Size Effects in Semi-brittle Materials and Gradient Theories with Application to Concrete

Transcription

1 PhD presentaton on Sze Effets n Sem-brttle Materals an Graent Theores wth Applaton to Conrete by Antonos Trantafyllou Department of Cvl Enneern Shool of Enneern, Unversty of Thessaly Volos, Greee Avsor: Prof. Phlp C. Perkars Co-avsor: Prof. Antonos E. Gannakopoulos January, 5

2 INTRODUCTION Mrorakn: branhn an rak fae brn Imaes reproue from the book Frature Proesses of Conrete by Van Mer

3 INTRODUCTION Complexty of the observe behavor of mrorakn Moe I frature Imaes reproue from the book Frature Proesses of Conrete by Van Mer

4 INTRODUCTION Moe I rakn n ompresson Imaes reproue from the book Frature Proesses of Conrete by Van Mer

5 INTRODUCTION Latte moels: Moeln the full etals of the mrostruture most mportant: partle struture Imaes reproue from Frature Proesses of Conrete by Van Mer

6 INTRODUCTION Ablty to pret the omplex behavor of mrorakn Imaes reproue from Frature Proesses of Conrete by Van Mer

7 INTRODUCTION Dpolar elastty Stran raent elastty: (D, ν=) Hooke s law extene to the n spatal ervatve of the Cauhy stran Non-loal moels: xx ( ) Exx A lenth sale parameter s neee for moeln materals suh as onrete exhbtn sze effet Stress-stran softenn law: f Sze effet n elastty: ( / Stffer stat response than that prete by lassal elastty f > ( / What s a physal nterpretaton of the nternal lenth? ) ) Normalze stress, σ/ft Normalze stran ε/ε t

8 HOMOGENIZATION ESTIMATES OF THE INTERNAL LENGTH The smplest ase of omposte materal (D ase of rular nlusons) Input parameters Matrx an nluson elastty propertes Composton value, =a/b Effetve materal propertes of omposte materal μ = f (μ m, ν m, μ, ν, ) ν = f (μ m, ν m, μ, ν, ) Estmaton of the nternal lenth U l U r

9 HOMOGENIZATION ESTIMATES OF THE INTERNAL LENGTH Graent nternal lenth to nluson raus rato, l/α P.%.%.%.% / / / / / / m m m m m m Composton, 7 P r θ a Loan ase 6 Loan Case 5 4.%.%.% Composton, Graent nternal lenth to nluson raus rato, l/α

10 STRUCTURAL ANALYSIS USING A DIPOLAR ELASTIC TIMOSHENKO BEAM q Tmoshenko knemats Cauhy stresses z uz ux x y C.G. z ux uy uz = z = = ψ(x) w(x) xx E xx xz kg xz Total stran enery: U U tot l U l _ V U xx r xx _ xz xz yz _ U yz xx xx xz xz xx xx r x x x x z z V Non-zero strans: xx xz u x x xz z u x u z z x w Term that shoul not be omtte

11 STRUCTURAL ANALYSIS USING A DIPOLAR ELASTIC TIMOSHENKO BEAM L L L L L tot w kag w EI w kag w EA w kag EI w kag w EA w kag EI U 4 nepenent BC w, w, ψ, ψ (bol are the lassal elastty varables) 4 eneretally onjuate quanttes Q, Y, M, m (Y, m: ouble shear or b-fore an ouble moment or b-moment) L L L L L m w Y M w Q w q W

12 STRUCTURAL ANALYSIS USING A DIPOLAR ELASTIC TIMOSHENKO BEAM Q M I A q Q w Q Q L w Q Y L Q M I A M M L M m L Equlbrum equatons Bounary ontons Close-form soluton 4 x / x / x / x / 4 4 x 6EI kag x kag q x 4EI q e e e e x )x ( w(x) x e e x EI kag x 6EI q x) ( 4 x / x / 8 onstants to be etermne from 8 B.C. s (4 at eah en)

13 STRUCTURAL ANALYSIS USING A DIPOLAR ELASTIC TIMOSHENKO BEAM Normalze Defleton,8,6,4, Tmoshenko beam (L/h=) Bernoull-Euler beam Fnte Elements Comparson wth FEM results,,4 /L,6,8 + Effet of /h /L= Normalze axal stran /L=.5 /L=. /L=.5 /L=. Clampe en: Bounary layer effet as n BE beam Free en: zero axal stans x/l

14 NON-LOCAL DAMAGE MODEL Gbb s enery: G τ : C : τ λb : λ A λ τ B (/ ) C Mrorakn as flexblty: C C C Openn an losn of mroraks: C P : C : P (postve projetons) n ε C : τ P ( C : τ ε C : τ P ( C : ( τ) ) ) Mnmzaton problem soluton for a ven state of stress (a) (b) C C IT C IC Atve mroraks: ε ε I IT C () ()

15 NON-LOCAL DAMAGE MODEL ) ( T I T I τ C R ) ( C I C I τ C R Damae rules: an Irreversble harater of amae:. Internal lenth s assume to be a funton of amae: ) ( A ) ( : ) ( ) ( : ) ( : : : : I I I T C T τ C τ τ R τ τ R τ τ R τ Enery ensty sspaton:, Assoate amae surfae: ) ( t τ : τ τ : τ / postve efnte an C IT C IC R IT R IC,,,

16 APPLICATION TO 4-POINT BENDING for, E E D )E ( for, E Defnton of amae ) / ( ) / ( f Assume stress-stran law: for D for ) / ( ) / ( D Damae funtons: ) / ( f E Youn s moulus an sotropy: t ) / ( f ) / ( f t t t t t t 4-pont benn: an assume stress-stran law s suffent nput nformaton for amae haraterzaton

17 APPLICATION TO 4-POINT BENDING Lnear stran strbuton: xx Input: a ven value for urvature, k m kz Fn: ε m that satsfes equlbrum (stress-stran law) Output: M, k pont on the response aram (sze nepenent) Input: k-δ knemat relaton (bounary value problem) h / N b xxz M h / b h / h / Fnal output: Fore vs. mspan efleton aram (sze epenent) xx zz 7 65 Prete benn moment apaty, M (knm) Prete mspan urvature, k (mm - x -6 ) Prete apple loa, P (kn) Prete mspan efleton, δ (mm)

18 APPLICATION TO 4-POINT BENDING Non-loal pretons: The loal M vs. k s transforme to M vs. k ra (saln of urvature-elastty) The raent soluton of the bounary value problem s use to obtan the knemat relaton, δ ra /k ra = f(/l) Ths relaton s not onstant but evolves wth amae sne =(D) Internal lenth evoluton law: e nd Prete benn moment apaty, M (knm) Loal Non-loal Prete appe loa, P (kn) Loal Non-loal Prete mspan urvature, k (mm - x -6 ) Prete mspan efleton, δ (mm)

19 APPLICATION TO 4-POINT BENDING Objetvty of the moel: Aoust tensor postve efnte Goo areement wth AE fnns: Intaton of softenn (D>) an therefore mrorakn Intense loalzaton of AE enery release rate maro-rak formaton (D>.95) Objetve amae haraterzaton: (for both = an >)

20 EXPERIMENTAL PROGRAM Apple loa, P Steel plate Alumnum frame DCDT, DCDT SG SG SG DCDT h=l/ Leather pa Pn support L/ L/ L/ SG SG b=h Mspan ross-seton SG D eometral smlar spemens

21 MATERIALS Mx Quanttes (K/m ) Cement (a) Areates (b) Water (w/ rato) Atves () Dry ensty (K/m ) Slump (m) Arontent () (%) CM (.65).6 - e.5 LC (.78) NC (.64) MC (.45) MC (.46) Seve openn (mm) % passn LC NC MC MC CM Four onrete mxes max= mm Low-strenth (LC) Normal-strenth (NC) Meum-strenth (MC an MC) Cement mortar max= mm

22 CLASSICAL ELASTICITY PROPERTIES Youn's moulus, E (GPa) 5 5 Posson's rato, ν Unaxal Compresson Splttn tenson 5 Unaxal Compresson Splttn tenson (vertal SGs of F.) Splttn tesnon (horzontal SGs of F. ) Compressve strenth, f (MPa) Compressve strenth, f (MPa) Classal elastty propertes use n the analyss Mx LC NC MC MC CM E, GPa ν.

23 INTERNAL LENGTH AND SIZE EFFECT IN ELASTICITY Stffness rato, Kexp/Kl CM = mm =5 mm Defleton ata Curv ature ata Stffness rato, Kexp/Kl LC = mm =5 mm = mm Def leton ata Curv ature ata Stffness rato, Kexp/Kl NC = mm =5 mm = mm Def leton ata Curv ature ata Beam heht, h (mm) Beam heht, h (mm) Beam heht, h (mm) Stffness rato, Kexp/Kl MC =5 mm = mm =5 m Delf eton ata Curv ature ata Beam heht, h (mm) Stffness rato, Kexp/Kl MC = mm =5 mm = m Def leton ata Curv ature ata Beam heht, h (mm) Elastty: P=K δ & P=S k Sze effet n elastty f: K exp /K l > S exp /S l >

24 INTERNAL LENGTH AND MICROSTRUCTURE Conrete ontans fferent areate szes n fferent volume fratons ompressve fber tensle fber Castn reton Averae nluson sze ranes between to mm Graent nternal lenth, (mm) All ata Curvature ata Defleton ata Youn's moulus, E (GPa) Frature surfaes an frature areates

25 INELASTICITY: PEAK LOAD Mx Compresson Unaxal stress-stran law parameters Tenson Youn s moulus f (a) ε ε (b) b (b) f t () /fsp (a) b t () ε t () ε t () E (a) (MPa) (x -6 ) (x -6 ) (x -6 ) (x -6 ) (GPa) LC NC MC MC CM (e) Cylner mheht ompressve stran (μs) at a(%) x f a=55% a=4% Compressve strenth, f (MPa) Strans at.5 x fsp (μs) Tensle stran Compressve stran Splttn strenth, f sp (MPa)

26 INELASTICITY: PEAK LOAD Peak loa (kn) Measure Prete Peak loa (kn) Measure Prete Moel pretons orrespon to a sze nepenent flexural strenth LC NC Sze, h (mm) Sze, h (mm) Peak loa (kn) Measure Prete MC Peak loa (kn) Measure Prete MC Satter of prete values orrespons to a +/- 5% evaton of the assume materal s tensle strenth Sze, h (mm) Sze, h (mm)

27 INELASTICITY: SOFTENING BRANCH LC mx:.9d 7e (mm) MC mx: D 8e (mm) Apple Loa (kn) sze S sze S sze S Non-loal moel pretons Apple Loa (kn) sze S sze S sze S Non-loal moel pretons Mspan efleton (mm) Mspan efleton (mm)

28 INELASTICITY: UNLOADING PATH Normalze loa at unloan Normalze mspan eformaton after unloan P /( D) pl K MC mx Apple loa (kn) Apple loa (kn) Apple loa (kn) 6 Expermental results 4 Non-loal preton Mspan efleton (mm) 6 Expermental results Non-loal preton Mspan efleton (mm) Expermental results 6 Non-loal preton Mspan efleton (mm)

29 EVOLUTION LAW OF GRADIENT INTERNAL LENGTH Non-loal parameters Mx LC NC MC MC (mm) Normalze nternal lenth, (D)/ n n=.9 n=. n=.65 n=. Proressvely stffer response f Apple Loa (kn) /D> (sze effet n nelastty) Cauhy stffness ereases wth amae: K/D = -K < But, K r >K l Sze S Loal pre. Expermental results Moel pretons Damae, D n nreases as brttleness nreases n the mxes Non-loal pre Mspan efleton (mm)

30 STRAIN GAGE MEASUREMENTS SG5 SG, SG, SG4 m m For >, S exp >S l or k exp <k l (P=Sk) 5 Apple loa, P (kn) SG4,5 SG, SG, S=7.66 Nm Mspan urvature, k (/mm) [ν] [Εt] [Ε].5.5 [] Measure stran Iε yi (μs) 5 5 SG,4 SG, Measure stran ε x (μs) Apple stress, σ (MPa) SG SG 5 5 Measure stran ε x (μs) Apple stress, σ (MPa) SG SG Measure stran Iε y I (μs)

31 STRAIN GAGE MEASUREMENTS mm SG (z=8 mm) z x mm Normalze loa, P/Ppeak LC-S- LC-S- 6 4 z (mm) Axal stress (MPa) Normalze loa, P/Ppeak Damae haraterzaton MC-S Axal stran (x -6 ) - LC-S- LC-S- -5 Plast strans: verfaton of proressvely stffer response n nelastty -5 D( ) E( ) (,, xx ), xx Axal ompressve stran (x -6 )

32 SIZE EFFECTS Sze effet n elastty (>) Sze effet n nelastty (=(D) wth /D>) Sze effet on strenth Soures/Explanatons of sze effet on strenth. Frature mehans sze effet. Statstal. Stress restrbuton at meso-sale (Latte moels) 4. Mult-fratal sze effet. Dffuson phenomena. Hyraton heat mrorakn urn ooln. Wall effet ue to nhomoeneous bounary layer

33 SIZE EFFECT ON FLEXURAL STRENGTH Flexural strenth, σν (MPa) Measure Prete LC Sze, h (mm) Flexural strenth, σν (MPa) Measure Prete NC Sze, h (mm) Flexural strenth, σν (MPa) Measure Prete MC Sze, h (mm) 7 7. Measure Measure Statstal sze effet 6 6 Prete Prete Flexural strenth, σν (MPa) 5 4 MC Sze, h (mm) Flexural strenth, σν (MPa) 5 4 CM Sze, h (mm) N h -/8 Observe sze effet on flexural strenth annot be attrbute to statstal soures

34 SIZE EFFECT ON FLEXURAL STRENGTH LC NC MC MC LC NC MC MC..4. Normalze moulus of rupture fr/ft Normalze strenth, fsp/ft Nomalze sze, h/l Normalze sze, D yl /l Lmte sale rane, :.5 :, for examnn sze effet on strenth

35 CONCLUDING REMARKS. Expermental quantfaton of the nternal lenth assume by smplfe polar elastty. Physal orrelaton between the nternal lenth an materal s mrostruture 5. Use of SG s an ontnuous stran raent moels 7. Sze effet on strenth. Conrete an be seen as a moel materal for stuyn sze effets n elastty 4. Expermental verfaton of key assumptons 6. Further expermental work s neee

36 FUTURE WORK Introun sze effet on strenth n the propose moel for the ase of onrete Refnement of the moel Inlue effet of strbute amae to the Tmoshenko moel Sze epenent amae haraterzaton (Van Vlet & Van Mer) Damae haraterzaton not base on Cauhy strans alone: D(ε,ε,x ) Stran raent latte moels pretn sze effet on strenth?

37 FUTURE WORK Fber Renfore Conrete (FRC) 4-pont benn experments of FRC = 8 mm of matrx = 4 mm same orer of mantue as prete by the smplfe D ase 45 4 Graent nternal lenth, (mm) 8 5 FRC Youn's moulus (GPa) Apple loa (kn) Mspan efleton (mm) Very utle softenn response b t parameter of stress-stran n the rane of.5 For FRC, n. Therefore, (D)= (the thermoynam lmt of /D= s reovere)

38 FUTURE WORK Mro-nerta lenth an ynam loan: Hamlton s prnple: t t U tot W K t Eqs. of moton an raent lenth: stffness relate an nerta relate h A I M Q q h m Aw Q h Dmensonless phase veloty, /s A I h m Class, (+) sn Graent, /H=., h/=.9 Graent, /H=., h/=. Graent, /H=., h/= I Class, (-) sn Graent, /H=., h/=.9 Graent, /H=., h/=. Graent, /H=., h/= kr

39 ACKNOWLEDGMENTS Laboratory of Renfore Conrete Tehnoloy an Strutures (Dretor: Prof. P. C. Perkars) & Laboratory for Strenth of Materals an Mromehans (Dretor: Prof. A. E. Gannakopoulos)

40 PhD presentaton on Sze Effets n Sem-brttle Materals an Graent Theores wth Applaton to Conrete By Antonos Trantafyllou Department of Cvl Enneern Shool of Enneern, Unversty of Thessaly Volos, Greee Avsor: Prof. Phlp C. Perkars Co-avsor: Prof. Antonos E. Gannakopoulos