weak fom (viual wok pinciple), he momenum balance fo he mixue can hus be wien as:

3 Modelling Hydomechanical Dilaion Geomaeial σ ui xj + 2 ui d x = mix, g i u k j x i d + ( ι ui + T ι Dui )dŵ k whee ui is any kinemaically admissible viual displacemen field, σ is he Cauchy sess (oal sess), k is he double sess dual of he viual second mico kinemaic gadien, x i ae he cuen coodinaes, g i is he gaviy acceleaion, i is he exenal (classical) foces pe uni aea and T an addiional exenal i (double) foce pe uni aea, boh applied on a pa Ŵ of he bounday of D denoes he nomal deivaive of any quaniy q. In ode o use C 0 funcions fo he displacemen field (i.e. only fis deivaives of he displacemen), he following equaion wih λ Lagange muliplies is used. σ u v ( ) i x + d j x λ ui k k x v d j = mix, g i u ( i d + i ui + TDu ) i dŵ, Ŵ ( ) λ ui xj v d = 0 In volume, he liquid fluid mass is equal o M w, vapou fluid mass is equal o M w, = S v, vφ. 15 = S w, w, φ and he In weak fom, he mass balance equaion fo he fluid and wae vapou can hus be wien as: ( (Ṁw, + Ṁ v,) p w m w, i = Q w, p w d Ṁ w, = w, (ṗw, ( Ṁ v, = v k w Sw, Ŵ q ( 1 S w, m w, i = w, k kw,τ 1 µ W Wih Ṁ w, is he ime deivaive of he fluid phase, m w, i is he mass flow of wae, k w is he fluid bulk modulus, k is he ininsic pemeabiliy, k w, is he wae p x i ) d q w, p w dŵ, φ + Ṡ w, Ŵ φ S w, ) Ṡ w, φ ), ( p w, ) + w, g i x i ),

4 16 Y. Sieffe e al. elaive pemeabiliy, µ w is he fluid viscosiy, Q w, is a sink em and Ŵq is he pa of he bounday whee he inpu fluid mass pe uni aea q w, is pescibed. 2.1 Mechanical Consiuive Law In ode o epoduce he pogessive decease of he maeial sengh, he mechanic consiuive law used in his sudy is an elasoplasic sain-sofening Duke-Page model wih yield cieion given by he follow equaion (Banichon 1998): f = II σ mi σ + k = 0 6ccosφ c whee m and k is defined by: m = 2 sin φ c 3(3 sin φc ), k = 3(3 sin φc ). φ c is he ficion angle, c is he cohesion, I σ = σ ii is he fis invaian and 1 II σ = 2 σ σ is he second deviaoic invaian. A geneal non-associaed plasiciy famewok is consideed o define he ae of plasic flow as pependicula o he plasic poenial g : ε p g = λ σ and g = II σ m I σ = C 1 wih m = 2 sin ψ wih, ψ is he dilaancy angle and c 3(3 sin ψ) 1 is a consan. The sofening pocess duing plasic flow is inoduced via an hypebolic vaiaion of he cohesion beween iniial c 0 and final c f values as a funcion of he Von Mises equivalen plasic sain εeq p : c = c 0 + (c f c 0)εeq p β c +εeq p. 2.2 Model Paamees The Ducke-Page model pesens he advanage o use simple fomulaion and does no equie enough paamees. All paamee values ae pesened in Tables 1 and 2. Table 1 Mechanic paamees Geomechanical chaaceisics Young s elasic modulus (MPa) 300 Poisson s aio ( ) Iniial cohesion (kpa) c Final cohesion (kpa) c f 100 Sofening paamee ( ) β c 0.01 Ficion angle ( ) φ c 18 Dilaion angle ( ) ψ 10 Solid specific mass (kg/m 3 ) s 2,026 Second gadien paamee (N) D 150

5 Modelling Hydomechanical Dilaion Geomaeial 17 Table 2 Hydaulic and ai paamees Hydaulic chaaceisics Iniial poosiy ( ) φ 0.39 Ininsic pemeabiliy (m/s) k 10 7 Iniial elaive wae pemeabiliy ( ) 1 Wae specific mass (kg/m 3 ) w 1,000 Wae dynamic viscosiy (Pa s) μ w 10 3 Wae compessibiliy coefficien (MPa 1 ) 1/k w Ai chaaceisics Gas pessue (kpa) p v Caviaion Model In he model, befoe caviaion, he sess sae govens he poe pessue and he specimen is sauaed. Afe caviaion, a phase change akes place and he poe pessue is elaed o he elaive degee ( ( of sauaion as pe he caviaion equaion 1 s w, ) ) /100 below: p w, 0.02 = p c C 2 + C 2 exp fo p w, < p c wih C 2 = 40 kpa in his pape. 3 Resuls Visualizaion of he localizaion shea bands is pefomed by obseving he loading index of a Gauss poin fo a given ime sep. When a Gauss poin undegoes a plasic loading, a small squae is ploed. No make appeas if he elemen undegoes elasic loading o unloading (Fig. 1). The numeical esuls ae displayed in he fom of load-displacemen cuve and poe pessue inside he shea band cuve in Fig. 2. Fis, he biaxial es leads o a homogenous soluion, i.e. he plasiciy behavio is idenical in he enie sample (Fig. 1a). As all elemens of he model ene ino plasiciy, global dilaion akes place and poe pessue sas dopping seadily. The educion in poe pessue uns ino an incease of effecive sess. The homogeneous soluion is sable unil he poe pessue achieves he caviaion pessue. Then, he wae sas o change in vapou phase (Figs. 1b and 2). The effecive sesses can decease feely and localized bands ae obained. In accod wih he expeimenal esuls given by Mokni and Desues (1998), when he plasiciy is fis obained wih an homogeneous behavio, he caviaion igges sain localizaion and hen caviaion occus befoe localizaion.

6 18 Y. Sieffe e al. Fig. 1 Visualizaion of he shea bands using he loading index a he gauss poin a axial sain = 0.14 %, b axial sain = 1.8 %, and c axial sain = 2.5 % Fig. 2 Global cuves of he loading foce and wae pessue vesus he specimen shoening 4 Conclusion This pape pesens numeical invesigaions of sain localizaion fo a dilaan maeial. To ensue objeciviy of he fomulaion, i.e. no mesh dependency, a second gadien hydomechanical model is used. This pape gives a numeical implemenaion of he caviaion phenomena o esolve he challenge of he unealisic negaive poe pessue obained ypically wih a dilaan poous maeial which could lead a a global hadening esponse. Based on he numeical simulaion, he caviaion is capable of esoing he shea bands fomaion.

7 Modelling Hydomechanical Dilaion Geomaeial 19 Refeences Banichon J (1998) Finie elemen modelling in sucual and peoleum geology. Ph.D. hesis, Univesiy of Liège, Belgium Collin F, Chambon R, Chalie R (2006) A finie elemen mehod fo poo mechanical modelling of geoechnical poblems using local second gadien models. In J Nume Mehods Eng 65: Loe B, Pevos JH (1991) Dynamics sain localisaion in fluid-sauaed poous media. J Eng Mech 11: Mokni M, Desues J (1998) Sain localizaion measuemens in undained plane-sain biaxial ess on Hoson RF sand. Mech Cohesive-Ficional Mae Suc 4:

Orthotropic Materials

Orthotropic Materials Kapiel 2 Ohoopic Maeials 2. Elasic Sain maix Elasic sains ae elaed o sesses by Hooke's law, as saed below. The sesssain elaionship is in each maeial poin fomulaed in he local caesian coodinae sysem. ε