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Computational & Multiscale CM3 Mechanics of Materials www.ltas-cm3.ulg.ac.be Simulations of composite laminates inter- and intra-laminar failure using on a non-local mean-field damage-enhanced multi-scale method Ling Wu (CM3), L. Adam

  1. Computational & Multiscale CM3 Mechanics of Materials www.ltas-cm3.ulg.ac.be Simulations of composite laminates inter- and intra-laminar failure using on a non-local mean-field damage-enhanced multi-scale method Ling Wu (CM3), L. Adam (e-Xstream), B. Bidaine (e-Xstream), Ludovic Noels. (CM3) Experiments: F. Sket (IMDEA), J.M. Molina (IMDEA), A. Makradi (List) STOMMMAC The research has been funded by the Walloon Region under the agreement no 1410246-STOMMMAC (CT-INT 2013-03- 28) in the context of M-ERA.NET Joint Call 2014. SIMUCOMP The research has been funded by the Walloon Region under the agreement no 1017232 (CT-EUC 2010-10-12) in the context of the ERA-NET +, Matera + framework. EMMC15 7 - 9 September 2016, Brussels, Belgium CM3

  2. Content • Introduction – Failure of composite laminates – Multi-scale modelling – Mean-Field-Homogenization (MFH) • Micro-scale modelling – Incremental-Secant MFH – Damage-enhanced incremental-secant MFH • Multi-scale method for the failure analysis of composite laminates – Intra-laminar failure: Non-local damage-enhanced mean-field- homogenization – Inter-laminar failure: Hybrid DG/cohesive zone model – Experimental validation • Introduction of uncertainties – As a random field CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 2

  3. Failure of composite laminates • Difficulties Fiber rupture – Different involved mechanisms at different scales Debonding • Inter-laminar failure • Intra-laminar failure Pull out – Direct finite element simulation Delamination Bridging Matrix rupture On Micro-scale volume Not possible at structural scale CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 3

  4. Failure of composite laminates • Difficulties Fiber rupture – Different involved mechanisms at different scales Debonding • Inter-laminar failure • Intra-laminar failure Pull out – Direct finite element simulation is not Delamination Bridging Matrix rupture possible at structural scale – Continuum damage models do not represent accurately the intra-laminar failure • Damage propagation direction is not in agreement with experiments CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 4

  5. Failure of composite laminates • Difficulties Fiber rupture – Different involved mechanisms at different scales Debonding • Inter-laminar failure • Intra-laminar failure Pull out – Direct finite element simulation is not Delamination Bridging Matrix rupture possible at structural scale – Continuum damage models do not represent accurately the intra-laminar failure • Damage propagation direction is not in agreement with experiments • Solution: – Embed damage model in a multi-scale formulation – For computational efficiency: use of mean-field-homogenization – For macro cracks: using hybrid DG/Cohesive zone model CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 5

  6. Multi-scale modelling • Mean-Field-Homogenization – Macro-scale • FE model At one integration point e is know, s is sought • – Transition Downscaling: e is used as input of the MFH model • Upscaling: s is the output of the MFH model •  σ σ ε – Micro-scale  ε • Semi-analytical model • Predict composite meso-scale response • From components material models w I w 0 Mori and Tanaka 73, Hill 65, Ponte Casta ñ eda 91, Suquet 95, Doghri et al 03, Lahellec et al. 11, Brassart et al. 12, … CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 7

  7. Mean-Field-Homogenization • Key principles w I – Based on the averaging of the fields σ inclusions 1   w 0 a a ( X ) d V V V – Meso-response composite ?  v  • v 1 From the volume ratios ( ) 0 I      σ σ σ σ σ σ v v v v w w 0 I 0 0 I I 0 I matrix      ε ε ε ε ε ε v v v v 0 w I w 0 0 I I ε 0 I • One more equation required e  ε : ε B I 0 – Difficulty: find the adequate relations    σ ε f I I   e  σ ε ? B f 0 0 e  ε : ε B I 0 CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 8

  8. Mean-Field-Homogenization • Key principles (2) – Linear materials w I σ • inclusions Materials behaviours σ  : ε w 0 C I I I σ  : ε C 0 0 0 composite ?   ε ε • Mori-Tanaka assumption 0 • Use Eshelby tensor   e  ε ε B I , C , C : I 0 I 0 matrix B e      1 1 [ I S : C : ( C C )] with 0 1 0 ε CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 9

  9. Mean-Field-Homogenization • Key principles (2) – Linear materials w I σ • inclusions Materials behaviours σ  : ε w 0 C I I I σ  : ε C 0 0 0 composite ?   ε ε • Mori-Tanaka assumption 0 • Use Eshelby tensor   e  ε ε B I , C , C : I 0 I 0 matrix B e      1 1 [ I S : C : ( C C )] with 0 1 0 ε σ alg – C Non-linear materials I • Define a Linear Comparison Composite (LCC) inclusions • Common approach: incremental tangent composite   alg e C    ε ε alg alg B I , C , C : σ 0 matrix: I 0 0 I 0 ε    ε ε ε I 0 CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 10

  10. Content • Micro-scale modelling – Incremental-Secant Mean-Field-Homogenization (MFH) – Damage-enhanced incremental-secant MFH CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 11

  11. Incremental-secant mean-field-homogenization σ • Material model – Elasto-plastic material   σ ε ε el pl • C : ( ) Stress tensor       s   σ σ eq Y • f , p R p 0 Yield surface  f     ε pl • N p N Plastic flow &  σ    σ ε alg • C : Linearization el C ε ε pl CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 12

  12. Incremental-secant mean-field-homogenization σ • New incremental-secant approach inclusions – Perform a virtual elastic unloading from previous solution composite • Composite material unloaded to reach the matrix stress-free state • Residual stress in components New Linear Comparison Composite (LCC) ε    ε ε unload ε unload unload I 0 CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 13

  13. Incremental-secant mean-field-homogenization σ • New incremental-secant approach inclusions – Perform a virtual elastic unloading from previous solution composite • Composite material unloaded to reach the matrix stress-free state • Residual stress in components New Linear Comparison Composite (LCC) ε – Apply MFH from unloaded state    ε ε unload ε unload unload I 0 • New strain increments (>0) σ inclusions      ε ε ε r unload I/0 I/0 I/0 composite • Use of secant operators   e    ε r Sr Sr ε r matrix B I , C , C : Sr C I 0 I 0 I Sr C 0 ε    ε r ε r ε r I 0 CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 14

  14. Incremental-secant mean-field-homogenization σ inclusions • Zero-incremental-secant method – Continuous fibres composite • 55 % volume fraction • Elastic matrix Sr C – I Elasto-plastic matrix – For inclusions with high hardening (elastic) • Model is too stiff Sr C 0 ε Transverse loading Longitudinal tension    ε r ε r ε r 3 12 I 0 FE (Jansson, 1992) 2.5 10 Sr C 0       s   σ σ eq Y f , p R p 0 8 2 s/s Y0 s/s Y0 σ eq 1.5 6 is underestimated FE (Jansson, 1992) 1 4 Sr C 0 0.5 2 0 0 0 0.003 0.006 0.009 0 0.002 0.004 e e CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 16

  15. Incremental-secant mean-field-homogenization σ inclusions • Zero-incremental-secant method (2) – Continuous fibres composite • 55 % volume fraction • Elastic matrix Sr C – I Elasto-plastic matrix – Secant model in the matrix • Modified if negative residual stress Sr C 0 ε Transverse loading Longitudinal tension    ε r ε r ε r 3 12 I 0 FE (Jansson, 1992) σ 2.5 10 Sr C 0 inclusions S0 C 0 8 2 s/s Y0 s/s Y0 composite 1.5 6 FE (Jansson, matrix Sr C 1992) I 1 4 Sr C 0 S0 C 0 0.5 2 S0 C 0 ε 0 0 0 0.003 0.006 0.009 0 0.002 0.004    ε r ε r ε e r e I 0 CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 17

  16. Incremental-secant mean-field-homogenization • .100 Verification of the method e 13  e 23 – Spherical inclusions .080 e 33 2 e 11 2 e 22 • 17 % volume fraction .060 e • Elastic – Elastic-perfectly-plastic matrix .040 – Non-proportional loading .020 .000 0 10 20 30 40 t FFT FE (Lahellec et al., 2013) MFH, incr. tg. 120 MFH, var. (Lahellec et al., 2013) 30 80 MFH, incr. sec. s 13 [Mpa] s 33 [Mpa] 5 40 -20 0 FFT FE (Lahellec et al., 2013) MFH, incr. tg. -45 -40 MFH, var. (Lahellec et al., 2013) MFH, incr. sec. -70 -80 0 10 20 30 40 0 10 20 30 40 t t CM3 EMMC15 - 7 - 9 September 2016, Brussels, Belgium - 18

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