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Eindhoven University of Technology

MASTER

Experimentally supported modelling analysis of continuous glass fibre reinforcedpolypropylene composites

Beemsterboer, P.B.F.

Award date:1994

Link to publication

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https://research.tue.nl/en/studentthesis/experimentally-supported-modelling-analysis-of-continuous-glass-fibre-reinforced-polypropylene-composites(e0ce97f6-6b75-4f71-a6a6-dc400d056358).html

Experimentally supported Modelling Analysis of Continuous Glass Pibre Reinforced Polypropylene Composites

P.B.F.Beemsterboer Graduate report; WFW 94.077

June 1994

Department of Mechanical Engineering Centre for Polymers and Composites University of Technology Eindhoven

in co-operation with

Ecole Normale Superieure de Cachan Laboratoire de Mecanique et Technologie Secteur "Structures & CMAO"

Je voudrais remercier T.Peijs, O.Allix, P.Ladevèze et H.Meyer qui mónt permis de travailler au

Laboratoire de Mécanique et Technologie et au Polymère Laboratoire Eindhoven. Je tiens également à

remercier M.Brekelmans, C.Cluze1, M vld Oever, A.Gasser, M.Arzt et les autres du Polymère

Département Eindhoven et Secteur Structures & CMAO de mávoir aide avec des problèmes plus

quotidiens et rendu ma formation très agréable.

I would like to express my gratitude to T.Peijs, O.Allix, PLadevZze and "eyer who made it possible

for me to work in the Laboratoire de Mécanique et Technologie as well as in the Polymer Laboratory

Eindhoven. Furthermore I would like to thank M.Brekelmans, C.Cluze1, M vid Oever, A.Gasser, M.Arzt

and the rest of the Polymer Department Eindhoven and Secteur Structures & CMAO for helping me with

more "dai1y"problems and maaè my graduation a pleasant one.

RESUME

Le travail présenté dans ce rapport concerne une étude sur le comportement mécanique du composite en matrice polypropylene renforcé par des fibres longues de verre. Dans les matériaux composites le phénomène mécaniquel le plus important est l’endommagement, autrement dit le développement plus OU moins progressif de microvides OU de ~ Z ~ E Q ~ ~ S S W ~ S . Afin d’approcher ce phénom&ne, une mdélisatim propode par Ladevezv et al. est étudiée. Cette modélisation décrit l’endommagement d’un composite au niveau de la couche élémentaire. Les variables d’endommagement sont associées aux variations de rigidité des matériaux. La procédure de l’identification du modèle est expliquée en détail. I1 s’agit d’exécuter quelques essais de traction avec différentes séquences d’empilement. De cette identification on peut conclure que le modèle n’est pas suffisant pour décrire le comportement mécanique, et un modèle modifié est introduit. Ce modèle modifié est bas6 sur le précédent. Et c’est la raison pour laquelle les principes essentiels restent les mêmes. Les paramètres du nouveau modèle sont identifiés et les simulations sont comparées avec les résultats expérimentaux. On peut conclure que les performances du modèle modifié permettant de décrire le comportement mécanique des composites verre/PP sont améliorées. Néanmoins, d’autres travaux de recherche semblent être nécessaires.

ABSTRACT

In this report a study is made of the mechanical behaviour of continuous glass fibre reinforced polypropylene composites. For composites the main mechanical phenomenon is obviously damage i.e. the gradual deterioration of a material due to the initiation and growth of micro-cracks and micro-voids. Therefore a model introduced by kadevèze et al., which uses a damage theory is disci?ssed. This model describes the composite damage at the elementary ply. Damage variables are associated with the relative material stiffness reduction. The model identification procedure is detailed explained. It consists of performing several tensile tests on laminates with different stacking sequences. From this identification the conclusion is made that the original proposed model is not capable enough to predict the mechanical behaviour, and a modified model is introduced. This modified model is based on the former one and therefore the main principles remain valid. The parameters of the new model are identified and simulations using this model are compared with experimental results. It can be concluded that the ability of the modified model in describing the mechanical behaviour of the glass fibre reinforced polypropylene composites improved, but more research seems to be necessary.

CONTENTS

Symbols

1 Introduction 1

2 . Theoretical modelling of the elementary ply

2.1 Damage concept 2.2 Damage kinematics of the elementary ply 2.3 Damage evolution law 2.4 Inelastic deformations 2.5 Identification of the model

3 . Experiments and first check of the model

3.1 Introduction 3.2 The description of the tensile test machine 3.3 The description of the specimen 3.4 The description of the performed tests

3.4.1 Damage- and plastic strain measurements 3.4.2 Transformation from measurements to ply properties

3.5.1 Rupture prediction 3.5.2 Tensile test on a [&45l2, laminate

3.6.1 Tensile test on a [014 laminate 3.6.2 Tensile test on a [0,90], laminate 3.6.3 Tensile test on a [k45I2, laminate

3.6.4 Tensile test on a [k67.5]2, laminate

3.5 First check of the model

3.6 Analysis of the tested laminates

3.6.3.1 Identification of the plasticity development law

2

2 2 3 4 5

6

6 6 6 7 7 7 8 8 10 13 13 14 15 15 16

4 . The modified model

4.1 Introduction 4.2 Modifications to the previous model

4.2.1 Modelling of the [&45l2, laminate 4.2.2 Modelhg of the [&67,5]2, laminate

4.3 Simulations using the modified model

5 . Micromechanics

Conclusions

References

Appendices

Appendix A ; Fibre volume fractions and dimensions Appendix B ; Damage- and plastic strain measurements Appendix C ; Results of the tests Appendix D ; Occurred problems

20

20 20 20 22 23

27

29

30

31

31 32 33 36

Symbols

a P E = strain c5 = stress V = Poisson's ratio 2) = off-axis angle

= power coefficient of the plasticity law = linear coefficient of the plasticity law

d d' E = Young's modulus G = shear modulus

P w = equivalent effective stress a = anelastic coupling parameter b

= damage parameter in transverse tension = damage parameter in shear tension

= accumulated effective plastic strain

= coupling parameter between the two damage parameters

Super- and subscripts

L T

= parallel to the axis of specimen = perpendicular t~ axis ~f specimen = rupture = elastic = plastic = initial value = parallel to fibre = perpendicular to fibre = effective = rate

1 Introduction

1 INTRODUCTION

The development of thermoplas tic resins as matrices for continuous fibre reinforced composites has offered considerable advantages over thermoset polymers in composite structures. These advantages are a better damage tolerance, environmental resistance, repairable and recycling. Until now, however, most research has been devoted to high- performance speciality resins like polyetheretherketone (PEEK), polyethersulphone (PES), and polyetherimide (PEI). These resins combined with carbon or aramid fibres provide ~ ~ m p ~ s i t e systems with excellent properties. Yet applications of suck high-price resins are limited to relatively small markets. To penetrate high volume markets (such as automotive) low-cost materials should be used. These systems can be found in a combination of glass fibres as a reinforcement combined with polypropylene (PP), polyethylene terephthalate (PET), or polyamide (PA) as matrix. Nowadays, these thermoplastic 'bulk' composites have two established technologies. First, short fibres reinforced injection moulding compounds and secondly, glass mat thermoplastics (GMT's), being stampable sheet products based on relatively long fibres in random array. To fiil the gap between high-performance composite systems as described above, and GMT's with relatively limited performance, activities move towards the field of continuous glass fibre reinforced composites based on commodity thermoplastics. The system described in this report is such a system, namely a polypropylene matrix with continuous glass fibres as reinforcement. To be successful in implementing materials such as polypropylene-glass fibre composite, the mechanical behaviour of the material should be known. Composites, in particular fibre reinforced laminate materials, subjected to mechanical loads show a continuous degeneration of structural properties. These are originated by a variety of damage mechanisms such as matrix failure, fibre failure, delamination, debonding and fibre pull-out. These phenomena are difficult to detect and measure and even more difficult to predict. The underlying mechanical models are complex by heterogeneity and anisotropy while, moreover, only little knowledge is available on the laws governing the evolution of damage quantities. In this report a model, using a damage theory, proposed by Ladevèze [5][6] is tested to describe the mechanical behaviour of composites. The composite may be defined by two elementary constituents:

The characteristic length here is the thickness of the elementary layer, and the damage state is taken to be uniform throughout the thickness of the elementary ply. This model describes the mechanical behaviour of composites by splitting up the different layers into elementary plies. In this study it is assumed that the interface between two elementary plies is perfect, and therefore it is not discussed in the modelling. The reasons for applying this model are; first, this model has proven its value for material- systems such as carbon/epoxy like IM6/914 and T300/914 or 3-D carbon/carbon composites. Secondly, the similarities of the ratio's (EJE2) and (G12/E2) between the above mentioned systems and polypropylene-glass fibres system. Also the main degradation mechanism seems to be the same in both systems. [1][2][3][4] In Chapter 2 the theoretical modelling of the elementary ply is explained. In Chapter 3 the description is given for the production and testing of the chosen laminates. Also in Chapter 3 a fiist quick check of the proposed model is discussed. The results of this first check is that a modified model is necessary. Before introducing this modified model the identification of all parameters is explained. Chapter 4 presents the modified model. Also in this Chapter the results of the comparison between experiments and (modified) model predictions are given. A short micromechanical study is presented in Chapter 5. Finally, some conclusions are discussed.

- a single layer; - an interface which is a mechanical surface connecting two adjacent layers.

1

2 Theoretical modelling of the elementary p ly

2 THEORETICAL MODELLING OF THE ELEMENTARY PLY

2.1 Damage concept

Damage i.e. the more or less gradual deterioration of a material due to the initiation and growth of micro-cracks and micro-voids, is obviously the main mechanical phenomenon for composite materials. The idea of damage mechanics is that the deterioration of a material can be described by its effects on the elastic characteïistics. The classical theory of isstíopic damage is not sufficient to study composite materials, since there are several damage mechanisms. In the proposed model it is assumed that only the transverse modulus E2 and the shear modulus G12 are influenced by the damage. The other independent elastic characteristics El and u12 remain constant, and are not effected by the damage state. This has been confirmed by experimental observations. To describe the damage state of the material two internal variables are introduced. It is clear that the relative variation of the Young's modulus E2 and the shear modulus GI2 are the characteristic parameters of the damage. Since we assume that the interlaminair-interface does not play an important rule, only the modelling of the elementary plies is sufficient to describe the mechanical behaviour of the laminate. [ 1][5][6] In 8 2.2 the damage kinematics of the elementary ply are discussed. In 0 2.3 the damage evolution law is introduced and in Q 2.4 plasticity. Paragraph 0 2.5 explains how to identify the model.

2.2 Damage kinematics of the elementary ply

The identification and the modelling of single layers are carried out with the assumption of in-plane stresses. The subscripts 1, 2 and 3 designate the fibre direction, the transverse- direction inside the layer, and the normal-direction respectively (figure 1).

3 k 2

Figure 1 : The ply references

The damage-strain energy is written in the following form :

with : <a>+ = a <a>- = a

if a 2 O ; otherwise <a>+ = O if a I O ; otherwise <a>- = O

where d and d' are two scalar variables which are constant throughout the thickness. They define the damage state of the single layer. If the transverse micro-cracks are loaded in compression, they close up and then have no effect on the transverse-direction behaviour.

2

2 Theoretical modellinx of the elementary ply

This explains splitting the transverse energy into 'tension' and 'compression' energy. The damage elastic law can be defined by :

or in components :

e 0 1 2

2G ?2(l-d ) E12 =

Associated forces Yd and Y,. are analogous to energy-release rates, and they govern damage development, just as the energy release rate governs crack propagation. The conjugated variables associated with the mechanical dissipation are :

where w is the free-energy density, and [ ] denotes the mean value throughout the thickness of the elementary ply.

2.3 Damage evolution law

Much of the mechanical behaviour is determined by the evolution of d. Therefore, the damage evolution law plays a dominant role in this model. The damage of a ply is assumed to be governed by the variable Y :

Y = Yd (t) + bYd (t) (5)

where b a coupling parameter is between the two damage variables and :

The damage-development laws are very simply written as follows :

3

2 Theoretical modelling of the elementary ply

The coupling of the two damage variables is written as :

d ' = b d

In figure 2 an example is given of the master-damage curve for a carbon/epoxy system. This figure shows that the damage evolution, for this material, linear dependent on is.

0.00 0,05 0, lO 0,15 0,20 0,25 . 0,30 0,35

Figure 2;

Remark;

The muster-damage curve for the carbon epoxy system T3001914

It is also necessary to include a rupture criterion for the fibre. Since only tension is applied, the following criterion is used ; O < 0 1 1 e

2.4 Inelastic deformations

The micro-defect, i.e. the damage, leads to sliding with friction and thus to inelastic strains. In order to model the inelastic strains introduced by damage, we build a plasticity model based on the following effective quantities:

- the effective stress 5, - the effective plastic strain rate

which verify :

T r [ g g p ] = T r [ d $ l

A particular choice for& is :

(9)

The effective stress 5 is written as :

4

2 Theoretical modelling of the elementary ply

* 011

The hardening is assumed to be isotropic. The elasticity domain is defined by the function f such that :

where the threshold R a function of the accumulated plastic strain p is. P => (R(p) + Ro ) is a material-characteristic function, and a2 is a material-characteristic constant. Under the yield conditions f= O andf= O it is possible by using the normality rules to write for the effective plastic strain rate :

= O

The model assumes that no plastic yield exists in the fibre direction (E fll = O). Furthermore, in the expression for the function f, we assume that the stress Oll has no effect on the plasticity development. Squaring eqn (13) and summing them, we obtain an expression forp in terms of the effective plastic-strain rate :

2.5 Identification of the model

To identify the model it is proposed to perform tensile tests on the following laminates : O 0 - [o14 : determination of E u12and O:l

- [*4512, - [k67,5]2,

: determination of G p2, R(p) , R o, YO and Y, : determination of E :, b and a2

In respect to the progressive damage the identification of the [*4512, laminates is the main test. That is why this test is described more precisely in the following chapter.

5

3 Experiments andfirst check of the model

3 EXPERIMENTS AND FIRST CHECK OF THE MODEL

3.1 Introduction

The first aim of experimental testing is to verify the modelling of the elementary ply. These experiments are also done with the purpose of finding the moduli, the Poisson's ratio, the rupture values and the evolution of the damage and plasticity :

v n o o - the elastic constants E - the hardening curvep => R(p) + R 0 and the constant a2 - the b, Yo, Y, ; constants which define the damage evolution laws - the rupture stress

E 2, G 12, v12

To determine the elastic constants, tensile tests on unidirectional, [014, and cross-ply [0,90], laminates are performed. Also the rupture stress can be determined in a tensile test on a [0,90], specimen. For the determination of the evolution of the damage and hardening, the main test is a tensile test on a [&45]?, laminate. A complementary test, for example tests on [*67,5]2, specimens, is used to identify the constants a and b, which specify the coupling between the shear stress and the transverse stress. First in 8 3.2 and 0 3.3 descriptions are given of the tensile test machine and the specimens. In 9 3.4 some general tools are introduced. A quick check of the model is given in 0 3.5. In the last paragraph, paragraph 9 3.6, the analysis of the tested laminates is explained.

3.2 The description of the tensile test machine

The tensile tests are performed on a hydraulic machine of 100kN, driven on displacement, controlled by a computer (an Apollo station). The tabs are clamped by the clamps of the tensile machine at a constant pressure of 2 MPa. The displacement speed is chosen in such a way that the strain rates are about 1E-5 [ l/sec]. The strain is measured every ten seconds, with strain gages as well as with an extensometer.

3.3 The description of the specimen

Unidirectional laminates were manufactured using the film stacking method. This is done by winding fibres on a rectangular mandrel with alternating layers of polypropylene (figure 3). Since the behaviour of glass fibres is highly moisture sensitive, the mandrel plus lay-up was placed in an oven for four hours at 90°C. To avoid the material from adhering to the mandrel, release coated Mylar@ sheets were used. Metal stop-strips were used to control the thickness of the laminate. Impregnation was achieved by applying heat and pressure as illustrated in figure 3.

C o m p r e s s i o n moul

I l-----l I

I t

ding

Fi b r o s c i t m

Figure 3; The production of the laminates.

6


The compression moulding temperature was limited to 200°C since at higher temperatures degradation of the polymer might take place. The pressure was stepwise risen to 20 bar and kept at 20 bar for 1 to 4 hours (depending on the thickness of the laminate). After compression moulding, the mandrel was cooled down slowly. Non-unidirectional laminates were produced in several steps. First, the needed numbers of layers with the alternating layers of polypropylene were produced. Subsequently, this ply was moulded between other single layers of fibres in such way that the desired laminate was obtained. [ 31 [4] a?;e edges cf the [O-OOI,, [k6l2,, and [f6?,512, specimens were grinded in order to remove cutting irregularities. Aluminium tabs were adhesively bonded to the specimens ends. The dimensions and fibre fractions of the specimens are given in appendix A.

3.4 The description of the performed tests

3.4.1 Damage- and plastic strain measurements

In order to measure the damage and the plastic strain, it is strictly necessary during the course of the test to load and unload. Nevertheless the number of cycles should not exceed six or seven in ordes to stay in a domain where low-cycle fatigue phenomena are negligible. Figure 4 illustrates the type of experiments which should be performed (for more information see appendix B)

E.

9

Figure 4; The tensile tests for damage- and plastic strain measurements.

Only for the [k45),, laminates the tests are not carried out until rupture. Due to the rupture strain which is about 35%, the strains for this particular laminate are to large for the strain gages. [3][4] For detailed information about the measured stress and strains, see Appendix C. In appendix D some occurred problems during the experiments are discussed.

3.4.2 Transformation from measurements to ply properties

The identification of the constants is based on the measured references or Paminate references shown in figure 5.

rn

Figure 5; he laminate references.

7

3 Experiments and first check of the model

The transformation from the measured properties (laminate references) to the ply references is given by :

from these equations we derive for an elementary ply :

EL cos2(8) + ET sin2(8) ~ 2 2 = EL sin2(8) + ET cos2(8) ~ 1 2 = (ET -EL) sin(8) cos(8)

and taken into account that E 1 >> (U P2)2E i we obtain for the stresses :

and :

Cll + C22 = OL a11 sin2($) + 0 2 2 cos2(8) + 2012 sin(8) cos(8) = O

3.5 First check of the model

The purpose of this first check is to discover or the proposed model is capable to predict the mechanical behaviour of the composites or not. The main curve in the model is the damage- master curve (d versus - ). This measured curve is the basis for the damage evolution law. This damage evolution can easily be found by performing tests on a [1145l2, laminate. This will be done in fi 3.5.2. First in fi 3.5.1, the rupture prediction will be explained by an example of a [&45l2, laminate.

3.5.1 Rupture prediction

In our model, rupture is expressed as a verification of an instability criterion (figure 6). This shall be explained by the derivation of the critical value for d (d,) for a [k45]2s laminate.

d

Figure 6; The instability criterion

8


A simple calculation shows that [&45l2, laminate behaviour in tension is essentially governed by ply shear behaviour; the strains Ell and &22 are negligible with respect to the shear strain &12. Also as will be explained in 0 3.5.2 ; The shear yield condition is then written as :

B

O12-R(p) -Ro=O

and we extract the following relationship :

Attempting to express the shear effective stress rate as a function of the effective shear stress :

Remember the damage development law eqn (7) :

As will be shown in 0 3.5.2 is can be reduced to :

B (for the shear case), therefore the expression for Y

(23) 2 G ,O,(l-d ) 2

dT = sup T l t .Iyd = sup

With this we derive for the damage rate :

‘d

2 4 m Y d =

The quantity Y d is expressed in terms of ci12 and d (see 8 2.2), so the rate :

9

3 Experiments andflrst check of the model

and substituting eqn (23) and eqn (25) in eqn (24) we extract d as a function of Ol2 :

=i 2 d = F( 1 -d ) d x d x

where :

0 1 2 F = l - .r, Y 42GP,(l-d ) 2

Then, substituting eqn (26) in eqn (21) and the resulting equation in eqn (20) and using

El, = &le2 + &f2 we have the following law :

The instability criterion is verified when the shear stress reaches a maximum, in other words, when :

Ol2 = o and E12 + O

From eqn (4) we make Yd appear in the expression for F, and we can calculate the limit shear damage value dc when F equals zero :

d c = z ( 1 i--) YO YC

YO YC

The t em - is generally small, therefore the value for dcis close to 0.5. In other words, when d reaches a value of about 0.5, rupture occurs.

3.5.2 Tensile test on a [f45I2, laminate

In figure 7 a typical test on a [f45I2, laminate is shown.

10


t"

X-axis : Strain (-)

Y-axis ; Stress (MPa)

Figure 7; A typical test on a [f45]2s laminate.

Using the transform equations (15-18) the stress ply variables can be defined by the measured references :

and the strain variables :

From these equations we obtain directly the shear modulus :

Because O:2 B Oi2 the reduced expression for fi is (eqn (23)):

With this equation and :

11


we are able to determine the damage master curve for the [*4512, laminate (figure 8).

- test A45 - test B45 - test C45 O 0,2 0,4 0,6 0,8 1

Figure 8; The identified damage-master curve on [245],, lamhates.

Remember our first assumption of d being linear dependent on fi ( Q 2.3), and the critical value dc of about 0.5 ( 5 3.5.1). Figure 8 shows that the measured relationship between d and fl is a non-linear relation. Also the values for d observed in the experiments are higher then the highest possible value obtained by theory (dc). Nevertheless some simulations are made, in order to verify our expectations. Simulations are made by using a program called Dam-Lam (Damage of Laminates). The simulations are performed under the assumption that the damage has a linear relation to n. This linear function is presented in figure 9. Since in our model rupture occurs at a value for d of about 0.5, and in reality this value can reach 0.8 (figure 8,9) we expect a premature rupture.

I I I I I I 1

O 0,2 0,4 0,6 0,8 1

model - - - - - - - -

0 test A45

A testB45

0 test C45

Figure 9; The values for this curve are :

The modelled damage-master curve. Y , = 0.0025 MPa, Yc = 0.2025 MPa

00 With Yo, Yc,R(p), Ro, and G :!obtained by the performed test and E 1, E 2, II:~,..;~, a and b taken from other tests, which will be introduced in the next paragraph, simulations are made for the [k45I2, laminate. The simulation is shown in figure 10.

12

3 Experiments andfrrst check of the model

I : , ; ; : ; : ; 4 3 2 1 1 2 3 4

- : Model - - - _ : Experiments

X-axis ; Strain (-)


L I

Figure I O; The simulation for the [545],, laminate.

Figure 10 shows the earlier mentioned premature rupture. Rupture occurs when d reaches a value of 0.47, which corresponds with our predictions. Therefore this model is not sufficient enough to predict the mechanical behaviour of the laminates, and a modified model is introduced. Before introducing this new model the determination of E R(p), Ro, a and b will be discussed in the rest of this chapter.

o O 0 E 2 , v12,

3.6 Analysis of the tested laminates

3.6.1 Tensile test on a [ O ] , laminate

Because the relations between measured and ply quantities are for unidirectional Iaminates straight forward, the longitudinal modulus E l and the Poisson's ratio u:2 are directly determined in the tests :

EL = E , = 40.5 GPa =I$! = 0,3698

A typical stress-strain curve of a unidirectional laminate is given in figure i 1.

13


X-axis : Strain (-)


Figure I 1 ; The typical stress strain curve for a [014 laminate.

Remark: From tests on cross-ply laminates it is also possible to derive the longitudinal modulus just as it is possible to derive the rupture stress from the unidirectional laminate tensile test. The comparison of the two tests give the same coeficients.

3.6.2 Tensile test on a [0,90], laminate

The following relationship allows us to reconstitute the fibre rupture :

with the assumptions of :

this leacis ûs to :

O:l = E i&; = 658.9 MPa

In figure 12 the result is given of a performed test on a cross ply laminate.

and

14


250 1

X-axis : Strain (-)


J

Figure 12; The typical stress strain curve for a [0,90], laminate.

3.6.3 Tensile test on a [f45I2, laminate

In fi 3.5.2 the identification of the shear modulus GI,, using the [rt,L45I2, laminate has been performed. With this modulus the damage-master curve and thus the damage evolution law could be identified. The tensile tests on [f45I2, laminates are also used to identify the plasticity development law, this will be discussed in the next paragraph.

3.6.3.1 Identification of the plasticity development law

The threshold values (Np) + Ro ) are obtained by eqn (12) :

f = + a2 - ~ ( p ) - R, (12)

As shown in fi 3.5.2 is 0 1 2 D 0 2 2 and E12 B E22. This means that if a2 < 1 the evolution of the plasticity is defined by the effective shear stress 012 :

For the accumulated plastic strain rate p :

I I

In the last equation it is possible to replace t with any other monotonous increasing variable. In this case we replace t by &f2 The accumulated plastic strain, p , is then calculated for each

15


1,s

i x 0 3

O

test by the integration of the curve 2(1-4 versus && (figure 13) :

-A 0 test A45

l--& 0 A testB45

-- O 9 8 A o A 0 test C45

O 0

o o oA

1 I

I I

i I I

Figure 13; p can be obtained by integration of the 2(1 -d) versus E; curve.

leading to the plasticity master curve :

loo T n

e & 80 e-

E test A45

$ 40 __b__ test B45 - test c45

60 O

2 20

O O 0,005 0,Ol 0,015

p (-)

Figure 14; The identified plasticity-master curve.

The functionp ->R(p) is chosen to take the form; R(p) = B(p )" with Ro = O MPa

8 = 5500 MPa a = l (The dashed line in figure 14)

3.6.4 Tensile test on a [+67,512, laminate

The transverse modulus related to the measured quantities is :

16


In order to calculate the non-initial transverse modulus (E2) it is necessary the know the value for d . It is possible to obtain this value by calculating the value for Y and then using the damage evolution law :o detemine d. However €QT the cahdation of Y, b should be known. With d it is possible to calculate E2 and therefore d', and b can be derived. This iteration process is scheduled as :

First estimation forb t t t 1 1 ?-

T

?-

?-

Calculate Y by equation (40)

Calculate E2 by equation (38)

Calculate b by equation (43)

Calculate d by equation (41)

Calculate d' by equation (42)

+ + + finalvalueforb 1 + +

Remember the equation for Y (eqns (43)) :

1 r q2>: 1 + b 3 1 lol,] Y = Y d (t) + bYg (t) = 2

G P2(1-d ) 2 E ;( 1-d' ) 2

Expressed as a function of laminate references, we Write :

The value of the damage variable d is eqn (7) :

With these values ford substituted in eqn (39), we can calculate the transverse modulus. As soon as the transverse modulus is determined, the value for CE can be calculated and then b :

This modified value for b is used as a new Initial value for the iteration. The value of 1,9

17


was found for b. For the determination of the value for a2 the following already introduced equations are used eqns (12), (13) and (14) :

0 2 2 %2= a2p R ( p ) + R ,

n e s e equations allow us to separate a2 :

2022&1;3

Expressing this equation in laminate references :

Taken into account that &i >> &; and cos2(9) B sin2(9) (9=67.5") it is possible to simplify equation (46) to :

G f2(1-d' ) E :(i-d )

a2 = (47)

A value of 0.1 was found for a. A typical stress strain curve for the [f67,5]2, laminate is presented in figure 15.

18


X-axis ; Strain (-)


I 1 2 3 4 5 6

I

R(P) yo y c a b

1.9 E , E 2 1”2 u102 011 40.5 5.462 1.709 0.3698 658.9 O+ 0.0025 0.2025 0.1 GPa GPa GPa MPa 5500p MPa MPa A

All parameters discussed in the preceding part are summarised in table 1.

Table I ; Parameters found in identijìcation of the model.

19

4 The modified model

4 THE MODIFIED MODEL

4.1 Introduction

Because the previous model was not suffic-snt enough to predict or describe the mechanical behaviour of our laminates, a modified model will be introduced. The demands on this new model are ;

The model will be based on the previous model, meaning that the main principles remain valid.

- - an 'easy' damage evolution law.

point of instability ( -9 dc = I);

4.2 Modifications to the previous model

Because of the predominant role of the damage master curve and the inability of the old model to describe this curve correct, a new damage evolution law is introduced. The variable Y will now be dependent on real properties instead of effective properties (eqn (4)(5)) and therefore the damage evolution law will be based on real variables :

Hence the damage evolution law :

Since this model is meant to be done 'by hand' we assume for simplification :

d = d' therefore b = 1 (50)

The damage master curve of this new model will also be identified on a test on a [k45]2s laminate.

4.2.1 Modelling of the [+4512, laminate

Because B 012 the reduced expression for fi is :

With this equation we are able to identify the new damage master curve (figure 16).

20


- test B45

test C45

-A- test B45 extenso.

T 3

I 1 I I I

fi

O 5 10 test C45 extenso.

-0- 15

Figure 16; The mod@ed damage master curve of the [f451,, laminates.

According to figure 16 we can predict the damage parameter d as :

where Y, = 6.25 MPa YC=64MPa (The dashed line in figure 16)

The modelling of the plasticity is checked by comparing model predictions and experimental measurements. This comparison is presented in figure 17. Because we want to verify the plasticity, the x-axis contains the plastic shear strain E& For the y-axis the variable d is chosen, because it is modelled as well as measured. Figure 17 shows good agreement between experiments and model predictions. This indicates a correct modelling for plasticity.

t

- : Model _ - - - : Experiments

X-axis ; Plastic strain (-)

Y-axis ; d (-)

I I F

Figure 17; Comparison of the model predictions and experiments for the plasticis, on [45.5J2, laminates.

21


4.2.2 Modelling for the [f67,5I2, laminate

Using the transformation equations (see 0 3.4.2) and eqn (50) we can write ford :

u - \ - - I

As soon as the value ford is known the calculations for the transverse and the shear stresses can be performed :

Using eqn (48), the damage master-curve can also be identified on [f67,5]2, laminates. This identified relation between the damage and Y should be identical with the one identified on the [&45]% specimens. The identified damage master curve on [_+67,5]~ laminates is given in figure 18. The dashed line is the proposed modelling of the damage evolution law identified on [k45I2, (eqn (52)). As can be observed in figure 18 the two curves are identical, indicating a correct identification for the damage evolution.

0.8 '1

0- '- - 0 test C67.5 O 2 4 6 8 10

Figure 18; The identified damage-master curve on [&57,5],, laminates.

Since the assumption d =d' allows us to determine the stresses and strains, it is also possible to obtain the plasticity master curve for [&67,5l2,. This plasticity curve should be identical with the one identified in chapter 3 (figure 14) on [f45]&. Let us recall the equation for the accumulated effective plastic strain p (eqn (14)) :

The value for p can be obtained by numerical integration of the former equation. The equivalent effective stress R is derived from eqn (12) :

~ ( p ) + R, = I/ + a2

which can be calculated directly. As a consequence, the plasticity master curve can be

22


E , E ; 1"2 UP2 qi R(P) yo Y C a

40.5 5.462 1.709 0.3698 658.9 O+ 6.25 64 0.33 GPa GPa GPa MPa 5500p MPa MPa

&

identified (figure 19). In the identification of the plasticity on a [+45]2, laminate, the anelastic coupling parameter a is absent. Therefore this parameter can be used to fit the two identified plasticity graphs. A value of 0.33 was found for a. The dashed line in figure 19 represents the modelling for the plasticity as identified on [+4512, in chapter 3.

1 - 8 ,

O

' 4 p A test B67.5

W

A P' 0 test A67.5

* * I I I o I- I I I I 0 test C67.5

O 0,0005 0,001 0,0015 0,002

p f - j

Figure 19; The identified plasticity master curve of [&57.512s.

All parameters discussed in this chapter are summarised in table 2.

Table 2; Parameters found during the identification of the modified model.

4.3 Simulations using the modified model.

Using the values found in the identification of the second model (table 2) comparisons are made between experimental results from the tensile tests and model predictions. The promising modified model as presented in the preceding part is implemented in Dam-Lam and simulations are made. Figure 20 and figure 21 show the comparison between tests and model predictions for the [014 and [0,90], laminates. These figures show good agreement between experiments and simulations for both behaviour development and rupture prediction. Due to differences in the fibre volume fraction (Appendix A) of the specimens the experimental results and the simulations may differ a little.

23

4 The modì$ed model

,.i * io'

Figure 20;

: Model _ - _ _ : Experiments -

KIE X-axis ; Strain (-)

Y-axis : Stress (MPa)

I I

Comparison of the model predictions and experiments for the [0j4 laminates.

X-axis ; Strain (-)


I Figure 21 ; Comparison of the model predictions and experiments for the [0,90l, laminates.

The simulations of the former laminates are hardly influenced by the differences between the original and the modified model (because the damage parameters do not play an important role in these laminates). This however is for the simulation of the [&45l2, specimens completely different. In figure 22 the simulations using the modified model are compared with the experimental results of the tensile tests. The problem of the premature rupture has indeed disappeared (see figure 10). Figure 22 shows good agreement between experiments and simulations for behaviour development. For the rupture prediction it is more difficult. Since the experiments for this type of specimen are not carried out until rupture, no information is available for the rupture. Therefore the simulations are also not carried out until rupture, Nevertheless when the highest measured values for d (~0.8) are compared with the simulated values €or d and almost no differences occur. This indicates a correct modelling.

24

4 The modijìed model

-- . Model - - - _ : Experiments

X-axis ; Strain (-)


, I

Figure 22; Comparison of the model predictions and experiments for the [24551,, lm*nates.

Since in the simulations of the [f45I2, laminates the coupling parameters a and b are not used, these values can be verified in the simulation of the [f67,5],, laminates. Also the assumption of d = d' can be checked by this simulation. Figure 23 shows the comparison between the tests and the simulations for [+67,5]2s laminates. The comparison shows good agreement between experiments and model predictions for behaviour development. Since in the modified model no rupture criterion is installed (except for the critical stress O;l) the simulations are interrupted when d reaches the maximum value measured in the experiments (= 0.67, see figure 18). As shown in figure 23 the value for d in the simulations is in accordance with the values obtained by experiments.

I

Figure 23;

- : Model : Experiments - - - -

I

X-axis ; Strain (-)


Comparison of the model predictions and experiments for the [&57,.512, laminates.

25

4 The mod@ed model

X-axis : Strain (-)


A complementary tensile test is performed on a [+56]3 laminate. These experimental results and simulation using the modified model are compared in figure 24. As can be observed in this figure, the behaviour development nor the rupture prediction shows reasonable agreement with the experiments.

X-axis ; Strain (-)


I : Model : Experiments

- o1 o O, o2 o3 - - - _ o2

I

X-axis : Strain (-)


I I

Figure 24; Comparison of the model predictions and experiments for the laminates.

The reason for this might be that the assumption of d = d' is correct on [&67,5]2, laminates but not on [+56],, laminates. The reason for assuming this is that the value for Y is in both simulations more or less the same. This means that the relation between Of2 and 022 is not correctly modelled. From figure 24 it can be seen that b should decrease, to improve the predictions for [&56],, laminates. If b decreases, the value for Y decreases and therefore the influence of the damage. In figure 25 is shown the results of b = 0.25 on [f56]2, and [&67,5l2, laminates (For [&4512, laminates the influence of b is negligible). As can be seen in this figure the results of decreasing b has positive effects on the [+56]2, laminates but negative effects on the [f67,5I2, laminates, and nothing has actually improved.

2

26

5 Micromechanics

5 MICROMECHANICS

The material has been examined under a Scanning Electron Microscope (SEM). From Contant [4] and v/d Oever [3] it is known that for [014 and [0,90], laminates the main damage mechanism is failure at the fibre/matrix interface. This was concluded from SEM- micrographs of the fracture surfaces of failed specimens. The micrographs presented in figure 25 and 26, are SEM-micrographs taken from the fracture surfaces of failed specimens. The ty-pe of laminates are respectively [&&I2, and [&67,5]2, laminates. As can be observed in figure 25 the fibres are completely free of matrix material. This indicates a failure at îke interface fibre/matrix.

Figure 25; SEM-photograph of the fracture surface of a [&4.5J2, laminate.

At the fracture surface of the [&67,512,, matrix is to some extend still adhering to the fibres, indicating a better bonding between matrix and fibre (figure 26). This is an indication of a combined failure of the matrix as well as the fibre/matrix interface.

Figure 26; SEM-photograph of the ffacture surface of a [267,.5J2, laminate.

27

5 Micromechanics

From these micrographs it is possible to conclude that (at least) two different failure mechanisms play a part in the mechanical behaviour of the PP/glass composites. From the tests on [+5612, laminates (see figure 24) we know the modified model is still not capable (although much improvements were made) to predict the mechanical behaviour of the composites. A possible solution might be to uncouple d and d' from each other, and derive for both of them a damage evolution law.

28

~ ~

CONCLUSIONS

In the model for the prediction of the mechanical behaviour of laminates introduced by Ladevèze et al. some general tools and concepts have been proposed for developing specific material damage models. This proposed model has proven not to be capable of describing the mechanical behaviour of continuous glass fibre reinforced polypropylene. Nevertheless, this damage mechanics approach seems to be a powerful tool for the prediction of the mechanical behaviour of composites. Therefore a modified model based on the model of Ladevèze was introduced.

During the identification of the modified model, promising results have been obtained. For the [&45]2, laminates a well defined continuous damage evolution law was determined. This damage evolution law was verified on [I67,5]2, specimens and an identical evolution law was found. For plasticity, the modified model remains the same as the or@d model. The master plasticity curve was verified on [k45I2, laminates, using a comparison between model predictions and experimental results .

Comparisons of the modified model predictions and the experimental results from the tensile tests were made. These comparisons showed good agreement for both behaviour development and rupture prediction in case of the [O],, [0,90], laminates. For the [&45]2, laminates the simulations also show good agreement of the behaviour development, but the rupture prediction for this type of laminate remains a problem. Since the experimental measurements were only accurate until strains of about 3 a 4%, no information is available on the rupture (Er = 35%). In comparisons between the modified model predictions and the experimental results for [&67,5l2, laminates, the new model proved to be capable to predict the mechanical behaviour of the investigated material.

In comparison of simulations and experimental results of an additional [&56]2s laminate, the modified model seemed not to be adequate to predict the mechanical behaviour. The reason for this is still not clear, but a proposal of uncoupling the two damage parameters d and d' is discussed. TO verify this proposal, further research seems to be necessary.

29

REFERENCES

c11

P I

131

~41

[51

Une nouvelle approche des composites par la mécanique de l'endommagement, Allix O.; Engrand D.; Ladevèze P.; Perret L., Avril 1993 Cachan

Reinforcements for PP: technology developments, Reinforced plastics volume 36-3 march 1992

Fatigue behaviour of continuous glass fibre reinforced polypropylene composites, van den Oever M.J.A., Graduate report Eindhoven University of Technology, November 1993

Influence of maleic anhydride modified polypropylene on mechanical properties of continuous glass fibre reinforced polypropylene composites, Contant M., Graduate report WFW 92-092 Eindhoven University of Technology, August 1992

Damage mechanics of composite materials, Ladevèze P., reprinted from Composite Materials Series 9,1993

Damage modelling for the elemantary ply for laminated composites, Ladevèze P.; le Dantec E., Composites Science and Technology 43 257-267,1992

Encyclopedie d'analyse des contraintes, Avril J., Micromesiares, 98 Boulevard Gabriel-Peri 92240 Malakoff France

Module-9 Composieten, Peijs T., Technische Universiteit Eindhoven

Modélisation du comportement mécanique des composites Carbone/carbone 3D EVOLUTIF, Allix O.; Cluzel C.; Ladevèze P., Rapport de contrat C.N.E.S. no.840/CNES/92/1854/00, Mai 1993

Logiciel DAM-LAM Post-processem danalyse a la rupture des composites stratifies pour des chargements thenno-mecaniques. Laboratoire de Mecanique et Technologie ENS Cachan 1

30

Appendices

Appendix A ; Fibre volume fractions and dimensions

I Type of laminate Fibre volume fraction (Vf) 35.5 - 40.1 %

Table 3; Fibre volume fractions of different laminates

Table 4; Specimens dimensions for different laminates

31

Appendices

Appendix B ; Damage- and plastic strain measurements

The chosen way to measure the damage- and plastic measurements is shown in figure 26:

Figure 26; Damage- and plastic strain measurements

32

Appendices

Appendix C ; Results of the tests

Tensile test on a [O], laminate

Tensile test on a [0,90], laminate

33

Tensile test on a [I45I2, laminate

34

Appendices

Tensile test on a [+67,512, laminate

initial 4,378

~~

W a l 3,671 0,194 0,1193 -0,0232 0,0193 -0,0033 3,269 0,191 0,1673 -0,0319 0,0286 -0,0053

35

Appendices

Appendix D ; Occurred Problems

D.1 The problem of the load cell

Since the tensile machine can handle ZOOkN it is necessary to change the load cell in order to gain aceuraey. At the EMT the smallest load cell is one with a maximum ~f 10kN. FQI- a tensile test on a [k67,5I2, laminate (with a surface of about 40 mm2) 300N would be sufficient (the ratio; 300/100OO = 0,03). For this reason, a great loss in accuracy occurred.

D.2 The possible influence of the strain gages

Since for specimens such as [k67,5], the stresses are small (-8 MPa) there might be an influence of the local added strain gages with the glue. The thickness of this glue layer is normally about 0,04 mm, with a modulus of 20 GPa. [7]

D.3 The problem of bonding the strain gages QII the specimen

Polypropylene is known for its chemical inertness. For this reason it is difficult to bond anything on the surface of the specimen. In our case an epoxy based adhesive was used. Another problem is that the surface of the specimen is not very smooth, and this is even getting worse during the test. Especially for the [k45I2,laminate the problem is really serious. The angle of the fibres is in the beginning of the test 45", whereas after failure (= without the elastic strain) the angle between the load axis and the fibres is about 35".[4]

D.4 Differences between the strain gages and the extensometer

The strain is measured three times; two strain gages (on each surface one) and an extensometer. Between the two gages the difference is relatively small, from which one couId conclude that the gages are well in line with the loading axis, and there is little bending. The difference between the gages and the extensometer is however quite large.

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