Mechanics of heterogeneous media Method of inclusions and its - - PowerPoint PPT Presentation

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

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 2

F rom E shelbyprinciple to equivalent stiffness of an inclusion assembly…

D

• D

D

• D

ij

• m
• m
• ij
• eff

ijkl

C

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

… applied to random fibre reinforced composites …

eff ijkl

C

• m

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 4

… and to textile composites

eff ijkl

C

• m

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 5

G eneral 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

eff ijkl

C

• m

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 6

Method of inclusions

• Eshelby transformation principle for an assembly of inclusions
• Mori-Tanaka algorithm
• Self-consistent algorithm

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 7

R eminder: E shelbytransformation principle for O N E inclusion

• m
• m
• ij
• ij

ij ijkl

C

m ijkl

C

m ijkl

C

• The solution (disturbance fields)

for the elasic problem

• f 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

• x
• 1

: ; : where and are Eshelby tensors and 1 2

ijkl ijkl

ij ijkl kl ij ijkl kl ijkl ijkl m kl klmn mn kl klmn mn kl m m ij kl

S const D S D C S C S C

• x

x x C S I C S C C x

• ,

, , ,

1 2

m mn mn ik lj jk li m klmn mn ik lj jk li

G G d C G G d

• x

x x x x x x x x x x

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 8

E shelbytransformation 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

m ij klmn mn ik lj jk li ij

C G G d V

• x

x x x x x average

• ,

, , ,

, 1 1 2

• r

; 1 2 For : is Eshelby tensor

ij m klmn ik lj jk li mn ij ijkl kl m ijkl mnkl ik lj jk li ijkl

d C G G d d V S S C G G d d V S

• ✁
• 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

• m
• m
• ij
• ij

ij ijkl

C

m ijkl

C

m ijkl

C

• 1...M

ij

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 9

E quations for disturbance strains and eigenstrains

• 1

1 1,

Summing up disturbance strains for all the inclusions, 1 1 ,

• r in tensor notation

, 1... (1) Stresses i

M M ij ij ij ij M

d d V V M

• x

x x x S S

• 1

n the inclusions , 1... (2) Total disturbance strains =0 (3) where is the disturbance strai

m M m m m

M c c

• C

C average

• 1

1 1 1

n in the matrix: 1 , are the volume fractions of the matrix and the inclusions: = , 1... ; 1 ; 1

M

m ij ij M V m M M m m

d V V c c V V c M c c c V V

• ✄
• x

x

• m
• m
• ij
• ij

ij ijkl

C

m ijkl

C

m ijkl

C

• 1...M

ij

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 10

Image strain and the mean field assumption

• m
• m
• ij
• ij

ij ijkl

C

m ijkl

C

m ijkl

C

• 1...M

ij

• 1,

, 1...

M

M

• 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 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)

M im m im im m m

• S

S S S S S

Pedersen, O.B. Thermoelasticity and plasticity of composites - I. Mean field theory Acta Metallurgica Materials 31(11): 1983 1795-1808.

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 11

S train 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 inclusion

m

• A

A A

• 1

1 1 1 1 1

s with the matrix strain (**) (5) Proof: (3) =0 1 (*),(**)

m m M m m m M m m M M m m m m m m

c c c c c c c c c

• A

A A I A A A

1 1 1

,

M M m m m m m m

c c c QED

• A

A A I A

• m
• ij

m m

• A

m

• A

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 12

H

• mogenised stiffness of the composite
• 1

1 1 1 1 1

; ; (*) (3) (*)

eff M m m m m m M eff m m m M m m M M m m m M m m ef

c c c c c c c c c c c c

• 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 M f m M eff m m

c c c

• C

I A C A C C C C A

• m
• ij

ijkl

C

m ijkl

C

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 13

Mori-T anaka method

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

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

• m
• ij

ijkl

C

m ijkl

C

• 1

1 1

Then the concentration tensors matrix inclusion are calculated as (7) and the composite stiffness is calculated with (6): where (5):

m m m M eff m m m m

c c c

• A

I S C C C C C C C A A A I

1 1 M m

• A

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

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 14

Mori-T anaka method: effective stiffness

Calculation 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 M

• S

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

• S

S S

• 1

1 1 1 1

e system .

• 3. Calculate strain concentration tensors:

,

• 4. Calculate effective stiffness of the composite:

glob M m m m m m m M eff m m

CS c c where c

• A

A I A A I S C C C C C C C A

• m
• ij

ijkl

C

m ijkl

C

• CS

x y z

• glob

CS xyz

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 15

Mori-T anaka method for effective stiffness: N

• tes

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

• n 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).

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 16

T he stiffness does not depend on the inclusions siz e, positions...

= = = =

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 17

… but depends on the volume fraction, shape, orientation

• =

volume

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 18

Mori-T anaka method: average strains in the inclusions

1

Calculation of average strains and stresses in the inclusions and the matrix: 1 ; , where ;

M m m m m m m m

c c

• A

A A I A C C

• m
• ij

ijkl

C

m ijkl

C

• CS

x y z

• glob

CS 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.

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 19

S elf-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

• nly.
• m
• ij

ijkl

C

eff ijkl

C

• CS

x y z

• glob

CS 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

• C

C C C S C A

• 1

1 1 1 1

,

• 4. Calculate new effective stiffness of the composite:
• 5. Check convergence of

. If not converged, go to step

M eff eff m m m m M eff eff eff eff

c c where c

• A

I A A I S C C C C C C C A C 2.

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 20

E xample: porous material with spherical pores (1 )

Isotropic material with shear module 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 1 15 1 2 5 1 15 1 4 5 1 15 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

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 21

E xample: porous material with spherical pores (2)

• 1

1 1 1 1 1 1 1 1

=

pore pore m pore m m pore pore pore m m pore m m pore m pore m

c c c c c c c

• 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 pore pore pore m m pore m m m 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

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 22

E xample: porous material with spherical pores (3)

• 1111

1111 1111 1111 1111 1111 1212 1212 1212 1221 2112 1212 1212 1212 1221 2112

1 1/ 2 2 2 1 / 2

eff m m eff ijpq pqkl m pqkl m ijpq pqkl pqkl eff eff m m eff eff eff m m m eff eff m 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 c c

• C

I S C I S

• 1

2 2 2 1 1 1 1/ 4 1/ 4 1 / 4 / 4 2 1

pore m m pore pore eff m m m m m pore

c c c c c c c c c c c

• 1111

1122 1133 1212 1221

7 5 1 15 1 2 5 1 15 1 4 5 1 15 1 4 S S S S S

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 23

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.

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 24

R andom 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

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 25

O rientation vector

• 2

1 3 p

dp d d

• 1

2 3 2 2

1 sin cos sin sin cos 0, , 0,2 ( ) , sin sin 4 p p p f d d f d d d d

• p

p p p

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 26

O rientation D istribution F unction

• Orientation distribution function (ODF):

, : / 2 / 2 cos / 2 cos cos / 2 , sin Symmetry:

• r

, , Normalisation: 2 1 , sin d d P d d d d P d d d

• p

p p p p

2

1

• dp

d d

• 2

1 3 p

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 27

O rientation D istribution F unction –examples (1 )

All the fibres are oriented in one direction

• ,

* *

• Uniform ODF in 3D space
• 1

, 4

• Planar orientation
• 2

, , 2 ; 1 d

• Uniform ODF on 2D plane

1 2

• 2

1 2

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 28

O rientation D istribution F unction –examples (2)

Almost planar distribution

• cos

2 cos 2 3

, 0,cos * , 1 cos 0,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

• 2

1 2

• 2

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 29

O rientation D istribution F unction –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).

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 30

E xample of fibre orientation distribution

http://www.moldflow.com/

Injection moulded part, glass/PP primary fibre

• rientation colour

code

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 31

O rientation 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

• 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 ijkk

a a

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 32

O rientation tensors –examples

All the fibres are oriented in x direction

• ,

2 1 sym

• 2

a

Planar orientation in the given direction

• 2

2

, * 2 cos * cos *sin * sin * sym

• 2

a

Uniform ODF in 3D space

• 1

, 4 1/3 1/3 1/3 sym

• 2

a

Uniform ODF on 2D plane

• 1

, 2 2 1/ 2 1/ 2 sym

• 2

a

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 33

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: 1 3 1 7 1 35 ... 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

• p

p xpansion of an arbitrary function ...

ij ij ijkl ijkl

F f f

• p

p p

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 34

R ecovery of O D F from orientation tensors

• Deviatoric orientation tensor:

1 3 1 7 1 35 ... 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

• p

p p

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 35

L

• ss of information for lower order tensors
• 2

Consider, as an ultimate example, : fibres parallel to -axis, and second-order orientation tensors: 1 2 / 3 0 ; 1/ 3 1/ 3 7 5 , sin cos 12 2 x sym sym

• 2

2

concentrated distribution a b

• 2

1/ 3 1/ 3 ; 1/ 3 1 , 4 sym sym

• 2

2

Uniform distribution a b

• 0.2

0.2 0.4 0.6 0.8

• 90
• 60
• 30

30 60 90 fi, ° psi

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 36

R eal planar distributions (compression moulded composites)

[Advani & Tucker] almost uniform distribution concentrated distribution

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 37

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 averaging T p p T p ion average of in an assembly with an arbitrary ODF ( ) is ( ) ( ) ’ stiffness matrix thermal conductivity ... d

• T

p T T p p p Examples of T s

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 38

O rientation 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 a stiffness tensor (

ij i j ij ij i j ij ij ij ij

T A p p A T A p p A Aa A T

• 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

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

jk

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 39

N

• tes 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

• f 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

• rientation tensor, may be the same for assemblies with similar, but not exactly

equal real ODFs. For example, -distribution and a bell-shaped distribution with standard deviation of about 30°have the same 2nd order orientation tensor.

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 40

G eneration of R VE

• f 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 d P d d d d l P l dl l l dl l dl Vf

• p

er (cylindrical fibres) How to generate an istance of RVE of the assembly, containing N fibres? d

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 41

D efining the R VE volume, placing the fibre centres

2 1/3

; 4 ; cubic RVE: , using uniform random number generator for all three coordin

mean L fibres mean fibres RVE RVE

d l l dl V N l V V a V Vf

• Calculate the RVE volume

Place fibre centers randomly in the cube ates

1 1

• f()

100 fibre centres (shown as small cylinders), Vf=10%

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 42

G eneration of a random value with a given distribution

F(x)

• x=F-1()

R N G

• ( )

0,1

x

F x d rand

• When used for , then

, 0,2 When used for , then cos , [ 1,1] sin x x x dx d

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 43

R ejection algorithm (von N eumann)

• min

max max

1 Choose , 2 Choose = 0, 3 If , then accept ; else go to 1° x rand x x rand x x

• x

The same algorithm may be used for a vector of the random variables R N G R N G

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 44

A ssign 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

• algorithm and the given

length 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%

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 45

O rientation distribution: S ummary

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.

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 46

Application: random fibre reinforced composites

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 47

R andom 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% 4 3 1 2

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 48

A ssumptions 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.

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 49

Monte Carlo and averaging

F lowchart

Generate random realisation of RVE Fibre length distribution Fibre orientation distribution Mori – Tanaka homogenisation Fibre stiffness Matrix stiffness Shape and

• rientation of

the individual fibres in RVE Fibre volume fraction Homogenised stiffness Algorithm for non-elastic matrix: see test problem #3 in the section 2

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 50

E xample: glass/polypropylene (P P ) 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.

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 51

E xample: glass/polypropylene (P P ) composite (2)

fibre debonding

S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 52

R andom 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 and macro structural analysis