Kinking Cracks in Composites
Kinking Cracks in Composites a b Brian Nyvang Legarth and Qingda Yang a Department of Mechanical Engineering Solid Mechanics Technical University of Denmark b Mechanical & Aerospace Engineering University of Miami Florida, USA Kinking Cracks in Composites – p. 1/21 Motivation • Experiments: Debonding crack kinks into the matrix • Typically at an angle of 70o relatively to uni-axial load (Paris et al. (2007), Vajari (2014), IMDEA (yesterday) ) 70o 40o Vajari, 2014 Legarth and Kuroda, 2004 Kinking Cracks in Composites – p. 2/21 The problem Motivated by some zones being fiber rich and some being matrix rich the following problem is considered: x2 AV s˙ AV 22 = κ s˙ 11 A B η2 AV s˙ AV 11 s˙ 11 βi 2bc ξ1 η1 D C Inclusions Volume fraction: f = 2ai 2bi πai bi 4ac bc x1 ξ2 AV s˙ AV 22 = κ s˙ 11 2ac Plane strain assumption Kinking Cracks in Composites – p. 3/21 Periodic Boundary Conditions Compatibility gives u1 (η1 ) = u1 (η2 ) − uB1 and u2 (η1 ) = u2 (η2 ) − uB2 u1 (ξ1 ) = u1 (ξ2 ) − uD1 and u2 (ξ1 ) = u2 (ξ2 ) − uD2 In addition, some conditions for the macroscopic average problem, denoted by ( )AV , must also be specified: AV 6= 0 S11 AV = 0, S22 AV 6= 0) (Uniaxial plane strain tension with S33 AV =0 S12 AV 6= 0) (⇒ F12 and AV AV =0 = ∆1 , F21 F11 AV 6= 0) (⇒ S21 AV = 0: Cell side parallel to x1 remains that on average F21 Kinking Cracks in Composites – p. 4/21 FE models Standard FE cannot track the kinking crack, as the exact initiation point and path are not known it advanced. • Cohesive elements between regular elements (Xu et al., 1997) • Local mesh refinement (Rashid, 1998) • eXtended FEM (XFEM) (Belytschko et al., 1999) • Augmented FEM (AFEM) (Ling et al., 2009) XFEM vs. AFEM • XFEM: Brittle fracture by the asymptotic solutions • AFEM: Not ideal for brittle fracture. Mesh-dependent stress concentrations are predicted. Ductile fracture: Embedded cohesive elements yield good results with low mesh-dependence Kinking Cracks in Composites – p. 5/21 AFEM - used for matrix cracking Matrix cracking: • Initiation: Average element stress • Direction: "Maximum principal stress" criterion • Propagation: Embedded cohesive elements • Crack-tip only at element side - not within the element AFEM is relatively simple to implement into existing standard FE-codes as standard shape functions are used. The embedded cohesive elements results in new element stiffness matrices with displacement DOF’s. Cost: Approxiamtely 60% increase in CPU-time over standard FEM. Kinking Cracks in Composites – p. 6/21 AFEM - element formulation • Continuous tractions - discontinuous displacements • Discontinuous displacements fully described by • • • • superposition of 2 standard elements (ME1 / ME2) Extrapolated displacements in Ωe1 /Ωe2 to ME1/ME2 Both MEs have identical geometry to Ωe , but with different material allocation. Additional nodes ( )’ introduced for interpolation User-Element (UE) routine for ABAQUS 6.12 Ling et al. (2009) Kinking Cracks in Composites – p. 7/21 Fiber-matrix debonding Same as used for AFEM. Same in normal and shear. σmax Traction G = 21 δmax σmax = GI + GII Work of fracture, G ≃ 0.01 · δmax Separation δmax Simplified version of Yang and Thouless (2001). UE in ABAQUS. Kinking Cracks in Composites – p. 8/21 Results - parameters and meshes √ Fiber radius: ri = ai bi =5µm (perfectly stiff) Volume fraction: f = 20% Matrix elasticity (isotropy): E = 4 GPa, ν = 0.33 Matrix strength: σmax = 100 MPa, δmax = 0.02ri / 0.2ri Interface strength: σmax = 60 MPa, δmax = 0.02ri ac bc =1 ai /bi = 1 ai /bi = 2 and βi = 30o Kinking Cracks in Composites – p. 9/21 AV Results - ai/bi = ac/bc = 1 (S22 = 0) 70 Debonding Matrix cracking 60 Perfectly stiff fiber Stress, σ1 50 55o 40 30 20 ǫ1 = ∆1 2ac 10 σ1 = F1 2bc 0 0 Magnification: x3 (animate) 0.005 0.01 0.015 0.02 Strain, ǫ1 Uniform interface strength: Debonding initiates ramdomly Uniform matrix strength: Crack kinks out at 55o < 70o Non-linearity of stress-strain response before debonding Kinking Cracks in Composites – p. 10/21 AV ai/bi = 2 and ac/bc = 1 (S22 = 0) 70 60 Perfectly stiff fiber βi = 0 o 30o Decreasing βi Stress, σ1 50 15o 0o 40 30 60o 20 10 Perfectly stiff fiber βi = 60o 0 0 50o Magnification: x3 0.005 0.01 Strain, ǫ1 0.015 0.02 Lower transverse stiffness for larger βi Lower failure strain for smaller βi Small load carrying capacity: βi = 0o Angle below 70o Kinking Cracks in Composites – p. 11/21 Effect of multiple fibers - kinking angle • Only fiber at center debonds • Periodic boundary conditions ǫmax ǫmax 55 o 55 ǫ1 = 0.010 o ǫ1 = 0.018 • Strain field disturbed to the left/right than above/below • Same angle of kinking - but at lower overall strain, ǫ1 Kinking Cracks in Composites – p. 12/21 Multiple fibers - 10x stronger matrix ǫmax 1.03 0.10 0.09 0.08 0.08 0.07 0.06 0.05 0.04 0.03 0.03 0.02 0.01 0.00 ǫmax 1.54 0.10 0.09 0.08 0.08 0.07 0.06 0.05 0.04 0.03 0.03 0.02 0.01 0.00 o 55 55 o Magnification: x3 (animate) • Single fiber with single crack • 9 fibers with dual cracks • Same (or sligtly smaller) kinking angle Kinking Cracks in Composites – p. 13/21 Ongoing and future work Hexagonal distribution of fibers Will the crack kinking angle be affected? How about overall failure? Random (real?) fiber distribution? More debonds? Kinking Cracks in Composites – p. 14/21 Concluding remarks • Unit cell analyses with periodic BC’s to study debonding and matrix cracking • AFEM - Augmented FEM • Uniaxial loading with geometrical anisotropy • Experiments: Crack kinks out at approximately 70o • Simulation: Crack kinks out at an angle below 70o (55o for plane strain tension) • Multi fiber cell: Same kinking angle but at lower strain Kinking Cracks in Composites – p. 15/21 Thank you! Acknowlegdment: This work was supported by the Danish Center for Composite Structures and Materials for Wind turbine (DCCSM), supported by The Danish Council for Strategic Research grant no: 09-067212 Kinking Cracks in Composites – p. 16/21 AV ai/bi = ac/bc = 1, Plane stress (S33 = 0) Plane stress studies generally don’t represent geometry well. However, in experiments is only the surface visible - and the surface can be represented by plane stress 45o AV =0 κ = 0.5 S22 Plane stress yields smaller angle than plane strain. κ = 0.5 still yields a matrix crack not starting at interface. Kinking Cracks in Composites – p. 17/21 Results - ai/bi = ac/bc = 1 (κ = 0.5) 80 70 Stress, σ1 60 50 40 30 20 10 0 0 0.005 0.01 0.015 0.02 0.025 0.03 Strain, ǫ1 Magnification: x3 • Biaxially loaded • Significantly debonded before matrix crack • Matrix crack not at tip • Angle below 70o Kinking Cracks in Composites – p. 18/21 Effect of multiple fibers - stress/strain Preliminary results 70 60 50 40 30 20 10 0 1 cell 9 cells (coarser mesh) 25 cells (coarser mesh) 9 cells 0 0.005 0.010 0.015 0.020 Post failure behavior not well captured. Failure strain seems to be reduced for multiple fiber case. Kinking Cracks in Composites – p. 19/21 Applications of Composites Airbus A350 (53%) Boeing 787 (50%) Kinking Cracks in Composites – p. 20/21 o AV ai/bi = 2 (βi = 30 ) and ac/bc = 1 (S22 = 0) 70 1 60 1s t c rack 4 2 5 3 40 30 crack Penetration 20 1s t c 2nd rack t ip Stress, σ1 50 Magnification: x3 (animate) 10 0 0 0.01 0.02 0.03 0.04 0.05 Strain, ǫ1 1) First debond 2) First crack 3) Second debond 4) Second crack 5) Too large penetration (distorted solution) Kinking Cracks in Composites – p. 21/21
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