Explicit expressions for the crack length correction parameters for the DCB, ENF, and MMB tests on multidirectional laminates

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1 1/17 Explicit expressions for the crack length correction parameters for the DCB, ENF, and MMB tests on multidirectional laminates Stefano BENNAT, Paolo FSCARO & Paolo S. VALVO University of Pisa Department of Civil and ndustrial Engineering Largo LucioLazzarino 5616 PSA (P) taly Web:

2 Standard mode and mode delamination tests Double cantilever beam (DCB) End notched flexure (ENF) /17 AECMA pren 6033:1995: Determination of interlaminar fracture toughness energy. Mode G c. SO 1504:001: Determination of mode interlaminar fracture toughness, G c, for unidirectionally reinforced materials. ASTM D558-01(007)e3: Standard Test Method for Mode nterlaminar Fracture Toughness of Unidirectional Fiber-Reinforced Polymer Matrix Composites. JS K : Testing methods for interlaminar fracture toughness of carbon fibre reinforced plastics. AECMA pren 6034:1995: Determination of interlaminar fracture toughness energy. Mode G c. CompTest 013 (Aalborg, April 4, 013)

3 Standard / mixed-mode delamination test Mixed-mode bending (MMB) 3/17 ASTM D6671/D6671M-06: Standard Test Method for Mixed Mode -Mode nterlaminar Fracture Toughness of Unidirectional Fiber Reinforced Polymer Matrix Composites. CompTest 013 (Aalborg, April 4, 013)

4 Simple beam theory (SBT) model Double cantilever beam (DCB) End notched flexure (ENF) 4/17 Mode energy release rate G C 1P a SBT = 3 B Exh Specimen s compliance 8a 3 SBT DCB = 3 BExh Mode energy release rate G C 9P a 16B Exh SBT = 3 Specimen s compliance 3a + l 3 3 SBT ENF = 3 8BExh CompTest 013 (Aalborg, April 4, 013)

5 Corrected beam theory (CBT) model Double cantilever beam (DCB) End notched flexure (ENF) 5/17 Mode energy release rate 1P G a h CBT = ( + χ 3 ) B Exh Mode crack length correction parameter Mode energy release rate 9P G a h 16B Exh CBT = ( + χ 3 ) Mode crack length correction parameter χ E Γ x = [3 ( ) ] 11Gzx 1+ Γ where Γ = 1.18 E E / G x z zx χ = 0.4χ CompTest 013 (Aalborg, April 4, 013)

6 Laminated specimens Double cantilever beam (DCB) End notched flexure (ENF) 6/17 Mode energy release rate P G a h CBT = ( + χ ) B D1 Mode crack length correction parameter χ =? Mode energy release rate P A h G a h CBT = 1 ( ) 16B D1 A1h + 4D + χ 1 Mode crack length correction parameter χ =? CompTest 013 (Aalborg, April 4, 013)

7 Enhanced beam theory (EBT) model Mixed-mode bending (MMB) Hypotheses: i) specimens split into two sublaminates having same extensional, shear, and bending stiffnesses; ii) general stacking sequence allowed, but no shear-extensionand no bending-extension coupling; iii) sublaminates connected by an elastic interface, which transmits both normal and tangential stresses; iv) negligible non-linear effects. 7/17 Results: i) complete, exact analytical solutionto the differential problem; ii) simplified, approximate expressions for the specimen s compliance, energy release rate, and mode mixity; iii) solutions for the DCB and ENF testsare obtained as special cases. CompTest 013 (Aalborg, April 4, 013)

8 Enhanced beam theory (EBT) model Exact analytical solution 8/17 Mode and energy release rates G EBT σ τ = = k 0 EBT 0, G z kx nterfacial stresses at the crack tip P ( λ1 λ )( λ1 tanh λb λ tanh λ1b) σ 0 = [ + B D ( λ1 + λ )(1 sech λ1b sech λb) + λ1λ a D λ1λ tanh λ1b tanh λb λ1λ a ], D P A h 1 sinh λl τ λ λ = [ (1 + 5a coth 5b) ], Bh A1h + 4D1 sinh λ5b where D = ( λ + λ ) tanh λ b tanh λ b λ λ (1 sech λ b sech λ b) 1 1 CompTest 013 (Aalborg, April 4, 013)

9 Enhanced beam theory (EBT) model Approximate expressions 9/17 Mode and energy release rates G G P 1 1 EBT ( a + + ) B D1 λ1 λ P A h EBT 1 ( a + ) 16B D1 A1h + 4D1 λ5 1 Roots of the characteristic equations of the governing differential equations kz C1 λ1 = (1 + 1 ) C k D 1 z 1 kz C1 λ = (1 1 ) C k D 1 z 1 1 h λ5 = kx ( + ) A 4D 1 1 CompTest 013 (Aalborg, April 4, 013)

10 Enhanced beam theory (EBT) model Crack length correction parameters 10/17 Mode and energy release rates P G a h EBT ( + χ ) B D1 P A h G a h EBT 1 ( + χ ) 16B D1 A1h + 4D1 Crack length correction parameters 1 D D χ = + h C k χ = h 1 h kx( + ) A 4D z 1 1 CompTest 013 (Aalborg, April 4, 013)

11 Application: unidirectional (UD) specimens Carbon/PEEK composite (Reeder and Crews, 199) 11/17 Specimen sizes L = 100 mm, B = 5.4 mm, H = h = 3 mm Ply elastic constants nterface elastic constants E = 19 GPa, E = E = 10.1 GPa, G = 5.5 GPa x y z zx Stacking sequence [01 // 01 ] k x = N/mm, k = 3150 N/mm 3 3 z Crack length correction parameters according to CBT model χ = 1.747, χ = Crack length correction parameters according to EBT model χ = 1.731, χ = CompTest 013 (Aalborg, April 4, 013)

12 Application: unidirectional (UD) specimens Comparison between CBT and EBT models 1/17 4 CBT EBT 4 CBT EBT 3 3 χ, χ χ χ, χ χ E x [MPa] χ χ G zx [MPa] CompTest 013 (Aalborg, April 4, 013)

13 Application: unidirectional (UD) specimens Comparison between CBT and EBT models 13/17 4 CBT EBT 3 χ, χ χ 1 χ E z [MPa] CompTest 013 (Aalborg, April 4, 013)

14 Application: multidirectional (MD) specimens Glass/epoxy composite (Pereira & de Morais, 006) 14/17 Specimen sizes L = 100 mm, B = 0 mm, H = h = 6 mm Ply elastic constants nterface elastic constants E = 33 GPa, E = 19 GPa, E = 8 GPa, G = 4.8 GPa x y z zx Stacking sequence [(0/90) 6/0//(0/90) 6/0] Sublaminate extensional, shear, and bending stiffnesses A = N/mm, C = N/mm, D = Nmm k x = 6147 N/mm, k = 4578 N/mm 3 3 z Crack length correction parameters according to EBT model χ = 1.153, χ = CompTest 013 (Aalborg, April 4, 013)

15 Application: multidirectional (MD) specimens Carbon/epoxy composite (Pereira & de Morais, 008) 15/17 Specimen sizes L = 100 mm, B = 0 mm, H = h = 6 mm Ply elastic constants nterface elastic constants E = 130 GPa, E = E = 8. GPa, G = 4.1 GPa x y z zx Stacking sequence [(0/90) 6/0//(0/90) 6/0] Sublaminate extensional, shear, and bending stiffnesses A = N/mm, C = 9130 N/mm, D = 7550 Nmm k x = 1735 N/mm, k = 7765 N/mm 3 3 z Crack length correction parameters according to EBT model χ = 1.903, χ = CompTest 013 (Aalborg, April 4, 013)

16 Experimental validation (work in progress) 16/17 Double cantilever beam (DCB) End notched flexure (ENF) 00 Specimen #5 750 Specimen #5 Load, P [N] Load, P [N] Opening displacement, δ [mm] Mid-span deflection, δ [mm] 1.0 Specimen # Specimen #5 Compliance, C [mm/n] EXP EBT SBT Compliance, C [mm/n] EXP SBT EBT Delamination length, a [mm] Delamination length, a [mm] CompTest 013 (Aalborg, April 4, 013)

17 References On the EBT model of the mixed-mode bending test 17/17 BENNAT, Stefano; FSCARO, Paolo; VALVO, Paolo Sebastiano(013): An enhanced beamtheory model of the mixed-mode bending (MMB) test -Part : literature review and mechanical model, Meccanica, 48(), p URL: (Erratum: BENNAT, Stefano; FSCARO, Paolo; VALVO, Paolo Sebastiano(013): An enhanced beamtheory model of the mixed-mode bending (MMB) test -Part : applications and results, Meccanica, 48(), p URL: (Erratum: On the estimation of the elastic interface constants BENNAT, Stefano; VALVO, Paolo Sebastiano(013): An experimental compliance calibration strategy for estimating the elastic interface constants of delamination test specimens, AMETA 013 XX Congresso Nazionale dell Associazione taliana di Meccanica Teorica e Applicata (Turin, taly, September 17 0, 013). URL: VALVO, Paolo Sebastiano; CORNETT, Pietro(013): Energetic estimation of the elastic interface constants for delamination modelling, AMETA 013 XX Congresso Nazionale dell Associazionetalianadi MeccanicaTeoricae Applicata(Turin, taly, September 17 0, 013). URL: CompTest 013 (Aalborg, April 4, 013)

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