A J integral approach for the determination of mixed mode cohesive laws

Transcription

1 A J iegral approach for he deermiaio of mixed mode cohesive laws Be F. Sørese Maerials Research Deparme Risø Naioal Laboraory Demark Torbe K. Jacobse LM Glasfiber A/S Demark EFP projec "Improved basis for desig of large wid urbie blades for large wid urbies of fibre composies phase 2" parly sposored by he Daish Eergy Agecy J. o. 1363/

2 Oulie 1. Wha is a cohesive law? 2. Approach for deermiaio of cohesive laws 3. Tes specime ad ex fixure 4. Example of measureme - delamiaio of a uidirecioal fibre composie

3 Wha is a cohesive law? Disribued ad localized pheomea

4 Fracure resisace Effec of fibre cross-over bridgig Log failure process zoe Icreasig fracure resisace Fracure Resisace J R (J/m 2 ) J 0 Iiiaio of crackig (o fibre bridgig) J ss Seady-sae crackig Opeig (m)

5 Cohesive law - a mahemaical descripio of he failure process zoe σ() Γ loc x 2 Γ ip x 1 σ J loc-jip 0 J loc = σ + 0 ( ) d Jip

6 Coceps of mixed mode cohesive laws ( ) σ = σ ( ) σ = σ σ σ Cohesive Zoe

7 Eergy poeial Φ( ) Assume ha he cohesive sresses ca be derived from a eergy poeial ( ) ( ) ( ) ( ) σ σ Φ = Φ =

8 J iegral aroud he cohesive zoe ( ) ( ) ( ) ip ip loc J J d d J + = Φ + + = 0 0 σ σ ( ) ( ) ( ) ( ) R R J J σ σ = = σ σ Γ loc

9 A mixed mode fracure specime Double cailever beam loaded wih ueve bedig momes (DCB-UBM) 2H M+M 1 2 M 2 J ex ( ) ( 2 2 ) 2 21 M + M = ν 4B 2 H 3 6M1M E 1 2

10 Trasverse Beam A P C P P B P D 2 1 x 2 DCB-specime x 1 Tes fixure - applyig ueve bedig momes

11 Mixed mode esig

12 Measuremes of ed-opeigs Exesomeer Δ E

13 Measuremes of ed-opeigs Exesomeer Δ E LVDT s L ad Δ 1 L Δ 2

14 Daa aalysis: deermiaio of ad Upper Crack Face Lower Crack Face H 2 θ 4 Neural Axis Δ 2 L Δ 1 L θ 5 H+Δ E θ 4 D Neural Axis H 2 Δ θ4 = ½ Arc a θ L 2 Δ D L 1 L π Δ1 + Arcsi 2 H + Δ ( θ ) 5 = θ 4 cos 2 E 4 = ( E H + Δ ) siθ5 H cosθ 4 ( E + ) cosθ5 = H Δ

15 Tes idea Tageial ed-opeig ϕ 0 0 J ss Perform ess wih differe load-raios o obai differe J R - - daa (differe opeig pahs) Normal opeig

16 Tageial Opeig (m) =-0.97 =0.94 =0.87 =0.50 =0.25 Ed-opeigs Examples =-0.52 Normal Opeig (m)

17 Tageial Opeig (m) =-0.97 =0.94 =0.87 =0.50 =0.25 Ed-opeigs Examples =-0.52 Normal Opeig (m) =0.94 =-0.87 Fracure resisace Fracure Resisace J R (J/m 2 ) =-0.97 =0.50 =0.25 =-0.52 Magiude of Opeig (m)

18 Experime measure M 2 Δ E L Δ Δ L 1 2 Daa aalysis Daa aalysis calculae J R Smooheig averaged daa ses Daa fiig polyomial surface J = f( ) R Cohesive laws Parial differeiaio J σ = R J σ = R

19 Daa fiig Fiig a surface (sum of double Chebyshev series) o J R - - daa; J R = J R ( ) J R k l ( x y) = a T ( x) T ( y) i= 0 j= 0 a ij are foud as he oes ha miimize he sum of squared derivaios (solvig he ormal equaios) ij i j

20 Examples - aalysed Normalized Fracure Resisace JR /J max 1.0 Normalised Normal Sress Normalized ageial opeig 0.8 σ /σ max 0.4 / max 0.2 Normalized ormal opeig / max Normalized Tageial Opeig Normalised Shear Sress 4 σ /σ max / max Normalized Tageial Opeig Normalized Normal Opeig / max / max Normalized Normal Opeig / max

21 Summary Deermiaio of mixed mode cohesive laws A ew J iegral based approach was developed Delamiaio of glass fibre/polyeser composie - he maximum ormal sress is relaive low (a few MPa) σˆ - he maximum shear sress is a order of magiude higher σˆ