1. No category

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