Modeling issues of the FRP detachment phenomenon

Transcription

1 Modeling issues of the FRP detachment phenomenon Elio Sacco in collaboration with: J. Toti, S. Marfia and E. Grande 1 Dipartimento di ngegneria Civile e Meccanica Università di Cassino e del Lazio Meridionale

2 Outline of the presentation Motivations Position of the problem Body nonlocal damage and plasticity model nterface damage model Coupled interface-body nonlocal damage model Numerical applications Conclusions 2

3 Motivations Croci, G., et al. (1987), ABSE Symp., Tokyo, Japan. Schwegler, G. (1994), 1th European Conf. Earthquake Engng, Vienna, Austria. Applications after the Marche-Umbria earthquake Advantages: weight/strength, easy installation, Careful design, new modeling problems. No ideological approach: no fan, no enemy! 3

4 Problem in the use of FRP: detachment phenomenon (sudden and brittle: undesired) FALURE MECHANSM Peeling of a thin layer of cohesive material from the external surface of the support free end detachment intermediate detachment 4

5 h Ω 1 Ω 2 F Scheme for detachment test: detachment force effective length. DOUBLE LAP TESTS b ROUND ROBN TEST 12 laboratories involved 28 tests performed Valluzzi et al. 212 Materials and Structures SNGLE LAP TESTS 5

6 RESULTS: Bond Resistance and Failure Modes F*max [kn] UNCAS / DL11 UNCH / DL11 CUT / DL55 UNPG / DL55 UNSALENTO/ SL SPAN/ SL CUT/ SL UNROME2 / DL11 UNNA / DL11 UNPD / DL55 UNPD / SL UPATRAS / SL UNROME3/ SL UNMNHO/ SL SNGLE LAP - DOUBLE LAP average: 5.3 stand. dev.: average: 4.7 stand. dev.: GLASS average: 7.1 stand. dev.: 1.2 CARBON G1 G2 G3 G4 G5 C1 C2 C3 C4 C5 S1 S2 S3 S4 S5 B1 B2 B3 B4 B5 specimens average: 8 stand. dev.: 1.8 STEEL BASALT 6

7 Available experimental results published before 211 with 12 mm Γ k k = f FC b G Fd bm btm f Specific fracture energy CNR-DT2 k G [mm] c [MPa] RRT Rilem importance of virtual testing 7

8 Position of the problem Assumptions: x N Ω 1 SUPPORT Ω 2 1) Ω 2 : linear elastic behavior FRP nteraction zone x T 2D plane strain 2) Ω 1 : nonlinear behavior: damage & plasticity 3) : nonlinear behavior: damage, unilateral contact & friction 8

9 Cohesive model for the body Ω 1 nonlocal damage and plasticity model Why a new model for cohesive material? damage: microcracks unilateral effect: opening and closure of microcracks plasticity: irreversible inelastic strains different behavior in tension and in compression nonlocal constitutive law, different material lengths in tension and in compression simple: governed by few parameters 9

10 Body nonlocal damage and plasticity model Constitutive law Stress (nominal) ( ) ( ) Ω σ = σ (1 Dt ) H sgn tr( e) ( e ) ( ) Ω + (1 D )(1 H sgn tr( ) c tension compression damage variables Heaviside function Effective stress σ = C( ε π) = Ce elasticity tensor plastic strain 1

11 Plasticity Yield function evolution law f branch of modified hyperbola ( σ) = A ( σ σ )( σ σ ) 1 y 2 + B + f t π = λ κ = dt π σ ( σ ) ( ) 1 σ y σ2 σ y y 2 2 Kuhn-Tucker conditions ( σ) λ f ( σ) λ, f, = 11

12 yield stress in compression σ Y = A + σ σ Y 2 Y σ y σ 2 σ 2 A σ 1 yield locus σ 1 B >.5 (, σ ) Y Y P σ P σ, σ ( ) Y Y 12

13 Damage in compression D Ω 1 c Ω Ω Ω 2 3 Dc = max min 1, Dc Dc = κ + κ history κ κ { { }} 3 2 with 3 2 nonlocal accumulated plastic strain κ 1 ( x) = c ( ) ( ) d ψ d ψ κ Ω Ω Ω x y y Ω c ( x y) u weight function u κ final accumulated plastic strain (compressive damage D Ω = 1 ) u c 13

14 Damage in tension { { }} Ω Ω Ω D = max min 1, D with D = nonlocal equivalent strain ε 1 eq ( x) = t ( ) eq ( ) d ψ d ψ ε Ω Ω Ω x y y Ω equivalent strain t D Ω 1 t t t history ( x y) t weight function ( k ( )) eq ε ε exp ε ε eq ε eq Condition D Ω t D Ω c 2 2 ε eq = e + e principal elastic strains damage in compression induces damage in tension Mazars, Pijaudier-Cabot, 1989 J. Engng Mech, ASCE 14

15 4 Tensile response Compressive response σ [MPa] σ [MPa] ε damage σ 2 σ [MPa] σ ε ELASTC DOMAN Ciclic response damage-plasticity ε

16 nterface model AA xn damaged part thickness = (1 D ) A D A Relative displacement s( x ) = u 2 ( x ) u 1 ( x ) xt2 xt1 undamaged part Constitutive law Alfano, Sacco 26 nt J Num Meth Eng ; ( ) d τ = τ + (1 D ) K c+ p ( D ( )) d τ = K s c+ p K N sn K = = H( sn ) = c p KT pt stiffness matrix contact vector sliding friction vector Sacco, Toti 21 nt J Comp Meth Engng Scie Mech 16

17 Evolution law of the inelastic slip if D > Yield function evolution law Kuhn-Tucker conditions ( τ ) φ = µ τ + τ = µτ + τ d d d d d N T N T p = ξ φ d τ T ξ, φ, ξφ = ( d) ( d τ τ ) 17

18 D Damage evolution law (Mode, Mode, Mode ) { { D }} = max, min 1, history D Y = Y 1 ( 1 η ) s s Y = Y + Y + Y Y = Y = Y = T1 T2 N N T1 T2 with T1 T2 N st1 st2 sn s N + T1 T2 2 N 2 T1 2 T2 η = s η + s η + s η s s s s s τ s s τ s s τ η η η N N N T1 T1 T1 T2 T2 T2 N = = T1 = = T2 = = f f f sn 2GcN st1 2GcT 1 st2 2GcT 2 first cracking relative displacements peak stresses full damage relative displacements fracture energies 18

19 D Damage evolution law (Mode, Mode, Mode ) { { D }} = max, min 1, history D Y = Y 1 ( 1 η ) Y = Y + Y + Y N T1 T N + T1 T2 2 N 2 T1 2 T2 η = s η + s η + s η s s s Mode τ G N τ T cn T1 Mode G ct 1 τ T τ 2T Mode G ct 2 s N f s N s N f s T 1 s T s T f 1 s T 1 s T 1 f s T 2 s T s T 2 f s T 2 s T 2 19

20 nterface-body damage coupling Ω 2 REA third layer glue glue inside the pores of Ω 1 REA thin layer Ω 1 Ω 1 combined damage from the body and interface microcracks due to the body damage microcracks due to the interface damage 2

21 Conjecture Damage in the third layer made of cohesive material due to: - detachment action D - degradation of the cohesive support material D Ω t x N Ω 2 x T the interface damage does not influence the body damage Ω 1 the body damage influences the interface damage Freddi, Frémond 26,. J. Mech. of Mater. and Struct. 21

22 Model 1 Body damage at the interface nterface damage D Ω t D coupled interface-body damage ( x) = max ( x), ( x) { } t D D D Ω c Marfia, Sacco, Toti, 212 Comp. Mech. 22

23 Model 2 Body damage at the interface nterface damage coupled interface-body damage D Ω t D W ( ) REA third layer c = D + 1 D τ τ τ Ω Ω A ( ) W KN sn c N τ = W A = Ω D A Ω ( 1 ) A = D A τ = K s D ( c+ p) W τ τ no analytic expression for D c 23

24 Numerical applications Application 1 : Tensile response: comparison of the two coupling approaches Application 2 : Evaluation of the detachment force F max as function of the adhesion length L b and of the initial damage state of the body Ω 1 Application 3 : Comparison between experimental and numerical data Application 4 : Analysis of a FRP-strengthened panel Geometry, boundary and loading conditions of the first application 1) Body Ω 1 Material properties E = 153 MPa ν = ε =.29 R= 15. mm G =.95 N/mm c 2 Ω 2 2) nterface K 3 N = KT = 27 N/mm µ =.5 N σt GcN GcT σ = = 4.7 MPa = =.34 N/mm 3) Body Ω 2 E = 16 MPa ν =

25 N Case 3 Case 2 Case 3 Case 2 Case 1 Application 1: tensile response Case 1 a) b) Response of whole structure nterface response σ [MPa] σ N [MPa] Case 1 Case 2 Case v [mm] Case 2 Case 3 Case 1 Case 3 Case 2 Case 1 Case 1 Case 2 Case 3 nterface constitutive law a) b) Case 3 Case 1 Case 2 c) Body response σ [MPa] s N [mm] Case 1 Case 2 Case 3 Case 3 Case 2 c) Case 1 ε =.16 Case 2 ε =.26 Case 3 ε =.36 1 Case ε x 1-4 Model 1 Model 2

26 Application 2 Maximum adhesion force (uncoupled and coupled theory) U=uncoupled theory F Ω 2 4 mm L b Ω 1 55 mm C=coupled theory 1 mm 25 mm 25 mm C1 U2 U3 C2 1 D Ω = % F MAX [N] U1 U4 C3 C D Ω = 5% D Ω = 7% D Ω = 9% L b [mm] L e (C4) 26

27 Application 3 Experimental campaign Geometry 4 15 F bonded zone X= 36 unbonded zone 244 F.22 u A 4 15 u B Adopted models: elastic support; uncoupled damage model; coupled Model 1; coupled Model 2. 27

28 Application 3 F u A u B Comparison experimental-numerical results 28 Carrara et al. 212, Composites B

29 Application 3 F u A u B Comparison experimental-numerical results ε[µε] S1(a) S1(b) S1(c) S1(d) S1(e) M1(a) M1(b) M1(c) M1(d) M1(e) ε[µε] S1(a) S1(b) S1(c) S1(d) S1(e) M2(a) M2(b) M2(c) M2(d) M2(e) x[mm] x[mm] Model a) 1 Model b) 2 Strain in the FRP along the bonded surface 29

30 Experimental failure mode 3

31 Application 4 Qualitative analysis of a FRP-strengthened panel F q base shear [kn] unreinforced panel reinforced panel (Model 1) reinforced panel (Model 2) reinforced panel (uncoupled model) reinforced panel (perfect masonry-frp adhesion) displacement [mm] 31

32 Conclusions Coupled model more reliable with respect to the uncoupled damage one Degradation process faster and smoother for the second approach of coupling than for the first one Coupled model able to reproduce experimental data regarding the detachment of FRP from the masonry support Future developments Comparisons with other experimental tests (panels, structures) Thermodynamic consistency 32 Thanks for your attention