Graphical representation of constitutive equations

Transcription

1 Graphal representaton of onsttutve equatons Ger Guehus, o. Prof. em. Dr.-Ing. Dr. h.. Professor Emertus Unversty of Karlsruhe Insttute of Sol Mehans an Rok Mehans Engler-unte-Rng Karlsruhe, Germany E-mal: Ger.Guehus@bf.un-karlsruhe.e Tel: , Fax: Dav Mašín, RNDr., M. Phl., Ph. D. Senor Leturer harles Unversty n Prague Faulty of Sene Insttute of Hyrogeology, Engneerng Geology an pple Geophyss lbertov Prague 2, zeh Republ E-mal: masn@natur.un.z Tel: , Fax: July 9, 28 Number of wors: 975 plus Referenes an fgure aptons Number of tables: Number of fgures: 7 epte verson of a Tehnal Note for Géotehnque orresponng author

2 Keywors onsttutve relatons, elastty, plastty, shear strength, stffness Introuton Many theores, base on fferent mathematal approahes, emerge from the researh on sol onsttutve moellng. Even though the moels may be evelope to be easy to use, wth low number of parameters an well-efne albraton proeures, ther mathematal formulatons are often rather omplex an only few sentsts, workng on ther evelopment, unerstan t nto etal. In ths Note we propose a way for graphal (as oppose to algebra) representaton of the most mportant sol propertes apture by the moels, whh eases jugment of ther response apabltes an reveals ther physal grouns wthout nee for unerstanng etals of ther mathematal formulatons. We fous on a polar representaton of tangental stffness for fferent stran rate retons an on representaton of state lmts, efne as states attane asymptotally by proportonal eformaton paths (.e., eformaton paths wth onstant stran rate retons). onsttutve relatons of rate type σ = f(σ, e, ǫ) () are stue, wth σ beng effetve auhy stress, e vo rato an ǫ the stran rate. In ths Note we restrt our attenton to ylnral symmetry. Moreover, rate-nepenene s presume,.e. the funton f satsfes f(σ, e, λ ǫ) = λf(σ, e, ǫ) (2) for any λ >. We onser purely frtonal materal whh annot attan tensle stresses. For ylnral symmetry the stress state may be fully haraterse by the effetve stress omponents σ an σ 2 = σ 3, an the stran rate by omponents ǫ an ǫ 2 = ǫ 3. To represent stress oblquty an stran rate reton, we efne an angle ψ σ n the plane σ vs. 2σ 2 (Fg. a), an angle ψ ǫ n the plane ǫ vs. 2 ǫ 2 (Fg. b). The followng speal retons of ψ σ an ψ ǫ wll be stngushe: Dreton whh orrespons to sotrop loang (σ = σ 2 an ǫ = ǫ 2 ) wth ψ σ () = an ψ ǫ () =, retons ± whh orrespon to rtal state ontons n ompresson an extenson wth ψ ǫ () = 9, ψ ǫ ( ) = 9 an ψ σ (±) relate to the rtal state frton angle ϕ through ( ψ σ (±) = ±tan 2 ) 2 snϕ (3) 3 snϕ an unattanable lmt bouns ± whh orrespon to axal splttng wth σ 2 = an ǫ = an sng wth σ = an ǫ 2 =,.e. ψ ǫ () 44.7, ψ ǫ ( ) 25.3, ψ σ () 54.7 an ψ σ ( ) Graphal representaton of tangental stffness Guehus (979) propose so-alle response envelopes as a powerful tool for representng onsttutve equatons of rate type. The response envelopes are polar agrams for unt stran rates,.e. for D = ǫ ǫ2 2 = (4) as plotte n the plane σ vs. 2σ 2 (Fg. 2). Response envelopes vsualse the tangental stffness prete by a moel for fferent ǫ retons (the larger the stane from the

3 orgn, the hgher the stffness), ther shape an sze epen on the onsttutve moel an on the state varables σ an e. To enhane the nformatve value of the polars, labels an graphal symbols are ae for partular retons of ǫ as efne n the Introuton. Response envelopes are often mappe from the stress rate spae nto the stress spae, whh reveals the nfluene of the stress state on the tangental stffness. Response envelopes for the Mofe am lay moel by Rosoe an urlan (968) are plotte n Fg. 3 n orer to emonstrate the representaton. The stress states orrespon to fferent postons wth respet to the ellpt yel surfae of the moel, whh s also nate n the fgure. Response envelopes for states nse the yel surfae (o an o2), orresponng to a hypoelast response, are ellpt an entre about the referene stress state. The elast shear an bulk moul G an K are reveale by the sze of the envelopes, ther rato s represente by the slenerness of the envelopes. The envelope for state o ( kpa) s twe as bg as envelope for state o2 (5 kpa), but ther aspet ratos are the same, showng that both G an K are proportonal to the mean stress an ther rato s onstant. The response envelope for an sotrop normally onsolate state () s ompose of two ellpt setons. oth are entre about the referene stress, one orrespons to elast unloang, the seon to elasto-plast loang. The response envelope s ontnuous thanks to the elasto-plast onssteny onton. The loang seton has another aspet rato than the unloang seton, showng a smaller bulk moulus for sotrop loang than for unloang. The envelopes for rtal states n ompresson () an extenson (-) are also ompose of two ellpt setons. The seton representng elasto-plast loang s now reue to a straght lne, revealng that the moel prets zero stress rate for elasto-plast loang at a rtal state. Graphal representaton of state lmts State lmts are efne as states approahe asymptotally by monotonous eformatons wth onstant ψ ǫ. They may be enote as attrators of sol behavour as they an be attane rrespetve of the ntal state. Smlar asymptot propertes were ntroue as swept-out-memory (SOM) by Guehus et al. (977), more etals an be foun n Guehus (28). State lmts represent funamental haratersts of sol behavour whh shoul be apture by onsttutve moels. They an be represente by plots as shown n Fg. 4. In the p vs e plane, physally amssble states are boun by the lmt vo ratos e an e. The lmt vo ratos erease affntvely wth nreasng p (Fg. 4a). relatve vo rato r e may be efne as r e = e e e e (5) wth e an e orresponng to the urrent mean stress p aorng to Fg. 4a. For state lmts r e epens on ψ σ as shown n Fg. 4b, an ψ ǫ epens on ψ σ as plotte n Fg. 4. For an sotrop ompresson () the vo rato e for a gven p s the hghest possble one, r e s maxmal an ψ σ an ψ ǫ are equal to from ther effntons. For proportonal strethng along ontratant paths the stress an stran rate angles ψ σ an ψ ǫ are n the setors between an, an r e exees. The rtal vo rato e that orrespons to strethng wth onstant volume (ψ ǫ (±)) s n between e an e. The two ψ σ for rtal states (labelle an ) are gven by Eqn. (3), relate ψ ǫ were also gven above. The same e s presume for an an a gven p, thus r e = hols by Eqn. (5). For latant paths the stress state s outse the setor between an, wth monotonous strethng the vo rato s nreasng an p s ereasng (therefore these states annot be approahe from a stress-free state), e < e hols an thus r e <. These state lmts ome lose to peak states, states of maxmum ψ σ an ψ ǫ aheve n shear tests wth onstant Verson wth onstant Posson rato ν s use, the elast shear stffness prete by the moel s therefore proportonal to the mean stress p = (σ + 2σ 2)/3 2

4 ψ σ (efne equvalently to ψ σ an ψ ǫ ). xal splttng wth σ 2 = an ǫ = an sng wth σ = an ǫ 2 = (labelle an ) are not attanable lmt bouns. They are relate wth a lower boun vo rato e,.e. wth r e =. omparson of onsttutve moels In the followng, we show how the propose graphal representaton an be use for omparson of response apabltes of fferent onsttutve moels. Three moels of fferent mathematal an same physal bases have been hosen the alreay mentone Mofe am lay moel (Rosoe an urlan 968), a hypoplast moel for granular materals (von Wolffersorff 996) an a hypoplast moel for lays (Mašín 25, Mašín 27). The moels have all been albrate to represent the behavour of one spef sol (eauare Marl, see Mašín et al. 26); the albraton an parameter values are not etale here, as they are not mportant for qualtatve omparson of the moel responses. Response envelopes by the two hypoplast moels are shown n Fg. 5. They are plotte for the same ntal states as use n Fg. 3, thus enablng a ret omparson wth the response by the Mofe am lay moel. Fgure 5 emonstrates that the response envelopes prete by the hypoplast moels are ellpt. Unlke n elasto-plastty, however, the ellpses are not entre about the referene state, so fferent tangental stffnesses are prete for fferent strethng retons (thus these moels are nrementally nonlnear). It may be note that the translaton wth respet to the referene state ereases wth nreasng overonsolaton from, through o to o2, representng the nfluene of overonsolaton rato (or relatve ensty). The nfluene of the mean stress, smlar to the Mofe am lay moel n Fg. 3, s reveale by the sze of the envelopes. It may also be seen that for rtal states (±) the response envelopes are shfte n suh a way that they touh the referene state n the pont orresponng to ψ ǫ (±) (see Fg. 2 for labels). The moels therefore pret zero stress rates for onstant volume strethng, as requre for rtal state ontons. Usng a representaton as n Fg. 4, state lmts prete by the three onsttutve moels are shown n Fg. 6. (vo rato lmts as by Fg. 4a are omtte). s the Mofe am lay an Mašín s moels o not spefy a lower boun e for the vo rato, r e n Eq. (5) was substtute by r e = e/e. Note that n the ase of these two moels the graph of Fg. 4b s ue to the logarthm ompresson law not unque for fferent p. These moels are formulate n suh a way that the state lmts an be represente unquely n an alternatve graph (not presente here) wth r e substtute by p e /p, were p e s Hvorslev s (937) equvalent pressure. It may be note that the urve by the Mofe am lay moel s not boun by retons an of ψ σ from Fg. a as s the urve by the Mašín s moel. Ths s beause the ellpt yel surfae of the Mofe am lay moel oes not exlue tensle stresses (see Fg. 3). lso, ths ellpt yel surfae mples an overpreton of the rtal state frton angle n extenson, whh s manfeste by fferent ψ σ for ponts n Fg. 6b for the Mofe am lay an other two moels, whh assume the same ϕ for ompresson an extenson. The urves n Fg. 6a are smlar for the three moels. Ths nates that the stran rate reton for state lmts, mple by the flow rule an elastty n the ase of the elasto-plast Mofe am lay moel, s prete smlarly by the hypoplast moels. Fg. 6 shows also that the ψ ǫ retons an of Fg. b o not orrespon exatly to retons ψ σ as efne n Fg. a an to r e =. Thus the unattanable lmts ± are not norporate presely nto the three stue onsttutve moels. To emonstrate asymptot propertes of the three onsttutve moels onsere, approahes to state lmts prete by them are plotte n Fg. 7 n planes of stress omponents (a,, e) an vo rato versus mean pressure (b,, f) for three stran paths retons. Dreton s ontratant an orrespons to K ontons wth ψ ǫ () 54.7, reton 3

5 s sohor an orrespons to reton wth ψ ǫ () = 9, reton s latant wth ψ ǫ () =. Intal states o not orrespon to state lmts. The state lmts aorng to Fg. 6 are nlue as thn otte lnes n Fg. 7. They are attrators,.e. they are reahe by proportonal strethng (onstant ψ ǫ ) nepenently of the ntal state. Fg. 7 shows how paths prete by the moels ten to these asymptotes. onlung remarks In ths Note, we propose graphal representatons of bas propertes of onsttutve rate equatons, namely tangental stffness an state lmts. The propose approah an be use to represent fferent onsttutve moels wthout nee for unerstanng etals of ther algebra representatons, an thus smplfes jugement of ther response apabltes. Usng the propose representaton we have emonstrate that the Mofe am lay moel an the selete hypoplast moels, though beng funamentally fferent algebraally, have the same physal grouns. knowlegement The seon author hghly appreates fnanal support by the researh grants GV I2765 an MSM Referenes Guehus, G. (979). omparson of some onsttutve laws for sols uner raally symmetr loang an unloang. In Pro. 3 r Int. onf. on Numeral Methos n Geomehans, pp ahen. Guehus, G. (28). Physal Sol Mehans. Sprnger, erln (n prnt). Guehus, G., M. Golsheer, an H. Wnter (977). Mehanal propertes of san an lay an numeral ntergraton methos: some soures of errors an bouns of auray. In G. Guehus (E.), Fnte Elements n Geomehans, pp Wley. Hvorslev, M. J. (937). Über e Festgketsegenshaften gestörter bnger öen. Ph. D. thess, Danmarks naturvenskabelge samfun, Københaven. Mašín, D. (25). hypoplast onsttutve moel for lays. Internatonal Journal for Numeral an nalytal Methos n Geomehans 29(4), Mašín, D. (27). hypoplast onsttutve moel for lays wth meta-stable struture. anaan Geotehnal Journal 44(3), Mašín, D.,. Tamagnn, G. Vggan, an D. ostanzo (26). Dretonal response of a reonsttute fne grane sol. Part II: performane of fferent onsttutve moels. Internatonal Journal for Numeral an nalytal Methos n Geomehans 3(3), Rosoe, K. H. an J.. urlan (968). On the generalse stress-stran behavour of wet lay. In J. Heyman an F.. Leke (Es.), Engneerng Plastty, pp ambrge: ambrge Unvesrty Press. von Wolffersorff, P.. (996). hypoplast relaton for granular materals wth a preefne lmt state surfae. Mehans of ohesve-frtonal Materals,

6 Lst of Fgures Defnton of angles ψ ǫ an ψ σ an speal retons,,, an Response envelope of stress rates ue to unt stran rates. Labels,,, an orrespon to retons of ψ ǫ as efne n Fg. b Response envelopes by the Mofe am lay moel for fferent ntal states 7 4 Graphal representaton of state lmts. mssble states le n grey zones. 7 5 Response envelopes for von Wolffersorff s (a) an Mašín s (b) hypoplast moels. Intal states the same as n Fg State lmts as prete by selete onsttutve moels, r e n (b) for the am lay an Mašín s moels plotte for p = kpa. Labels,,, an orrespon to retons of ψ ǫ as efne n Fg. b symptot propertes of the Mofe am lay moel (a, b), von Wolffersorff s moel (, ) an Mašín s moel (e, f). State lmts orresponng to strethng retons, an nlue as thn otte lnes, vo rato lmts e, e (an e ) by thn otte lnes wth labels

7 σ amssble state ε ψ σ 2 ψ ε 2 2ε 2 (a) 2σ 2 (b) Fgure : Defnton of angles ψ ǫ an ψ σ an speal retons,,, an. ε D = σ σ 2 2 2ε Fgure 2: Response envelope of stress rates ue to unt stran rates. Labels,,, an orrespon to retons of ψ ǫ as efne n Fg. b. 6

8 2 σ [kpa] o o2 yel surfae Fgure 3: Response envelopes by the Mofe am lay moel for fferent ntal states e r e ψ ε 44.7 e 9 e 35.3 e ψ σ (b) p [kpa] (a) () ψ σ Fgure 4: Graphal representaton of state lmts. mssble states le n grey zones. 7

9 2 σ [kpa] o o2-2 3 (a) 2 σ [kpa] o o2-2 3 (b) Fgure 5: Response envelopes for von Wolffersorff s (a) an Mašín s (b) hypoplast moels. Intal states the same as n Fg. 3. 8

10 ψ ε [ ] Mo. am lay von Wolffersorff Masn ψ σ [ ] (a) r e [-] Mo. am lay von Wolffersorff Masn ψ σ [ ] (b) Fgure 6: State lmts as prete by selete onsttutve moels, r e n (b) for the am lay an Mašín s moels plotte for p = kpa. Labels,,, an orrespon to retons of ψ ǫ as efne n Fg. b 9

11 4.8 e e σ [kpa] 2 e [-] (a) p [kpa] (b) e e 4.8 σ [kpa] 3 2 e [-].75.7 e () p [kpa] () e e σ [kpa] (e) e [-] p [kpa] (f) 2 3 Fgure 7: symptot propertes of the Mofe am lay moel (a, b), von Wolffersorff s moel (, ) an Mašín s moel (e, f). State lmts orresponng to strethng retons, an nlue as thn otte lnes, vo rato lmts e, e (an e ) by thn otte lnes wth labels.