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

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Explicit expressions for the crack length

correction parameters for the DCB, ENF, and

MMB tests on multidirectional laminates

Stefano BENNATI, Paolo FISICARO & Paolo S. VALVO University of Pisa

Department of Civil and Industrial Engineering Largo Lucio Lazzarino – 56126 PISA (PI) – Italy

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Standard mode I and mode II delamination tests

AECMA prEN 6033:1995: Determination of

interlaminar fracture toughness energy. Mode I G_Ic. ISO 15024:2001: Determination of mode I

interlaminar fracture toughness, G_Ic, for unidirectionally reinforced materials.

ASTM D5528-01(2007)e3: Standard Test Method for

Mode I Interlaminar Fracture Toughness of Unidirectional Fiber-Reinforced Polymer Matrix Composites.

Double cantilever beam (DCB)

JIS K 7086-1993: Testing methods for interlaminar

fracture toughness of carbon fibre reinforced plastics.

AECMA prEN 6034:1995: Determination of

interlaminar fracture toughness energy. Mode II G_IIc.

End notched flexure (ENF)

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Standard I/II mixed-mode delamination test

ASTM D6671/D6671M-06: Standard Test Method for Mixed

Mode I-Mode II Interlaminar Fracture Toughness of

Unidirectional Fiber Reinforced Polymer Matrix Composites.

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Simple beam theory (SBT) model

2 2 SBT I I 2 3 12 x P a G B E h =

Double cantilever beam (DCB)

End notched flexure (ENF)

2 2 SBT II II 2 3 9 16 _x P a G B E h =

Mode I energy release rate Mode II energy release rate

3 SBT DCB 3 8 x a C BE h = SBT 3 3 ENF 3 3 2 8 _x a C BE h + = ℓ

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Corrected beam theory (CBT) model

2 I [3 2( ) ] 11 1 where 1.18 / x zx x z zx E G E E G χ = − Γ + Γ Γ = 2 CBT I 2 I 2 3 I 12 ( ) x P G a h B E h χ = +

Mode I crack length correction parameter Mode I energy release rate

Double cantilever beam (DCB)

II 0.42 I χ = χ 2 CBT II 2 II 2 3 II 9 ( ) 16 _x P G a h B E h χ = +

Mode II crack length correction parameter Mode II energy release rate

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Laminated specimens

Double cantilever beam (DCB)

End notched flexure (ENF)

2 CBT I 2 I 2 I 1 ( ) P G a h B χ = + D II ? χ = 2 2 CBT II 1 2 II 2 2 II 1 1 1 ( ) 16 4 P h G a h B h χ = + + A D A D I ? χ =

Mode I crack length correction parameter Mode I energy release rate

Mode II crack length correction parameter Mode II energy release rate

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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-extension and

no bending-extension coupling;

iii) sublaminates connected by an elastic interface, which transmits both normal and tangential stresses;

iv) negligible non-linear effects.

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 tests are obtained as special cases.

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Exact analytical solution

Enhanced beam theory (EBT) model

Mode I and II energy release rates

2 2 EBT 0 EBT 0 I , II 2 _z 2 _x G G k k σ τ = = 2 2 I 1 2 1 2 2 1 0 2 2 1 2 1 2 1 2 1 2 1 2 1 2 2 5 II 1 0 2 5 5 1 1 5 2 2 1 2 1 2 1 2 ( )( tanh tanh ) [ ( )(1 sech sech ) 2 tanh tanh ], sinh 1 [ (1 coth ) ], 4 2 sinh where ( ) tanh tanh 2 (1 sech P b b B D b b a D b b a D P h a b Bh h b D b b λ λ λ λ λ λ σ λ λ λ λ λ λ λ λ λ λ λ λ λ τ λ λ λ λ λ λ λ λ λ λ − − = + + − + − = + − + = + + − − ℓ A A D 1bsechλ2b)

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Mode I and II energy release rates 2 EBT I 2 I 2 1 1 2 2 2 EBT II 1 2 II 2 2 1 1 1 5 1 1 ( ) 1 ( ) 16 4 P G a B P h G a B h λ λ λ ≅ + + ≅ + + D A D A D 2 1 1 1 1 2 1 2 1 1 2 5 1 1 2 (1 1 ) 2 (1 1 ) 1 2 ( ) 4 z z z z x k k k k h k λ λ λ = + − = − − = + C C D C C D A D

Roots of the characteristic equations of the governing differential equations

Enhanced beam theory (EBT) model

Approximate expressions

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Enhanced beam theory (EBT) model

Mode I and II energy release rates

2 EBT I 2 I 2 I 1 2 2 EBT II 1 2 II 2 2 II 1 1 1 ( ) ( ) 16 4 P G a h B P h G a h B h χ χ ≅ + ≅ + + D A D A D

Crack length correction parameters

I II ₂ 1 1 1 1 1 1 1 1 2 ( ) 4 1 2 z x h _h k h k χ χ = = + + A D D D C

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Carbon/PEEK composite (Reeder and Crews, 1992)

Specimen sizes

100 mm, 25.4 mm, 2 3 mm

L= B = H = h=

129 GPa, 10.1 GPa, 5.5 GPa

x y z zx

E = E = E = G =

Ply elastic constants

12 12 [0 // 0 ]

Stacking sequence

Crack length correction parameters according to CBT model

I 1.747, II 0.734

χ = χ =

Crack length correction parameters according to EBT model

I 1.731, II 0.541

χ = χ =

Application: unidirectional (UD) specimens

Interface elastic constants

3 3

31550 N/mm , 23150 N/mm

x z

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Application: unidirectional (UD) specimens

Comparison between CBT and EBT models

0 50000 100000 150000 200000 0 1 2 3 4 E x[MPa] χ I , χ II χ I χ II CBT EBT 0 5000 10000 15000 20000 0 1 2 3 4 G zx[MPa] χ I , χ II χ_I χ_II CBT EBT

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0 5000 10000 15000 20000 0 1 2 3 4 E z[MPa] χ I , χ II χ_I χ II CBT EBT

Comparison between CBT and EBT models

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Glass/epoxy composite (Pereira & de Morais, 2006)

Specimen sizes

100 mm, 20 mm, 2 6 mm

L= B= H = h=

33 GPa, 19 GPa, 8 GPa, 4.8 GPa

x y z zx

E = E = E = G =

2 6 2 2 6 2

[(0 /90) /0 //(0 /90) /0 ] Stacking sequence

Sublaminate extensional, shear, and bending stiffnesses

1 =86400 N/mm, 1 =10170 N/mm, 1 =66785 Nmm

A C D

Application: multidirectional (MD) specimens

I 1.153, II 0.541

χ = χ =

3 3

6147 N/mm , 4578 N/mm

x z

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Carbon/epoxy composite (Pereira & de Morais, 2008)

Specimen sizes

100 mm, 20 mm, 2 6 mm

L= B= H = h=

130 GPa, 8.2 GPa, 4.1 GPa

x y z zx

E = E = E = G =

2 6 2 2 6 2

[(0 /90) /0 //(0 /90) /0 ] Stacking sequence

Application: multidirectional (MD) specimens

I 1.903, II 0.569

χ = χ =

Sublaminate extensional, shear, and bending stiffnesses

1 =280380 N/mm, 1 =9130 N/mm, 1 =227550 Nmm

A C D

3 3

12735 N/mm , 7765 N/mm

x z

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Experimental validation (work in progress)

Double cantilever beam (DCB)

End notched flexure (ENF)

Specimen #5 0 50 100 150 200 0 5 10 15 20 25 30 Opening displacement, δ [mm] L o a d , P [ N ] Specimen #5 0.0 0.3 0.5 0.8 1.0 0 20 40 60 80 100 C o m p li an c e , C [ m m /N ] EXP EBT SBT Specimen #5 0.000 0.005 0.010 0.015 0 10 20 30 40 50 C o m p li an ce , C [ m m /N ] EXP SBT EBT Specimen #5 0 250 500 750 0.00 2.00 4.00 6.00 8.00 10.00 12.00 Mid-span deflection, δ [mm] L o ad , P [ N ]

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On the EBT model of the mixed-mode bending test

BENNATI, Stefano; FISICARO, Paolo; VALVO, Paolo Sebastiano (2013): An enhanced beam-theory model of the mixed-mode bending (MMB) test - Part I: literature review and

mechanical model, Meccanica, 48 (2), p. 443-462. URL: http://dx.doi.org/10.1007/s11012-012-9686-3 (Erratum: http://dx.doi.org/10.1007/s11012-013-9697-8).

BENNATI, Stefano; FISICARO, Paolo; VALVO, Paolo Sebastiano (2013): An enhanced beam-theory model of the mixed-mode bending (MMB) test - Part II: applications and results,

Meccanica, 48 (2), p. 465-484. URL: http://dx.doi.org/10.1007/s11012-012-9682-7

(Erratum: http://dx.doi.org/10.1007/s11012-013-9696-9).

References

BENNATI, Stefano; VALVO, Paolo Sebastiano (2013): An experimental compliance calibration strategy for estimating the elastic interface constants of delamination test specimens, AIMETA

2013 – XXI Congresso Nazionale dell’Associazione Italiana di Meccanica Teorica e Applicata

(Turin, Italy, September 17–20, 2013). URL: http://www.aimetatorino2013.it.

VALVO, Paolo Sebastiano; CORNETTI, Pietro (2013): Energetic estimation of the elastic interface constants for delamination modelling, AIMETA 2013 – XXI Congresso Nazionale

dell’Associazione Italiana di Meccanica Teorica e Applicata (Turin, Italy, September 17–20,

2013). URL: http://www.aimetatorino2013.it.