Multiscale Approach to Damage Analysis of Laminated ... · Analysis of Laminated Composite Structures Problem: Numerical modeling of failure and damage of laminated composite structures:

Multiscale Approach to Damage Analysis of Laminated Composite Structures

Darko Ivančević, Ivica Smojver

University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture,

Department of Aeronautical Engineering

3DS SIMULIA COMMUNITY CONFERENCE MAY 15 – 17, 2012 – PROVIDENCE, RI, USA

Introduction

Micromechanical model

Abaqus implementation

Numerical model

Results

Conclusion

2D. Ivančević , I. Smojver: Multiscale Approach to Damage

Analysis of Laminated Composite Structures

Introduction Micromechanical model Abaqus implementation Failure criteria Results Conclusion



Problem:

Numerical modeling of failure and damage of laminated composite structures:

• failure modes of composite materials are closely related to the material microstructure• analyses on the constituent level using micromechanical principles• modeling of damage progression on the micro-level• in order to solve engineering problems analyses have to be performed on two length scales

Solution: • High Fidelity Generalized Method of Cells (HFGMC)• Abaqus/Explicit • multiscale approach



• reformulated version1

• unit cell approach• fibers extend in the x1 - direction and are arranged in a doubly periodic array in the x2 and x3

directions

HFGMC:

• HFGMC model• heterogeneous material is discretizedby Nβ x Nγ subcells• subcells can be either fiber or matrix subcells• belongs to the group of micromechanical models which originated from Aboudi’s Method of cells theory2



1. Bansal, Y. and M.-J. Pindera, “Efficient Reformulation of the Thermoelastic Higher-Order Theory for FGMs,” NASA/CR –2002 – 211909, 2002.2. Aboudi, J, A., “Closed Form Constitutive Equations for Metal Matrix Composites,” International Journal of Engineering Science, vol. 25, no. 9, pp.1229-1240, 1987.

unit cell morphologies investigated in this work(75 x 75 subcells)




( , ) ( , )ε A ε

• strain tensors of each subcell: strain concentration tensors ( , )A

• displacement field within each subcell – Legendre type polynomials

2 2

( , ) ( , ) ( ) ( , ) ( ) ( , ) ( )2 ( , ) ( )2 ( , )

(00) 2 (10) 3 (01) 2 (20) 3 (02)

1 1+ 3 3

2 4 2 4i ij j i i i i i

h lu x W y W y W y W y W

• the stiffness matrix of each subcell can be divided into axial L and transverse K stiffness matrix3

( , ) ( , )( , ) ( , )2 211 12 13 14 661 1

2 221 22 23 24 66 121 1

3 331 32 33 34 55 131 1

3 341 42 43 44 661 1

0

02

0

0

L L L L Ct u

L L L L Ct u

L L L L Ct u

L L L L Ct u

( , )2

11 12 15 16 17 182

221 22 24 25 26 272

233 34 35 36 37 383

243 44 45 46 47 483

351 52 53 54 55 562

361 62 63 64 65 662

371 72 73 74 73

3

3

0 0

0 0

0 0

0 0

0 0

0 0

0 0

K K K K K Kt

K K K K K Kt

K K K K K Kt

K K K K K Kt

K K K K K Kt

K K K K K Kt

K K K K Kt

t

( , )( , ) 2

2

2

2

2

3

2

3

3

2

3

2

37 78 3

381 82 83 84 87 88 3

12 22 23

12 22 23

44

44

44

44

13 23 33

13 23 3

0 0

0

0

0 0 0 2

0 0 0

0 0 0 2

0 0 0 2

0

u

u

u

u

u

u

K u

K K K K K K u

C C C

C C C

C

C

C

C

C C C

C C C

( , )

11

22

33

23

3 0

3. Bansal, Y. and M.-J. Pindera, “A Second Look at the Higher-Order Theory for Periodic Multiphase Materials,” Journal of Applied Mechanics, no. 72, pp. 177-195, 2005

global system of equations:

21211 12 111

31321 22 221

L L Δc 0u

L L 0 Δcu

211 12 13 1111 13 14 2

224 2222 23 24 3

334 3331 32 33 2

341 42 43 2341 42 44 3

ΔC ΔC ΔC 0K 0 K K u

0 0 0 ΔC0 K K K u

0 0 0 ΔCK K K 0 u

ΔC ΔC ΔC 0K K 0 K u



2 2 elementsN N N N 4 4 elementsN N N N

• highly sparsed systems of equations


2 ( ) 2 ( 1 )

3 ( ) 3 ( 1)

0

0

β,γ β+ ,γ

i i

β,γ β,γ+

i i

t t

t t

2 ( ) 2 ( 1, ) 2( 1, )

3 ( ) 3 ( 1) 3( 1)

β,γ β γ β γ

i i i

β,γ β,γ β,γ

i i i

u' u' u'

u' u' u'

1,2,3i

1,2,3i

• traction continuity conditions:

• displacement continuity conditions:

• periodicity conditions at unit cell boundaries

• additional 4 displacement components are fixed to prevent rigid body displacements of the unit cell


( , ) ( , ) ( , )σ C ε

( , ) ( , )

1 1

1N N

h lhl

σ C A ε

* ( , ) ( , )

1 1

1N N

h lhl

C C A

• solution of the global system enables calculation of microvariables W which determine subcellstrain field

• after calculation of the subcell strain field, a numerical procedure is used to determine the strain concentration tensors A(β,γ)

• stress field

• average unit cell stress tensor:

homogenized material elasticity tensor



( )

11 11

( ) ( ) ( ) ( )

22 22 2(10) 2 2(20)

( ) ( ) ( ) ( )

33 33 3(01) 2 3(02)

( ) ( ) ( ) ( )

12 12 1(10) 2 1(20)

( ) ( ) ( ) ( )

13 13 1(01) 3 1(02)

( )

23

3

3

13

2

13

2

β,γ

β,γ β,γ β,γ

β,γ β,γ β,γ

β,γ β,γ β β,γ

β,γ β,γ γ β,γ

β,γ

ε ε

ε ε W y W

ε ε W y W

ε ε W y W

ε ε W y W

ε

( ) ( ) ( ) ( ) ( ) ( )

23 2(01) 3 2(02) 3(10) 2 3(20)

13 3

2

β,γ γ β,γ β,γ β β,γε W y W W y W




• user material subroutine VUMAT• HFGMC is programmed as a subroutine called from VUMAT

SDV’s:•ply level failure criteria • equivalent properties• maximal values of micromechanical failure criteria within the unit cell

for future work:• damage variables• equivalent properties of damaged material

macroscopic strain tensor

HFGMC output:- equivalent material properties- solution dependent state variables

HFGMC inputvariables




preprocessor

application of periodicity,traction and displacement continuity conditions

• call to HFGMC subroutine: input variables

maximal value of the subcell is returned to VUMAT as SDV

Math Kernel Library (MKL) solving of sparse system - dss_solve_real:- enhanced computational time- higher unit cell refinement (Nβ x Nγ ) - solved problem with memory storage of large arrays

future work – inclusion of damage effects - Continuum Damage Mechanics principles

A(β,γ) are stored as

COMMON blocks




• ply level: - programmed in VUMAT subroutine in order to use the results for the micromechanical analysis- maximal values are stored as SDV’s

2 2 2

1 1 2 2 6 6 11 1 22 2 66 6 12 1 2 16 1 6 26 2 62 2 2 1F F F F F F F F F

1. Tsai-Wu criterion

2. Tsai-Hill criterion2 2 2

1 1 2 2 12

2 2 2 21

X X Y S

3. Hashin failure criteria

2 2

11 12

12

1tX S

2

11 1cX

2 2

22 12

12

1tY S

2 2 2

22 22 12

23 23 12

1 12 2

c

c

Y

S S Y S

4. Puck’s failure model

1f fTX 1f fCX

2 2

221

2 22 1A

p p

R RR

22

21 2 1

( )2 121

12 1 S

C

C D

Y

Yp

fiber tensile fiber compressive

matrix tensile matrix compressive

tensile compressive

22

21 2 1

( )2 121

12 1 S

C

C D

Y

Yp

fiber failure:

inter-fiber failure:

Mode A Mode B Mode C




• micromechanical failure criteria : - calculated within the HFGMC subroutine- maximal value within the unit cell is stored as SDV

1. 3-D Tsai –Hill criterion to predict matrix failure4

2 2 2 2 2 2

( , ) ( , ) ( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ) ( , ) ( , )11 22 33 12 13 23 211 22 11 33 22 33

2 2 2 mdY Y T

2. Maximal strain criterion for fiber4

2( , )

211

fU

ft

d

11 0

3. Maximal stress criterion for fiber

( , )2

( , )11

11

1 11

f f f fT C T C

4. Micromechanics of failure criterion (for fiber)5

5. Micromechanics of failure criterion (for matrix)5

( , )2

( , )

1

1 11VM

m m m m

IT C T C

for

2( , )

211

fU

ft

d

( , )

11 0 for

4. Pineda, E. J., A. M. Waas and B. A. Bednarcyk, “MultiscaleModel for Progressive Damage and Failure of Laminated Composites Using an Explicit Finite Element Method,” 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2009.5. Sun, X. S., V. B. C. Tan, and T. E. Tay, “Micromechanics-Based Progressive Failure Analysis of Fibre-Reinforced Composites with Non-Iterative Element-Failure Method,” Computers and Structures, no. 89, pp. 1103-1116, 2011.




• the multiscale technology has been tested on a very simple finite element model• CFRP plate 0.3 x 0.2 m (T300/5208)• ply layup [90/45/45/90] t = 0.125 mm• T300/5208 with 70% fiber volume fraction•homogenized properties calculated HFGMC (100 x 100 unit cell, single fiber at unit cell centre)

• finite element model:- 150 S4R elements- 176 nodes- 1056 degrees of freedom

• 3 unit cell morphologies have been evaluated:

• effects of unit cell refinement have been studied:- Nβ x Nγ = 20 x 20, 50 x 50, 75 x 75

• explicit analysis:- static tensile load case- tensile force F= 5500 N (distributed along nodes of shorter edges)

Example 1:




• comparison of failure initiation criteria - Puck's Mode A IFF criterion and micromechanical matrix failure criteria calculated on a 5-fiber unit cell with 75 x75 subcells:

Unit cell

type:CRITERION:

20X20 50X50 75X75

90° 45° 90° 45° 90° 45°

3D TH 2.249 1.931 4.173 3.406 3.622 2.980

MMF 3.500 3.130 8.798 6.841 6.963 5.571

3D TH 2.243 1.926 4.171 3.405 3.201 2.660

MMF 3.488 3.122 8.791 6.837 5.754 4.747

3D TH 0.951 0.921 0.961 0.929 1.042 1.065

MMF 1.197 1.241 1.219 1.264 1.414 1.460

• Maximal values of micromechanical matrix failure criteria in the FE model:




• effect of unit cell morphology on 3D Tsai Hill criterion (Nβ x Nγ = 75 x 75)

• effect of unit cell refinement on distribution of Micromechanics of Failure matrix damage initiation criterion

20 x 20Nβ x Nγ : 50 x 50 75 x 75




• stress distribution within the unit cell [GPa] - Nβ x Nγ = 100 x 100

22σ33σ

23σ 13σ 12σ




• composite panel with stringer reinforcements• T300/5208 composite system with 70% fiber volume fraction• equivalent properties calculated by HFGMC • unidirectional CFRP plies, t= 0.125 mm• unit cell morphology with a single fiber

[90/+45/-45/0]S

[(+45/-45)3/06 ] • finite element model:- 5270 S4R elements- 5418 nodes- 32508 degrees of freedom

• explicit analysis:- static tensile load case- F = 255.2 kN (distributed along the panel edges)- boundary conditions which replicate experimental conditions used for axial loading of reinforced panels

Example 2:




• comparison of macroscopic failure criteria (maximal through thickness values are shown)




max = 0.655

max = 1.219

• maximal through thickness values of 3D Tsai Hill micromechanical criterion and distribution of the criterion within the unit cell

• maximal values of Micromechanics of Failure theory matrix failure criterion in the unit cell.




• the HFGMC implementation in Abaqus/Explicit is in the initial phase• HFGMC model has been used to calculate equivalent composite properties and micromechanical matrix failure criteria• static tensile load case - future work: impact loading, blast loads and similar high velocity / high strain phenomena• micromechanical matrix failure criteria show dependence on the unit cell morphology and refinement, although for all morphologies Vf = 70%• effect of unit cell refinement on the values of failure criteria is not fully understood• finer levels of the HFGMC mesh result in improved distribution of the micromechanical stresses thereby improving the distribution of failure criteria - more accurate modeling of damage processes as the continuation of ongoing work• micro-level doesn't necessarily indicate failure of the complete ply• reasonable correlation between failure prediction on the micro-level and ply-level based failure criteria or experimental results is expected after inclusion of damage processes in the HFGMC model• influence of the interphase at the matrix/fiber boundaries will have to be included in the micromechanical analysis - effects at the interphase play a significant role in damage mechanisms of composite materials• maximal values of failure criteria are predicted in matrix subcells which surround the fibers, additionally indicating the importance of correct modeling of this area of the unit cell•despite all problems encountered during the current work, the HFGMC presents a highly promising model for further development




Thank you!

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Multiscale Approach to Damage Analysis of Laminated ... · Analysis of Laminated Composite Structures Problem: Numerical modeling of failure and damage of laminated composite structures: