The influence of the interface on the transverse properties of carbon fibre reinforced composites

The influence of the interface on the transverse properties of

carbon fibre reinforced composites

Citation for published version (APA):

Kok, de, J. M. M. (1992). The influence of the interface on the transverse properties of carbon fibre reinforced composites. Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1992

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Computational Mechanics

/

The influence of the interface on

the transverse properties

of

carbon

fibre reinforced composites

CIP-DATA KONINKLIJKE BIBLIOTHEËK, DEN HAAG- ~

Kok, J.M.M. de

The influence of the interface on the transverse

properties of carbon fibre reinforced composites

/

J.M.M. de Kok.

-

Eindhoven : Instituut Vervolgopleidingen, Technische Universiteit Eindhoven. - 111.

With index, ref.

ISBN 90-5282-1 82-8 bound

Subject headings: interfaces

/

carbon fibre ; transverse

prC!pe!ties.

@ 1992,Ir. J.M.M. de Kok, Eindhoven. ~ -

Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt door middel van druk, fotokopie, microfilm of op welke andere wijze dan ook zonder voorafgaande schriftelijke toestemming van de auteur.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including

The

infuenee

si the

interface om

the transverse properties of carbon

fibre reinforced composites

J.M.M. de Kok Computational Mechanics

/

Department of Engineering Fundamentals Centre for Polymers and Composites

Eindhoven University of Technology May 1992

This work was supervised by: Prof.dr.ir. H.E.H. Meijer

Ing. A.A.J.M. Peijs Dr.ir. P.J.G. Schreurs

Content

Abstract i

1 Introduction. 1-1

2 Review on micromechanical analyses

of

transversely

loaded

2-1

unidirectional composites.

2.1

2.2 Perfectly bonded fibres.

Models based on a regular filament packing arrays. 2.2.1 Linear elastic deformation.

2.2.2 Plastic deformation.

2.2.3 Matrix crack initiation and propagation. Debonding due to transverse normal loading. 2.4.1 Cylindrical voids.

2.4.2 Interface as a contact surface. 2.3

2.4 Composites with debonded fibres.

2.5 Interface regions. 3 Strategy.

3.1 Method of investigation.

3.2 Material choice to study the influence of the interface.

4 Numerical analyses.

4.1 Micromechanical modelling of the fibre-matrix interface. 4.1 .I Introduction of an interface element for micro-

mechanical analyses.

4.1.2 Influence of the interface properties on the transverse deformation.

4.1.3 Analyses with a debonding interface. Analysis with not bonded fibres.

Analysis with perfectly bonded fibres. 4.2

4.3

5 Experiments.

5.1 Sample preparation. 5.2 Mechanical testing.

5.2.1 Three point bending. 5.2.2 Failure detection.

5.3 Fibre surface characterization.

2-2 2-3 2-4 2-5 2-7 2-8 2-1 1 2-1 1 2-12 2-14 3-1 3-1 3-3 4-1 4-1 4-1 4-3 4-7 4-1 0 4-1 3 5-1 5-1 5-3 5-3 5-4 5-4

5.4.1 Mechanical properties. 5.4.2 Damage development. 5.4.3 Failure modes.

5.4.4 Fibre surface characterization.

Results from the composites with the silicon oil treated fibres and the high-strength fibres.

5.5.1 Mechanical properties. 5.5.3 Faiiure mode.

5.5

5.5.2 DaiÌÌage develupliielit.

6 Confrontation of numerical

and

experimental results.

6.1 6.2

The influence of the interface on the transverse strength. The influence of the interface on the transverse failure

s t ra i n. 5-5 5-6 5-8 5-1 0 5-1 1 5-1 1 5-1 3 5-1 4 6-1 6-1 6-3 7 Conclusion. 7- 1 8 References. R-1

Abstract

The influence of the fibre-matrix interface on the transverse properties of unidirectional composites has been studied using a combination of experiments and numerical micromechanical analyses. A finite element micromechanical model is developed to analyze the influence

of

the interface on the transverse properties of continuous carbon fibre reinforced epoxy. Elastic springs are introduced as interface elements to attain an 'interphase', displaying mechanical characteristics close to those of the matrix and enabling separation of fibre and matrix. Since these elastic springs represent the chemical bonds formed at the interface as a result of fibre surface treatments, the micromechanical modelling can directly be related to these treatments. For verification of the numerical analyses the influence of the interface is determined experimentally by transverse testing of carbon fibre reinforced composites, incorporating carbon fibres subjected to different levels of surface treatment.

Numerical results show that the interface modulus neither affects the overall composite modulus nor the stress concentrations in the composite. These observations are confirmed by experiments which also indicate that there is no effect of the fibre-matrix interfacial bond strength on the transverse composite modulus.

With respect to the transverse composite strength the numerical analysis predicts a transverse stress and strain to failure proportional to the number of chemical bonds formed at the interface. As a result of the fibre treatment the amount of reactive oxygen at the fibres surface increases, leading to an increase in bonds at the interface. Experimental validation showed that the transverse composite strengths can be fitted to the predicted linear relationship with oxygen concentration.

However, the numerical calculations indicate that the proportionality of the strength with the interface properties is unlimited. Consequently, the micromechanical modelling is extended with an upper an lower limit with respect to transverse strength. A minimum and maximum transverse strength is predicted for composites with respectively not bonded and perfectly bonded fibres due to a change in failure mode from debonding to matrix cracking. These predicted ultimate strength values are also experimentally verified.

Finally, it can be concluded that micromechanical modelling is useful for material development, since the models predict the ultimate transverse properties as obtained after surface treatment and also the interface properties necessary to reach these limits are indicated by the modelling.

1

Introduction.

Composite materials with continuous unidirectional fibres are frequently used in high performance applications. The stiffness and stress to failure of unidirectional composites are extremely high in the longitudinal direction but the transverse properties are very low. To compensate for the low transverse stiffness and strength, usually stacked plies with different fibre orientations are used for structural applications. By applying such a laminate a good stiffness and strength in every direction can be realized. However, even in these laminates the

low

transverse strain to failure causes problems, since very low off-axis strains will lead to premature failure in the individual layers. Therefore, it is mainly the low transverse strain to failure of the unidirectional composites that limits the application of these materials. An improvement of this transverse property may lead to a substantial expansion of their use.

Most of the recent research on the transverse properties of composites focused on the improvement of the transverse stress to failure. Both experiments and numerical analyses of micromechanical models of composite cross-sections show that the low transverse strength of Unidirectional composites is a result of considerable stress concentrations in the transversal plane. These stress concentrations lead to damage initiation at relatively low loads. With a poor adhesion between the fibres and the matrix, separation occurs upon transverse loading. With a sufficient high adhesion the fibres and matrix can be loaded until one of the two constituents fails. Matrix failure is the most common, but fibre splitting may also occur with highly anisotropic fibres possessing a relatively low transverse strength. The numerical analyses can be used to predict the transverse stress-strain curves and transverse strengths of unidirectional composites. Usually the predictions have been found to agree fairly well with experimental results.

To achieve an improvement of the transverse strength most of the current research is directed towards postponing the described failure modes to higher transverse loads. This can be achieved by modification of the fibre surface to improve adhesion, by modification of the fibre to improve its transverse strength or by toughening of the matrix, resulting in plastic behaviour under transverse loading. Unfortunately, modifications of fibres and matrix do not always lead to the expected improvement of the transverse behaviour. The reason for this is that the various parameters that influence this behaviour, effect the transverse properties in a very complex manner and the parameter variation has not been performed efficiently.

To arrive at an appropriate improvement of composite materials it is necessary to combine experiments and numerical micromechanical analyses. For material development an intensive study is required, varying the parameters that may have any influence on the transverse properties. Using numerical analyses such a parameter variation is rather easy. However, the influence of these parameters on the transverse behaviour has to be validated by experiments. For these experiments it is necessary to use a certain composite system that allows for the desired parameter

variatim.

A zon:iûntatiûn

ui

the yiûbâi experimentai and numerical results is necessary for the vaiidation

of

the numerical analyses. The confrontation of microscopic damage analyses with the stress concentrations from the numerical analyses should lead to a better understanding of the deformation and failure in composites under transverse loading and the influence of the investigated parameters. Finally, a better understanding might also lead to guidelines for the development of materials with improved transverse properties. One of the parameters that has the most significant effect on the transverse behaviour is fibre-matrix adhesion. In this report it is explained how the previous route described

is

used to study the influence

of

the interface on the transverse properties of unidirectional composites. In chapter 2 the various micromechanical models developed for modelling composites with different kinds

of

fibre-matrix adhesion are described. In chapter 3 it is explained how experimental and numerical techniques are combined to investigate the influence of the interface, by using carbon epoxy composites as a model composite system. Both numerical analyses and experiments on this composite system follow in chapter 4 and 5

respectively. In the confrontation between the calculations and experiments in chapter 6 all results are discussed. Finally, in chapter 7 it is evaluated how the investigation can be used for material development.

2

Review on

micromechanical analyses

of

transversely loaded unidirectional composites.

Because the low transverse strength of unidirectional composites limits the applicability of these materials, a lot of effort has been put in the understanding and improvement of this specific materia! ~hzr~cfsristlc. In c~n attempt ta come

tu

ci

better Understanding of the problem, several investigators have used various theoretical analyses to study the influence of various parameters on the transverse composite properties. The more sophisticated methods for modelling unidirectional composites subjected to transverse normal loadings are generally based on finite element analyses or finite difference analyses to describe the mechanical behaviour of a transverse cross section of a unidirectional composite. Usually a repeating unit of a regular packing array is used as a representative volume unit.

With the aid of the micromechanical models mechanical properties like the transverse modulus can be predicted. Theoretically predicted stress-strain curves and transverse strengths have been found to agree fairly well with experimentally found data. The analyses

also

have been used to investigate damage phenomena like matrix cracking and debonding. Further details about the modelling will follow in section 2.1.

Experiments show that the transverse mechanical behaviour of fibre reinforced composites is significantly affected by the nature of the bond between the fibres and the matrix [10,20,25,26,27]. The role of the fibre-matrix interface on the transverse behaviour is very dependent on the composite system, and for each composite system the interface properties are strongly controlled by the fibre and matrix treatment. Therefore it is very important to adjust the interface conditions in the micromechanical model to the composite materials under consideration.

In the next sections it will be explained how different interface conditions can be used in the micromechanical analyses. In section 2.2 analyses of composites with perfectly bonded fibres

will

be described. How the effect of debonding and debonded fibres on the mechanical properties of composites can be analyzed with a micromechanical model, is explained in section 2.3 and section 2.4 respectively. Section 2.5 describes the effect of an interface region on the stress and strain state in the composite and on its overall behaviour.

2.1

Models based on a regular filament packing array.

To analyze the transverse behaviour of composite materials using a finite element method or a finite difference method a model geometry is required. The model geometry is based on the micromechanical model of a composite cross section. Although photomicrographs of transverse cross sections of actual composites show that the filament packing is completely random, most existing models assume a regular filament array. Utilizing such a regular packing array the following assumptions are made [4]:

-

The fibres and the matrix are homogeneous.

-

Fibres and matrix are free from voids.

-

The fibres are regularly spaced.

-

The fibres within the composite are perfectly straight.

-

Any mechanical process like yielding or fracture takes place along the Commonly used packing arrays are the rectangular, the square and the hexagonal filament packing (figure 2.1).

whole length

of

the fibres.

By assuming a regular packing array, a fundamental or repeating unit can be isolated, as indicated in figure 2.1 [2,3]. Because of the symmetry in the arrays along the x and y axes, only one quadrant of the repeating unit needs to be considered. When a composite is subjected to transverse normal loading a complex state of stress is induced in the material. This is the result of the dissimilar material properties of the fibre and matrix and also because of interactions between the adjacent fibres. The stress distribution along the boundaries of the repeating unit will not be uniform.

O 0 0

_-

_.

Oi810

O 0 0

Square A

O 0 0

O B 0

O 0 0

Rec tangu I ar

0 ; a O

O

00°

Hexagonal , Midpoint

However, because of symmetry it is required that the repeating unit of the rectangular and the square array remain rectangular when normal stresses are applied. The same conditions of boundary displacement are applicable for the hexagonal array, except for the side CD in figure 2.1. For this side a special displacement of the points on the boundary is determined: points equidistant from the midpoint will have an equal and opposite displacement from the midpoint

[2].

Thus for the rectangular, square and hexagonal array the boundary conditions can

be fûrmüiáieci in terms

of

aispiacements. interactions between adjacent filaments, which induce non uniform stresses at the boundaries of the representative volume element, are taken into account automatically and correctly.

Due to the random filament packing in real composites, unidirectional composites are usually transversely isotropic. However, the regular packing arrays described are not transversely isotropic. The rectangular and square array are highly anisotropic and only the hexagonal array is close to transverse isotropic. Actually the hexagonal array model is physically a rather realistic packing array, since it resembles the most to the random filament packing in real composites. However, although the hexagonal array seems the best packing array to use, mechanical properties like transverse stiffness in uniaxial tension usually are better predicted with the square array [3,5]. The square packing array seems to account for the randomness of the filament spacing between the nearly hexagonal stacked fibres in real composites. Models that are based upon a partially random hexagonal packing

[5] or a totally random packing [6,7] have an even better agreement, as expected. However, these statistical approaches are not often utilized because of their complexity and additionally they are very time consuming even for simple predictions of the transverse modulus.

2.2

Perfectly bonded fibres.

When composites are transversely loaded three distinct phenomena can be seen

[I]. First, a linear elastic response up to an elastic limit. Beyond the elastic limit inelastic (plastic) behaviour may occur until first failure is initiated. The third event will be crack propagation until total failure of the composite. In some cases of course no inelastic behaviour will occur at all and sometimes a failure mechanism, which is not catastrophic, may develop before plastic deformation. In the following sections it will be described how these three distinct phenomena have been analyzed previously for composites with perfectly borided fibres.

2.2.1

Linear elastic deformation.

Mechanical properties such as transverse stiffness and Poisson’s ratio can be calculated by applying a unidirectional load normal to one boundary of the repeating unit. This results in an elongation in the loaded direction and contraction in the other direction, A lot of work has been done to determine these linear elastic properties. One of the best known modelling is done by D.F. A d a m and D.R. Doner p j utilizing a square array. They have determined the transverse modulus for various composites with different fibre/matrix stiffness ratios and filament spacings (figure 2.2).

Figure 2.2: Normalized composite stiffness] E/E, for various composites

Pl.

They also determined the normalized maximum principal stress (stress concentration) within those composites, suggesting a change in transverse strength depending on the various composite parameters (figure 2.3). The calculated stiffnesses agree fairly well with the experimental values found for composites with isotropic fibres like glass and boron. Predicted values for carbon-epoxy composites differ from the experimental values, since anisotropic fibres were not modelled.

2.2.2

1 2 4 6 8 10 20 40 60 100 200 400 600 1000

CONSTITUENT STIFFNESS RATIO, Ef/E,

Figure

2.3: Normalized maximum principal stress in matrix for various

composites subjected to a transverse normal loading [3].

Plastic

deformation.

The linear elastic analysis described before,

is

only

useful

if

the matrix behaves

linearly. However, matrix materials

with

a

high

strain capacity are far more

interesting for application

in composite materials. Beyond the elastic limit these

materials

will show plastic deformation. Due to

this

yield behaviour

a

plastic

deformation analysis

is

required. The analysis can be

divided

into two parts.

The

first

part is

a

linear elastic analysis

to determine the elastic limit. The second part

is

the elastic- plastic analysis

in

which the composite array

will

be

loaded further

incrementally.

In

a linear elastic analysis it is possible to determine at which load

plastic deformation will start. The analysis shows at

which

point

the matrix stresses

are maximum. Because the response is totally linear

it

is directly known

how

much

the applied load

should

be increased

so

that the highest stresses

will

reach the

elastic limit. When the composite

is

loaded further, plastic deformation

will

start at

this

point and

with

increasing load it

will expand to a larger area. Since further

loading leads to

a

highly

inelastic response within the composite, the load

has to

be

incremented

in

reasonably small steps. The loading

will

be continued

until

the

stress

(or

strain

)

to failure is reached

in

the composite and first failure will start.

35

2 SquarearraY

Fig u re 2.4: Boron-aluminum composite, comparisons

of

predicted and experimental stress-strain behaviour [8].

Strain -Z. (percent1

Figure 2.5: Boron-aluminum composite, predicted stress-strain curve and corresponding octahedral shear stress contours 181.

One of the earlier elastic-plastic analyses have been carried out by D.F. Adams [8] and R.L. Foye [9]. In their work a uni-axial stress-strain curve for the ductile matrix was used to describe the plastic deformation within the composite. Their analyses lead to stress-strain curves of composites in transverse loading. Adams used a rectangular array that was based on photomicrographs. His numerical results lead

to a good agreement with stress-strain curves obtained experimentally (figure 2.4).

Although some of the matrices were highly inelastic, the stress-strain curves of the composites exhibited very little of the latent non-linearity. This is possibly due to the stress concentration which give rise to localisation of plastic deformation (figure 2.5).

2.2.3

Matrix crack initiation and propagation.

It is obvious that failure will occur when a composite is highly loaded. With the analyses mentioned before (the linear elastic as well as the elastic-plastic) it is possible to determine when failure initiates if a suitable failure criterion for the matrix is available. After a crack has initiated it will propagate through the matrix. Crack propagation in a composite with a ductile matrix has been described in literature [1,10,11]. D.F. Adams started to model the crack propagation in fibre reinforced aluminum composites using a finite element method. His criterion for crack propagation was simply based upon the stress and strain situation in the matrix. When in an element the principal stress

or

principal strain exceeded stress

or

strain to fracture of the matrix material, the element was removed by changing its stiffness to zero (figure 2.6).

Figure 2.6: Failed elements and octahedral shear stress contours

Although the analysis carried out by D.F. Adams showed a fairly good agreement with experimental results, his modelling is probably oversimplified. The finite element grids he used were quite course and finer grids might give different results.

A precise development of the crack growth is unfortunately not possible because the course element meshes lead to a very jumpy stress-strain curve (figure 2.7).

20 is 16 14 ïñ y 12

-

Y vt 10 I- ' " a o W 5 6 n 2 4 2 O A EXPERIMENTAL W T A 61 TRW TEST SPECIMEN 4 f e 4 2

I

I I I I I I I I O O1 O2 0.3 04 0.5 0.6 0.7 0.8 0.9

COMPOSITE STRAIN (PERCENT)

Figure 2.7: Predicted and experimental stress-strain curves [I].

2.3

Debonding due to transverse normal loading.

When a unidirectional composite is loaded transversely, debonding may occur if the stress at the interface has reached the interfacial bond-strength. With the analyses mentioned before (the linear elastic as well as the elastic-plastic) it is possible to determine initiation of the debonding if a failure criterion for the interface is known. Initiation can depend on the interfacial shear stress and the normal stress. Usually, the debonding will start between the fibres, where the tensile normal stress is the highest [21]. After debonding has initiated it will propagate along the interface or through the matrix.

Crack propagation at the interface of fibre and matrix in a transversely loaded composite has been mode!!ed several times by usirig special spring elements at the interface 121,221. Special springs are used (called gap elements) to connect the

fibre and the matrix and enable the separation of these two constituents. These spring elements consist of two nodes, which are initially coincident. Compressive stress is transferred directly, but tensile stress and shear stress will lead to extension and shear of the interface. The conditions across the interface can be expressed as

om = of = k,(u,-u,) i f u, 2 u,,

o, = o, and u, =

u,

otherwise, (2-2)

tm = tf = kT( vm-

v,).

(2-3)

where o and t are the interfacial normal and shear stress and u and v are the

displacements respectively perpendicular and parallel to the interface. The subscribes ‘MI and ‘f’ stand for the quantities in respectively the matrix and the

fibre. The constants k, and k, are the coefficients of the springs. The addition of

(2-2) insures that the model will not allow the two constituents to penetrate at the interface. The interface conditions (2-I), (2-2) and (2-3) were used between all points, which are initially coincident when the interface is intact. If however, due to interfacial failure, the surfaces of the fibre and adjacent matrix are not in contact the interface conditions will be

o, = of = t, = t, =

o

(2-4)

The material properties of the composite being modelled surely depend on the stiffness of the interface springs. The interface stresses are highly dependent on the flexibility of the interface. By using infinitely stiff springs the mechanical behaviour will be the same as that of composites with perfectly bonded fibres. At the interface of the bonded fibre a crack can be obtained by removing springs at a part of the interface by reducing the their stiffness to zero. At the crack-tip a singularity will occur and a stress or strain criterion cannot be used for crack propagation. Because of the singularity, the stresses at the crack-tip will be sufficient to ensure further propagation and the interface is rapidly totally debonded

[21,22]. When the interface is not infinitely rigid, no singularity will occur and a critical stress, critical strain or critical energy density criterion can be used to describe the crack propagation. This is done by determining the crack propagation criterion at the crack-tip as a function of the crack length.

By using a normalized energy density criterion for crack propagation at the interface it can be seen that a propagating crack can arrest. First, upon initiation, the cracked interface is unstable, since the crack growth leads to an increase of the near-tip energy density. But after this increase the energy density decreases with the crack length and the crack propagation will arrest, independent of the interface stiffness. 0.55 0.50 - 0.45 -

i

0.10

i

- k = 10 - k = 1 0.00

!

I I I I

1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 2.9: Normalized energy density at the crack tip vs crack length [22].

With the aid of the analysis of the partially cracked (debonded) composite a stress- strain curve can be made of a debonding composite. The mechanical properties are influenced by the interfacial stiffness before debonding, because a lower stiffness of the interface leads to a lower transverse stiffness of the composite. The moment when debonding starts is very much influenced by the interface properties. After the debonding however, the transverse modulus is decreased strongly and independent of the interfacial stiffness [22,23]. Because

of

the decrease in stiffness after debonding, not debonded fibres near a debonded fibre cause a lower load at the debonded fibre, which leads to a stability

of

the crack at a smaller crack-length [24]. On the other hand, the neighbouring fibres have higher interfacial stresses and are also expected to debond. Since the matrix gets highly loaded after debonding, failure of the composite can be expected.

1.2 D A b 0 0 b \ b O

Figure 2.1 O: Effective stress-strain relation for cracked and uncracked fibre- reinforced composite

1221.

2.4

Composites with

debonded

fibres.

To study the transverse behaviour of composites with not bonded fibres, micromechanical models have been developed in which no tensile stresses are transferred at the fibre-matrix interface. To obtain this specific interface condition fibre-like-voids are introduced or interfaces are considered as contact surfaces. In the next sections further details are given about the modelling of composites with not bonded fibres.

2.4.1

Cy1

i

nd

ricai

voids.

Composites with debonded fibres are easy to model, because if the fibres

are

already loose within the matrix, it

is

not necessary to describe the debonding process itself. Because of the debonding, it is likely that the fibres don’t carry any transverse load and that the transverse mechanical properties will mostly depend on the matrix [ ! 5 ] . This

iriakec

it possible to model composites with debonded fibres as a matrix with cylindrical voids or fibres without stiffness. This assumption

leads to a simple model that can be analyzed in the same way as was described in section 2.2. The transverse modulus can also be predicted fairly well with the Halpin-Tsai equation [16] with a fibre modulus equal to zero given as

Ef

Ern

Ef

-

+ 5

E m

-

-1

=

2

for cylindrical fibres

where

iEf

and E, are the fibre and matrix modulus, respectively and cf is the fibre volume fraction. By assuming the debonded fibres to be cylindrical holes in the matrix, an expression for the transverse strength [lil may be given as

where om is the matrix ultimate strength.

When a composite with debonded fibres is modelled with loose fibres in a matrix, special care has to be taken to describe the contact between the fibre and the matrix, to avoid penetration of the two constituents in finite element analyses. Usually gap elements are used to connect the fibre and matrix (see section 2.4.2). Although this modelling is physically more realistic than modelling cylindrical holes, the results are quite the same. Both formulas for the modulus and the strength lead to similar results as the more sophisticated numerical analysis [I 81.

2.4.2

Interface as a contact surface.

In the case of a perfectly bonded interface a finite element model simply uses a common set of nodes along the interface to describe both the matrix and the fibre elements In the case of an unbonded interface, or an interface without tensile strength, the interface is modeled as a contact surface. A contact boundary

condition is necessary, which prohibits nodes associated with the matrix from penetrating the fibre and vice versa. The only stresses that can be transferred between fibre and matrix are compressive stresses or shear stresses governed by e.g. a Coulomb friction law. Under normal tensile stress the fibre and the matrix will separate at the interface. When the interfaces are well-bonded, the transverse load-displacement behaviour and effective elastic properties can be formulated by application of the classical elasticity theory. However, when the interface

is

u G u u I ~ ~ ~ ~ , LIK aridiysis is mucn more compiex. Because of the

loss

of the fibre-

matrix displacement continuity, the problem is non-linear and an iterative numerical solution

is

required.

,.Jmh-mA..cl &L., - - _ I . . - : -

To model the interface contact behaviour with finite element analyses a special two-dimensional gap element can be used between the fibre and the matrix. This element is defined by two nodes, which are initially coincident, and the direction of the slip plane. With the nodes in contact, compressive nodal force components normal to the slip surface are transferred across the element. When the nodes are not in contact, no loads are transferred through the element. Loads parallel to the slip surface may also be transferred across the gap element by Coulomb friction if a non-zero friction coefficient is defined as a element material property.

Analysis of unidirectional composites show that the transverse tensile modulus is much lower when the fibres are debonded. In compression however, the modulus is almost the same for debonded fibres and perfectly bonded fibres [18,19]. It has been shown that the transverse modulus under tension and compression are not significantly influenced by the interface coefficient of Coulomb friction, in a range from 0.0 to 10.0.

Because thermally induced residual stresses, due to manufacturing of the composite, may lead to compressive stresses at the interface, an initial modulus may be found equal to the transverse modulus with perfectly bonded fibres. At higher loads this compressive stress will be overcome, leading to fibre/matrix separation and reduction of the slope in the stress-strain curve [19,20]. This reduction leads to a bilinear stress-strain curve with a distinct knee, even under the assumption of linear-elastic material behaviour.

2.5

Interface regions.

Although it is often assumed that a fibre reinforced composite consists of only two

distinguishable materials, it is not unlikely that an interface region or "interphase" exists between the fibre and the matrix. The chemical reaction at the interface while curing an epoxy resin may lead to an reaction layer at the interface

[12].

Such an interfacial region might not only occur by incident, but can also be applied aeiiberaieiy, to infiuence the overall mechanical properties of the composite. It is very difficult to determine the mechanical properties sf the very thin reaction layers that can occur, but fibre coatings which are applied usually have well known properties. These kind of coatings are often rubbery elastomers, applied to lower the stress concentrations that occur within the composite. It is possible to determine the influence of such

an interphase on the mechanical properties by

micromechanical analysis after introducing a third constituent between the

two

basic constituents, the fibre and the matrix. The micromechanical modelling remains exactly the same as previously described.

Figure 2.1 1 : Crack paiterns in composites with perfect and 25% degraded interfaces

[ I I].

The reaction layer, that is very likely to exist in various fibre reinforced composites

with polymer matrices, is probably dependent on the pre-treatment of the fibres and the sizing that is used. Although the mechanical properties of reaction layers are unknown, D.F Adams tried to model carbon fibre reinforced composites with a reaction layer to combine his finite element analyses with experiments done with carbon fibres with various sizings and environmental treatments [I I]. He assumed a degradation after houwet treatment that may occur at the

interdace

leading to a

adjacent to the interface. The failure mechanism described changed from matrix cracking to debonding (figure 2.1 1). However, no experimental evidence has been shown, that changes in mechanical properties due to the environmental treatment are a result of debonding within the composite, instead of changes of the matrix material properties. Moreover, the kind of sizing that had been used did not make any difference concerning the mechanical properties at all. The analyses lead to several stress-strain curves for composites with perfect ad hesion and degraded interiaces. The curves obtained by modelling degraded interfaces did not show any similarity with stress-strain curves that were experimentally obtained. The author assumes that this great difference is a result of the chosen 25% degradation, which seems to be too high. Even if it is sure that debonding leads to the observed changes, conclusions still cannot be made on how the interface should be changed for the analysis. Also with this application of crack propagation (like in section 2.2.3) the correctness is doubtíul, and mechanically irrelevant changes like element sizes may lead to different results.

10 r

Figure 2.12: Optimal inferlayer thickness [14].

Coatings that are applied at the fibre interface usually have, in contrast to reaction layers, well known mechanical properties. The main interest in coating filaments is to change the stress-strain situation in the composite. T. RiccÒ et. al. [I31 showed that a very thin elastomer coating on a hard spherical particle has a very strong effect

on the stresses in the surrounding matrix, leading to a stress-strain state

similar to that in the matrix filled with rubber particles. When unidirectional fibres are coated with a rubbery layer, the stresses in the composite are

also

changed and the stress concentrations between the fibres are strongly reduced. The failure

the rubber interlayer, because of its low strength. The performance of a composite is of course dependent on the thickness and the stiffness of the coating. By assuming that the optimal thickness for transverse strength improvement is the one that will cause the lowest normal stress at the matrix/coating interface, the optimal coating thickness can be calculated, depending on the stiffness of the applied coating. Figure 2.1 2 shows the relationship found between the optimal thickness and stiffness for a 50% volume glass fibre epoxy composite, when an embedded composite cyiinaer moaei is used [14].

3

Strategy.

The low transverse strain to failure of composites is one of the key issues in composite materials, which limits the use of these materials. However, most studies devoted to the improvement of the transverse properties focused on the modification of the fibre-matrix interface and/or the matrix to obtain a higher failure stress. Sometimes also a slight higher failure strain will be obtained this way. However, to obtain a considerable increase in the transverse failure strain, a different approach than the usual straight forward methods must be followed. An appropriate improvement of the composite materials can only be achieved if the relationship between the composite micro-structure and its transverse properties is known. Consequently, it is necessary to investigate how various material parameters influence the transverse composite behaviour in general and the transverse failure strain in particular. This can be done in an empirical way via numerous experiments on all kinds of composites with a variety in parameters, However, these experimental methods are very time consuming and have not led to significant improvements so far, because of the complexity of the problem.

Numerical analyses, based on micromechanical models, can also be used to study the influence of various parameters. Numerical analyses allow for a more easy parameter variation and may lead to a better understanding of the material behaviour.

Of course, there is a need for experimental validation

of

the numerical results. Therefore, a combination of experiments and numerical analyses is essential for the appropriate development of new composite materials with improved properties. Next sections will show how the experimental and numerical techniques should be used for a systematic study on the influence of several parameters on the transverse properties. Previous research of several investigators will be used to choose a particular model composite system for the investigation. It will

also

be indicated how some parameters can be studied with this composite system.

3.1

Method of

investigation.

Because of the complex relationships between structural parameters and the composite transverse properties it is essential to study the influence of these parameters very systematically following a relevant parameter variation. This can

be done by investigating the transverse behaviour as shown in figure 3.1, where the experimental and numerical techniques are com bined. To investigate the influence of one parameter, it is necessary to work on a well characterized composite system that allows for a quantitative variation of this parameter, without changing any of the other parameters. If, for example, the influence of fibre-matrix adhesion is studied, the level of adhesion should be varied without influencing anything else, such as the type of fibre, the matrix, the fibre volume fraction, the

test temperature ana cure cycie or even the method of fibre surface treatment. This demand requires a deliberate choice of a (model) composite system (a) before starting the experiments. Previous research of other investigators can be very helpful to find a appropriate system.

Material Choice (a)

I Experiments (b)

I

Failure Analyses ( c )

I

Material Development (f) Development Mi cromechan ical Model (d) I

~ Numerical Analyses (e)

Figure 3.1 : Material development using experimental and numerical

techniques.

The composite system that has been chosen, has to be tested experimentally to determine the composite’s transverse properties such as Young’s modulus, failure strain, failure stress and how they are effected by the parameter being studied (b). During and after the mechanical testing the damage development and failure mode should be determined respectively (c). The failure phenomena that are observed may help to develop a physical relevant micromechanical model for the numerical

analyses (d). In the numerical analyses large variations in parameters are possible (e). This might help to understand the influence of these parameters and to predict or extrapolate the mechanical behaviour of fictitious composites, that have not been tested yet. The obtained understanding and/or the predictions may lead to suggestions for new composite microstructures and guidelines for the development of new composite systems with the desired properties

(9.

With a good integration between experiments and analyses a better material might be developed.

In this paper the method mentioned before has been used to determine the influence of the fibre-matrix interface on the transverse properties of unidirectional fibre reinforced composites. The next section will show which model composite system has been used to investigate the influence of the interface. Furthermore it is indicated how the micromechanical modelling has to be developed to study the influence of the interface by numerical analyses.

3.2

Material choice to study the influence of the interface.

Unidirectional composites are generally composed of two constituents, namely the fibres and the matrix. All fibres are continuous and oriented in one direction for reinforcement of the matrix material. The matrix behaves as an adhesive material between the fibres. Because of this role of the matrix material it is obvious that the performance of unidirectional composites is strongly controlled by the matrix adhesion capability. Therefore, a lot of effort has been put in the improvement of the fibre-matrix bonding and in the study of the influence of fibre-matrix bonding on composite properties.

To obtain a better adhesion, the fibre surface can be made rougher or even chemical reactive to the matrix. For glass fibre reinforced thermoset matrices of epoxy or polyester it has been found that the fibre-matrix bonding is mostly determined by the chemical reactivity of the fibre surface [28]. The reactivity of the glass surface can be increased using silane sizings containing organic sites that are reactive with the matrix material. An increase of the surface reactivity of other fibres usually also leads to an increase of the fibre-matrix adhesion [27,30,31].

The fibre-matrix adhesion is often characterized by determining interface dominated properties of the unidirectional composites [25,26,27,29]. It is generally found that the fibre-matrix bonding does not significantly effect the longitudinal properties of the unidirectional composites, because these properties are mainly fibre dominated

[25]. However, the bonding has a very strong effect on the transverse and shear properties [26,27]. Surface treatment leads to a slight change of the transverse modulus and an increase of the transverse stress and strain to failure. Upon surface treatment an optimum will occur in transverse strength, because above an optimum level of surface treatment no further improvement in the transverse strength will be obtained due to a change in failure mode from debonding to other fracture phenomena like matrix [26] or fibre

[271

dominated failure modes.

TG study

the

influence of

the

interface on the transverse properties of unidirectional fibre reinforced plastics, various high-performance fibres such as glass, carbon, aramid and polyethylene fibres can be used. With all these fibres the level of adhesion can be controlled by fibre surface treatments [25-321. As was mentioned before, adhesion to glass fibres can be improved using silane sizings. Different reactivities of the fibres can be obtained by changing the organic compound on the silanes [28,31]. Other organic groups lead to other reactivities. However, this chemical reactivity cannot be quantified. Adhesion in the polymeric fibre based composite systems can be obtained by oxidation of the fibre surface [27,30,31]. For these fibres, different reactivities are obtained by varying the treatment time. Longer fibre treatments lead to more reactive sites [26,29,31]. Because all active sites will be nearly identical, the fibre reactivity is proportional to the number of sites. The amount of active sites can be determined by measuring the amount of surface oxygen, being characteristic for the active sites [30,31,33]. Consequently, these polymeric fibre are excellent to use in model composite systems. Because polyethylene and aramid fibres are less common for structural composite applications, carbon fibres are chosen for this investigation.

Together with the fibre a matrix material is required that exhibits a chemical reactivity with the carbon fibres. Thermoset matrices baaed on epoxy resins obtain this reactivity, leading to good or even perfect adhesion. It is important that a

homogeneous epoxy system is used, since the fibre surface treatment may influence the morphology of heterogeneous matrix systems, leading to different matrix properties [34]. By choosing a fairly brittle epoxy with a nearly linear elastic behaviour the micromechanical modelling can be restricted to reasonably simple linear elastic analyses on carbon epoxy composites.

The composites are studied using numerical analyses and transverse tests. Literature indicates that for the numerical analyses the number of chemical bonds and the interface strength must be included in the micromechanical model. Considering the observed failure phenomena, fibre-matrix separation must also be

are studied experimentally by transverse testing of carbon fibre epoxy composites with carbon fibres subjected to different levels of surface treatment, leading to a variation of the active sites. The experiments are described in chapter 5. The two most important issues being investigated in this study are the influence of interface on the overall transverse properties and the changes in failure modes caused by surface treatment.

4

Numerical analyses.

Generally, when effects of the adhesion level on the transverse properties are studied using micromechanics, the analyses are restricted to only two levels, viz. no bonding and perfect bonding. This implies directly that only the extreme types of composites are studied. However, in our study we are not only interested in these extreme bonding types, but especially in the relation between the level of adhesion and the transverse properties. Therefore, a micromechanical model has been developed which includes the interfacial bond strength. In the next sections it is described how the fibre-matrix bonding is incorporated in this a micromechanical model and how the model is used to study the effects of the interface on the transverse properties of carbon fibre reinforced epoxy composites. In addition, analyses are performed with not bonded fibres and perfectly bonded fibres to study the changes in failure modes caused by surface treatment.

~

4.1

Micromechanical modelling of the fibre-matrix interface.

Most fibre treatments lead to an increase in reactive sites at the fibre surface and an increase in chemical bonds [30,31]. This chemical bonding is particularly relevant to the interface properties and moreover to the transverse properties. Consequently, by including the chemical bonding in micromechanical analyses the effect of surface treatment on the transverse properties can be investigated. In section 4.1.1 an interface model is presented that allows variation of the fibre- matrix bonding. In the next sections it is described how this interface model is used to study the influence of the interface on the transverse deformation and failure.

4.1

.I

Introduction of an interface element for mìcro-

mechanical analyses.

The effect of the interfacial bond strength on the transverse strength of unidirectional composites can be investigated if an 'interphase' is incorporated in a micromechanical model. By combining such an interface model with the level of the surface treatment, the effect of surface treatment on the transverse properties can be evaluated. Most fibre treatments, such as the oxidation of carbon fibres, lead to an increase in chemical bonds at the interface. Separation of the interfaces of the two constituents implies breakage of the covalent bonds. This implies that in

micromechanical analysis the interface should be modelled by describing these chemical bonds. Such bonds can be represented by elastic springs. However, it would be very impractical to represent each bond by one single spring, since the number of chemical bonds is rather high. Consequently, it seems more useful to let one spring describe several bonds. If it is assumed that the chemical bonds behave linear elastically their mechanical characteristic can be given by

f =

k s u (4-1

9

where f is the force transferred through the bonds, k is a spring constant and u is the displacement between the end points. By multiplying the left and the right side of 4.1 with the number of bonds per area N a constitutive equation of the interface will be obtained, represented as a relation between the Cauchy stress os and the linear strain es:

- -

os=-.- n f - n k u - N k L * c s

A A (4-2)

where n is the number of bonds being presented per area A and L is the length of the springs in the undeformed initial situation. With 4.2 a specific interface modulus can be introduced, defined by

E s = N k L (4-39

The mechanical properties of the interface springs depend on the number of bonds they represent. Their modulus is directly proportional to the number of bonds. With respect to the mechanical properties of such an interface, it can be assumed that the interface has mechanical properties close to those of the matrix, since interfacial bonds are formed via reactions with this matrix.

The interface properties can be introduced in a finite element analysis by using one dimensional truss elements with the mechanical characteristics of (4-2) and (4-39 to connect the interfaces of fibre and matrix. The cross section of the elements has to be equal to the interface surface, Separation of the interfaces of the two constituents (fibre and matrix) implies breakage of the covalent bonds, which can be obtained by lowering the modulus of the truss elements to zero, which means that the interface modulus E, is reduced to zero. The influence of the interface on the transverse behaviour can be investigated by varying the interface properties in the micromechanical modelling.

4.1.2

Influence of the interface properties on the transverse

defsr

mati

o

n.

To study the effects of the interface on the transverse behaviour of unidirectional composites, the interface elements previously described are introduced in a finite element analysis of a transverse cross-section. The finite element analysis is based on a two-dimensional generalized plane strain model of a unidirectional composite.

By

assuming a regular square packing array, the model geometry can be reuucea to a repeating unit of a quarter fibre in a block of matrix (figure 4.1). For regularity, loads such as axial stress

q,

or strain eo should be applied in combination with boundary conditions of equal displacements perpendicular to the boundaries. The mesh that has been used is shown in figure 4.1. Because of the high stress gradients expected at the interface it is better to use eight node quadratic elements [21]. However, using quadratic elements will lead to a more complex interface modelling. Therefore, four node linear elements have been used for the fibre and the matrix. The interfaces are connected with 33 truss elements that are equidistant and have the same cross-sectional area. However, because of symmetry, the cross-section of the truss elements at the boundaries (where

4

equals to zero and ninety degrees in figure 4.1) is half of that of the other elements. The total cross- sectional areas are equal to the average of the surfaces of the fibre and the matrix interface.

The analysis is performed for a unidirectional carbon fibre reinforced epoxy composites with a fibre diameter of 6pm and a fibre volume fraction of 50%. The mechanical properties of the carbon fibre and the epoxy resin used are listed in table 4.1. The carbon fibres are considered to be transversely isotropic. To investigate the influence of fibre treatment on the performance

of

transversely loaded unidirectional composites, the interface modulus is varied. In the case

of

poor adhesion, interface properties such as stiffness and strength will presumably

h e !WJ csmpared tû

those

of

the matrix.

Therefwore,

in tnis anaiysic, reiativeiy

low

values

for

the interface stiffness and strength are used.

EPOXY 3.2 GPa 0.37 -

-

-

- . . . . _ _ . . . - . . . Carbon Fibre -

-

305 GPa 14 GPa 0.20 0.25 15 GPa

1 able 4.1 : Mechanical properties of the materials.

Material Young's modulus Poisson ratio Long. modulus Transv. modulus E Y E' E T

Long. Poisson ralja vLT Transv. Poisson ratio Y,.-,. Shear modulus GLT

To describe the chemical covalent bonds correctly in a micromechanical model, the interface spring elements should be very small compared to the total mesh size. The size

of

a C-C bond is 1.5

A,

which is only 4*10- times the mesh size. However, if the interface thickness

is

related to the crosslinks of the epoxy matrix, the elements must be approximately 30

A,

which is only 8*10- times the mesh size. For a physically correct model the choice of the interface thickness might be very relevant. To investigate if the interface thickness is important with respect to the modelling two interface thicknesses have been used, viz. I O e 4 and times the mesh size respectively.

1 1 +.++ *+b..r, I I I I I I I

/jy

I 4 ’ I I I I I I /

P

1 -I I I Interface Thickness: -+- 16-4 - 0 - 16-3 O 3 6 9

\

12 E m

Log Interface Modulus [Pa]

Figure 4.2: Influence

of

the interface properties on the transverse modulus.

Figure 4.2 shows the influence of the interface modulus on the composite transverse modulus. If the interface modulus is very low, practically

no

stresses are transferred to the fibre, resulting in a low transverse modulus. If however the interface is reasonably stiff, a higher transverse modulus is obtained. Figure 4.2

shows that when the interface modulus is close to that of the matrix, the transverse modulus is neither influenced by the interface modulus nor by the interface thic knesc. L

5

Interface Thickness: -+- 16-4 - 0 - 16-3 O 3 6 9

\

12 E m

Log Interface Modulus [Pa]

Figure 4.3: influence of the interface properties on the interface stress at

When the interface modulus is close to that of the matrix, the interface stresses in are not effected by the interface modulus or thickness. This can be seen in figure

4.3 for the calculated interface stress concentration at (I = O ((I is defined in figure 4.1). The size of the interface elements has no effect on the results

of

the analysis, since this relatively small change in size does not disturb the stress situation in the composite. Because deformations of the very small truss elements only result in small displacements of the truss ends, the absolute value of these displacements is

iïieiwmi iû the matrix and fibre defûiriiatiûri.

0.1 0.01 0.001 I I I I 7 Interface Thickness: -+- 18-4

-*-

l e - 3 O 3 6 9 \ 12 E m

Log Interface Modulus [Pa]

Figure 4.4: Influence of the interface properties on the matrix singularity ratio.

However, the size and the stiffness

of

the interface elements cannot be varied unlimited. If the elements are too small or too stiff a numerical solution of the finite element problem cannot be obtained. This is because very small or very stiff interface elements lead to a ill-conditioned stiffness matrix as shown in figure 4.4.

(D o i3 .IL -c (D c

-

2 ' 1 - O -1

Figure 4.5:

m t t t

\H

& i & & &

I -A-

-.-

-'I-

-*-

0.5 G P a I .O GPa 2.0 GPa 4.5 GPa 0 [degrees]

Normalized interface stresses along the fibre-matrix inte dace.

T h e objective of t h e micromechanical model is to study debonding of fibre a n d

matrix. However, before t h e debonding c a n

b e

studied it is n e c e s s a r y to determine

w h e r e a n d w h e n debonding will initiate. T h e o n s e t of debonding c a n

b e

determined

by analyzing t h e interfacial s t r e s s situation. Figure

4.5

s h o w s t h e s t r e s s e s a t t h e

interface

for various

interface moduli

as

a

function

of

t h e position along t h e fibre-

matrix interface

@

(defined in figure

4.1).

Variations in t h e interface stiffness h a v e

n o influence o n t h e normalized s t r e s s a t t h e interface, because t h e s t r e s s situation

in

t h e

composite

is

hardly influenced

by

t h e interface properties. Debonding will

initiate a t t h e highest s t r e s s along t h e interface, being between two

fibres

a t

4

= O.

In section

4.1.3

t h e debonding will

be

studied further.

4.1.3

Analyses with a debonding interface.

To study debonding with t h e micromechanical model, failure h a s to

be

included in

t h e mechanical behaviour of t h e interface. B e c a u s e t h e mechanical behaviour

of

t h e interface

is

related t o t h e number of identical chemical bonds,

all

interfaces in

t h e model

are

considered

as having t h e s a m e strain t o failure. This implies that

both t h e interface modulus a n d t h e interface s t r e s s to failure

are

proportional

to t h e

number

of

bonds. By a s s u m i n g a linear elastic behaviour until failure a t

a

critical

strain

e C R ,

fibre surface treatments will

lead

to variations of t h e interface behaviour

as

indicated

in

figure 4.6. Debonding

will be

studied by casing t h e stress-strain

curves

of

figure

4.6.

Instead

of

incorporating t h e s e curves in t h e modelling,

debonding will be obtained by removing interface elements by lowering their stiffness to zero (figure 4.6). Debonding propagation will depend on the stress or strain situation at the 'crack-tip' as a function of the crack-length C.

00' € 0 Carbon Fibre INTERFACE ELEMENTS REMOVED 0.5 GPa 1 .O GPa 2.0 GPa 4.0 GPa INTERFACE STRESS

-

7--rTT

"o' € 0 INTERFACE STRAIN

Figure 4.6: Stress-strain curves

o f

the interfaces being studied and modelling

o f

debonding propagation.

In the previous section the position of debonding initiation is predicted. By lowering the modulus of the truss element to zero, at $ = O, a crack is initiated. Initially, this

leads to an increase of the stress in the neighbouring element, being the crack-tip (figure 4.7) above the failure stress. This leads to an unstable propagation of the debonding. Crack-growth will continue until the stress concentration finally decreases below the critical value. Figure 4.7 shows that also in the case of crack propagation, the stress situation at the interface is not affected by the modulus of

the interface.

-

I I B 07 0 u) u) 2!

2

Y

o

Q .- _c o a L Figure 4.7: -A- 0 . 5 G P a - 0 - 1 . 0 G P a -v- 2 . 8 G P a -+- 4 . 0 G P a I -2

'

I 0.0 1.5 3.0 4.5 Crack-length C [pm]