Mechanics of heterogeneous media Method of …...S.V. Lomov - Mechanics of heterogeneous media - 3....

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 1

Mechanics of heterogeneous media

Method of inclusions and its applications for random fibre composites

Stepan V. Lomov


From Eshelbyprinciple to equivalent stiffness of an inclusion assembly…

Dw of

:D

Dw of

:D

ijH

D:m:

D:m:

ijDH

effijklC


…applied to random fibre reinforced composites …

effijklC

D:m:


…and to textile composites

effijklC

D:m:


General scheme of application of the inclusion method

1. The heterogeneous medium should constitute a homogeneous matrix with a second (discontinuous) phase, or more phases of reinforcement embedded in it

2. Build a geometrical model of the RVE of the reinforcement

3. Subdivide the reinforcement into elements, which somehow could be represented as ellipsoids.

4. Consider the assembly of the ellipsoidal inclusions in the matrix

5. Using properties of the reinforcement, assign stiffness tensors to the inclusions (micro-homogenisation may be performed on this step)

6. Apply the inclusion theory to calculate the equivalent stiffness of the RVE

effijklC

D:m:


Method of inclusions

• Eshelby transformation principle for an assembly of inclusions

• Mori-Tanaka algorithm

• Self-consistent algorithm


Reminder: Eshelby transformation principle for ONE inclusion

D:m:

D:m:

ijDH

ijH f

ijH f

ijklCD

mijklC

mijklC

� ��

The solution (disturbance fields) for the elasic problem

of an anisotropic ellipsoidal inclusion

in an anisotropic matrix

with the strain at infinity is given by

ijkl

ijkl

ij ij ij ij

m

ij

C

C

D D

D

H H H H

H

6 f

f

�

x � ��

� � � ��

� �

1

: ;

:

where and are Eshelby tensors and

12

ijkl ijkl

ij ijkl kl

ij ijkl kl

ijkl ijkl

mkl klmn mn kl klmn mn kl

m m

ij kl

S const

D

S D

C S C S

C

D DD

D DD D

DD DD

D D D D D D

D DD D D

D

H HH O H

OH H H H H

H H

H

f f

� f

�: �:

� � � �ª º� � � �¬ ¼

�

x

x x

C S I C S C C

x � � � � � ��

, ,

, ,

12

mmn mn ik lj jk li

mklmn mn ik lj jk li

G G d

C G G d

D

D

H

H

:

:

c c c cª º� � � ¬ ¼c c cª º � � � �¬ ¼

³³

x x x x x x

x x x x x


Eshelby transformation principle for an assembly of inclusions

� � � � � �

� � � �

, ,

Consider disturbance field produced by inclusion (or by the source domain ):

1,

2

The disturbance in induced by the inclusion :1

mij klmn mn ik lj jk li

ij

C G G d

V

�

E E

D

D ED

EE

H H

EH

:

c c cª º: � � � �¬ ¼:

: :

³x x x x x x

average

� ��

� � � � � � � � � �

, ,

, ,

,

1 12

or

;

12

For : is Eshelby tensor

ij

mklmn ik lj jk li mn

ij ijkl kl

mijkl mnkl ik lj jk li

ijkl

d

C G G d dV

S

S C G G d dV

S

�

� ��

E

E D ED

D E DE E

DE E DD

DE

H

H

H H

D E

:

: :

: :

: ½° °c c cª º � � � � : : �® ¾¬ ¼: ° °¯ ¿

: c c cª º � � � � : :¬ ¼:

³³ ³

³ ³

x x

x x x x x x

x x x x x x

for an isolated inclusion.

NOTE: From now on we consider strains in inclusions and the matrix.averaged

D:m:

D:m:

ijDH

ijH f

ijH f

ijklCD

mijklC

mijklC

E:

E:1...MD

ijEH


Equations for disturbance strains and eigenstrains

� � � � � � � � � �1 1

1,

Summing up disturbance strains for all the inclusions,

1 1,

or in tensor notation

, 1... (1)

Stresses i

M M

ij ij ij ij

M

d dV V

M

� �

D E D ED D E E

D DD D DE E

E E D

H H H H

H H H D

: :

z

: :: :

��

¦ ¦³ ³

¦

x x x x

S S

� � � �

1

n the inclusions

, 1... (2)

Total disturbance strains

=0 (3)

where is the disturbance strai

m

Mm

m

m

M

c c

D D D D D

DD

D

V H H H H H D

H H

H

f f

��

�¦C C

average

� � � �

� � � �11

1

1

n in the matrix:1

, are the volume fractions of the matrix and the inclusions:

= , 1... ; 1 ; 1

M

mij ijM

V

m

M

M

m m

dV V

c c

VVc M c c c

V V�

�

D

D

D

DDD

D DD

H H

D

�

� :

� : ¦

:: � �

³¦

¦ ¦

x x

D:m:

D:m:

ijDH

ijH f

ijH f

ijklCD

mijklC

mijklC

E:

E:1...MD

ijEH


Image strain and the mean field assumption

D:m:

D:m:

ijDH

ijH f

ijH f

ijklCD

mijklC

mijklC

E:

E:1...MD

ijEH

1,

, 1...M

MD DD D DE E

E E DH H H D

z �� ¦S S

The second term is called image strain and accounts for the additional (in comparison with the isolated inclusion case) strain, that the inclusion D receives due to interaction with other inclusions.

Mean field assumption

Image strain could be approximated by its mean value which is the same everywhere – in all the inclusions and in the matrix

� � � �

1,

1 (4)

Mim

m im

im m

m

D DD D DE E DD D

E E D

D DD D DD D

D DD D

H H H H H

H H HH H H H H HH H H

z

f6

f 6 6

� 6 6

� � � � � � � � � � � � � � � � � � �

¦S S S

S S

SPedersen, O.B. Thermoelasticity and plasticity of composites - I. Mean field theory Acta Metallurgica Materials31(11): 1983 1795-1808.


Strain concentration factors and homogenised stiffness

Strain concentration tensors relate full strains in the inclusions with the applied strain:

(*)

Dilute strain concentration tensors relate full strains in the inclusionm

D

D D

DH H f6 � �

A

A

A

� � � �

� � � �

1

1

1

1 1

1

s with the matrix strain

(**)

(5)

Proof:

(3) =0

1

(*),(**)

mm

M

m m m

Mm

m

M Mm

m m

mm m

c c

c c

c c c c

c

D D

D D EE

E

DD

D

DD D

D D

E E

H H

H H H H

H H H

H H

6 6

�

f f6 6

f

6 6

�6 6

� �

§ · �¨ ¸© ¹

� � � ��

ª º� �� ¬ ¼

¦

¦¦ ¦

A

A A I A

A A1

1 1

,M M

m mm m m mc c c QEDD D D E

D ED E

H�

6

§ ·� � � � �¨ ¸© ¹¦ ¦A A A I A

D:

m:

D DH H Hf6 �

ijH fm mH H Hf6 �

DAmDA


Homogenised stiffness of the composite

� � � � � � � ��

� � � �

1

1

1

1 1

1

;

;

(*)

(3) 0

(*)

eff

Mm m m m

m

Meff m m

m

Mm

m

M Mm

m m

Mm

m

ef

c c

c c

c c

c c c c

c c

D DD

D

D DD

D

DD

D

DD D

D D

DD

D

V H H HV H H

H H H H

H H H H

f

f

f f f6

f f f f

� � � � � � � �

§ · � � � � � � �¨ ¸© ¹ �

� ��

� � � �

� �

�

¦¦

¦¦ ¦

¦

C

C x x C x A x

C A C A A

C C A C A

A A

A I A I

A A I

C

� �1 1

1

(6)

M Mf m

Meff m m

c c

c

D D DD D

D D

D DD

D

§ · � �¨ ¸© ¹� � �

¦ ¦¦

C I A C A

C C C C A

D:m:

ijH fijklCD

mijklC

E:


Mori-Tanaka method

Mori, T. and K. Tanaka Average stress in matrix and average elastic energy of materials with misfittinginclusions Acta Metall.Mater. 21, 1973 571-574

Consider average strains in the inclusions and the matrix and adopt mean field assumption.

D:m:

ijH fijklCD

mijklC

E:

� � � ��

11

1

Then the concentration tensors matrix inclusion are calculated as

(7)

and the composite stiffness is calculated with (6):

where

(5):

m mm

Meff m m

m m

c

c c

D DD D

D DD

D

D DE

��

oª º � �« »¬ ¼

� �

�

¦

A I S C C C

C C C C A

A A I1

1

M

mE

E

�

§ ·¨ ¸© ¹¦ A

Equation (7) is derived from the Eshelby transformation principle and mean field assumption, using (2) and (4).


Mori-Tanaka method: effective stiffnessCalculation of the effective stiffness of composite with inclusions:

1. Calculate Eshelby tensors for the individual inclusions, using formulae for exterior points of the ellipsoids (lecture on E

MDDS

shelby theory).

is defined by the matrix properties. Tensor is expressed in coordinate

system , aligned with the axes of the inclusion .

2. Transform the tensors in the global coordinat

CS

DD DD

D

DDD

S S

S

� � � ��

111

1

1

e system .3. Calculate strain concentration tensors:

,

4. Calculate effective stiffness of the composite:

glob

Mm m

m m m m

Meff m m

CS

c c where

c

D D E D DD DE

E

D DD

D

� ��

§ · ª º � � �¨ ¸ « »¬ ¼© ¹

� �

¦

¦

A A I A A I S C C C

C C C C AD:

m:ijHf

ijklCD

mijklC

E:

CS x y z

� � � �

globCS xyz


Mori-Tanaka method for effective stiffness: Notes

1. Interactions between the inclusions are taken into account in the simplified manner: mean field assumption. Each inclusion feels the presence of other inclusions indirectly through the total strain of the matrix.

2. Eshelby tensors used are those for eigenstrains inside the inclusions, and depend on the properties of the matrix, NOT the inclusions.

3. For isolated inclusion the strain inside it is constant. It is no more true for the assembly of inclusions (see the slide on Eshelby transformation principle). The strains inside the inclusions are averaged in Mori-Tanaka theory.

4. The strain in the matrix is also averaged.

5. The composite stiffness, calculated using the mean field assumption, depends for the given matrix/inclusions material combination on the volume fraction, shape and orientation of inclusions ONLY.

6. The composite stiffness DOES NOT depend on the size of the inclusions, nor on their positions (for the given orientations).

7. Practically, Mori-Tanaka method gives fairly good predictions of the composite stiffness. However, it is sometimes criticized as it leads to physical inconsistencies for certain non-trivial orientation distributions of non-isotropic inclusions (see, for example, “Papers for review”, Freour et al).


The stiffness does not depend on the inclusions size, positions...

= =

= =


…but depends on the volume fraction, shape, orientation

z= volume

z


Mori-Tanaka method: average strains in the inclusions

1

Calculation of average strains and stresses in the inclusions and the matrix:

1; , where

;

Mm m m

m

m m m

cc

D D DD

DD D D

H H H H

V H V H

f f

§ · �¨ ¸© ¹

¦A A A I A

C CD:

m:ijHf

ijklCD

mijklC

E:

CS x y z

� � � �

globCS xyz Notes

1. These are average stresses and strains.

2. Staying inside Eshelby approach and the mean field assumption, it is possible also to evaluate interface stresses between the inclusions and the matrix.


Self-consistent method

Still using mean field approximation, replace the matrix with a medium having properties equal to those of the composite itself. This means that the Eshelby tensors represent the constraining effect of the composite as the whole rather then the matrix only.

D:m:

ijHfijklCD

effijklC

E:

CS x y z

� � � �

globCS xyz

� � � �The calculation scheme:

1. Set

2. Calculate Eshelby tensors for the individual inclusions.

These tensors depend on !3. Calculate strain concentration tensors:

eff

eff m

eff

D D D D

DD

D

H H H H Hf f � � � � � � ��

C C

C C

S

C

A � � � ��

111

1

1

,

4. Calculate new effective stiffness of the composite:

5. Check convergence of . If not converged, go to step

Meff eff

m m m m

Meff eff eff

eff

c c where

c

D E D DD DE

E

D DD

D

� ��

§ · ª º� � �¨ ¸ « »¬ ¼© ¹

� �

¦

¦

A I A A I S C C C

C C C C A

C 2.


Example: porous material with spherical pores (1)

Isotropic material with shear module P and Poisson coefficient 1/5.

1. Eshelby tensor [Mura, p.79, eq (11.21)]

� ��

� �

1111 2222 3333

1122 2233 1133 2233 2211 3311

1212 1221 2112

2323 3223 2332

3131 1331 3113

7 5 115 1 2

5 10

15 1

4 5 115 1 4

all other components are zero

S S S

S S S S S S

S S S

S S S

S S S

QQ

QQ

QQ

� ��

� �



� � � � � ��

� � � �

11 1

11 1 1

1 1

0

=

pore

pore m pore mm

pore pore porem m pore m m pore

m pore m

c c c c

c c c

��

��

� �

ª º � � �« »¬ ¼

ª º � � � � ¬ ¼ª º � � �¬ ¼

C

A I S C C C I S

A A I A I S I I S

I S I I S

2. Strain concentration tensor

3. Effective stiffness

� � � ��

1

1

eff m pore m pore m porepore pore

mm pore m

mm m

c c

c c c

c c

�

�

� � � � � � � �

C C C C A C I A

C I S I I S

C I S I S



� � � ��

� ��

� �

1111 1111 1111 1111 1111 1111

1212 1212 1212 1221 2112

1212 1212 1212 1221 2112

1 1/ 22 2

1 / 2

effm m

effijpq pqkl m pqkl m ijpq pqkl pqkl

eff effm m

eff eff effm m

m

eff effm

m

c c

C I c S c C I S

C c C S c C C S

C c C S c C S

c C C S C S

cc

O P O P O

� ��

� ��

� �

��

C I S C I S

� � � �12 22 1

11 1/ 4 1/ 41 / 4 / 4 2 1

porem

m pore

poreeff mm

m m m pore

ccc c

ccc

c c c c

P O P

P P P P

��

� ��

1111

1122 1133

1212 1221

7 5 115 1 2

5 10

15 1

4 5 115 1 4

S

S S

S S

QQ

QQQQ

� ��

� �


Orientation distribution of a random assembly of particles

• Orientation distribution function

• Orientation tensors

• Generation of an RVE of a random assembly or oriented particles

Advani, S.G. and C.L.I. Tucker 1987 The use of tensors to describe and predict fibre orientation in short fibre composites Journal of Reology 31(8) 751-784.


Random assembly of particles

Consider an RVE of material containing randomly oriented particles. The orientation of each particle is assumed to be characterised by orientation of one axis, i.e., by a unit vector p of this axis.

Examples:

• a composite reinforced by straight glass fibres;

• grains in steel, characterized by the axis of orthotropy

The orientation of the particles is stochastic and is described by the orientation distribution function (ODF).

Note. The ODF dose not refer to a particular RVE, but to a random assembly as a whole. A particular RVE (which contains a limited number of particles) constitutes a random realisation of the assembly. The size of the realisation (number of particles in RVE) is a subject of an arbitrary choice.

particle = fibre


Orientation vector

4M

2

1

3p

dp

dMd4

> @ > @� �

1

2

3

2

0 0

2

0 0

1

sin cos

sin sin

cos

0, , 0,2

( ) , sin

sin 4

p

p

p

f d d f d

d d d

S S

S S

T MT MT

T S M S

M M T T T

M T T S

� �

³ ³ ³³ ³ ³

p

p p

p

�

�


Orientation Distribution Function

� � � ��

� �

� � � � � � � �

� � � �0

Orientation distribution function (ODF):

, :

/ 2 / 2

cos / 2 cos cos / 2

, sin

Symmetry:

or , ,

Normalisation:

0 21

0

, sin

d dP

d d

d d

P

d d dS

\ \ M TM M M M M

T T T T T\ M T T T M

\ \ \ M T \ M S S T

M ST S

\ M \ M T T T

� � � � ½° ° ® ¾� � � �° °¯ ¿

� � �

� � ½ �® ¾� �¯ ¿ ³

p

p p

p p2

0

1S ³ ³�

dp

dMd4

4M

2

1

3p


Orientation Distribution Function –examples (1)

All the fibres are oriented in one direction

� � � � � �, * *\ M T G M M G T T � �Uniform ODF in 3D space

� � 1,

4\ M T S

Planar orientation

� � � �� 2

0

, ,2

; 1d

M

S

M M M

S\ M T G T \ M

\ M \ M S \ M M

§ · �¨ ¸© ¹ � ³

Uniform ODF on 2D plane

� � 12M\ M S

M2S

12S


Orientation Distribution Function –examples (2)

Almost planar distribution� � � � � ��

cos

2

cos 2

3

, 0,cos * ,

1 cos0,cos * exp

2 cos *

NB: 1° cos

2 We assume * / 2, at least * / 6, then the normalisation error is less then 0.1%

N

N

p

T M

T

\ M T T \ MTT S T

TT S T S

§ · �¨ ¸© ¹

q ��

M2S

12S

T2S


Orientation Distribution Function –notes

1. ODF is a complete, unambiguous description of the fibre orientation

2. Definition of an ODF from experimental data requires certain approximation of the function. It is easy in case of an assumed distribution: uniform, normal…

3. The real distributions are often not uniform and not normal. For example, in Sheet Moulding Composites a sheet with initially uniformly distributed fibres is subject to flow, which distorts this distribution locally

4. It is desirable to have more concise characterisation of the orientation distribution

5. We will use ODF as a static characteristic of the (local) fibre orientation. ODF can be also used dynamically, to simulate changing fibre orientation during flow of the material (see Advani & Tucker).


Example of fibre orientation distribution

http://www.moldflow.com/

Injection moulded part, glass/PP

primary fibre orientation colour

code


Orientation tensors

� ��

Consider diadic products of the orientation vector ,weighted by the distribution function and averaged:

Symmetries and normalisation:; ...

ij i j

ijkl i j k l

ij ji ijkl jikl

i

a p p d

a p p p p d

a a a a

a

\\

³³

p

p p pp

p p pppp

�

�

1 (summation)

Tensors of the 4th order provide full information of the tensors of 2nd order:

Note: it is possible also define planar orientation tensors, see Advani & Tucker.

i

ij ijkka a


Orientation tensors –examples

All the fibres are oriented in x direction

� � � �,2

1 0 00 0

0sym

S\ M T G M G T§ · �¨ ¸© ¹ª º« » « »« »¬ ¼

2a

Planar orientation in the given direction

� � � �2

2

, *2

cos * cos *sin * 0sin * 0

0sym

S\ M T G M M G T

M M MM

§ · � �¨ ¸© ¹ª º« » « »« »¬ ¼

2a

Uniform ODF in 3D space

� � 1,

41/ 3 0 0

1/ 3 01/ 3sym

\ M T S ª º« » « »« »¬ ¼

2a

Uniform ODF on 2D plane

� � 1,

2 2

1/ 2 0 01/ 2 0

0sym

S\ M T G T S§ · �¨ ¸© ¹

ª º« » « »« »¬ ¼2a


Basis functions –tensors of the increasing order

Onat, E. T., and Leckie, F. A. (1988) Representation of mechanical behavior in the presence of changing internal structures, J. Appl. Mech., 55, 1-10

� ��

Orthogonal basis functions (spherical harmonics), expressed as tensors of increasing order:

13

17

135

...E

ij i j ij

ijkl i j k l ij k l ik j l il k j jk i l jl i k kl i j

ij kl ik jl il jk

f p p

f p p p p p p p p p p p p p p p p

G

G G G G G G

G G G G G G

� � � � � � � �

� � �

p

p

� � � � � �0

xpansion of an arbitrary function

...ij ij ijkl ijklF f fD D D � � �p p p


Recovery of ODF from orientation tensors

� ��

� � � � � �

Deviatoric orientation tensor:13

17

135

...Expansion of the ODF

1 15 315...

4 8 32

ij ij ij

ijkl ijkl ij kl ik jl il kj jk il jl ik kl ij

ij kl ik jl il jk

ij ij ijkl ijkl

b a

b a a a a a a a

b f b f

G

G G G G G G

G G G G G G

\ S S S

� � � � � � � �

� � �

� � �p p p


Loss of information for lower order tensors

� � 2

Consider, as an ultimate example, :fibres parallel to -axis, and second-order orientation tensors:

1 0 0 2 / 3 0 00 0 ; 1 / 3 0

0 1 / 3

7 5, sin cos

12 2

x

sym sym

\ M T TS S

ª º ª º« » « » �« » « »�« » « »¬ ¼ ¬ ¼ � �

2 2

concentrated distribution

a b

� �

2

1 / 3 0 0 0 0 01 / 3 0 ; 0 0

1 / 3 0

1,

4

sym sym

M

\ M T S

ª º ª º« » « » « » « »« » « »¬ ¼ ¬ ¼

2 2

Uniform distribution

a b

-0.2

0

0.2

0.4

0.6

0.8

-90 -60 -30 0 30 60 90

fi, °

psi

M

\


Real planar distributions (compression moulded composites)

[Advani & Tucker]

almost uniform distribution

concentrated distribution


What order is sufficient? –the case of the orientation averaging

Consider a tensor property ( ), assosiated with a unidirectional microstructure,aligned in the direction of . is trasvesely isotropic, with the axis of symmetry .The orientat

Orientation averagingT p

p T pion average of in an assembly with an arbitrary ODF ( ) is

( ) ( )

’ stiffness matrix thermal conductivity ...

d

\\ ³

T p

T T p p p

Examples of T s

�


Orientation averaging of a tensor of second and fourth order

� � 1 2

1 2 1 2

Transversly isotropic tensor of second order:

Orientation average:

Transversly isotropic tensor of fouth order, having the symmetry of astiffness tensor (

ij i j ij

ij i j ij ij ij

ij

T A p p A

T A p p A A a A

T

G

G G

�

� �

p

� � � � � ��

� � � �1 2 3

4 5

1 2 3

4 5

= = = ):kl jikl ijlk klij

ijkl i j k l i j kl k l ij i k jl i l jk j l il j k il

ij kl ik jl il jk

ijkl ijkl ij kl kl ij ik jl il jk jl il jk il

ij kl ik jl il

T T T

T B p p p p B p p p p B p p p p p p p p

B B

T B a B a a B a a a a

B B

G G G G G GG G G G G G

G G G G G GG G G G G G

� � � � � � ��

� � � � � � ��

p

� �Orientation average of an n-th order tensor property is fully defined by the n-th order orientation tensor, even if the ODF is not reconstructed from it exactly

jk


Notes on the sufficiency of the fourth order orientation tensor

1. The theorem on the exact prediction of the stiffness by the 4th order orientation tensor is valid only for the orientation averaging algorithm, which per se is not exact. When more complex and more precise methods are used, e.g. Mori –Tanaka, the conclusion does not hold rigorously. However, in heuristic sense it is still likely, that the 4th order orientation tensor leads to a good approximation of the stiffness.

2. When an orientation tensor is calculated for an assembly with a certain ODF, and then the ODF is reconstructed using the orientation tensor, some information (higher “harmonics”) is lost, and the ODF is changed.

3. The possible error in predictions of stiffness, based on orientation tensors, could be understood from the following: ODF, reconstructed from the orientation tensor, may be the same for assemblies with similar, but not exactly equal real ODFs. For example, G-distribution and a bell-shaped distribution with standard deviation of about 30°have the same 2nd order orientation tensor.


Generation of RVE of a random assembly of fibres

� � � ��

� � ^ ` � �

Consider an assembly of fibres characterized by: 1° ODF

, :

/ 2 / 2, sin

/ 2 / 2

2 Length distribution

: / 2 / 2

3 Volume fraction 4 Fibre diamet

L L

d dP d d

d d

l P l dl l l dl l dl

Vf

\ \ M TM M M M M \ M T T T MT T T T T

\ \

� � � � ½ ® ¾� � � �¯ ¿

q� � � �

qq

p

er (cylindrical fibres)

How to generate an istance of RVE of the assembly, containing N fibres?

d


Defining the RVE volume, placing the fibre centres

2

0

1/3

; 4

; cubic RVE:

,using uniform random number generator for all three coordin

mean L fibres mean

fibresRVE RVE

dl l dl V N l

VV a V

Vf

S\

[

f � �

³

Calculate the RVE volume

Place fibre centers randomly in the cubeates

0 1

1

[

f([)

100 fibre centres (shown as small cylinders),

Vf=10%


Generation of a random value with a given distribution

F(x)[

x=F-1([)RNG

� �� 0

( )

0,1

x

F x d

rand

\ ] ]

[

³

> @� �

When used for , then , 0, 2

When used for , then cos , [ 1,1] sin

x

x x dx d

M M M ST T T T

� � � �


Rejection algorithm (von Neumann)

� ��

� �min max

max

1 Choose ,

2 Choose = 0,

3 If , then accept ; else go to 1°

x rand x x

rand

x x

[ \[ \

q qq �

[

xThe same algorithm may be used for a vector of the random variables

RNG

RNG


Assign fibre directions and length

� �1° For each fibre assign the angles and using the rejection algorithm and the

given ODF , . Use and cos as independent random variables.

2 For each fibre assign the length using the rejection

M T\ M T M T

q algorithm and the givenlength distribution function ( ).

Note: some fibres may protrude from the RVE volume (centre is inside)

L l\

100 fibres

l/d is uniformly distributed in (1,10)

Random orientation distribution

Vf=10%


Orientation distribution: Summary

1. Orientation distribution of slender fibres or anisotropic grains is the major structural parameter of random heterogeneous materials.

2. ODF is a complete, unambiguous description of the fibre orientation.

3. Orientation tensors provide concise, albeit approximate (especially 2nd order tensors), characterisation of orientation distribution.

4. Theoretically, the 4th order orientation tensors are sufficient if orientation averaging method is employed for prediction of the material stiffness. This becomes an heuristic evaluation for more complex homogenisation methods.

5. When ODF of fibres is given, together with fibre volume fraction and length value or distribution, it is easy to generate a random instance of RVE.


Application: random fibre reinforced composites


Random fibre reinforced material

Input data

• fibre diameter

• distribution of the fibre length

• ODF or orientation tensor

• mechanical properties of the (anisotropic) fibres

• tensile diagram and Poisson coefficient of the isotropic matrix

Elastic matrix:

calculate homogenised stiffness matrix of the composite

Non-linear matrix:

calculate tensile diagram of the composite 5%431 2


Assumptions and simplifications

1. Fibres are elastic, but may be anisotropic (carbon; homogenised fibre bundles).

2. Matrix may be non-elastic, characterised by tensile diagram

3. Mori – Tanaka scheme is used for homogenisation, hence mean field assumption.

4. Fibres are straight, approximated by slender ellipsoids. Typical fibre length (glass) from 0.5 to 10 mm, fibre diameter ca 0.01 mm, elongation from 50 to 1000.

5. The bonding fibre-matrix is perfect. Possible extension of the method to calculation of the debonding.


Monte Carlo and averaging

Flowchart

Generate random realisation of RVE

Fibre length distribution

Fibre orientation distribution

Mori – Tanaka homogenisation

Fibre stiffness

Matrix stiffness

Shape and orientation of the individual fibres in RVE

Fibre volume fraction

Homogenised stiffness Algorithm for non-elastic

matrix: see test problem #3 in the section 2


Example: glass/polypropylene (PP) composite (1)

Jao Jules, E., S.V. Lomov, I. Verpoest, P. Naughton, A.W. Beekman and R. Van Daele Prediction of non-linear behaviour of discontinuous long glass fibres polypropylene composites in Proceedings of the 15th International Conference on Composite Materials (ICCM-15): 2005 Durban CD edition.


Example: glass/polypropylene (PP) composite (2)

fibre debonding


Random reinforced composites: Summary

1. Method of inclusions is effectively used for calculation of elastic stiffness and non-linear deformation diagrams of random fibre reinforced composites

2. The micromechanical calculations using the method of inclusions are build in the multi-level simulations, combining flow simulation, micromechanics andmacro structural analysis

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