“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