Damage Meso-model for Laminated

Composite Structures with Material Non-linearity

PM Mohite

Department of Aerospace Engineering

Indian Institute of Technology Kanpur

• Introduction to damage mechanisms in laminated composites

• Damage meso-model

• Material non-linearity

• Key results

• Conclusions

Layout of Presentation

Introduction

• Composite Material: Excellent properties (tailorable)

• Critical component applications: space, marine, automobile

• Modeling structural response: global, micromechanical,

inelastic, damage

• 3D elasticity: intractable analytically

• Challenge: Heterogeneity and orthotropy

• Alleviation: reduced models and Finite Element Method

Failure Mechanisms in Fibrous Laminated Composites

• Micro-level failure

• Macro-level failure

• Micro-level Failure Mechanisms - failure mechanisms at fibre-matrix level

• Fibre-level mechanisms

• Matrix-level mechanisms

• Fibre-matrix coupled mechanisms

• Macro-level failure Mechanisms

• Delamination

• Longitudinal splitting

Failure Mechanisms in Fibrous Laminated Composites

1. Intralaminar damage mechanisms

A. Fibre Breaking

B. Fibre-Matrix Interphase Debonding (Diffuse Damage)

C. Transverse Matrix Microcracking

Fibre-level Failure Mechanisms

Fibre breakage Fibre kinking

Fibre bending Fibre splitting and radial crack

Matrix-level Failure Mechanisms

Matrix cracking

Matrix interface cracking

Fibre-Matrix Coupled Failure Mechanisms

Fibre pullout Fibre breakage, interfacial debonding

Transverse matrix cracking

Fibre failure due to matrix cracking Interfacial shear

Macro-level Failure Mechanisms

Delamination

Longitudinal splitting

Failure Mechanisms in Fibrous Laminated Composites

• Interlaminar damage mechanisms

Delamination

• Interaction of all Intra and Interlaminar Damage Mechanisms

• Effective Stress

• Effective Strain/Modulus

Damage Mechanics Approach

dSSS −=

( ) ( )nn ddS

F

S

F

−=

−==

11

1

( )ndEE −==

1

Damage Meso Modeling

• Based on three foundations of:

1. Meso Scale: assumed as stacking of alternate ply and interphase

2. Internal Variable Approach: relates effect of damage to mechanical behavior

of material.

3. Method of Local States: relates damage to thermodynamic forces associated

with strain energy.

• Ladevèze et al, Composite Science and Technology, 1992

• Ladevèze et al, Composite Structures, 1992, 1993

Fibre Breaking Damage Modeling

• Strain energy of the damaged fibre (Transversely isotropic, in this case):

• Damage parameter: dF

•Thermodynamic force:

( ) ( ) ( )

( )

( )

0 012 12

0 0 01 1 1

11 110 2 20 223 13 2312 12

22 220 0 0 0 0 01 2 2 12 13 23

33 33002312

0 0 01 2 2

1

1 1 1

12

1

1

1

d

F F FT

F

F

F

E d E d E d

eE d E E G G G

E d E E

− −− − −

= − − + + +−

− −−

d

F

F

d

F

eY

d

= −

Fibre Breaking Damage Modeling

Damage evolution:

• Brittle type fracture criterion (hold true for Carbon and Glass fibres)

11

11

While and , 0

If and 0, then 1 (rupture in traction)

If and 0, then 1 (rupture in compression)

F F

F

F

T C

d F d F F

T

d F F

C

d F F

Y Y Y Y d

Y Y d

Y Y d

• =

• =

• =

Fibre Matrix Interphase Debonding Modeling

• Strain energy of the damaged interphase:

• Damage parameters:

• Transversely isotropic nature of damage

• Thermodynamic forces:

( )

( )

( ) ( ) ( )

0 0

12 12

0 0 0

1 1 1

11 1100

231222 220 00

221 22 133 33

00

2312

0 0 0221 2 2 1

2 22

13 2312

0 0 012 12 2312 2 13 2 23 3

1

12

1

1

1

1 1 1

d

T

I

E E E

eE EE d

E E E d

G d G d G d

− −

= − − − − − −

+ + +− − −

22,12, 23d

ij

I

dij

eY ij

d

= −

22 12 23, ,d d d

12 13d d=

Matrix Cracking Modeling

• Longitudinal and Transverse through thickness matrix cracking

Matrix Cracking Modeling

• Longitudinal through thickness matrix cracking

• longitudinal matrix cracking does not lead to complete reduction of the elastic

moduli and Poisson’s ratio

• This reduction due to transverse matrix cracking is less than 5% for the materials

studied (can go undetected) !

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

12 12

23 23

0

2 2 1 22 1

0

2 12 2

0

3 23 3

0

23 23 4 4

1 , 0 1

1 , 0 1

1 , 0 1

1 , 0 1

C C

C C

C C

C C

mc

E E d

G G d

G G d

d

= −

= −

= −

= −

Matrix Cracking Modeling

• Strain energy of the damaged ply due to longitudinal matrix cracking

• Thermodynamic forces

( )

( )

( )

( )

( )

( )

( )

( ) ( ) ( )

( )

( )

( )

( )

( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

0 0

12 12

0 0 0

1 1 1

11 1100

231222 220 0 0

1 2 1 22 233 33

00

2312

0 0 0

1 2 1 22 2

2 22

13 2312

0 0 0

12 2 12 13 23 3 23

1

12

1

1

1

1 1

C C

C C C

T

CCm

d C C C

CC

C C C

C C C

E E E

eE E d E

E E d E

G d G G d

− −

= − − −

− −

−

+ + +− −

22,12, 23d

ij

m

d

ij

eY ij

d

= −

All Damage Mechanisms Together Modeling

• Strain energy of the damaged ply with these three damages

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) ( ) ( ) ( )( )

( )

( )

( )

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

( )

0 0

12 12

0 0 0

1 1 1

11 1100

231222 220 00

221 22 1 1 2233 33

0 0

2312

0 0 0221 2 1 22 2 1

2

12

0

12

1

1 1 1

12

1 1 1

1

1 1 1

1

C C

C C C

F F FT

CC

d C CC

F

C

C C C

F

C

E d E d E d

eE d EE d d

E d E d E d

G

− − − − −

= − − − − − − −

− − −

+( )( ) ( ) ( ) ( ) ( )( )

2 2

13 23

0 012 12 232 2 12 13 2 23 3 3 231 1 1 1

C Cd d G d G d d

+ +

− − − − −

Delamination Modeling

• Interlaminar interphase is a thin layer of matrix material

Initiation: von Mises stress criterion

Propagation: strain energy of the damaged interphase

Thermodynamic forces:

( ) ( ) ( )

2 22 2

33 33 23 13

3 3 3 23 2 13 1

21 1 1

I

dEE E d G d G d

− += + + +

− − −

( ) ( ) ( )3 2 1

23 13

22 2

33 23 13

2 2 20 0 0

3 3 2 1

1 1 1, ,

2 2 21 1 1d d dY Y Y

E d G d G d

+= = =

− − −

Delamination Modeling

• Equivalent Thermodynamic force:

where, , and are material parameters.

( ) ( ) ( )

( ) ( )

1 22

1

1 2

sup

d d d

T

Y Y Y Y

Y T Y

= + +

=

( ) 0

01

n

C

Y YnW Y

n Y Y

−=

+ −

1 1

( ) ( )1 2 3

1 2 3

if 1

1 otherwise

d d d W Y W Y

d d d

= = =

= = =

Implementation

Implementation

• Solution algorithm for material non-linearity

• at (k+1) load step, the residue vector is defined as

• {u} – displacement vector, {f} – external load vector, {p} – incremental (nodal)

force vector

• Predictor Step:

• truncated Taylor series

• [K] – stiffness matrix,

• {Δd} – increment in displacement vector for corresponding external load vector

increment

Implementation

• This gives

• That is,

where,

• This is conventional incremental predictor step

Implementation

• Corrector Step:

• The displacement for step (k+1) {Δd} is

• with this solution stresses at (k+1) step are computed

• hence, internal forces can be obtained

• Then residue for (k+1) step, is obtained, which is not zero. Hence, a

corrector step is applied as

• Then displacement is corrected as

Implementation

• Corrector Step:

• Next load step when

Implementation

• Normal stress sign reversal:

• Volume average normal stress signs

• change their signs as load is increased. This will affect the

damage state in the load step.

• This can violate the principle that damage should be dormant in local compression

• If the tensile region becomes compressive then damage is set to be dormant.

• If the compressive region becomes tensile then damage is set accordingly.

Implementation

• Normal Stress sign reversal:

• Growth of dF – fibre breaking in bottom ply of [0/0] laminate

Results on Damage Growth (Material- M55J/M18)

• Growth of matrix cracking in bottom layer of [0/0] laminate

Results on Damage Growth

• Growth of fibre-matrix debonding in bottom layer of [0/0] laminate

Results on Damage Growth

• Growth of delamination in [0/0] laminate

Results on Damage Growth

Growth of matrix cracking in bottom ply of [45/-45] laminate

Results on Damage Growth

Growth of d22 in bottom ply of [45/-45] laminate

Results on Damage Growth

0 N/mm2 6 N/mm2

45 N/mm2

91 N/mm288 N/mm2

65 N/mm2

• Delamination growth

Results on Damage Growth (contd…..)

155 N/mm2 190 N/mm2

350 N/mm2

420 N/mm2375 N/mm2

300 N/mm2

• Material non-linearity due to damage is taken care.

• Damage growth in prominent modes like fibre breakage,

matrix cracking, fibre-matrix debonding and delamination are

modeled and implemented

• Results are as expected.

Conclusions

Thank You

• Meso-constituents: Ply and interface

• Internal Variable Approach: relates effect of damage on

mechanical behaviour of material

• Method of Local States: relates damage to thermodynamic

forces associated with strain energy

• Damaged strain energy density

Ply Meso-Model

+−

+−

+−

−= 3322

2

23

3

323311

1

13

3

312211

2

21

1

12

1

2

11

11)1(2

1

o

o

o

o

o

o

o

o

o

o

o

o

oDE

v

E

v

E

v

E

v

E

v

E

v

EdE

++

−++

+

−

+

−+

23

2

23

13

2

13

1212

2

12

3

2

33

2

2

22

222

2

22

)1()1(2

1

o

ooooo GGdGEEdE

• No healing of damage

• Evolution of damage in Fiber/matrix interface tensile failure

• Evolution of damage in Fiber/matrix interface tensile failure

Meso-Model of Laminates (contd…..)

0.1else,0.1)ˆ(ifˆ

)ˆ( 2222

22

22

2222 =−

==+

dYY

YYYd

c

o

0.1else,0.1)ˆ(ifˆ

)ˆ( 1212

12

12

1212 =−

==+

dYY

YYYd

c

o

))((max)()),((max)( 12122222

YtYYtYtt

==

• Damaged strain energy density

• Thermodynamic forces

• Equivalent damage force

• No healing

Interface Meso-Model (contd…..)

−+

−+

−

+

= +−

)1()1()1(2

1

31

2

31

32

2

32

33

2

33

33

2

33

IIIIII

I

Ddkdkdkk

E

2

33

2

33

)1(2

1

I

Idk

Y−

= +

2

32

2

32

)1(2

1

II

IIdk

Y−

=

2

31

2

31

)1(2

1

III

IIIdk

Y−

=

1

21 ])()([ IIIIIIe YYYY ++=

))((max)(

et

e YtY

=

• Material function

• Damage evolution

Interface Meso-Model (contd…..)

n

o

e

c

e

o

eee

YY

YY

n

nY

−

−

+= +

1)(

0.1)(if)( === eeIIIIII YYddd

otherwise0.1=== IIIIII ddd