The stiffness of short and randomly distributed fiber composites

The stiffness of short and randomly distributed fiber composites

E. SIDERIDIS, J. VENETIS, V. KYTOPOULOS

School of Applied Mathematics And Physical Sciences, Section Of Mechanics National Technical University of Athens

5 Heroes Of Polytechnion Avenue, Gr – 15773 Athens GREECE

Telephone: +302107721251 Fax: +302107721302

Email: siderem@mail.ntua.gr, johnvenetis4@gmail.com, victor@central.ntua.gr

Abstract: In this work, analytical calculations are described to estimate the elastic moduli of polymer composite materials consisting of short fibers, by extending and generalizing a reliable model. The fibers may have a finite length and an orientation characterized as random. An effort was made to complete and improve a previous procedure concerning transversely isotropic composites. To this end, a cylinder model with a short fiber in the centre of the matrix is considered. The elastic moduli were expressed in terms of the fiber content and fiber length. The obtained representations were used to evaluate the moduli of a randomly oriented short fiber composite. A comparison was made with theoretical values derived from two other authors who established trustworthy and accurate models. Finally, our theoretical values were also compared with obtained experimental results.

Keywords: Fiber composite materials, short fibers, unit cell, elastic constants, model of equivalent fiber, fiber contiguity

1 Introduction

It is well known that from the commercial standpoint the composite materials are made from matrices of epoxy, unsaturated polyester and in addition of some other thermosets and a f ew thermoplastics. The reinforcements are glass, graphite, aramid, thermoplastic fibers, metal and ceramic. The reinforcement may be continuous, woven, or chopped fiber and a typical commercial composite contains from 20% to 50% by weight glass or other reinforcement. The percentage of reinforcement in advanced composites can be as high as 70%; these materials usually use epoxy as the matrix material, and graphite is the most common reinforcement. Glass fibers are usually used for low cost and wherever high stiffness is not necessary.

The purpose of adding reinforcements to polymers is usually to enhance mechanical properties.

One of the most useful forms of composites for the construction of high – performance structural elements, is the type of panels made from aligned fibers containing polymerized matrix. They can also be made with woven fibers or randomly oriented chopped fibers; these have inferior stiffness and cannot have such high fiber materials. In order to perform stress and stiffness analyses of the chopped – fiber resin composites, it is essential that the elastic properties be known.

In reality, these materials are anisotropic and heterogeneous. Evidently, their properties depend on the elastic properties and volume fractions of the constituent materials, the fiber length or aspect ratio, the degree of the alignment, the adhesion between fibers and matrix and finally they are affected by the fabrication techniques.

Nevertheless, one should primarily elucidate that chopped fibers, flakes, particles, and similar discontinuous reinforcements may enhance short – term mechanical properties, but these types of reinforcements are usually not as ef fective as continuous reinforcements in increasing creep strength and similar long – term strength characteristics, since continuous reinforcements serve to distribute applied loads and strain throughout the entire structure.

If the fibers are randomly distributed with respect to orientation and position, then samples of material which contain statistically significant numbers of fibers seem to be isotropic. If several such samples are compared to one another, the material can be considered as h omogeneous. I f the fibers have a specific orientation, then the composite will be anisotropic. Besides, in case the statistical distribution of fillers’ orientation varies with position, then the composite becomes inhomogeneous even if the fiber density is uniform.

Mostly, theoretical studies for the stiffness and strength of such materials have been concerned

with the aligned case, or with the situation of cross – ply laminates of continuous unidirectional fibers. The application of materials reinforced with short randomly – oriented f ibers is not relatively new. For some reasons, the fibers may be arranged such that they have partially aligned whereby they are constrained to lie in a plane but are otherwise randomly oriented.

Generally, the elastic moduli of a chopped – fiber composite can be derived from those of a unidirectional discontinuous fiber composite through a normalized integration process. There have been several previous studies of the random fiber case. T he first such study was apparently that due to Cox [1] who studied the stiffness of cellulose fiber materials by assuming that the external load is carried entirely by the fibers. The result was the Cox formula i.e.

3 f

3

f

D

U

E = E

where

E

_fis the fiber modulus and

U

_f the its

volume fraction. The corresponding result for the two dimensional case was found to be

2 f

3

f

D

U

E = E

The first consideration of fiber – matrix effects was provided by Tsai and Pagano [2], in the context of laminated plates. Nielson and Chen [3]

used computational techniques to predict the modulus of the composite. Further work along the lines using the “Laminate analogy” was given by Halpin, Jerina and Whitney [4].

Lees [5], in order to explain his experimental data concerning the longitudinal modulus derived a formula for the elastic modulus of short fiber composites assuming that the fiber and the matrix have different longitudinal strain under axial loading. One of the most important theories developed is that of Christensen and Waals [6], who provided a method for determining random orientation properties using a geometric averaging method. Later, Christensen [7], thought that there is a need to obtain analytical forms which directly admit physical interpretation. Assuming that the fiber phase is much stiffer than the matrix one he simplified the expressions based on the classical results of Hill [8,9] and Hashin [10,11] on unidirectional transversely isotropic composites which after a normalized integration process yield the elastic moduli of a randomly oriented fiber composite. The results for the three and two

dimensional case respectively are represented as follows

f m f f

f f

D

U U E

U U

E E

_^









 + +

+ −

= 1 4 6

1 1 6

2

3 ;

m f m f f

f m f m f

f f

D f

E

U E U E

U E U E E

U U E E

) 1(

) 1(

) 1(

) 1(

27 19 3

1

2 3 − + +

− + + +

+ −

=

where

E

_m is the matrix modulus

The procedure presented in Ref. [6] was used by Eisenberg [12] in order to derive expressions which take into consideration the fabrication induced anisotropy of chopped – fiber resin composites. Except for the theory of Halpin and Pagano [2, 4], the above discussed theories do not consider the influence of fiber length or aspect ratio and therefore they are applicable for continuous fiber composites.

There are several theoretical formulae appearing in the literature to study the elastic moduli of unidirectional short fiber composites. Ogorkievicz and Weidman [13] idealized the composite consisting of a polymeric matrix containing unidirectionally aligned discontinuous glass fibers into a prism of the polymeric material and within it, a prism of glass. The volume of the glass prism, expressed as a fraction of the volume of the larger prism is equal to the volume fraction of the glass fibers and its proportions are fixed by the aspect ratio of the fibers. Some assumptions and a strength of materials approach led to Eq. (4) for the modulus.

A theory to enable prediction of moduli from basic matrix and fiber properties is due to Krenchel [14] who considered the composite to comprise a number of plies of uniaxially aligned fibers with the fibers of each ply being aligned in a different direction. The fiber – ply contributions are summed together with a contribution due to the matrix to give the continuous fibers length correction factors have been proposed to account for the reduction in composite stiffness that occurs with fibers of finite length. The factor proposed by Cox [1] is presented in the previous equations in a form which takes into account the wide distribution of fiber lengths. Halpin et al [15] adopted a more rigorous approach towards the prediction of the anisotropy of stiffness due to fiber orientation. They considered the composite to consist of a number of plies, each one containing matrix and uniaxially aligned fibers, again with the fibers of each ply to be aligned in a different direction. The moduli of each ply can be estimated. The simple relationships in these

equations are based on a generalization of the classical Rayleigh and Maxwell results, which may also describe short fibers with finite aspect ratio



_f

/d

_fwere used by Charrier and Sudlow [16] to calculate the moduli of short fiber systems.

The fibers may have a finite aspect ratio and their orientation is characterized by one of the several parameters, intermediate between randomness and complete alignment in one direction.

The longitudinal modulus can be evaluated via an expression derived in Ref. [17] and has been used by Berthelot [18] along with Hashin and Hill expressions for the other moduli to estimate the effective elastic properties of a u nidirectional fiber composite with misaligned discontinuous fibers. It was assumed that they are planar uniformly oriented between directions

θ

_{1 and -}

θ

1 where

θ

₁ denotes the misalignment angle.

Amongst the proposed formulas concerning unidirectional fiber composites, one can distinguish Halpin – Tsai formulas since they appear to be capable to account analytically for the influence of fiber length on elastic properties.

A successful theoretical investigation on the elastic moduli of randomly oriented short – fiber composites was carried out by Weng and Sun [20]. In this work, the longitudinal modulus and Poisson ratio of such a composite were found in terms of fiber content and the tip to tip spacing of the fibers. The proposed simulation was a composite cylinder model with a short cylindrical fiber, embedded in the centre of the matrix.

Here, these expressions are modified to account for the influence of volume fraction and aspect ratio of the filler. These results together with Hashin – Hill formulas for the other moduli were used in Christensen and Waal’s normalized expressions to calculate the elastic modulus and Poisson ratio of such a composite in terms of the fiber content and its aspect ratios.

Facca et al [21] applied six micromechanical composite models (theoretical and semi – empirical) to predict the properties of fibre reinforced composites. Kalaprasad et al [22] also selected a number of micromechanical composite models to predict the properties of the composites with longitudinally as well as randomly oriented fibres. The Hirsch and the Bowyer – Bader models were found to forecast the rates of elastic modulus of the composites with both types of fiber distribution most accurately, whereas both the rule of mixtures or parallel model and the inverse rule of mixtures or series model failed to predict it.

The Halpin – Tsai model [23,21] is also a theoretical model, which except the elastic models of the constituent materials includes a geometrical parameter i.e. aspect ratio of the fiber. The model has a complicated mathematical structure, but still fails to meet the observations made in Refs.

[21,22] satisfactorily. The developed semi – empirical modified Halpin – Tsai model [22, 23]

however, is in good agreement with the experimental results of Facca et al.

On the other hand, the experimental results of Kalaprasad et al [22] were described satisfactorily by another semi – empirical model namely Bowyer – Badel model [24].

From the literature data [21,22,25], it has become evident that the theoretical models which do not contain any adjustable parameter, usually fail to predict the modulus of elasticity of fiber reinforced composites, and to predict the modulus satisfactorily and therefore one has to apply a relation with at least one adjustable parameter, which provide the model with semi-empirical nature. As there is no escape from the use of an empirical relation, it is easier to adopt some predictive model expressed in terms of mass fraction instead of volume fraction.

A few years ago Mirbagheri et al. [26] conducted intensive research on hybrid composites consisting of ternary mixture of wood flour, kenaf fiber and polypropylene, and found that the rule of mixtures could successfully describe the modulus of elasticity of the polymer composites.

Meanwhile, Fu et al. [27] applied two approaches i.e. the rule of mixtures along with laminate analogy approach in order to describe the elastic modulus of a ternary mixture of particle – fiber – polymer. Besides, Islam et al [28] performed a thorough analysis on t he existing theoretical models to predict the elastic modulus of short fiber reinforced polymer composites, whereas in Ref. [29] an Elasticity approach was adopted to predict the elastic constants of fibrous composites, reinforced with transversely isotropic fibers.

In the present article, by extending and completing the theory of equivalent fiber based on a composite – cylinder model with a short cylindrical matrix the moduli of a unidirectional, short fiber composite were derived in terms of fiber content and aspect ratio of the fibers. These results were used to calculate elastic modulus and Poisson ratio of a randomly oriented short – fiber composite in terms of fiber content and aspect ratio. The theoretical results were compared with those derived from two other dominant theories on this subject. Moreover, tensile experiments were carried out on p olyester resin – randomly

oriented short fiber composites to determine mechanical properties and compare experimental results with theoretical values.

2 Theoretical Considerations

In this unit, by adopting a composite cylinder model in accordance with the theory of elasticity the longitudinal and the transverse elastic modulus, along with the longitudinal Poisson ratio of a composite with unidirectional continuous fibers will be derived. The theoretical analysis is based on the following assumptions:

i) T he matrix and the fibers are isotropic and linearly elastic.

ii) The composite reinforced with unidirectional fibers is linearly elastic, macroscopically homogeneous, transversely isotropic and without voids.

iii) The adhesion between the matrix and the fibers is perfect.

iv) The possible interaction amongst the fibers is neglected.

Here we should elucidate that the aforementioned suppositions will be taken into account in the derivation of the elastic moduli of a unidirectional continuous fibrous reinforced composite material.

a) Longitudinal elastic modulus

In order to evaluate the longitudinal elastic modulus, let us consider a representative layer of the composite material reinforced with continuous fibers, as it can be seen in Fig. 1.

Fig.1 Representative layer of the composite Then, let us concentrate on a cross – sectional area of the above model as it can be illustrated in Fig. 2.

Fig. 2 Cross section of the model

By the use of Airy stress – function the compatibility equation in cylindrical coordinates is expressed as follows

4 3 2

2

4 3 2 2 3

2 1 1 0

d d d d

dr r dr r dr r dr

Φ Φ Φ Φ

∇ Φ = + − + = (1)

Evidently, this ordinary differential equation belongs to Euler’s type and the its solution is given as

2 2

1ln 2 ln 3 4

c r c r r c r c

Φ = + + + (2)

Each one of the phases of the composite material is characterized by a corresponding stress function.

Thus, the expressions for the stresses in each one of the aforementioned phases are

( )

, 2

1

^f

1 2ln 2

r f

d A B r C

r dr r

σ Φ

= = + + +

(3)

( )

2

,_f d 2^f A B2

3 2ln

r

2

C

dr r

σ

θ

Φ

= = − + + +

₍₄₎

( )

, 2

1 ^m 1 2ln 2

r m d F G r H

r dr r

σ

= ^Φ = + + + ₍₅₎

( )

2

,_m d 2^m F G2

3 2ln

r

2

H

dr r

σ

θ

= Φ = − + + +

₍₆₎

To avoid infinite stresses atr =0 both constants A και B should vanish.

Thus, Eqns. (3) and (4) respectively, become

, , 2

r f θ f C

σ

=

σ

=

For the matrix material, it can be shown that 0

G = and therefore

, 2 2

r m F H

σ

= r + and _,m F₂ 2H

θ r

σ

= − +

Next, let us apply a tensile stress

σ

_c, which is exerted at the direction of z axis.

The equilibrium of forces at this direction yields the following relationship

c c m m f

f

⋅ A + σ ⋅ A = σ ⋅ A

σ

(7) with

, , ,

z f f z m m

σ

=

σ σ

=

σ

where

c m

f

A A

A

, , is the area of the fiber, matrix, and composite respectively.

Now, if one puts

σ

_f =s_1,

σ =

_m s_2,

σ =

_c s _and divide by the term F_c the above relationship becomes

s U s U

s

₁^⋅ _f ⁺ ₂^⋅ _m⁼ (8) where

2 2 m f

r

f

U

⁼

r

and ₂

2 2

m f

m m

r

r

U

⁼

r

⁻ are the fiber and

matrix content respectively Evidently,

U

_f +

U

_m =1

Moreover, from the stress – strain relationships we deduce that

( )

[

₁

]

,

1 2 C U 2 C s

E

_f ^f

f

r

= − +

ε

(9)

( ) ( ) _ ^

  + − − −

=

_{2 2}

,

1 1 U 2 H 1 U U s

r F

E

_m ^{m m m}

m

ε

r ₍₁₀₎

( )

[

₁

]

,

1 2 C U 2 C s

E

_f ^f

f

= − +

ε

θ (11)

( ) ( )

_^

 + − − −

= _{2 2}

, 1 1 U 2H1 U U s

r F

E_m ^{m m m}

m

εθ (12)

[ s U C ]

E

_f ^f

f

z

1 4

1

,

= −

ε

(13)

[ s U H ]

E

_m ^m

m

z 1 4

2

, = −

ε

(14) where Ef and Em are the elastic moduli of fiber and matrix respectively

Meanwhile, the radial displacements are given as

, ,

r f f

u

=

r

ε

θ _and u_{r m,}

=

r

ε

θ_,m

In continuing, the boundary conditions are formulated as follows

At r r= _f

, ,

2

2

2

r f r m

f

C F H

σ = σ → =

r

+

(15) At r r

=

_m

,

0

2

2 0

r m

f

F H

σ = →

r

+ =

(16) At r r= _f

⇒

=

_{r f}

f

r

u

u

_{, ,}

( )

[ ] ( ) ( )











− + + − −

=

−

− _{1 2} 1 2 1 ₂

1

2 U H U U s

r E F s U U C

E _{m m m}

f f f f

m (17)

Since the axial strains in matrix and fiber coincide, it implies that

) 4 ( ) 4

( _{1 f f 2 m}

m

s CU E s HU

E

^{− = −} (18)

The solution of the system of eqns. (17) and (18) for the terms s₁ands₂respectively yields

m f m f f

m f f m

f U

U E U E

U HE U CE

s1⁼sE ⁺⁴⁽ ₊ ^{− )} (19)

m f m f f

m f f

m

f

U

U E U E

U HE U

CE s sE

+

−

= −4( )

2 (20)

A substitution of the above data back into Eqn.

(18) gives

f m m f f m m

m m m f f f f m m f f

f v EU EU E

r C E F U v E U v E v U E U E v

f

) )(

2 1(

) (

) )(

1(− + − + =− ₂ + +

[ ]

C E sW U v E U v E v U E U E C v

H

f m f m f m f m m m f f

m)( ) ( ) 2

1(− + − + +

+ (21)

A

where νf and νm are the Poisson ratios of fiber and matrix respectively

From the solution of the system of eqns. (15), (16) and (21) one estimates the rates of the

constants F,H,Cas follows U s

K P L K F Wr

f f

) ( ) (

2

− +

= + (22) U s

K P L K H WU

f f

) ( ) 2 (

− +

= + (23)

U s K P L K

U C W

f f

) ( ) (

) 1 2 (

− + +

= − (24)

with

f m m f f m f f m m f f

f EU EU v EU v EU v E

v

K=((1− )( + )−2 ( + )) (25)

f m m f f

m

E U E U E

v

L = ( 1 + )( + )

(26)

f f m m m f f m m m f f

m

E U E U v E U v E U v E

v

P

=((1− )( + )−2 ( + )) (27)

(

f m

)

f m

W = v −v E E (28) A combination between the system of eqns. (25) to (28) and the system of eqns. (19) and (20) results in the following explicit expressions for the quantities s₁ands₂respectively

( )

( )

^_^{ ⇒}







− + + +

+

− + ⋅

= + s

U K P L K U E U E

v U E v E U U W U E U E s E

f m

m f f

m f f f m f m m

m f f

f

) ( ) ( ) (

) 1(

2

1

s

s

₁

⁼ η ^⋅

(29)

and

( )

(

+ + −

)

^⇒

+

+

−

− ⋅

= +

f m

m f f

m f f f m f f m

m f f

m EU EU K L P KU

v U E v E U U W U E U E

E s s

) ( ) () (

) 1(

2 2

s

s

₂ ⁼

ξ

^⋅ (30) The longitudinal modulus EL of the composite can be estimated by the equalization of the overall deformation energy with the sum of corresponding ones for the constituent materials.

Hence we can write out

( )

∫

∫ ^{= + + +}

f

c V

f f z f z f f f r f r V

c L

dV E dV

s

, , , , , ,

2

2 1 2

1 σ ε σ

_θ

ε

_θ

σ ε

( )

∫ ^{+ +}

Vm r_,m r_,m _,m _,m z_,m z_,m

dV

m

2

1 σ ε σ

_θ

ε

_θ

σ ε

(31)

Since we have initially proposed a modified version of Hashin cylinder model, the last relationship can be equivalently recasted as follows

( )

( )

2

, , , , , ,

0

, , , , , ,

1 2 1 2

2 2

1 2

2

f

c m

f

r

r f r f f f z f z f

L V

r

r m r m m m z m z m

r

s rhdr rhdr

E

rhdr

θ θ

θ θ

π σ ε σ ε σ ε π

σ ε σ ε σ ε π

= + +

+ + +

∫ ∫

∫

(32)

Finally, by substituting the above expressions for both stresses and strains back into eqn. (32) one finds

( )

( )

2 2

2 2 2

1 1 8 (1 ) 8

1 2 8 (1 ) 8 (1 )

f f f

L f

f m m f

m

C v Cv U

E E

F U H v Hv U

E

η η

ξ ξ

= − − +

+ + − − + −

(33)

b) Transverse elastic modulus

Next, in order to evaluate the transverse elastic modulus of the composite let us consider a cross – sectional area of this material under the following loading condition, as it can be seen in Fig.3, where, p2 is the applied external pressure and p1 denotes the common stress at the interface.

Let us consider the stress function in eqn. (1) and the described relationship in eqns. (2) – (6).

Fig. 3 Loading for the specification of transverse modulus

The boundary conditions are formulated as follows

r r= f: _{r f, r m, 1 2}

2

₁

f

p F H p

σ = σ = − →

r

+ = −

(34)

r r

=

m: _{r m, 2 2} 2 ₂

m

p F H p

σ = − → +r = − (35) Hence the constants F and H are estimated as

(

2 1

) ( )

²

2 2

f m

m f

p p r r

F r r

= −

−

(36) and

(

1 ² 2 ²

)

2 2

2 ^{f m}

m f

p r p r

H r r

= −

− ⁽³⁷⁾

Moreover, in the present analysis we have also assumed that the axial strains are a priori negligible, and therefore

( )

,

0

, , ,

z f z f vf r f θ f

ε = → σ = σ + σ

(38a)

( )

, 0 , , ,

z m z m vm r m θm

ε

= →

σ

=

σ

+

σ

(38b)

Besides, from the stress – strain relationships we have

(

²

)

, ,

2 1 _f 2 _f

f r f

f

C v v

θ E

ε

=

ε

= ^{− −} (39a)

( ) (

²

)

, 2

1 1 2 1 2

m m m m

m

F v H v v

E r

ε

θ = ^− + + − − ^^(39b)

( ) (

²

)

, 2

1 1 2 1 2

r m m m m

m

F v H v v

ε =

E r

^   + + − − ^  

^(39c)

Concurrently, the following relationships hold

, ,

r f f

u =r

ε

θ _, u_{r m,} =r

ε

θ_,m

Also, at r r= _f it implies that

, ,

r f r m

u

=

u (40) Consequently, we infer

(

²

)

2

^{( )} (

²

)

2 1 _f 2 _{f m f} 1 _m 2 1 _m 2_m

f

C v v E E F v H v v r

 

− − = − + + − − 

 

  (41)

Apparently, if one substitutes the rates of the quantities F and 2H back into eqn. (41) a linear algebraic expression between the magnitudes of the pressures p₁and p₂arises.

Hence we can write out

1 2

p

= λ

p with

(

²⁽¹

)

^{)(1 ) 2}

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

m m f

m m f f f f f m

v v E

v v U E v U E

λ ν

+ −

= + − + + − − − (42)

The transverse elastic modulus is calculated from the eq. (31), by setting on its left member instead of the integral

∫

Vc

dV

c

E s

 2

2

1

the terms in the

quantity

∫

Vc c

c

K p

₂²

dV 2

1

_{, where}

p

2 is the initially imposed pressure and

K

_c denotes the bulk modulus of the composite, which is evaluated by

means of the following expression

 



 

 − −

=

L LT T

TT c

E v E v

K

₂

2 2 1

1

Since

( )

_z

L y LT TT x T

x

v v E

E σ σ σ

ε

_



 



−

−

= 1

( )

_z

L x LT TT y T

y

v v E

E σ σ σ

ε

_



 



−

−

= 1

( )

1

z z LT x y

L

E v

ε

= ^

σ

−

σ

+

σ

^

2

x y p

σ

=

σ

= − and

ε =

_z

0

whereE_T denotes the transverse elastic modulus.

and v_LT,

v

TT are the longitudinal and transverse Poisson ratio respectively

Thus, by substituting the resultant relationships for the strains, the stresses and the unknown constants together with the auxiliary term

λ

back into eqn. (32) which is modified properly in order to include at the left member the expression with Kc and p2 one obtains the following explicit expression for the transverse elastic modulus

2 2 2 2

2

2 2 2

2

(1 2 ) (1 2 )( 1)

1

(1 ) (1 ) (1 )

(1 )(1 ) 2

(1 ) (1 ) (1 )

f f f m m f

T f T T m f TT

f m LT

m f T T L T T

U v v v v U

E E v E U v

U v v

E U v E v

λ λ

λ

− − − − −

= + +

− − −

+ −

− − + −

(43a)

c) Longitudinal Poisson ratio

Next, the longitudinal Poisson ratio can be evaluated by means of the following reasoning.

It is valid that

r m

LT

z

v

= −

d r

ε

where d_r denotes the radial displacement in the cylindrical area of the composite and

ε

_{z is the} axial displacement.

Thus

2 2

,

, 2

/ (1 ) 2 (1 ) ( ) /

4

m m m m m

r m r r m LT

z m m

F r v H v v s

u r

v ε s Hv

= − + + − − 

= − =

− (43b)

A substitution of eqns. (22), (23), (30), back into (43b), leads to the following closed – form relationship

(43c)

d) Transverse Poisson ratio

Moreover, the transverse Poisson ratio of the composite can be estimated by means of Halpin –

Tsai equation which is presented as follows:

f m f

TT

U

v U

v

− ⋅

⋅

⋅

= +

1 1 1

1 1

η η

ξ

(44)

where:

m f

m f

v v

ν ξ η ν

1

1

+

= −

and

...

3 , 2

1

= ,1

ξ

e) Shear Modulus

On the other hand, to define shear modulus one can take into account Halpin – Tsai formula i.e.

f m f

LT

U

G U

G − ⋅

⋅

⋅

= +

2 2 2

1 1

η η

ξ

(45)

where

m f

m f

G G

G G

2

2

ξ

η +

= −

and

ξ

₂ is a constant term which depends on the fiber volume fraction and arises from the following empirical relationship

10f

2

1 40 U

ξ = + ⋅

Here

G

_f,

G

_mare the shear moduli of fiber and matrix respectively.

Besides, Hashin and Rosen proposed the following expression concerning the shear modulus of a composite was proposed

] ) (

) [(

] ) (

) [(

f m f m f

f m f m m f

LT

G G G G U

U G G G

G G

G

+ − − ⋅

⋅

− +

= + (46)

In continuing, to introduce the discontinuity of fibers we will use the model of equivalent fiber, which is described as follows:

Let us suppose a composite material consisting of unidirectional discontinuous fibers

Fig. 4 Arrangement of the short fibers in the matrix

Next, let as focus on the cross – sectional area of a representative volume element of this material.

Fig. 5 Cross – sectional area of an arbitrary short fiber of the composite

We transform the above volume element in order to create another volume element consisting of an equivalent fiber

Fig. 6 Cross – sectional area of the equivalent fiber

Here, we should emphasize that the real system consisting of short fibers and matrix is

characterized by the elastic

constants

E

_f,

E

_m,

U

_f,

U

_m whereas the equivalent system consisting equivalent fiber and matrix, is described by the constants

E

_feq,

E

_m,

U

_feq,

U

_meq. In this context, let as c alculate the fundamental elastic properties of the composite.

Primarily, we have to estimate the filler content which in the real system is given as

l l d U d d l

d l

U

^f _f ^{f f}

f

f

⇒ = ⋅

⋅

⋅

⋅

= ⋅

_{2 2}²

2

4 4 π π

(47)

and in the equivalent system

2 2 2

2

4

4 d

U d d l

d l

U

_feq ^f

f

f

⇒ =

⋅

⋅

⋅

= ⋅ π π

(48)

Hence,

f f

feq

U

l l

U = ⋅

(49)

To evaluate the elastic modulus of equivalent fiber, one may adopt the Strength of Materials approach according to which the system filler – matrix is equivalent with a system of two ideal springs connected in series. Evidently, this implies that the exerted force is the same both for matrix and filler, while their lengthening differs, as it can be seen in Fig. 7

Fig. 7 S trength of materials approach for the simulation of equivalent fiber

f m

F F= =F and

∆ = ∆

l

( ) ( )

l _f

+ ∆

l _m Moreover the following relationships hold,

F = ⋅ Α

σ

, F_f =

σ

_f ⋅ Α_f

m m m

F

= σ ⋅ Α

,

σ

=E_feq⋅

ε

f Ef f

σ

= ⋅

ε

,

σ

_m

=

E_m

⋅ ε

_m

( ) ∆

l m

= ε

m

⋅

lm

and therefore,

( ) ∆

l f

= ε

f

⋅

lf

f f m m f f m m f f m m

l l l l l V l V V V

ε⋅ = ⋅ + ⋅ ⇒ ⋅ = ⋅ ⋅ + ⋅ ⋅ ⇒ = ⋅ + ⋅ ⇒ε ε ε ε ε ε ε ε

f f m m

feq f m

l l

E E l E l

σ σ

σ ⋅ ⋅

= +

⋅ ⋅

(50) Besides, referring to the equivalent fiber the following equality holds

f m

σ

=

σ

=

σ

Consequently we find

m f

feq f m

m f

E l EE El E

l l

= ⋅

⋅ + ⋅

(51)

To calculate Poisson ratio of the equivalent fiber we may consider without violating the generality of our presented mathematical formalism, that the overall lengthening and the dilatation of this hypothetical fiber are equal with the sum and the average of the real fiber and matrix respectively.

The overall lengthening is

1 1 f m

f f m m f m

l l

l l l

l l

ε ⋅ =ε ⋅ +ε ⋅ ⇒ =ε ε ⋅ +ε ⋅ (52) whereas the dilatation is given as

2 f m

f f m m

l l

v v

l l

ε = − ⋅ ε ⋅ − ⋅ ε ⋅

(53)

Since ²

1

vfeq

ε

= −

ε

_{we have}

1

1

m f f m

f feq

m f

f

E E l

v l

E E l l

ν ν  

⋅ + ⋅ ⋅    −   

=  

+ ⋅   −  

 

(54)

Here, let us introduce a parameter named R which performs the ratio of the distance between two neighboring fibers and their length as it can be seen in Fig. 8

Fig. 8 Dimension of the fibers in the composite

This aforementioned parameter arises from the following expression

1

m

f f

l l

R

=

l

=

l

−

(55) Hence, after the necessary algebraic manipulation we find

m f f m

feq

m f

E v E v R

v E E R

⋅ + ⋅ ⋅

= + ⋅

⁽⁵⁶⁾

( 1 )

f m

feq

m f

E E R

E E R E

⋅ ⋅ +

= + ⋅

⁽⁵⁷⁾

( R )

U

U

_feq

=

_f

1 +

(58)

Moreover, to examine the influence of aspect ratio of the fibers we introduce another parameter named

a

and given as

f f f f

r l d

l

a = = 2

(59) However, we have already found that

d S U d l l d U d

d U d

f f f f f

feq f

⋅

=

⇒

⋅

=

=

2 2 2

2 2 2

where l^f S

=

l

Therefore the latter relationship yields

⇒

⋅

⋅

=

⋅

⇒

⋅

=

⋅

⇒

⋅

=

d l S l U l

d S U l d

l l U d

f f

f f f

f f f

2 2 2 2 2 2

2 2 2 2 2 2 2

2

a

a

2 3 2

a2 



 



= 

⋅

d

S l

U

_f (60)

Here, one may additionally suppose, without violating the initial hypothesis concerning the randomness of fiber distribution and orientation, that the distance of two neighboring fibers in the longitudinal direction coincides with the distance of two neighboring fibers in the transverse direction as it can be seen in Fig.9.

Fig. 9 Simplified approach of the fiber spacing

Thus we can write out

f f

l l− = −d d =C Hence it follows

1

^{f f}

f f

l d l d d l l

d d d

= − + ⇒ = + −

(61)

Since

S U d

d d

l d

l

_{f f}

f f

f

= ⋅ = a

, eqn. (61) yields

( )

S U S

d l +

f

⋅ a − 1

=

(62) After an elementary algebraic manipulation, eqn.

(62) results in the following quadratic equation

( )

^{2 1 a 22 2}

( ) (

^{a 2}

)

⁰

f f f f f

U S⋅ − ⋅ − ⋅ ⋅S U − ⋅S U ⋅ + + − ⋅ ⋅S U S U = (63) Evidently, the roots of the above polynomial equation are

( )

( 1 )

a ⋅ +

⋅

−

= ⋅

S U

U S U S

f

f

f

or

( )

(

1

)

a ⋅ −

⋅

−

= ⋅

S U

U S U S

f

f f

According to the nature of the examined physical problem, it is obvious that one should reject beforehand the first solution since it leads to unrealistic results.

Thus, it im plies that the quantities S and R should be related as follows

1

S

1

=

R

+

and therefore

 

 



 −

+

 

 



 ⋅

− + + ⋅

=

1 1 1

1 1 1

1 a

U R

R U R U

f

f f

(64)

3 Evaluation of the moduli by applying the model of equivalent fiber

Now, we apply the model of equivalent fiber in the relationships of the elastic constants which were previously derived, using Airy stress function. By this way, we introduce the discontinuity of fibers through the parameters

R

and

a

.

Here, one may distinguish and examine the following three cases which have been motivated by the consideration of the aspect ratio

f f

d

= l a

∞

a =

(continuous fibers)

• a =10 (discontinuous fibers)

• a =0 (particles)

For each case, one can estimate the elastic constants of the composite material by