ELASTO-PLASTIC AND ELASTO-DAMAGE MODELS BASED ON STRAIN GRADIENT ELASTICITY WITH EVOLVING INTERNAL LENGTH

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1 ELASTO-PLASTIC AND ELASTO-DAMAGE MODELS BASED ON STRAIN GRADIENT ELASTICITY WITH EVOLVING INTERNAL LENGTH Triantafyllou 1,, A.E. Giannakopoulos T. Papatheoharis 1, P.C. Perdikaris 1 1 Laboratory of Reinfored Conrete Tehnoloy and Strutures Laboratory for Strenth of Materials and Miromehanis ICCES1 CRETE

2 STRAIN GRADIENT MODEL USED: We employ a simplifiation of form II Midlin s strain radient elastiity of just one internal lenth parameter. Cauhy stress tensor: τ = + λδ pq ii µ pq Double (non-lassial) stress tensor: m rpq = λδ pq ii x r + µ x pq r λ µ : internal lenth, and :Lame s onstants and : the strain tensor WHY THIS MODEL? It is the simplest possible strain radient model that an apture size effet of omposite materials suh as onrete of low porosity. ICCES1 CRETE

3 Should the internal lenth evolve with damae or plastiity? Internal lenth used by radient theories is assoiated with the mirostruture of the material. Plastiity and damae alter the inner struture of the material Material onstant Movement of disloations Mirorakin Sine in the inelasti reion the mirostruture evolves, then the internal lenth should evolve as well. ICCES1 CRETE

4 ICCES1 CRETE Free enery potential: ( ) +Ξ Ψ = Ψ p p r p r q x q x τ τ τ, ;,,, ;, = Ψ r pq r pq r qq r pp pq pq jj ii x x x x µ λ µ λ is the elasti strain enery: Ψ q: internal plasti variable and : plasti potential funtion Ξ PLASTICITY:

5 ICCES1 CRETE Clausius-Duhem dissipation inequality ( nd law of thermodynamis): + Ψ+ r pq rpq x m τ In the ase of plastiity, we have two internal variables: the internal lenth and the total plasti strain Dissipation rate (thermodynami flux rate of hane of the int. variable): Ξ + Ξ Ψ Ξ + Ξ + Ψ p p p p q q q q τ τ τ τ * Stamoulis and Giannakopoulos, 1

6 Experimental proof (e.. Voyiadjis and Abu Al-Rub, 5) Miro-torsion experiments Miro-bendin experiments Internal lenth law assumed ICCES1 CRETE

7 Why onsider a strain radient damae model? 1. Materials that exhibit strain softenin in uniaxial tests, i.e. after the maximum load is reahed there is no yield plateau but the load delines at inreasin displaements, are size sensitive.. The inelasti response of suh materials that manifests itself throuh mirorakin ouht to be non-loal. (FPZ: Frature Proess Zone) ICCES1 CRETE

8 Plastiity Damae Stiffness is unaffeted Loss of stiffness Plasti strain: Closure effet ICCES1 CRETE

9 Should the internal lenth inrease or derease with damae? Peerlins et al, 1996: Constant int lenth ; FPZ inreases with inreasin the int. lenth (FEM estimates) Geers et al, 1998: An inreasin int. lenth with an end value after a ertain level of strain is neessary in order to obtain preditions with finite elements of a damae zone with finite width. Paudier-Cabot et al., 4 : Aousti emission experiments; miromehanial aruments ICCES1 CRETE

10 Extend Ortiz, 1983, rate independent damae model to inlude radient effets. Gibb s enery for isothermal proess: G= 1 1 τ : C : τ+ τ C : τ A : free enery required to form the miroraks (distribution) Assume that the elasti omplianes have an additive struture: C = C + C A C o : elastiity tensor of the un-raked material C : added flexibility due to ative miroraks Damae rule (assumed): (all miroraks are ative) C = µ R( τ ) = µ I + II τ ( R ( τ ) R ( )) Experimental neessity µ : salar parameter whih is a measure of umulative damae R(τ ) : a response funtion of the material whih determines the diretion in whih damae takes plae and the open/lose mode of raks ICCES1 CRETE

11 When does a mirorak beome ative? Miroraks follow a tortuous path Cross effet: Craks an open when ated upon by ompressive stresses Craks ated upon ompressive stresses an remain losed: (d): mode I openin or tensile mode (a): mode II openin or ompressive mode due to ross effet Stiffenin effet (spindle shape hysteri loops in yli bendin) ICCES1 CRETE

12 We assume two internal variables, the damae parameter and the internal lenth, plus a onstraint that the internal lenth should be a funtion of damae: (µ) µ Dissipation inequality demands: 1 1 d = τ : C : τ+ τ C : τ+ 1 ( ) τ C : τ A d Rate of hane of the internal lenth: = µ d µ d C d = τ : RI : τ + τ : RII : τ + τ RI : τ+ τ : τ * µ A dµ E The irreversible harater of damae neessitates: µ and sine R and C are positive definite, it follows that: = d / dµ rit =.15 The internal lenth = 5mm should inrease with E= 45MPa damae µ / = / rit ν =.8 Suested (empirially) by Geers et al., ICCES1 CRETE

13 τ τ University of Thessaly Dept. of Civil Enineerin Estimation of A (D ase) from miromehanis Enery release rate: Chane of G due to inrease of damae: Averae*: Chane of G due to inrease of damae: Averae*: G = dg da dg da dg da dg da K I πτ = E + K E πτ = E 9E II sin ( ϕ) 8 dτ = a sin 4 dy 1 dτ = a 3E dy ( ϕ) * uniformly distributed anle φ E E for plane stress = E /(1 ν ) for plane strain ICCES1 CRETE

14 (µ The inelasti free enery is a funtion of damae: A = A ) = µ ( da / dµ ) A 1 t( ) da π = t( µ ) + µ a dµ 3 x µ=πa µ = a / H / A (3D) (D) = ( µ ) = ξ ( a) ICCES1 CRETE

15 Define a damae funtion: F ( τ) = τ : RI : τ + τ : RII : τ = F I + F II f t Example: F ( τ) = τ : τ + FI FII f τ : τ = + F Assume assoiated damae rule: RI τ) = I F ( R τ) = II + + II ( τ τ τ τ Damae surfae: π Φ ( τ ) = F ( τ) t ( µ ) = Instead of tryin to define the tensorial quantity R, it is easier to define the salar funtion F. ICCES1 CRETE

16 How to determine the evolution law of the internal lenth? Compression experiment (absene of radient effets) Τάση - σ (MPa) Μτακίνηση μβόλου - stroke (mm) Determine the relation between the strain and the damae parameter Stress-strain relationof Smith and Youn, 1955: τ = f ( / )exp( 1 / ) τ = 1/ E + µ t( µ ) =τ( ( µ )) ICCES1 CRETE

17 . Bendin experiment (the radient effet is oupled with the effet of damae) Inrease of int. lenth Inrease of stiffness Inrease of damae Derease of stiffness 35 Φορτίο - Ρ (kn) The total Stiffness is a funtion of both phenomena whih at simultaneously but an be separated in the analysis Μτακίνηση μβόλου - stroke (mm) ICCES1 CRETE

18 Le Belleo et al., 3 Size effet learly deteted Gradient damae model: Constant internal lenth & Monotoni bendin Calibration laked objetivity A monotoni experiment will not yield any information about the internal lenth evolution law. Unloadin and reloadin at different levels of damae is neessary in order to fully desribe a strain radient damae model for onrete ICCES1 CRETE

19 Future Work Use experiments on eometrially similar speimens of Fiber Reinfored Conrete (FRC) to alibrate and verify the strain radient damae model ICCES1 CRETE

20 Thank you for your attention,, Aknowledements We are thankful to Hrakleitos II projet of the Greek Ministry of National Eduation for basi researh on size effet phenomena. ICCES1 CRETE