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“STRENGTH PREDICTION IN OPEN HOLE LAMINATED COMPOSITES BY USING REGULARIZED X-FEM”
Michael Swindeman2,Endel Iarve1,2,
David Mollenhauer1, Stephen Hallett3,
Robert Brockman2
1Air Force Research Laboratory, Materials and Manufacturing Directorate, AFRL/RX, Wright-Patterson Air Force Base, Ohio 45433
2 University of Dayton Research Institute, Dayton OHAF Contract FA8650-10-D-5011
3University of Bristol, UK
5th International Conference on Composites Testing and Model Identification
Lausanne, Switzerland
February, 2011
Contents
•Motivation
•Method Description– X-FEM and Regularized X-FEM
•Results– Quasi-Isotropic open hole laminate
Motivation
• Composite failure is dominated by interactions between matrix cracks & delaminations
• Strength of notched and unnotched composite laminates can be predicted accurately by modeling critical events involving matrix crack patterns and delamination interactions
Carlos DáVila, “The Long Road To Virtual Testing of Composite Structures, Are We There Yet?,” Keynote Address at 2nd ECCOMAS, London, April 2009.
Goal: Discrete modeling of matrix cracking and delamination networks
General approach based on X-FEM ideas (Moes, et. al., 1999, IJNME).
1) preserves the kinematics of true displacement continuity
2) allows direct application of fracture mechanics criteria for propagation
Modifications needed to accommodate cracking and delaminationinteraction
Emerging Modeling Techniques
Modeling Goal
[1] Van der Meer F P and Sluys L J, (2nd ECCOMAS, 2009)[2] Qingda Yang and Brian Cox, (CompTest, 2008)[3] Iarve et al. (Composites A, 2005; IJMS, 2007)[4] …..
x-FEM
• Moes, Dolbow, and Belytschko (1999)
• Hansbo and Hansbo (2004)
6
Nodes
Integration Points
Duplicated Nodes
V
aa dVWfHWfHW 21 ))(1()(
VS
dVdHdsdM
00)()(
u=H(fa) u1+(1-H(fa) )u2
e=H(fa) e1+(1-H(fa) ) e2 s=H(fa) s1+(1-H(fa) ) s2 H(fa)=0
H(fa)=1
- Strain Energy - Cohesive Energy
MIC – Mesh Independent Cracks based on Regularized X-FEM
x
( )H x( )H x
xElement Length
The Heaviside function is replaced by a continuous function
Crack location Crack location
V
aa dVWfHWfHW 21 ))(1()(
V
dVdHM
0
)(
Example of Regularized Step Function
Instead of a sharp transition, the crack is resolved within a band of width equal to the element diameter.
Connection Between Plies
• The original Gauss integration schema is preserved for any crack orientation
• Adjacent plies tied through node/and or surface element integration contact
• Propagation is through cohesive zone method
MIC & Delamination Interaction and Propagation
General Modeling Flow1. Step i=0 is thermal pre-stress
2. Add axial displacement increment
3. Perform Newton-Raphson iterations to converge damage variables in delam and MIC cohesive laws
4. Check matrix failure criteria
5. Add damage and repeat 2-5
Matrix Failure Criteria - Dávila, Camanho, and Rose, “Failure criteria for FRP laminates,” J. of Composite Materials, Vol.39 2005.
Cohesive Zone Propagation - Turon, Camanho, Costa, and Dávila, “A damage model for the simulation of delamination in advanced composites under variable-mode loading,” Mechanics of Materials, Vol.38, 2006.
Mesh Independent Cracks - Iarve, “Mesh independent modeling of cracks by using higher order shape functions,” Int. J. Num. Meth. Eng., Vol.56, 2003.
Numerical Model Details
Stress Based Failure Criterion Used for MIC Initiation
-Yc Yt
SMatrix failure
TensionCompression
Fiber failure TensionCompression
LaRC03- Dávila, Camanho, and Rose, “Failure criteria for FRP laminates,” J. of Composite Materials, Vol.39 2005.
nDun – normal displacement discontinuity vector
Du – total displacement discontinuity vector
|Du|
T=(1-d)K Du + nonpenetration
| T |
2
2
4
)(1
u
nn uu
B
B=1B=0
S
YtS
Yt
GIc
GIIc
- Initial stiffness
- Transverse strength
- Shear strength
-Mode I critical ERR
-Mode II critical ERR
-Mixed Mode test
-
Cohesive Model Used for Delamination and MIC Propagation
h
K
Turon, et al. Composites: Part A, 2007
Laminates Under Tensile LoadingModel Verification
1. Scaled Laminates
Hallett, et al. Composites Science and Technology(2008)
2. Stacking sequence and plyOrientation effects
[452/-452/902]s vs. [602/-602]s
Johnson and Chang, J Composite Mat. (2002)
3. Ply thickness effects
[25/-25/90n]s
Wang and Crossman, STP 775, 1982
Ply Thickness and Crack Density
"Reprinted, with permission, from ASTM STP 775 Damage in Composite Materials, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428."
T300/914
Wang, ASD and Crossman, STP 775, 1982, Reifsnider, Ed.
Ply Thickness and Crack Density Effect
[±25/908]s[±25/903]s
Delamination shape at the time step prior to global delamination
Tensile Strength Scaling in Quasi-Isotropic Composite Laminates
•Wisnom et al, Strength Scaling Studies•Quasi-Isotropic laminates with various numbers of sub-laminates (n) and blocked plies (m)
•All with same scaled dimensions W/D = 5, L/D = 20
[45m/90m/-45m/0m]ns
x
y
Experimental Data
Pull-outBrittle Delamination
Fiber Failure
As hole size increases, failure stress decreases
Delamination Failure
As hole size increases, failure stress increases
BG Green, MR Wisnom and SR Hallett, Composites Part A (2007)
Table of Models and Results
These cases were selected for study because they failed in delamination mode.
All cases contained only one sub-laminate (n=1)
CASE
No. Blocked
Plies
Ply Thickness
Overall thickness
Hole Diameter Failure Stress (MPa)
m Tply (mm) T (mm) D (mm) ExperimentCoarse
“C”
Fine
“F”
B2 2 0.25 2 3.175396
469448
C2
4 0.5 4
3.175 275 308
C3 6.35 285 318 29712.7 362 387 (344)C425.4 417 466 (424)C5
D28 1.0 8
3.175 202 211D5 25.4 232 276 (239)
4-Blocked Ply 6.35 mm Hole
Meshes
Hole Size Effect(4-Blocked Ply Cases)
0 0.002 0.004 0.006 0.008 0.010
50
100
150
200
250
300
350
400
450
500
3.175 mm, Coarse Mesh
6.35 mm, Coarse Mesh
6.35 mm Fine Mesh
12.7 mm, Coarse Mesh
25.4 mm, Coarse Mesh
Strain (mm/mm)
Ave
rag
e T
ract
ion
(M
Pa)
Hole Size Effects(4-Blocked plies)
0 5 10 15 20 25 30200
250
300
350
400
450
500Failure Stress for 0.5 mm Ply Thickness (m = 4) Cases
Experiment
Coarse Mesh
Fine Mesh
Hole Size (mm)
Fai
lure
Str
ess
(MP
a)
Ply Thickness Effect(3.175mm Hole)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1200
250
300
350
400
450
500Failure Stress for 3.175 mm Hole Diameter Cases
Experiment
Coarse Mesh
Fine Mesh
Ply Thickness (mm)
Fai
lure
Str
ess
(MP
a)
Response (4 blocked ply – 6.35 mm Hole)
0 0.001 0.002 0.003 0.004 0.005 0.0060
50
100
150
200
250
300
350
Strain
Av
era
ge
T
rac
tio
n (
MP
a)
Matrix Crac
k Initiation an
d Growth
Delamination Initiation and Spreading
Early Damage Progression
Red = [45/90] interface, Green = [90/-45] interface, Blue = [-45/0] interface
0 0.001 0.002 0.003 0.004 0.005 0.006
0
50
100
150
200
250
300
350
Strain
Av
era
ge
T
rac
tio
n (
MP
a)
Early Damage Progression
Red = [45/90] interface, Green = [90/-45] interface, Blue = [-45/0] interface
0 0.001 0.002 0.003 0.004 0.005 0.0060
50
100
150
200
250
300
350
Strain
Ave
rage
Tra
ction
(MPa
)
Start of Delamination Interaction
Red = [45/90] interface, Green = [90/-45] interface, Blue = [-45/0] interface
0 0.001 0.002 0.003 0.004 0.005 0.0060
50
100
150
200
250
300
350
Strain
Ave
rage
Tra
ction
(MPa
)
Late Damage
Red = [45/90] interface, Green = [90/-45] interface, Blue = [-45/0] interface
0 0.001 0.002 0.003 0.004 0.005 0.0060
50
100
150
200
250
300
350
Strain
Ave
rage
Tra
ction
(MPa
)
Late Damage
Red = [45/90] interface, Green = [90/-45] interface, Blue = [-45/0] interface
0 0.001 0.002 0.003 0.004 0.005 0.0060
50
100
150
200
250
300
350
Strain
Ave
rage
Tra
ction
(MPa
)
Near Failure
Red = [45/90] interface, Green = [90/-45] interface, Blue = [-45/0] interface
0 0.001 0.002 0.003 0.004 0.005 0.0060
50
100
150
200
250
300
350
Strain
Ave
rage
Tra
ction
(MPa
)
Peak Traction
Red = [45/90] interface, Green = [90/-45] interface, Blue = [-45/0] interface
0 0.001 0.002 0.003 0.004 0.005 0.0060
50
100
150
200
250
300
350
Strain
Ave
rage
Tra
ction
(MPa
)
Post Failure
Red = [45/90] interface, Green = [90/-45] interface, Blue = [-45/0] interface
0 0.001 0.002 0.003 0.004 0.005 0.0060
50
100
150
200
250
300
350
Strain
Ave
rage
Tra
ction
(MPa
)
Conclusions
•A finite element method and software implementing regularized X-FEM approach and allowing modeling of complex interactive networks of matrix cracks and delamination has been developed. •Effects of Hole Size and Ply Thickness have been simulated
– Simulation without preconceived knowledge of damage evolution– Strength produced with coarse and fine mesh agreed with the
experimental hole size effect trend– Delamination strength is proportional to the ligament width, which
explains the apparent strength increase for larger specimens
Acknowledgements
• NASA AAD-2 contract number NNX08AB05A-G• Special thanks to Dr. Cheryl Rose, Dr. Carlos Davila at
NASA Langley