P.Fernberg H Zrida

J Varna

Effe Z Ayadi

Many types of composite materials are today used in various types of load carrying structures, due to their excellent strength and stiffness to weight ratio. Simplicity, reliability and low cost of the material processing are important factors affecting the final selection.

With the textile reinforced composites, the cost-efficiency is reached by using dry preforms which are impregnated by resin infusion, resin transfer molding etc.; this have made a break-through and have been widely used.

Textile composites with bundle meso-structure have been studied in this thesis for elastic properties and damage investigations.

In chapter I a state of the art was conducted the different methods for elastic properties determination in composites with bundle mesostructure and mesostructure heterogeneity. The different damage features occurring in such composite materials are also investigated.

Chapter II of this thesis deals with elastic properties modeling for Non-crimp fabric (NCF) based composites for investigating the effect of meso-structure defects on mechanical properties degradation. The objective of the work is to formulate a model for the NCF composite mesostructure in an attempt to investigate the effect of the waviness on stiffness reduction. Moreover, the stiffness calculation methods for the complex geometry are explained and justified and finally, the different geometrical parameters changes are taken into consideration and included in the calculation.

The damage initiation and development is presented in chapter III, where woven fabric composites designated for high temperature application were investigated under severe thermal conditions to study their thermal stability and their resistance to thermal damage. The mechanical performance of the same composites was studied.

The effect of aging was also investigated. 3D models were realized with Finite elements in order to explain the edge effect on the evolution of the cracks observed during the tensile tests. In addition, the differences and similarities in cracking in v different layers were analysed using probabilistic approaches (a simple one as well as Mo te Ca lo si ulatio s ith Hashi 's a d also shea lag odel a d f a tu e mechanics arguments.

vi

Preface

The work presented in this thesis has been carried out within the Division of Materials Science in Luleå University of Technology in Sweden and in the Division of Mechanics of Materials (SI2M) in Jean Lamour Institute in University of Lorraine France during the period from October 2011 to December 2015. This thesis could not have seen the light of day without the precious support of many persons.

There is no way I can ever thank my both supervisors Professor Janis Varna and Professor Zoubir Ayadi for their unconditional and unswerving help, for their precious time, for their valuable advices, for their patience and for their willingness to listen and their whole-hearted readiness to help when I most need it. Words just fail me whenever I try to express my gratitude for them, thank you for making my research experience ever so unique.

I gratefully acknowledge my co-author Dr. Erik Marklund for his assistance in my research and for shearing his knowledge and for his contributions.

I would like to express also my sincerest gratitude to my co-author Docent Patrik Fernberg for his support and guidance and for his fruitful suggestions and remarks.

It is a pleasure to thank the Joint European Doctoral Program in Material Science and Engineering (DocMase) for financing part of this project. My acknowledgments go also to the local government Norbotten in Sweden for financing part of this research work and also Swerea SICOMP for their support. I would like to offer my special thanks to Professor Zouhir Fakhfakh from my home country (Tunisia) for his encouragement and his personal support. Special thanks go iii also to Professor Roberts Joffe and Professor Lennart Wallström for their kind help.

Many thanks go to my colleagues and my friends who were important to the successful realization of my thesis.

Finally, I owe my deepest gratitude to my relatives, my parents, my brother and my husband for their love, prayer, support, care and encouraging words that light my way and kindle my enthusiasm to bring out the best in me in all my academic endeavors.

Luleå, May 2016

Chapter I

State of the art 1. Introduction: Fibre reinforced composites

The quest of weight gain using materials with superior specific properties, engages the interest of many researches to explore composite materials. Each year, composites find their way into hundreds of new applications from golf clubs and tennis rackets to jet skis, aircraft, missiles and spacecraft.

Composite materials are widely used in different areas such as aeronautics (helicopter blades, pressure bulkhead, cargo door, etc), maritime transport (boats, etc), automotive industry (car roof, car carline, etc), electronic (insulation, mounting printed circuit, boards, etc), buildings (furniture, roofing, etc), industry (tanks, pipes, wind turbine blades, etc) and even in sports and entertainment (skis, fishing rods, helmets,

et …

A composite material consists of an assembly of different immiscible materials which complement each other and bring about a material whose physical properties are better than those of the individual constituents working separately.

Polymer composites are formed with reinforcement in the form of particles or fibres embedded in a matrix. The matrix is often a thermoplastic or a thermoset polymer.

The matrix preserves the geometric arrangement of fibres, protects them from the environmental attack and damage, to which the sample can be exposed. The fibres can have inorganic or organic nature such as carbon fibres, glass fibres, oxide fibres, silicon carbide fibres, etc. They can be continuous or discontinuous filaments. The high stiffness and strength of polymer composite materials stems from the high stiffness and strength of fibres. Their geometry allows them to have minimal defects and their composition gives them a high strength. The fibres work as reinforcements since the load is transferred to them from the matrix. The volume fraction of fibres for a structural composite is typically 45%-65%. The maximum theoretical value is 79% for square array and 91% for a hexagonal array [START_REF] Varna | Mechanics of fiber composite materials-micromechanics, laminate theory and failure analysis[END_REF].

The fibres can be arranged in different ways: homogenously or uniformly dispersed like in pre-impregnated tape laminates or in form of bundles like in the woven reinforced composites or the non-crimp fabric composites. The main advantage of this material is the high fibre volume fraction that can be obtained with well aligned fibres, showing an excellent in-plane stiffness and strength.

The problem with this material is that it is expensive. The high costs combine the high labour costs and the high storing costs (pre-impregnated tapes require low temperature to prevent curing). Another drawback is the sensitivity to inter-layer delamination cracking under impact loading due to their poor interlaminar fracture This composite has two-directional reinforcement and its manufacturing cost is lower than that of the pre-impregnated tape based composite. The woven structure shows high waviness in the out-of-plane direction. This waviness brings advantages to the material by improving its fracture toughness and its mechanical properties in the outof-plane direction. But, it brings also significant drawbacks to woven composite by reducing the in-plane properties. Some drawbacks of the pre-impregnated tape based composites and of the woven NCFs are relatively new class of textiles in which a fabric is constructed of layers of fibre bundles aligned in specific directions [START_REF] Hogg | Non-crimp thermoplastic composite fabrics: aerospace solutions to automotive problems[END_REF]. The layers of fibres are produced by laying tows next to each other in a specified direction and subsequently employing a secondary fine yarn knitted around the tows to hold the fabric in place. The use of through-thickness stitching allows for improvement in damage tolerance and in the interlaminar fracture toughness. For textile composites (woven and NCF), the manufacturing technique provides a dry preform (fabric) which can be used in complicated shapes before it is consolidated into the final composite by resin transfer moulding with low manufacturing costs comparing to the pre-impregnated tape based composite. NCF composites, ideally, would combine a good in-plane response, like preimpregnated tape based composites and the good through-thickness stiffness and strength due to the stitching in the thickness direction [START_REF] Mounitz | A review of the effect of stitching on the inplane mechanical properties of fiber-reinforced polymer composites[END_REF].

With bundle mesostructure, the woven and NCF composites show heterogeneities in the micro-and the meso -scale as shown in Figure I-4 [START_REF] Mattsson | Methodology of characterization of internal structure parameters governing performance in NCF composites[END_REF]. The micro-scale heterogeneity can be seen from the microstructure of the fibre-matrix inside the bundles, the meso-scale heterogeneity is due to the structure of layers where the fibre bundles are separated by the matrix.

Mechanical properties in composites with bundle mesostructure

In order to study the mechanical properties dependence on the architecture parameters, finite element analysis and theoretical analysis methods are more convenient than the experimental techniques because of the complex geometry and the numerous parameter controlling the mechanical behaviour of the textile composites [9-10].

Finite element method (FE) started with Ritz who developed an effective method for an approximate solution of problems in the mechanics of deformable solids [START_REF] Barkanov | Introduction to the finite element method[END_REF][START_REF] Ritz | über eine neue metode zur lösung gewisser variations problem der matematischen physik[END_REF]. FE is a numerical method for finding approximate solutions to boundary value problems for differential equations. It uses variational methods to minimize an error function and produces a stable solution. The problem is transformed to an equivalent one in terms of properties and geometry and the method is based on the discretization principle which is translated by the choice of a mesh that occurs by dividing the studied area into sub-domains with simple geometry (triangle, quadrilateral, etc) and with finite dimensions, hence the fi ite ele e t ethod's a e. Results a e determined for each element. The accuracy of the results is directly related to the mesh quality realized (number of element, their distribution in the structure, form of element, etc). In FE analysis many errors can be produced, such as wrong interpretations of the physical model, mechanical modelling errors due to many assumptions like geometric simplifications and resolution errors which are due to problems of numerical accuracy. The engineer must be aware of the existence of these errors and must be able to estimate a level of confidence in the results.

Many researches have been conducted for predicting the mechanical properties of textile composites. The basic principle to determine the elastic properties of a textile composite using FE is to divide the structure into unit cells and then mechanical properties are calculated for the unit cell.

Since the textile composites have a complex architecture it is not easy to incorporate all geometrical parameters and simplifications are needed in the FE modelling

The damage features in textile composites

In order to have confidence in the structural integrity of composite components,

designers

Matrix crack (Intralaminar cracking)

The stiffness and strength of fibre reinforced composites are higher in the longitudinal direction than in the transverse one. In addition, the stiffness of the reinforcing fibres is significantly higher than the matrix material. Thus, stress concentrations occur in the matrix when a ply is loaded in the transverse direction. The stresses at which the failure occurs in off-axis ply are lower than that of plies aligned in the loading direction.

In the off-axis plies, the cracks develop and run parallel to the fibres direction. These

Delamination (Interlaminar cracking)

The delamination is a longitudinal crack in the interface between two adjacent plies. Its

Fibres break

As its name says, it is the breaking of the fibres oriented parallel to the loading direction. In a unidirectional composite loaded in tension along the longitudinal direction, the fibres fail at their weak points and stress redistribution between fibres and matrix occurs, affecting other fibres and breaking more of them. An example of broken fibres is presented in proved that the ° la e 's da age does 't ha e a i po ta t effe t o the You g's modulus which decreases slightly, since the laminate modulus is mainly controlled by the 0° non-damaged laminate. However, it has a strong effect on the degradation of the Poisso 's atio e ause the t a s e se a ks i ease the strain in the load direction and reduce contraction in the transverse direction. The appearance of the new longitudinal cracks was explained by the stress concentration caused by the forced straightening of the 0° fibre bundles in tension.

More studies on damage accumulation in textile composites

The complexity in geometry of the damage and its quantification has made the progress in studying the effect of the damage accumulation on the mechanical properties in textile composites slower than in composites with uniform fibres distribution.

Gao et al. During composite manufacture, preforms are stacked in a mould and infiltrated by a thermoset resin to form the composite. Thus an NCF composite is created which is heterogeneous not only on microscale (fibres and resin) as for pre-impregnated based composites, but also on mesoscale due to the appearance of distinct fibre bundles and resin pockets. Since the bundle scale is much larger than the fibre scale, homogenization over the fibre/matrix scale is possible representing the bundle as transversally isotropic material.

Ideally, NCF composites would consist of perfectly aligned fibre bundles where the size of each bundle is determined by the stitching procedure. However, due to the complex manufacturing technique, NCF composites have both in-and out-of-plane waviness of the bundles which reduces the in-plane stiffness. In A different approach is using the assumption that the NCF composite stiffness problem a e edu ed to CLT p o le fo la i ate ith effe ti e elasti p ope ties of effe ti e flat layers. The effective stiffness is calculated considering an isolated curved beam (bundle, layer), replacing its interaction with the rest of the composite with proper boundary conditions. In [START_REF] Edgren | Approximate constitutive model for non-crimp fabric composites[END_REF] the effective modulus was calculated using a It has to be noted that the observed trends and obtained results in all the following may be of relevance not only for NCF composites with bundle waviness, but also for woven composites.

FE modelling procedure (Material, geometry and assumptions)

In the previous study, [23], a two blanket cross-pl NCF's ⁰-tow waviness was assumed sinusoidal and two possible geometrical configurations were analysed numerically: the in-phase ase, he the a es of the t o oute ⁰-layers are in phase, and the out-of-phase ase, he the a es of the oute ⁰-layers are out of phase. Those are the extreme cases and in a real composite the RVE is often much larger than the wave length of the individual bundle in one blanket because the blankets of the fabric are randomly shifted in horizontal directions.

In this study, we analyse the stiffness of a simple 0/90 unit of the NCF composite. The rest of the composite is roughly replaced with symmetry conditions. Two units are analysed, a u it ith a su fa e ⁰-layer, see Bundles in the NCF composite are considered as unidirectional (UD) composites with certain fibre volume fraction and their elastic properties may be calculated using hexagonal unit cell as it was explained in [START_REF] Marklund | Modeling stiffness and strength of non-crimp fabric composites semi-laminar analysis[END_REF] or simple rule of mixture based on isostrain assumption, Halpin-Tsai expressions etc.

Bundles or layers with out-of-plane waviness are referred to i follo i g as u ed u dles o u ed la e s . Fi st e assu e that the st u tu e of u ed u dles i a layer may be replaced by homogenized layer with average fibre volume content, which is kept 0.6. In the bundle case the fibre volume fraction was 0.7. The elastic constants of the layers and the constituents are given in Table II

Table II-1 Elastic constants of constituents and homogenized layers

The error introduced by replacing the layer with bundles by homogenized layer has been analysed before, for example, in In calculations both models had the same fibre content. Changing the amplitude A of the a i ess the a e age thi k ess of the ⁰-layer was not changed. The results

showed that the stiffness is just marginally affected by the mesostructure of the u dles a d the ⁰-layer homogenization is justified.

In the next step the CLT approach to NCF composite stiffness calculation is adapted, replacing the cu ed la e ith a st aight la e hi h has the effe ti e i -plane stiff ess of the u ed la e . Thus, the la i ate is ade of effe ti e la e s. The CLT approach for calculating the axial stiffness of the NCF composites is very attractive due to its simplicity in application. For a symmetric and balanced laminate the macroscopic in-plane stress-strain relationship is = + (II-1)

In (II-1) direction 1 is the axial (loading) direction. Focusing on the laminate axial stiffness element we will perform FE numerical analysis for plane strain case ( = ). In this loading case is obtained directly dividing the calculated axial average stress by the macroscopic strain applied. The average stress is axial force divided by nominal (average) thickness of the laminate ℎ. Hence

Effect of wavelength and amplitude on laminate stiffness

Results for CF/EP1 laminate with elastic properties in Table 1 are shown in

Predictions based on isolated curved layers with boundary conditions used in previous studies

The The effective in-plane axial stiffness of the curved layer is lower than the stiffness of a straight layer mainly because the fibres are not oriented in-plane. However, as shown in [23], the effective stiffness strongly depends also on the interaction with the adjacent material (tows of different orientations, resin).

Calculating according to (II-2) we use the total force which due to force balance is the same in any cross-section. For example, in Figure

II-3b = = ⁄ is equal to = = .
In other words, using (II-2) and FE it is not important in which cross-section the reaction force is obtained before dividing it with the average composite thickness ℎ .

If instead the CLT based (II-4) is used, the effective layer stiffness has to be found first.

Similarly as was done for the whole laminate stiffness case, we would for this purpose divide the calculated force acting on some arbitrary cross-section of the layer by its cross-section area and then by the applied strain. However, the force on a crosssection of a layer depends on which cross-section we consider. For example

° ≠ ° ° ≠ ° ° + ° = (II-6)
The numbers ° and ° in parenthesis indicate the layer under consideration.

Because of Equation ( 6) effective layer stiffness calculated on the left and the right edge of the model differ

° ≠ ° ° ≠ ° (II-7)
where

° = ° , ° = ° , ° = 9 ° 9 , ° = 9 ° 9 (II-8)
Due to interaction, layers are not subjected to uniaxial loading and strictly speaking the calculated numbers are not stiffness matrix elements. This explains the two different values. The situation will be similar analysing isolated curved layers with traction boundary conditions. Therefore, it has to be clarified which layer stiffness to be used in the CLT (II-4).

For this purpose we may formally write that the force is distributed between 0⁰-layer and 90⁰-layer according to: Averaging gives:

= + = 9 ° + 9 ° + ° + ° (II-12)
Substituting (II-12) in (II-2) and using (II-8): decrease due to the waviness described by A/t 0 for several values of L/t 0 .The difference between ° and ° calculated on both edges is rather small. In contrast the transverse effe ti e stiff ess pa a ete s of the ⁰-layer, and have very different trends: one (calculated at = ) is increasing, the other one is decreasing.

= 9 9 + 9 + 9 9 - 9 9 ℎ +
The combined effective stiffness (II-4), i hi h the ⁰-la e effe ti e stiff ess is assu ed e ual to the ⁰-layer material stiffness may be therefore be motivated:

9 - = 9 (II-16)
The accuracy of the simplification in (II-16) will be verified in following calculations.

Approximation of tractions at the 0/90-layer interface

In line with the objectives of this study, the task is to find the effective stiffness of the u ed ⁰-layer analysing an isolated curved layer subjected to relevant boundary conditions. These include not only end loads applied to the layer, but also surface Knowledge regarding the distributed load is conveniently obtained by analysing st esses at the ⁰-a d the ⁰-layer interface using FE. The results and the observed trends used to define shape functions for the interface stresses from fitting a limited number of FE-calculations are presented in this section. The methodology suggested for finding coefficients in these shape functions is described in more detail in Appendix. These fu tio s a the e used fo u ed ⁰-layer stiffness analysis with an arbitrary amplitude or wavelength.

Local stress distribution at the 0/90-layers interface has been analysed: The normal stress and the shear stress were determined in each node along a path on the interface. In the FE-analysis those stress components are slightly different on both sides of the interface. The value along the path on the interface which is given by the code is the average of the stresses calculated for the closest element to the interface Thus, performing FE-calculations for a limited number of L/t 0 and A/t 0 cases we can find approximate interface stress expressions for a given material to be used for any practical combination of these geometrical parameters.

The main objective of this part was to find simple expressions for further analytical application. The expressions for and dependence on L/t 0 and A/t 0 are given in Appendix.

Composite stiffness based on effective stiffness of an isolated 0⁰-layer with surface loads

In this subsection we use the calculated and approximated normal and shear stress Ve s all diffe e es i esults a e see he the ⁰-layer effective stiffness is represented by its material transverse stiffness, proving that this approximation may be used.

Effective stiffness determination using master curve approach

In this section, it will be demonstrated that the axial stiffness of a biaxial cross-ply NCF composite described by knock-down factor is similar to the knock-down factor for the

(II-30)

In fact none of these numbers is stiffness, because the loading to the layer is not uniaxial. For the purpose of employing the CLT, the effective layer stiffness These definitions together with (II-23) will be used in all calculations in this section.

NCF stiffness knock-down factor

In order to investigate which term ( or ∆ ) is affecting more the NCF knock down factor in (II-25), calculations are performed for two different ratios of wavelength L and the layer thickness t 0 : L/t 0 =10 and L/t 0 =15 and for both CF/EP and GF/EP In the following, the focus will be mostly on the determination of knock down factor of the ⁰-layer. An important observation deduced from Figure II-17 is that the curves of k 0 corresponding to L/t 0 =10 and L/t 0 =15 look similar and have similar shape. This leads to assu ptio that oth u es a oi ide pe fo i g p ope defo atio of both axis (changing scales). As a consequence the curve corresponding to L/t 0 =15 could be obtained as a part of the curve corresponding to L/t 0 =10. In the following discussion it will be shown that the knock-down factor for any L/t 0 ratio as a function of A/t 0 based on known relationship for one given L/t 0 ratio can be considered as a master curve. It is suggested to determine the master curve by performing accurate FE calculations for the case of short wavelength L/t 0 and to fit it with an analytical function.

Master curve approach

Knock-down factor curve and fitting function

The case of L/t 0 =6 is considered as extreme case where the same amplitude can lead to a steep a gle of the a i ess. The ⁰-layer knock-down factor is decreasing more than in cases with larger ratios. The knock-down factor dependence on the wave amplitude at L/t 0 =6 will be considered as master curve for the rest of calculations. The analytical fitting of the case of L/t 0 =6 will be used to calculate the knock-down factors for the other cases of the higher ratios of L/t 0 . The calculated values and the fitting curves are presented in

Validation and application of the master curve approach

In this section the aim is to demonstrate and identify the relationship between knockdown factor curves for different combinations of L/t 0 and A/t 0 . It will be explained how the aste u e is used to p edi t the ⁰-layer knock down factor degradation for any case of L/t 0 .

The procedure, as illustrated in

Development of the master curve approach

In the previous section the master curve approach was developed to calculate knock- ). This may be an indication that the longitudinal modulus of the corresponding UD material is the main parameter defining the c-parameter. It should be reminded that all results in this study are obtained assuming sinusoidal shape of the waviness.

Engineering approach for the amplitude and wavelength's of an equivalent cell

We In the above form the amplitude of the equivalent cell cannot be determined because depends on the length of the equivalent cell (equations (38), (39)) which is not known yet. However, (47) can be simplified: it was demonstrated that k NCF ≈ k , which means that one can assume k 9 ≈ . Similar derivation as show above leads to

⁄ = √ ∑ ⁄ = (Method 2) (II-48)
The same expression is obtained when → , which is the case when the contribution of the 90°-layer can be ignored.

Method 2 includes b i and also which are LTi dependent and therefore for the equivalent cell can not be determined at the current stage.

The wavelength of the equivalent unit is determined through the average waviness angle. This value of / is used in (II-50) to find the wave length / and then is found using (II-39). After that (II-48) can be used to refine the value of / etc. Obviously, Method 3 is the first iteration in a more complex routine for Method 2.

Stiffness of chain of cells

Axial stiffness of the meso-scale composite consisting of a chain of n=5 unit cells, see Here stiffness of each cell was calculated using the master curve: applying (II-24)

where the knock down factor was found with (II-38), (II-39).The excellent agreement with FEM for all analysed cases confirms the applicability of (II-52).

Next the concept of the equivalent unit cell representing the composite chain was investigated. The stiffness of the defined equivalent cell was calculated a) using FEM; b) using the master curve approach. The results of a) and b) calculations are almost coinciding. The procedure in the master curve approach is as before: the constant is calculated for the equivalent wavelength using expression (II-39), and then the master curve approach are also presented for all four cases.

Interaction effects of curved bundles in cells connected "in parallel"

The idealized meso-geometry of the NCF composite containing two symmetric layers (chains of units) is not typical in practice. It was shown previously that surface tractions due to interaction with the surrounding material strongly affect the knock-down factor of the 0°-layer. This means that in a multi-layer NCF composite with many embedded and surface unit cells the waviness parameters of 0°-layer in one cell can significantly affect the knock-down factor of the 0°-layer in a cell located on the top or below this cell.

The interaction effect on knock-down factor was investigated applying FEM on the models shown in The effect of the thickness of the ⁰-layer on the knock-down factor reduction of the ⁰-layer in Figure II-30 is not significant, the reduction being larger with increasing 9 ⁄ . The largest change is between t 90 /t 0 =1 and t 90 /t 0 =2. For the thickness ratios higher than 2 the knock down factor is almost the same as for the case 9 ⁄ = . and ⁄ = .

Table II-9.K o k do fa to of ⁰-layer in the embedded unit with ⁄ = .

and ⁄ = .

Increasing the wave amplitude of the neighbouring ⁰-layer, the knock-down factor of the ⁰-layer in concern increases in the out of phase case and decreases in the in phase ase. The thi k ess of the ⁰-layer in between two 0°-layers is the same and constant (t90⁰=2t0⁰) in average. The symmetric shape of the out of phase model compensates the e di g of the u it a d akes the ⁰-layer stiffer (the result is similar as using symmetry condition at the unit interface). The change (%) with respect to the knockdown factor value when the neighbouring layer is straight also shown in Tables II-8, II-9 is very small for GF/EP composite (about 2%) but larger for CF/EP (10% for surface u its a d % fo e edded u its . We o lude that the effe t of ⁰-layer waviness on the knock-down factor of the neighbouring one is rather small and, therefore, it could be accounted for using simple approximations.

To ide tif this si ple app o i atio , e o pa e the k o k do fa to of the ⁰layer in the meso-cell to the factor in the unit cell with t90⁰=2t0⁰ (as demonstrated above cells with t90⁰>2t0⁰ render the same result). For all the cases of out of phase meso-cells, the approximation of replacing the neighbouring a ed ⁰-la e a ⁰layer with t90⁰=2t0⁰ would lead to under-estimation of the knock-down factor by 2-10%. For the in phase meso-cell the same operation would lead to over-estimation of the k o k do of the ⁰-layer in concern by about 0.3-7%. In a real NCF composite, the unit cells are randomly shifted and, therefore, the in-phase and out-of-phase meso-cells represent extreme cases. Since the suggested approximation underestimates the factor for in-phase units and overestimates it for out-of-phase units, it may lead to satisfactory stiffness reduction predictions in NCF composites containing all these meso-cells. Certainly, more detailed and more accurate approximations are possible, for example applying different approximations for in-phase and out-of-phase meso-cells.

Finally, the approximation for knock-down factors suggested above was used for symmetric NCF composite consisting of 5 meso-cells with different amplitudes and for the 0°-layer interaction in the approximate way described above (using 9 = in the unit cell). In this routine the value of " " in (II-39) is taken from Table II-4 for case 9 = and the calculated " " is substituted in (II-38) to calculate the knock-down factor. After that the stiffness of the unit cell is calculated using (II-24) with the real thickness of the 90°-layer in the cell, 9 = . Then the stiffness of the chain of unit cells (surface chain is treated separately from the embedded chain) is calculated using The equivalent meso-cell with two unit cells in parallel was analysed using a) FEM; b)

analytically. The equivalent meso-cell is different for the in-and out-of-phase cases in The analytical solution for the equivalent meso-cell is based on the master curve e p essio s: a the k o k do fa to of ⁰-layer in each of the two equivalent cells is calculated using expressions (II-38),( II-39) (using " " in (39) from Table II-

Conclusions

The effe t of the ⁰-bundle waviness on stiffness of cross-ply NCF composites was modelled. Multiscale approach was used by first calculating the homogenized bundle material properties from its constituents by using an assumed hexagonal fibre packing.

The u dle st u tu e of the u ed ⁰-la e a d the u dle st u tu e of the ⁰-layer with varying thickness are then replaced with homogenized materials. Finally, the u ed ⁰-la e a d the ⁰-layer with varying thickness were replaced by flat layers with effective stiffness and classical laminate theory was used to calculate the macroscopic stiffness. The presented study focused on the axial macroscopic stiffness.

The macroscopic axial stiffness was expressed through the effective stiffness of the u ed ⁰-la e a d the effe ti e stiff ess of the ⁰-layer with varying thickness. It was shown that the effective stiffness of a layer in the composite can be calculated averaging the apparent stiffness, which are on the right and the left edges of the layer.

These two forces to layer are different due to varying layer thickness and due to the interaction between layers resulting in normal and shear stresses at the 0°/90°-layer interface.

Usi g this app oa h the effe ti e ⁰-layer stiffness was calculated analysing isolated u ed ⁰-layer subjected not only to end loading, but also to surface loading. To identify the surface loads to be applied, a detailed FE-analysis of the interface stresses was performed. It is concluded that these stresses can be approximated by a sinus shaped function with amplitude dependent on the wave length and amplitude of the curve normalized with respect to the layer nominal thickness. Fitting expressions for the stress amplitude dependence on these layer waviness parameters were obtained, and are given in Appendix.

The des i ed si us shaped su fa e loads e e applied to isolated u ed ⁰-layer FEmodel together with end loading showing that the calculated effective stiffness of the layer is in good agreement with the effective stiffness of the ⁰-layer extracted from the cross-ply composite analysis. This proves that further efforts are meaningful to develop analytical approximate models for curved beams with sinusoidal surface tractions.

Finally, the calculated effective 0°-layer stiffness was successfully used to calculate the macroscopic stiffness of three different composites proving validity of the used multiscale approach. It was also shown that without losing accuracy the effective stiff ess of the ⁰-layer (which generally speaking should be calculated numerically o side i g ⁰-layer with varying thickness and applied surface and end loads) can be epla ed the t a s e se stiff ess of the ho oge ized ⁰-layer material.

In another approach, numerical parametric FE-analysis showed that the axial stiffness reduction (knock-down factor) of a cross-ply type NCF composite made of carbon fibre (CF) or glass fibre (GF) bundles with out-of-plane waviness of a surface 0-⁰la e o embedded 0-⁰la e is al ost the sa e as the effe ti e stiffness reduction of the u ed ⁰-layer. This conclusion holds for a large variety of wavelength and wave amplitudes of imperfections.

Analysing the al ulated ⁰-layer knock-down factor versus the wave amplitude curves for NCF composites with different wavelength we observed that these curves for different wavelength can be reduced to one master curve by changing the scale of the wave amplitude axis.

It was found analysing results for CF and GF cases that the numerical data building the master curve can be fitted by a very simple one-parameter function. Thus, for a given material only a few FE calculations for a selected wavelength and amplitude are needed to construct the master curve. More calculations would serve the validation purpose.

To construct the ⁰-layer knock-down factor versus wave amplitude curve for a different wavelength only one FE calculation is required for arbitrary selected amplitude. Predictions performed for CF and GF NCF composites show a very good agreement with direct FE calculations.

Axial stiffness of a symmetric NCF composite, consisting of many 0°/90° unit cells with different amplitudes and wavelength of the 0°-layer, is calculated using FEM and also using analytical models with knock-down factors for curved 0°-layers.

Analytical expression for knock-down factor of the 0°-layer in the unit cell is presented.

It is simple but very accurate fitting expression based on extensive FEM parametric analysis and reflects the effect of the local amplitude and wavelength of the 0°-layer on the effective stiffness of the layer in the unit cell. There is only one parameter in these expressions which depends on the used material (values of this parameter are presented for typical GF/EP and CF/EP materials), which means that for a new material one FEM calculation for simple unit cell with fixed amplitude and wavelength is enough to obtain knock-down factors for any combination of amplitude and wavelength. It was found that the value of the knock down factor is affected by the presence and waviness of the 0°-layer in the neighbouring unit cell. In the current routine this effect is accounted for in a very simple way: assuming that the effect is the same as increasing the thickness of the 90°-layer. In the next step of simulations rule of mixtures and constant force assumptions are used when appropriate. Analytical modelling gives results in a good agreement with FEM.

An alternative approach, attractive for experimental characterization, is to use the information regarding the variation of local waviness to create an equivalent mesocell for the NCF composite. Three different methods to identify parameters of the equivalent meso-cell are discussed and good agreement with FEM is demonstrated.

Finally, Axial stiffness of a symmetric NCF composite, consisting of many 0°/90° unit cells with different amplitudes and wavelength of the 0°-layer, is calculated using FEM and also using analytical models with knock-down factors for curved 0°-layers (Master curve expression) which is simple but very accurate fitting expression based on extensive FEM parametric analysis and reflects the effect of the local amplitude and wavelength of the 0°-layer on the effective stiffness of the layer in the unit cell. There is only one parameter in these expressions which depends on the used material (values of this parameter are presented for typical GF/EP and CF/EP materials), which means that for a new material one FEM calculation for simple unit cell with fixed amplitude and wavelength is enough to obtain knock-down factors for any combination of amplitude and wavelength. It was found that the value of the knock down factor is affected by the presence and waviness of the 0°-layer in the neighbouring unit cell. In the current routine this effect is accounted for in a very simple way: assuming that the effect is the same as increasing the thickness of the 90°-layer. In the next step of simulations rule of mixtures and constant force assumptions are used when appropriate. Analytical modelling gives results in a good agreement with FEM.

An alternative approach, attractive for experimental characterization, is to use the information regarding the variation of local waviness to create an equivalent mesocell for the NCF composite. Three different methods to identify parameters of the equivalent meso-cell are discussed and good agreement with FEM is demonstrated.

Chapter III

Damage in Carbon fibre/polyimide laminates

Introduction

An increasing interest of aerospace industry in composite materials for applications in harsh environments has been driving improvement and development of new composites. In addition to good mechanical performance some components require materials suitable for the environment in which it would be functioning. For example, the composites designated for aero-engine applications considered in this chapter need to perform good at high temperatures. Thermal cycling (fatigue) and long

Objectives

The objective of this chapter is to study the thermal fatigue resistance and the thermal stability of quasi-isotropic CF/polyimide woven composite by analysing the Specimens were cut from 2 different plates labelled 432, 433. These plates were manufactured at Swerea SICOMP, Sweden, using resin transfer moulding (RTM) in stainless steel tool using a flow and pressure controlled injection piston. The material was initially cured at 340°C for 30min followed by 2.5h post cure at 370°C using 12 bar pressure. In the 433 plate, the MHT-R resin was mixed with 4% of an additive A57, which is a reactive diluent acting to lower the viscosity during processing. Both plates were cooled down during 12 hours and demoulded at 70°C then cooled down to 0°C for 2 hours.

The volume fraction of fibres is around 60%; the Tg is around 370°C; the uncured resin melts between 200°C and 250°C and the initiation of cross-linking happens above 320°C. Some mechanical properties of the T650/MHT-R quasi-isotropic laminate [START_REF] Tsa | Me ha i al pe fo a e of o el high Tg polyimide matrix carbon fibre-ei fo[END_REF] are given bellow:

E Tensile ≈ GPa / E Compression ≈ GPa.
Te sile st e gth ≈ MPa / Co p essi e st e gth ≈ MPa.

"t ai to failu e ≈ . % i te sio . % i o p essio .

Fi e failu e st ai ≈ 1.7%.

The matrix material is extremely expensive and the amount of material for testing was very limited. The composite exists only as a quasi-isotropic woven laminate and therefore properties of unidirectional composite that would be useful for modelling are not available.

The 8-harness satin meso-architecture allows approximating in theoretical analysis the o posite as a la i ate ith la e s of e tai u dle o ie tatio . The thi k ess of the laminate is 2.9mm which means that average thickness of one la e u dle is approximately 0.18 mm. The bundle width is about 1.5mm.

Experimental procedures

Two groups of specimens were cut from each plate (see Table III- The spe i e s of the se o d pa t of the plate ith lette A i the label were subjected to isothermal loading at 288°C for up to 960 hours (40 days) in air environment and cooled down to RT for inspection. Then, tensile tests with displacement rate 2mm/min were performed at room temperature (RT). Strains were measured using extensometer with gauge length 50mm. After reaching pre-described strain the specimen was removed from the machine and inspected using optical microscope.

Table III-1. "pe i e s' la els a d loadi g o ditio s

Fine polishing was performed after cutting using the following sequence of sand papers (P240, P600, P1200, P2500, P4000) followed by polishing using liquid diamond slurry (from 9 micron to 0.25 micron). After certain number of cycles (number in parenthesis in Table III where is the fiber orientation angle in the layer. stresses are obtained when the plates were gently cooled down to 0°C directly after manufacturing. We assume that the temperature difference causing the damage state after manufacturing is approximately 0-340= -340°C. The stress state in the bulk part of the composite plate can be calculated using the Classical Laminate Theory (CLT).

According to CLT, in a quasi-isotropic laminate the thermal transverse stresses in all layers are equal and the in-plane shear stress is zero. Since these two stress components are responsible for intralaminar cracking, we can say that conditions for initiation of intralaminar cracks in all layers are the same. Due to statistical distribution of failure properties in a layer the number of initiated cracks is limited but statistically it should be the same in all layers.

Cutting specimens from the plate was performed at room temperature (RT) which is about 22°C higher than the lowest temperature during the manufacturing cycle. This means that at RT the thermal stresses in the plate were lower by about 6-7 %. New cracks were not introduced during the cutting, especially because after cutting, about To explain these observations, 3-D Finite Element (FE) thermal stress analysis was performed to study edge effects in a specimen with finite width. The upper part of the laminate with [(+45/-45)/(90/0)] 2s configuration was modelled assuming that the bundle meso-structure can be homogenized in this evaluation which is more of a qualitative nature than a quantitative characterization of stresses. The assumed thermo-elastic constants of the UD composite are: E 1 =141.025GPa, E 2 =8.821GPa,

ν 12 = . , ν 23 =0.442, G 12 =3.62GPa, G 23 = . GPa, α L = , α T =25e -6 °C-1 .
The total number of SOLID186-type of brick elements was 33600. The simulated specimen was long in x-direction (for all edge effects from x=0, x=L to decay in the middle of the specimen) whereas the specimen width was 4mm and thickness 3mm.

These dimensions were selected to have even more interacting edge stresses than in the experimental cases where the specimen width was 7 mm for thermal cycling and 12 mm for aging.

The in-plane stress distributions normalized with the CLT transverse stress value are presented in Figure III-3. Even for this small width specimen, the transverse stress in the middle = ⁄ is close to the CLT value, which means that the same plateau region will be also in the experimental specimens which are wider. It also means that the magnitude of edge effects is not disturbed by overlapping the stress perturbations from both edges. The shear stress in layers, in Figure III-3b, is not negligible but still significantly lower than the transverse stress. In Figure III-3a we observe clear differences in the edge region between transverse stresses in ± -layers and in 90layers. In 90-layers the transverse stress is magnified by approximately 20% whereas in all 45-layers it is about 30% of the CLT stress. This means that after cutting the thermal stress state in the specimen is different than in the initial plate. For 90-layers the transverse stress is about 20% higher which is much larger change than the thermal stress reduction by 6-7% when the plate was brought from 0°C to RT. This means that additional cracks were initiated in the edge regions of 90-layers. Most probably they did not grow inside the specimen because the stress conditions there did not change after cutting. This hypothesis was not possible to check experimentally because the matrix limited amount of material available for testing. On the other hand, the transverse stress in the edge region of 45-layers is much lower than in the middle part which means that new cracks were not introduced in these layers by making specimens. The above analysis explains the difference between crack densities in 90and 45-layers shown in Figure III-2. However, it does not explain the much higher crack density in the surface 45-layer which will be discussed later. Most of the 90-layer cracks in plate 432 originate from interbundle voids (see Figure

III-1a).

The average void content was measured over the area of about 100mm 2 . From one specimen to another, the average void content varied from 0.3% to 2.4% in 432.

The measured void content is not too high; however, the voids were not uniformly distributed in the material. When measured in a smaller window of 1.15mm 2 (0.93mmx1.24mm), the void content varied from 0% to 13% in the plate 432. The void content in the plate 433, where diluent was used to reduce the viscosity, was almost zero.

General for both plates, it is very high variability between specimens. Therefore, at this stage solid conclusions regarding the relationship between performance and manufacturing routines cannot be drawn and all observed trends should be treated with caution. The void content in the middle of the 432-plate, see In fact, the role of voids is dual: they act as stress concentrators and first cracks appear at lower thermal load, but also they lead to larger stress relaxation after the crack is created, that may delay the appearance of new cracks and arrest the growth of the existing cracks. This phenomenon was described in To explain the higher crack density in the surface layers we have to distinguish two steps in crack evolution: initiation and propagation along fibres. The growth of a crack may be described by Linear Elastic fracture mechanics (LEFM) which states that the crack will propagate when the energy release rate (ERR) which is quadratic function of the stress reaches or exceeds the critical value (material fracture toughness which in composites may be different for different growth directions and depends on the fracture mode mixity).

Intralaminar cracks initiate in stress concentration regions (voids, edges) and in weakest positions where the local fibre concentration is high and/or the fibre/matrix i te fa e o the at i is i pe fe t fo e a ple, defe ts due to i p eg atio problems in high fibre content regions) characterized by certain defect state. The crack initiation process is stochastic and characterized by coalescence of fibre/matrix debonds leading to growth in the ply thickness direction. Most probably the thickness direction growth during initiation is the ply-scale thermal stress state governed, which is according to CLT is the same for all layers. However, at given thermal stress the ERR for defect growth in ply thickness direction is much higher for defects of the same size at the surface than in internal layers. This fact is well known for infinite or semiinfinite plates with cracks data in different layers for plate 432 and 433 in R2 ramp. ∆ is larger for R2 ramp than for R1 ramp (larger range of stress change between max and min temperature). Even if there is not enough data for quantitative parameter determination it is obvious that for plate 432 parameter is larger: larger difference between crack density development in R2 and in R1.

According to (III-10) in each ramp the average crack length which is proportional to the observed crack density grows linearly with the number of cycles. This trend can be seen in Figure III-5 for cracks in internal layers but the relationship seems to be different for cracking in the surface +45-layer. The crack density is also much higher in surface layers. The analysis given in this section is based on LEFM. In fact, it is not certain that in ramps with high upper temperature parameters and remain constant. More irreversible effects including aging can be expected at +288C which may lead to accelerated crack propagation when the lowest temperature is reached in the next cycle. Temperature gradients during the cooling down part of the cycle can lead to additional damage in surface layers.

Damage state at RT in aged composites

Specimens were exposed to 288°C in ambient air environment for certain amount of During the aging test the surface 45-layers are subjected to contact with the high temperature environment not only in the edge region but also over the whole surface.

Therefore more uniform material degradation can be expected in the layer leading to overall decrease in resistance to crack initiation and propagation. The stresses at the edge of surface layers after aging are as low as the stresses on the edge of the internal 45-layers but the available energy conditions are more favourable for a crack to reach the edge. Since the crack density on the edge of the surface layer is result of propagation from the inside of the layer, the crack density is lower than at the edge of 90-layers where the stress is magnified and were the initiation takes place.

The phenomenon of the aged zone propagating from the specimen surface explains also the peculiar data in Figure III-9 with the crack density in internal 45-layers apart from edges being in average several times higher than in 90-layers (the propagation conditions are similar for both types of layers). More detailed analysis reveals that the crack density in internal 45-layer is a function of the distance of this layer from the laminate surface. The crack density in the layer next to surface layer is much higher than in 45-layers close to the symmetry plane of the laminate where the crack density is very similar as in 90-layers. This means that not only the surface layer but also the next layer has aged, which is expected because the thickness of one layer (0.18mm) is much smaller that the size of the observed aged zone at the edge (about 1 mm), even if the oxygen diffusion rate could be higher in fibre direction.

The long exposure to high temperature in a direct contact with oxidative atmosphere where the thermal part remains unchanged whereas the mechanical part grows linearly with the applied strain. The possible problems with edge observations and also the possible effects of edge stresses on damage initiation were estimated analysing stress distributions and also comparing the damage states experimentally.

The simulated specimen was long in x-direction to have a region with x-independent stress state. The specimen width was varied between 4mm and 30 mm whereas the laminate thickness was 3mm. These dimensions were selected to have isolated effects from both edges as well as interactive ones.

According to CLT transverse thermal stress, ℎ for the assumed properties and ∆ = -= is 53 MPa. The CLT ℎ is the same for all layers in a quasi-isotropic  In the -45-layers there is a magnification about 10%.

The in-plane shear stress in all 45-layers is magnified in the edge region by 20 to 40%. Superposing the mechanical and thermal edge effects we can conclude that there is transverse stress magnification in 90-layer but not in 45-layers. The shear stress is magnified at the edge of 45-la e s hi h a o pe sate fo the lo e transverse stress in this region leading to rather uniform conditions for damage initiation in these layers. In terms of crack density there is a risk that in 90-layers more cracks are in the edge zone because of higher transverse stress making edge observations unreliable. In 45-layers the crack density on the edge may be more representative. However, edge delaminations linking cracks are increasing the significance of each crack in stress reduction and lowering the probability of new cracks initiating between two existing cracks, thus, working against the edge effect in stress distribution in 90-layers.

These questions were resolved experimentally comparing damage statistics on the edge with the damage state quantified (after polishing) at several distances from the edge. In this inspection one NA specimen after loading to 1% strain and one aged (A) specimen after loading to 0.7% strain were used. The delamination length normalized with the interface length is shown in Table III-2 for each interface in a NA specimen after loading to 1% strain. At the edge the normalized delaminations are larger (more than 20%) at interfaces closer to the laminate surface. However, already at = . from the edge the delaminations are much smaller and they practically disappear at = . . The uniform crack density in Figure III-14 means that most of the initiated cracks had enough of ERR to propagate along the fibre direction. The hypothesis is that the 45cracks initiated in the bulk part could reach the edge and the 90-cracks initiated at the edge could propagate through the bulk of the layer. However, at the edge of the 90layer we have stress concentrations but also edge delaminations relaxing stresses.

Table III-2 Normalized delamination length

Based on the data presented in this section we cannot resolve whether they are selfcompensating or not and whether the edge is or is not the initiation region for most of the cracks in the 90-layer. Further information on this subject will be given later

where Weibull analysis will be applied to 90-cracks and the results used to predict multiple cracking in 45-layers.

Nevertheless, the presented results give sufficient confidence in ability of edge observations to represent the damage state and in the analysis presented in next section.

Damage accumulation analysis

Measurements show that the growth of crack density is different in different layers.

We will first analyse the damage evolution in 90-and +45-layers comparing both composites before and after aging. Then we will analyse the behaviour of -45-layers which is rather different than the behaviour of +45 layers. But, we will start by presenting the theories that will be used in the analysis.

Theoretical background

Analysing multiple intralaminar cracking in laminates we have to distinguish two steps in evolution of each individual crack: initiation (growth mostly in ply thickness direction) and propagation along fibres in the layer. Some of the equations mentioned in section 4.1 are reused in this section in addition to others to explain the damage accumulation. The concept will be re-explained from the beginning to make a complete story. Utilizing Linear Elastic Fracture Mechanics (LEFM) concept, the crack will grow when the energy release rate (ERR) which is a quadratic function of stress, reaches or exceeds the critical value called fracture toughness . In composites the fracture toughness is different for different growth directions and depends on the fracture Mode mix. Since mechanisms in shear and transverse tensile failure are similar, we may expect that in a layer with -orientation shear stress, will facilitate cracking caused by transverse stress ; however, the role of tensile is dominant.

The stress state is a superposition of thermal stresses (zero shear stress in quasiisotropic laminates) and mechanical stresses. Unfortunately reliable mixed mode criteria are not available and are difficult to establish experimentally. Therefore, in cases when the shear stress is several times smaller than the transverse stress, as it is in this paper, only Mode I is often considered, keeping in mind that adding Mode II would increase the number of cracks.

According to LEFM growth conditions for a crack (notch) with a length are entirely different for a crack in the middle of an infinite plate and at the edge of a semi-infinite plate [66]: the critical stress level for growth is much lower for edge crack. The relationship between ERR for edge crack and internal crack growth at given stress is

= ( . √ ) (III-12)
In internal layer of a laminate the surface defect growth is suppressed by constraint from the adjacent layers and most probably defects from the middle of the layer will develop a crack. Therefore, for defect of a given size and at given stress the ERR for its growth in ply thickness direction is higher for defect on the free surface of the surface layer than for the defect in internal layers. After reaching certain size the crack growth becomes unstable and then slows down close to the interface with the next layer finally being arrested there The selected specimen has certain transverse strength distribution along the transverse direction of the layer. The first crack will appear in the element with the lowest strength. After that, for any fixed damage state with given spatial distribution of a ks, the st ess dist i utio is al ulated usi g so e odel shea lag, Hashi 's et and failure index (stress versus strength) is calculated in each element to find the position of the next crack. This procedure seems rather complex but in this way the interaction between cracks which weakens the stress field and delays cracking is accounted for.

In Hence, according to LEFM the propagation of initiated crack in the surface layer requires much lower stresses than in internal layer of the same orientation. Results

show that cracks in our composite have propagated across the specimen width, which means that an initiated crack is propagating at the same stress.

Summarizing the above discussion we can expect that a) the number of initiated cracks in surface layer is much higher; b) in our composite all cracks propagate as soon as they are initiated. Therefore, focus in this analysis will be on crack initiation which we will describe by the Weibull analysis discussed above.

Multiple cracking in 90-and +45-layers of not aged (NA) composites

Analysing cracking in the 90-layers of the 432 NA laminate, first, the crack density dependence on transverse stress was obtained from the applied strain (using CLT) and "i e the eal pa a ete s dete i ed f o Mo te-Carlo simulations are always related to some specific approximate stress model, and the goal of this paper is rather a ualitati e e pla atio of o se ed phe o e a the a ua titati e pe fe tio , we will use, when comparing damage development in different layers, different composites as well as the effect of thermal aging, the -approach because of its simplicity.

The Weibull parameters from Table III- These results show that the discussion regarding the role of the free surface on defect growth has been oversimplified. In fact, the = . √ effect has been proven by computation micromechanics simulations for the first crack only [START_REF] Herraez | Transverse cracking of cross-ply laminates: A computational micromechanics perspective[END_REF]. It is incorrect to assume that in the surface layer of the laminate all cracks will initiate at the free surface: even assuming that the defect state is uniform across the layer (it is just an assumption-in reality there could be resin rich region at the surface changing the statistics) only the first cracks will originate from the largest defects close to the surface, but after that the largest remaining defects are not on the surface of the layer and sooner or later it will be their turn to grow. Thus, the multiple crack initiation in the surface layer should be a mix of cracks coming from the surface and from the Comparing results for NA-432 (with voids) and NA-433 (no voids) presented in this section we do not see large differences. Even the Weibull parameters are very similar and the differences most probably are due to scatter and a different number of available data points. A real difference is in surface layers, where 433 laminate, without voids, has more cracks before mechanical loading and also during increasing mechanical loading. Indeed, the role of voids may be dual: they act as stress concentrators and first cracks appear at lower stress, but they also lead to larger stress relaxation after the crack is created and linked to the void (especially in a surface layer where one surface is free). That may delay appearance of new cracks. This phenomenon was described in [64 -65] showing that in mechanical fatigue the cracking starts first in layers with voids but with increasing number of cycles the difference in crack density diminishes.

Multiple cracking in 90-and +45-layers of aged (A) composites

The crack density in aged composites was analysed as described in section 4.5.2. First, the Weibull parameters, see Table III- Similarly as for NA specimens, the crack density in the surface layers could not be predicted: it is even outside the previously defined bounds. The latter means that the surface layer has different failure properties than the internal layers: it has aged and therefore is much higher than in NA case. It seems that the surface layer of the A-432 composite has aged more than that of the A-433 composite: the crack density at zero loads is higher and it is growing faster in A-432 composite. Partially it could be explained by voids in the 432 providing pathways for oxygen during the aging test. In these figures the -approach simulations, which describe cracking in +45-layers very well, are shown as solid lines. Crack density is shown separately in L3 layer which is close to the mid-plane and in L7 layer which is close to the surface.

For not aged specimens there is no systematic difference between L3 and L7 layers.

The crack density in -45-layers is similar as in +45-layers only in the very beginning of cracking. After 0.6% strain the crack density in -45-layers is almost two times larger than in +45-layers. The only explanation for this difference is in stress concentrations due to damage developing in neighbouring layers. An internal +45-layer has 0-and -45layers as adjacent layers. There are no cracks in 0-layers (except some thermal cracks)

and cracks in the adjacent -45-layer are perpendicular to expected cracks in the +45layerwith small effect of their stress concentrations. In contrary, the -45-layer is layers the cracking is in average the same as in not aged specimens, the crack density in L7 layers is much higher. An obvious explanation for that is aging: layers close to the laminate mid-plane have not aged and only the surface layer and the L7 -45-layer are.

Conclusions

Micro-damage evolution in two quasi-isotropic CF/polyimide composites for high temperature applications, with different resin composition, was investigated showing two phenomena: thermal fatigue and thermal aging.

Firstly, the increasing intra-bundle crack density during thermal cycling was studied

showing that the highest temperature in the cycle which has the lowest thermal stress level is of primary importance for damage developments. To separate the fatigue effects from possible material aging effects during the total time at high temperature, a separate group of specimens was aged for the time equal to the total time at the same highest temperature in the cycling test showing that aging is a part of the total degradation during cyclic loading.

Secondly, the composite failure resistance during aging was analysed using the aging time as a variable. The resistance to cracking reduces with aging time and the crack density after cooling down to room temperature is increasing. Results show a large edge effect in composite degradation: the number of cracks inside the specimen is much lower than on the edge or in surface layers. Results and observed trends are explained using 3-D FEM edge stress analysis and linear elastic fracture mechanics.

Afterwards, the intralaminar cracking in tensile loading of the same materials was investigated analysing effect of thermal aging at 288°C for 30days.

Numerical analysis and following fractography showed that the intralaminar cracking state on the edge gives a good representation of the damage inside the composite.

Cracking in different layers of a laminate was quantified and analysed using Weibull transverse strength distribution for crack initiation probability. A simplified parameter determination and cracking simulation scheme is suggested and results are compared with Monte-Ca lo si ulatio s ased o shea lag as ell as Hashi 's st ess odels.

Cracking in internal +45-layers could be adequately predicted based on Weibull analysis of 90-layers. The crack density in the surface +45-layer was much higher even for not aged specimens, which is explained by free surface effect on crack initiation.

Cracking in 45-layers adjacent to 90-layer is significantly accelerated by stress concentrations from cracks in the 90-layer.

Aging degraded about 1mm of the composite at the specimen edges (this region was removed before mechanical testing) and also the whole surface layer and the layer next to it whereas failure properties of central layers were not affected.

Chapter IV Summary, conclusions and perspectives

I the fi st pa t of this thesis, the effe t of the ⁰-tow waviness on axial stiffness of cross-ply non-crimp fabric (NCF) composites is analysed using multiscale approach. The In another approach, it is shown that the NCF composite knock-down factor characterizing the stiffness degradation has almost the same dependence on wave parameters as the knock-do fa to fo the u ed ⁰-layer. Numerical analysis sho ed that ⁰-layer knock-down factor versus amplitude curves for different wavelength can be reduced to one master curve which can be described by a oneparameter expression with the parameter dependent on the used material. This observation is used to obtain high accuracy for analytical predictions for knock down factors for cases with different wavelength and amplitudes based on two FE calculations only.

Finally, the axial stiffness of a symmetric NCF composite, consisting of 0°/90° unit cells with different amplitudes and wavelength of the 0°-layer, is calculated using FEM and also using analytical models with knock-down factors for curved 0°-layers which describes the effect of the local amplitude and wavelength of the 0°-layer on the effective stiffness of the layer in the unit cell. The dependence of the knock-down factor on the presence and waviness in the neighbouring unit cell is accounted for.

Rule of mixtures and constant force assumptions are used when appropriate to find the macroscopic stiffness. Analytical modelling is in a good agreement with FEM.

An alternative approach is suggested where one equivalent meso-cell represent the macro-stiffness of the NCF composite. Methods for geometrical parameter determination of the equivalent meso-cell are discussed.

As perspectives for this part, we can say that further efforts are meaningful to develop analytical approximate models for curved beams with sinusoidal surface tractions and also the observed features used in the master curve approach require further investigation regarding their mechanical origin and potential of application to more In perspectives, further study is needed concerning the aging effect on the stiffness and the strength of this material. More aging can be done for more time and higher temperature. The polyimide resin is continued to be developed for resistance to higher temperatures.

Titre

1 .

 1 Introduction ............................................................................................................. 2. FE modelling procedure (Material, geometry and assumptions) ............................ 3. Effective stiffness determination using isolated beam subjected to appropriate boundary conditions .................................................................................................... 3.1. Meso-scale homogenization ................................................................................. 3.2. Effect of wavelength and amplitude on laminate stiffness ................................... 3.3.Predictions based on isolated curved layers with boundary conditions used in previous studies ........................................................................................................... 3.4. The layers effective stiffness dependence on A/t0 and L/t0 ................................. 3.5. Approximation of tractions at the 0/90 layer interface ......................................... 3.6.Composite stiffness based on effective stiffness of an isolated 0-layer with surface loads ............................................................................................................................. 4. Effective stiffness determination using master curve approach .............................. 4.1. NCF stiffness knock down factor ......................................................................... 4.2. Master curve approach.......................................................................................... 4.2.1. Knock down factor curve and fitting function................................................... 4.2.2. Validation and application of the master curve approach ................................. 5. Master curve approach for effective stiffness of composite with more complex geometries .................................................................................................................... 5.1. Geometries ............................................................................................................ 5.2. Development of the master curve approach ......................................................... 5.3. Engineering approach for the amplitude and wavelength's of an equivalent cell 5.4. Stiffness of chain of cells...................................................................................... xii 5.5. Interaction effects of curved bundles in cells connected in parallel ..................... 6. Conclusions ............................................................................................................. Chapter III. Damage in carcon fibre/polyimide laminate ..................................... 1. Introduction ............................................................................................................. 2. Objectives ................................................................................................................ 3. Experiments ............................................................................................................. 3.1. Materials description ............................................................................................ 3.2. Experimental procedure ........................................................................................ 4. Results and discussions ........................................................................................... 4.1. Void content and initial damage state................................................................... 4.2. Damage state at RT in thermally cycled composites ............................................ 4.3. Damage state at RT in aged composites ............................................................... 4.4. Damage state in mechanically loaded composites ............................................. 4.5. Damage accumulation analysis .......................................................................... 4.5.1. Theoretical background ................................................................................... 4.5.2. Multiple cracking in 90-+45 layers of not aged (NA) composites .................. 4.5.3. Multiple cracking in 90-+45 layers of (A) composites ................................... 4.5.4. Extraordinary cracking pattern in -45layers .................................................... 5. Conclusions ........................................................................................................... xiii Chapter IV. Summary, conclusions and perspectives ..........................................

  The pre-impregnated tape laminate, shown in Figure I-1 [2], is a stack of unidirectional plies in which the layers are perfectly bonded to each other and the mechanical properties depend on their orientations.

Figure I- 1 .

 1 Figure I-1. Internal structure of pre impregnated tape based

toughness [ 3

 3 ]. This problem is solved with woven composites shown in Figure I-2 [4], the reinforcing fibres are assembled in bundles in different directions and form a fabric.

Figure I- 2 .

 2 Figure I-2. Images of E-glass (left) and basalt (right) woven fabrics [4]

  composites are overcome with new type of textile composite called non-crimp-fabric composites (NCF) presented in Figure I-3 [5].

Figure I- 3 .

 3 Figure I-3. Schematic diagram showing a multi-axial non-crimp fabric [5]

Figure I- 4 .

 4 Figure I-4. Hierarchical structure of the NCF composites [8]

  [START_REF] Tan | Modeling for predicting the mechanical properties of textile composites-a review[END_REF]. proposed a 2D crimp model for the elastic analysis of a 2D plain weave. The unit cell in this model was divided into sections and then the series-parallel models were used to estimate the lower and the upper bounds of the elastic constants.Ishika a a dChou de eloped the osai odel [17], the fibre u dulatio odel [18] a d fu the the idgi g odel [19] for analysing the elastic behaviour of woven hybrid composites. In these models a fabric composite was simply regarded as an assembly of blocks of cross-ply laminates neglecting the shear deformation in the thickness direction. Byström at al. [20] developed a homogenization method for stiffness matrix computation of woven composites; the method was called reiterated homogenization. The authors studied the linear elastic problems with periodic microstructure, which justifies the use of representative volume element which is enough to represent the elastic properties of the whole material. More recent study was performed by Riccio et al. [21] who developed a representative volume element (RVE) for NCF composites under tension loading taking into account the to 's a i ess a d the stit hi g. The classical micromechanical theory was used together with the stiffness averaging method. It was demonstrated that the exclusion of the to 's a i ess f o the odel leads to a ig e o of the stiff ess o pa i g to the experimental data. However, the lack of stitching in the model is less relevant leadi g to a s all e o that a e egle ted. The stiff ess depe de e o the to 's waviness was investigated and it was shown that this stiffness strongly depends on the waviness. A mesoscopic FE model of the NCF structure was realized by Drapier [22] in order to investigate the interlaminar shear behaviour of non-crimp fabric composites. The geo et i al hete oge eit as take i to a ou t. The to 's i p as o side ed large enough for the composite to be regarded between the pre-impregnated tapes and the woven structures. This waviness was assumed as sinusoidal shaped characterized by a wavelength and amplitude. The same assumption was used in [23] by Edgren et al. where the authors used Timoshenko beam theory in addition to FE to calculate the stiffness of a layer in the NCF composite considered like a single curved beam. In addition to this assumption, Mattsson et al. [24] demonstrated that the 90°layer with bundle mesostructure can be replaced by homogenized 90°-layer without losing accuracy in the NCF laminate stiffness investigation. One more important geometric parameter that should be taken into consideration while modelling textile composites, is the inter-strand gap between the bundles, it has been demonstrated in [25] that a change in the inter-strand gap width leads to a significant modification of the elastic properties (elastic modulus, shear modulus and Poisso 's ratios) because of the change of the fibres volume fraction and the matrix volume fraction inside the layers.

  must have a good understanding of the effect of stress concentrations which lead to the damage. Studies of initiation of cracks, their growth characteristics and their effect on the laminate properties belong to an active field of study called damage mechanics, playing a central role in the assessment of durability and damage tolerance of composite structures. Composite structures can undergo multiple micro-cracks before losing ability to carry the design loads. The damage mechanisms occurring in textile composites are basically the same as in pre-impregnated tape based composites. It mainly consists of matrix cracking, delamination and fibres fracture [26] presented in Figure I-5.

Figure I- 5 .

 5 Figure I-5. Damage mechanisms in laminates [27]

  cracks are usually the first mode of damage in fibre-reinforced composites. Such cracks are caused by tensile loading, fatigue loading, as well as by changes in temperature or by thermal cycling. Matrix cracks do not cause a total failure of the composite, but may lead to significant degradation of the elastic properties and to appearance of other damage modes. An example of matrix crack is presented in Figure I-6.

Figure I- 6 .

 6 Figure I-6. Matrix crack [28].

  propagation leads to the separation of the layers. This mode of damage causes the rapid deterioration of the mechanical properties and the total failure of the composite structure. An example of delamination is presented in Figure I-7 where it shows how it is starting from the matrix crack tip.

Figure I- 7 .

 7 Figure I-7. Delamination starting from a matrix crack tip [28]

Figure I- 8 .

 8

Figure I- 8 . 3 . 4 .

 834 Figure I-8. Broken fibres [28]

Figure I- 9 .Figure I- 10 .Figure I- 11

 91011 Figure I-9. Schematic showing the four crack types observed in NCF cross-ply

Figure I- 11 .

 11 Figure I-11. Examples of whole cracks running through two neighbouring

Figure I- 12 .Figure I- 13

 1213 Figure I-12. Examples of half cracks in 90°fibre bundles of the cross-ply NCF laminate

Figure I- 13 .

 13 Figure I-13. Examples of double cracks in single 90° fibre bundles of the cross-ply NCF

[ 32 ]

 32 have studied the relationship between the mechanical properties and the damage accumulation in woven fabric laminate under quasi-static loading. The shear lag analysis, originally derived for the pre-impregnated tape based laminate, were employed idealized laminate replacing the woven one (Figure I-14), since in the studied eight-hardness satin fabric, the inter-crimp distance is quite large. It has been de o st ated that the You g's odulus is affe ted little ith the a u ulatio of matrix cracks and crimp delaminations until the saturation of the crack density. The Poisso 's atio as much more sensitive to the damage.

Figure I- 14 .

 14 Figure I-14. Idealization of two layer woven fabric laminate [32]

  [22,40] experimental data regarding the measured out-of-plane waviness are discussed, and the waviness in terms of sinusoidal shape was used in a 2D FE-model of a composite with periodic structure in the thickness direction and a biaxial NCF as a repeating unit cell (RUC) in order to study the effect of the parameters defining the 2D mesoscopic model on the NCF compressive strength. It has been shown that the NCF compressive strength is o t olled the ⁰-tow geometrical instability and by the resin shear plastic flow. In the sense of the out-of-plane waviness the architecture of real NCF composite described above and shown in Figure II-1a has similarities to woven fabric composites; see Figure II-1b. Therefore, methods and theoretical models for woven fabric composites [41] have been applied also for NCF composites. For woven composites Ishikawa and Chou [17,18] proposed the mosaic and fibre undulation models. In these models, an assumed representative volume element (RVE) is divided into infinitesimal strips and the classical laminate theory (CLT) is used to calculate elastic properties of the strip. The mosaic model disregards the waviness of the bundle whereas the fibre undulation model also includes the waviness. In [42], the 3D RVE consists of flat matrix pockets as well as in-plane and interlaced bundle regions. The iso-strain assumption was used in the in-plane directions and constant stress assumption in the out-of-plane direction.

Figure II- 1 .

 1 Figure II-1. Edge view of the mesostructure: (a) NCF composite (b) Woven composite

  based methodology for axial stiffness calculation of imperfect biaxial NCF composites with fibre tow waviness using layers effective properties of idealized straight laminate (Figure II-2) calculated using 2-dimentional FE modelling.

Figure II- 2 .

 2 Figure II-2. NCF waved structure replaced by flat layers using effective stiffness while

Figure

  II-3a and II-3b, and another with a e edded ⁰-layer, see Figure II-3c. The units correspond to a particular case of cross-ply NCF composite with zero shifts between blankets.

I

  al ulatio s he e the ⁰-layer meso-structure with bundles, Figure II-3a, was odelled, the shape of the ⁰-bundle in Figure II-3a was changed in order to keep the same fibre volume fraction V f = . i the ⁰-layer for models with different wave amplitudes and/or wave length.

[ 24 ]

 24 sho i g that the a ial stiff ess of a flat ⁰la e does ot ha ge sig ifi a tl if the u dle esost u tu e of the ⁰-layer is replaced by homogenized layer with elastic properties corresponding to the average volume fraction of fibres in the layer. This esult justifies the use of a u ed ⁰-layer i stead of u ed ⁰-bundles in NCF composite stiffness investigation. The same study de o st ated that the ⁰-layer meso-scale details on the NCF laminate stiffness can be neglected in cross-ply NCF with st aight ⁰-u dles a d o sta t ⁰-layer thickness. The validity of the assumption, that, i NCF o posite ith u ed ⁰-layer, the ⁰-la e u dle st u tu e a e ho oge ized s ea ed out , was checked in the current work comparing axial stiffness of models in Figure II-3a and Figure II-3b with sinusoidal shape of the waviness.

Figure II- 3 .

 3 Figure II-3. NCF composites units (a) su fa e ⁰-layer with bundle structure (b) surface ⁰-layer with ho oge ized ⁰-layer (c) e edded ⁰-la e ith ho oge ized ⁰-

2 ) 9 - 4 .

 294 The laminate stiffness elements are related to the A-matrix of the laminate With and being the effective stiffness matrix of the layer in global coordinates and average layer thickness respectively. Using the CLT approach with effective layers in a cross-ply NCF composite we obtain is the effe ti e t a s e se stiff ess of the ho oge ized ⁰-layer with varying thickness ( 9 is the average thickness), is the effective axial stiffness of the u ed ⁰-layer. The problem now lies in the correct definition and determination of the effective stiffness. Two types of models were analysed. a) I di idual u ed ⁰-layer , see Figure II-4 b) NCF composite shown in Figure II-3a and waved laminate Figure II-3b The boundary conditions are presented in Figure II-3 and Figure II-For all models, symmetry condition is applied along the left vertical boundary and an x-displacement is applied along the right vertical boundary. The average strain in x-direction introduced by the applied displacement is equal to 1%. An additional symmetry condition was applied to the waved laminate along the bottom boundary which is the mid-plane of [0, 90] s NCF composite or the interface with the rest of the composite in a more general case. A isolated si usoidal ⁰-layer with constant thickness subjected to load in x-direction at ends and different combinations of loads on curved surfaces was also analysed using FE (only half of the wavelength was considered). The boundary conditions used a al si g isolated u ed ⁰-layers are shown in Figure II-4. I additio , fo the f ee e d ⁰-layer, the displacement in the z-direction for the middle node of the left e ti al ou da as set to ze o. Fo the u ed ⁰-la e ith fi ed e ds the sa e displacement was zeroed for the first and the last nodes along the bottom edge. For the ⁰-la e o igid fou datio , this displa e e t is ze o fo all odes elo gi g to the bottom boundary. Finally, for the model in Figure II-4d, the load was distributed alo g the otto su fa e of the u ed ⁰-layer following a sinus shape function.

Figure II- 4 . 5 )

 45 Figure II-4. Curved layer subjected to different boundary conditions

Figure II- 5 .

 5 Figure II-5. Axial stiffness comparison between NCF composite ith ⁰-layers u dles a d o posite ith ho oge ized ⁰-layer.

Figure II- 6 .

 6 Parameters , , are defined in Figure II-3b. In this figure is the thickness of the u ed ⁰-layer, 9 is the a e age thi k ess of the ho oge ized ⁰-layer, ℎ = + 9 . It is clear that is significantly reduced with increasing amplitude and decreasing wavelength of the waviness.

Figure II- 6 . 9 ):

 69 Figure II-6. Effect of the waviness on the CF/EP1 NCF composite axial stiffness =

  three curves in Figure II-7 showing the laminate stiffness were obtained using (II-4). The effe ti e stiff ess of the ⁰-layer was calculated for isolated curved layer shown in Figure II-4 using FE with boundary conditions (a), (b) and (c). The ⁰la e effe ti e stiff ess as assu ed e ual to the ⁰-layer material transverse stiffness using data in Table II-1. These curves may be compared with direct FE results (symbols in Figure II-7) taken from Figure II-6. The comparison demonstrates the failure of these boundary conditions used to give values of 0⁰-layer effective stiffness relevant for using in (II-4).

Figure II- 7 . 9 ) 3 . 4 .

 7934 Figure II-7. Effect of the waviness on the CF/EP1 NCF composite axial stiffness =

Comparing (II- 13 ) 9 (II- 15 )

 13915 with the CLT expression in (II-4) the correct expressions for effe ti e stiff ess of la e s o side i g the as isolated is gi e : Forces for using in (II-8) were calculated from the model in Figure II-3b by summing the reaction forces acting on the nodes along the corresponding edge of the layer. The reaction force on the node shared by both layers is replaced by half of the reaction force of the neighbouring node, by analogy with the reaction force on the upper-right node in the free upper boundary, which is also equal to half of the reaction force of its neighbouring node. Figure II-8 sho s ho the effe ti e stiff ess of the ⁰layer, a d the effe ti e stiff ess of the ⁰-layer 9 -

9 -

 9 calculated according to (II-15) decreases from about 9 GPa to 7 GPa.

Figure II- 8 .

 8 Figure II-8. Axial stiffness for (a) the ⁰-layer and (b) the ⁰-layer, CF/EP1 composite

Figure II- 9 .

 9 Figure II-9. Co pa iso et ee k o k do fa to of the ⁰-la e , the ⁰-layer and

  loadi g sho as dist i uted load in Figure II-4d. Only in the presence of dist i uted load fo es ° and ° may differ.

iFigure II- 10 .

 10 Figure II-10. Normal stresses at layer interface in CF/EP1 composite and fitting (a) for

Figure II- 11 .

 11 Figure II-11. Shear stresses at layer interface in CF/EP1 composite and fitting (a) for

  distributions along the layer interface as surface loads in a isolated u ed ⁰-layer model shown in Figure II-12. The approximate expressions for the normal and the shear stresses given in Appendix are used. The surface load is obtained by multiplying the stress components by the element length which is assumed equal to the distance dx between two neighbouring nodes. The calculated load is applied in the local coordinate system related to each element on the bottom surface of the curved layer. This new boundary condition is introduced in addition to the symmetry on the left edge and the displacement applied in the right edge. Since the loading applied to the ⁰-layer is not uniaxial the calculated using (II-14) is, strictly speaking, not the a ial stiff ess. Ne e theless, it ep ese ts the u ed ⁰-layer mechanical behaviour in the composite.

Figure II- 12 .

 12 Figure II-12. Appli atio of dist i uted load o the ⁰-layer surface

Figure II- 13 .

 13 Figure II-13. Effe ti e stiff ess of the ⁰-layer with different boundary conditions

9 -= 9 (

 99 which is not accurate, but simple for use). In Figure II-14 to Figure II-16 the laminate stiffness is presented as a function of A/t 0 for different L/t 0 . In Figure II-14a, II-15a and II-16a the effective stiffness of the ⁰-layer is used whereas in Figures II-14b, II-15b and II-16b the ⁰-layer with varying thickness is represented by its transverse stiffness 9 calculated using data in

Figure II- 14 .

 14 Figure II-14. CLT using effective stiffness compared to FE-analysis for CF/EP1

u 9 - 23 )(II- 27 )

 92327 ed la e stiff ess o tai ed i the sa e FE al ulatio . The a o el aster curve app oa h is p ese ted hi h allo s al ulati g the k o k do fa to of a u ed layer of arbitrary wave amplitude and length. The master curve for the given NCF composite material is obtained fitting FE knock-down factor values for one (preferably relatively short) wavelength and at least two values of amplitude. Only one additional FE calculation is required to cover all possible amplitude cases for a different wavelength. The curved layer knock-down factor is used to calculate the NCF composite knock-down factor. The very high accuracy of this approach is demonstrated in comparison with direct FE calculations. The CLT approach for calculating the axial stiffness of the NCF composite specimen is the same as in calculations for the laminate with straight layers. The stiffness elements of a laminate with flat layers (NCF composite with zero waviness) are related to the Amatrix of the laminate For the NCF composite represented by laminate with effective layers we have ℎ being the effective stiffness matrix of the layer in global coordinates, average layer thickness and average laminate thickness respectively. For the axial stiffness of a cross-ply laminate and the NCF composite analysed in this part the following relations are obtained: is the effe ti e t a s e se stiff ess of the ho oge ized ⁰-layer with varying thickness ( 9 is the average thickness) and is the effective axial stiff ess of the u ed ⁰-layer. I this se tio the i de LAM i di ates the NCF ith st aight la e s A= , a d the i de NCF efe s to NCF o posites ith a ed la e s. The stiffness knock-do fa to s fo the ⁰-la e , fo the ⁰-layer and for the NCF composite (represented by laminate) are introduced as follows Combining (II-22) and (II-23) gives Using (II-21) to replace , (II-24) can be rearranged in the form The second term in (II-25) depends on the layer stiffness and thickness ratio and on the difference between the knock-do fa to s fo the ⁰-a d ⁰-layer. It will be shown further that this term can be neglected without introducing any noticeable error. Certainly, to find these knock-down factors, we have to start with proper definition and methodology for determination of the effective stiffness. The axial stiffness of symmetric and balanced NCF composite can be obtained from the macroscopic in-plane stress-strain relationship = + (II-26) In (II-26) x-direction is the specimen axial (loading) direction, see Figure II-3. The NCF composite axial stiffness element is obtained performing FE-analysis for plane strain case ( = ). In this loading case is obtained directly dividing the calculated axial average stress by the macroscopic strain applied. The average stress is axial force divided by nominal (average) thickness of the laminate ℎ. Hence = ℎ The effective in-pla e a ial stiff ess of the u ed ⁰-layer is lower than the stiffness of a straight layer mainly because the fibres are not oriented in-plane. However, as shown in [48] and described in details in the previous section, the curved layer in addition to end loading (applied displacement) is subjected to large normal and tangential tractions at the layer interface. In other words the loading to the curved layer is not uniaxial and as a consequence the force calculated at the right end of the layer (index R) is not equal to the force on the left end (index L) , ° ≠ ° . This makes the determination of the effective stiffness uncertain, because different values are obtained using different cross-se tio s. The sa e applies to the ⁰-layer with varying thickness ° ≠ ° . The numbers ° and ° in parenthesis indicate the layer under consideration. Certainly, on both ends the total force is the same, for example ° + ° = (II-28) Be ause of the des i ed easo the effe ti e la e stiff ess al ulated o the left and the right edge of the model differs

-

and 9 -

 9 have to be defined in a way that using (II-20) gives exactly the same values for NCF composite as direct FE calculation.In the previous section, it was demonstrated that the following definitions of effective stiffness give this result -

  ate ials fo the u it ith su fa e ⁰-layer. Results are presented in Figure II-17. Although the knock down factors in the case of CF/EP are decreasing more than for the case of GF/EP, both figures present similar behaviour of the curves and the same conclusions can be deduced for both materials. The term ∆ in (II-25) can be neglected since it is almost equal to zero in the figures. Thus, the NCF composite knock down factor defined by (II-23) a e assu ed e ual to the ⁰-layer knock down factor defined by (II-23) and (II-31), ⋍ . In other words, in order to know NCF cross-pl o posite k o k do fa to , it is e ough to k o the ⁰-layer knock down factor.

Figure II- 17 .

 17 Figure II-17. Comparison of and ∆ in the expression of NCF knock-down factor in (II-25) fo u it ith su fa e ⁰-layer (a) CF/EP and (b) GF/EP

Figure

  II-18 for both materials and for the unit with surface ⁰-layer. The same fitting expression in (II-33) is found for both materials.

Figure II- 18 .

 18 Figure II-18. Fitting function for Master curve (L/t 0 = su fa e ⁰-layer) (a) CF/EP and

Figure II- 19 ,

 19 figure each knock-do fa to u e is diffe e tl defo ed i the ho izo tal direction according to the first term in (II-35). In new coordinates , the two considered points / * , * and / * , * coincide (the a -coordinate is the

Figure II- 20 .

 20 Figure II-20. Figure II-19 after calculation using (II-35) for the case of LM/t 0 =6 and Ln/t 0 =8, for CF/EP and GF/EP and for the unit with surface ⁰-layer.

Figure II- 21 .

 21 Figure II-21. Comparison of calculated using FE with calculated using master u e app oa h fo u it ith su fa e ⁰-layer (a) L/t 0 =10 and (b) L/t 0 =15.

Figure II- 22 .

 22 Figure II-22. Comparison of calculated using FE with calculated using master u e app oa h fo u it ith e edded ⁰-layer (a) L/t 0 =10 and (b) L/t 0 =15.

Figure II- 23 .

 23 Figure II-23. NCF o posite k o k do fa to ith su fa e ⁰-layer (a) CF/EP (b)

Figure II- 24 .

 24 Figure II-24. NCF o posite k o k do fa to ith e edded ⁰-layer (a) CF/EP (b)

Figure II- 25 .Figure II- 26 .Figure II- 27 .

 252627 Figure II-25. Meso-cells with in phase and out of phase combinations of surface and

- 38 )- 39 )

 3839 down factors for curved 0°-la e s. Nu e i al a al sis sho ed that the ⁰-layer knockdown factor versus amplitude curves for any wavelength can be reduced to one master curve by curve shifting. The master curve that shows the knock-down factor dependence on wave amplitude for a reference wave length is well described by a one-parameter expression with the parameter dependent on the used material k = +b A/ (IIThis expression together with the shifting procedure is used to obtain high accuracy analytical predictions for knock down factors for cases with different wavelength and amplitudes. For any new wave length one FE calculations is required. In other words, the fitting parameter b" is not only material dependent but also wavelength dependent. FE parametric analysis was performed to analyse b", see Figure II-28, and a power law is suggested to fit the b" parameter dependence on wavelength. b = c × L/ - (IIThe fitti g o sta t depe ds o the used ate ial. Values fo the used GF/EP a d CF/EP (CF/EP1) are given in Table II-4. They are different for surface and for embedded units.

Figure II- 28 .

 28 Figure II-28. Fitti g of the pa a ete b

(II- 41 )

 41 consider surface or embedded unit cells with different amplitudes and different wavelengths combined in chain (See Figure II-26) producing a NCF composite made of n unit cells. The approach is to replace these chains by one equivalent cell (correspondingly surface or embedded). The equivalent cell has to have an amplitude and wavelength that lead to the same stiffness as the initial chain of unit cells has. The objective is to find the equivalent amplitude and wavelength. This analysis has practical significance because this is the way how experimental (for example, optical) observations can be implemented in a model. The question is: what is the proper averaging procedure for observation data? The simplest and commonly used method is to calculate the average of the amplitude and of the wavelength by simple summation of the local amplitudes and wavelengths (II-40) taking into account the volume fraction Vi (II-41) of each unit which leads to rule of mixtures the number of unit cells in the chain, is the volume fraction of the i-th unit cell in the chain = FEM results in Figure II-29 discussed in detail later show that this simple way of averaging significantly overestimates the stiffness. A more accurate and mechanics based procedure is based on summation rule for stiffness of cells combined in chain. For the equivalent cell

In Method 2 /

 2 depends via / , see (II-48), on whereas for the equivalent cell depends on / . In other words both parameters are coupled and an iterative solution is one of the options: in first iteration we assume that and can be neglected in (II-47) leading to

Figure II- 26 , 5 . 3 . 6 .Table II- 5 .

 265365 Figure II-26, was calculated analysing separately chain of surface cells, Figure II-26a,

introduced in expression (II- 38 )

 38 together with the equivalent amplitude to calculate the k o k do fa to of the ⁰-layer in the equivalent unit cell. The knock-down factor is used in the CLT expression (II-24), to calculate the axial stiffness of the equivalent unit. Among the three methods defining parameters of the equivalent cell Method 1 based on simple averaging of amplitude and wavelength is the less accurate: for both materials and for surface as well as embedded composites chains it significantly overestimates the composite stiffness. The results in Figure II-29 show that the axial stiffness of the equivalent cell with average amplitude and wavelength calculated using Method 2 and Method 3 are in rather good agreement to the axial stiffness of NCF composite. Method 3 gives almost the same results as Method 2, which means, that ignoring the variation of the parameter bi does not lead to any considerable error in calculation of the average amplitude.

Figure II- 29 .

 29 Figure II-29. Axial stiffness of a combination of 5 cells of surface and embedded unit

Figure II- 25 .

 25 Figure II-3c are considered changing the thickness ratio between the layers as following: t 90 /t 0 = , , , . The k o k do of the ⁰-layer is presented as a function

Figure II- 30 .

 30 Figure II-30. Effe t of la e s' thi k ess atios o ⁰-la e 's k o k do fa to fo the unit cell with: a) "u fa e ⁰-layer b) E edded ⁰-layer.

Figure II- 31 .Figure II- 3 )Table II- 7 .

 3137 Figure II-31. Master curve Fitting

  same wavelength ⁄ = (Figure II-27). The stiffness of the two NCF composites shown in Figure II-27, calculated using FEM, is labelled in Figure II-32 as FEM F a a d FEM F espe ti el . The o posites in Figure II-27 were treated also analytically using the engineering approach based on the master curve. First the knock-down factors of all units were determined accounting

55 )

 55 The stiffness of both chains is used to calculate the stiffness of the NCF composite shown in Figure II-27. of the surface chain and embedded chain respectively. The NCF stiffness calculated using this technique is denoted in Figure II-32 as aste . The alues a e et ee the FEM values for in-phase (Figure II-27a) and outof-phase (Figure II-27b) composites. In the used approximation this technique does not distinguish between in-phase and out-of-phase composites. This is because calculating the effect of the neighbouring 0°-layer on the knock-down factor the 0°layer is replaced by 90°-layer which does not have any in-phase or out-of-phase waviness. An alternative for FEM as well as for analytical calculations is replacing the composite in Figure II-27 by an equivalent meso-cell consisting of two equivalent unit cells connected in parallel. The potential of this approach is analysed next. The equivalent a plitude of ⁰-layer in each equivalent unit cell is calculated separately using Method3 (II-51) with Vi being either or . The equivalent wavelength is calculated according to (50) which in the used calculation case (units of equal length) is equal to the length of the unit cell.

Figure II- 27 .

 27 Figure II-27. This is reflected in t o diffe e t FEM alues FEM F a-e a d FEM F27b-e i Figure II-32.

4 for case 9 =(II- 57 ) 27 . 3 .Figure II- 32 .

 95727332 Figure II-32. Axial stiffness of in phase and out of phase combinations of surface and

  exposure to elevated temperatures (aging) can cause changes in the morphology, strength and stiffness [49-51]. Therefore understanding the behaviour of a composite material under this type of environmental exposure is of great importance. Many researches are dedicated to study the thermal stability of the composite by quantifying the effect of thermal fatigue and aging on the mechanical properties and the damage development [52-54]. The effect of thermal cycling (-100°C,100°C) and thermal aging (250h, 400h at 110°C) on mechanical performance of carbon fibre/RTM6 resin and carbon fibre/BMI resin cross-ply laminates was studied in [55] showing that the intralaminar crack density is a clear function of aging time, indicating the matrix and interface degradation. The elevated temperature level during thermal cycling with a fixed lowest level has significant effect on the damage level [56-57]. At the highest temperature in the cycle the composite laminate is close to the stress-free state and, therefore, thermo-elastic analysis cannot explain the role of the highest temperature in the damage development. Using Paris law with energy release rate (ERR) change during the cycle to explain fatigue crack propagation, the highest temperature is of importance because the ERR change is larger. Nevertheless, it is only one of the possible reasons: during exposure to high temperature irreversible phenomena take place and when the lowest temperature in the cycle (with high stresses) is reached micro-damage evolves. Possible causes for degradation are the disintegration of the molecular network leading to degradation of the resin properties. In [51] authors demonstrated that this effect can be intensified and accelerated if an elevated pressure is added.The degradation may be quantified by the weight loss measurement either after thermal cycling or after isothermal aging [58-59]. It has been demonstrated that the isothermally aged woven composite made of carbon fibre/epoxy loses more weight than the thermally-cycled specimens throughout the entire range of the equivalent aging time at 170°C. This was explained by the volumetric relaxation of the polymer matrix after each cycle and by the diffusion rate of oxygen through the epoxy matrix during thermal cycling, which is considerably lower than the diffusion rate at isothermal condition. Another study[START_REF] Lévêque | Analysis of how thermal aging affects the long-term mechanical behavior and strength of polymer-matrix composites[END_REF] showed that the aging time of carbon fibre/epoxy (IMS/977-2) composite (at 217°C) has an effect on its viscoelastic properties. This effect is linked to the effect on the glass transition temperature. The chemical and physical changes occurring during aging are accompanied by a variation in molecular mobility, the glass transition temperature increases at the beginning of aging followed by a decrease.It is clea that the at i di tates the ate ial's espo se at high te pe atu e. The degradation of neat polyimide resin and CF/polyimide composite at (320°C, 360°C) was studied in[START_REF] Marceau | Thermal ageing of PMR15 polyimide matrix[END_REF]. The authors have proved that the composite weight loss comes only from the matrix degradation indicating that the carbon fibres in the composite have good thermal oxidation resistance. The choice of the appropriate resin is the most complicated issue for the high temperature composites. Development of high temperature polymeric resins was reported in [62] replacing a commonly used epoxy, with the limit of service temperature above 135°C, by polyimide resin in composites for service temperatures between 250°C and 350°C.

  initiation and evolution of micro-cracking due to thermal cyclic loading or after isothermal aging at 288°C and afterwards, to study the multiple intralaminar cracking and delamination growth during tensile loading of the same material. The experimental findings in different layers are explained analysing the micro-cracking initiation and growth using stochastic initiation strength distribution approach and fracture mechanics, focussing on  Differences in crack density growth in different layers  Comparing composites with two modifications of the polyimide  Analysing the effect of high temperature aging on intralaminar cracking The composite material studied in this chapter is designed for severe operating temperatures with a potential for aero-engine applications. Quasi-isotropic [(+45/-/ / ] s la i ates e e p epa ed usi g -harness satin weave based on Cytec T650 carbon fibres and thermosetting polyimide resin denoted NEXIMID® MHT-R (MHT-R). This resin is provided by Nexam Chemical AB; it is a low molecular weight phenylethynyl terminated polyimide that contains a combination of 4-(Phenylethynyl)Phthalic Anhydride (4-PEPA), end-group cross-linker and ethynyl bisphthalic anhydride (EBPA) main chain cross-linker. The use of optimized combinations of the 4-PEPA a d EBPA oth Ne a ide™ t pe e a les ulti ate Tg of a ou d °C.

1 ).

 1 The first group (75mmx8mm) was divided into two subgroups and subjected to two different thermal cycling loading sequences. The first subgroup was subjected to ramp (R1) between -60°C and room temperature (RT) whereas the second subgroup to ramp (R2) between -60°C and 288°C. The specimens were exposed to each temperature for 10 minutes and were subjected to up to 150 cycles. The fatigue behaviour was characterised by determining the micro-crack density in a layer by means of optical microscopy.

  The mesostructure and the damage state of all the specimens were analysed under microscope before any treatment. Some examples are presented in Figure III-1. The main difference between both plates is in void content. The void content was measured using image analysis software (NIS-elements BR) together with optical microscope. The void content was defined as the area fraction of voids from the observed surface.

Figure III- 1 . 4 . Results and discussions 4 . 1 . 2 .

 14412 Figure III-1. Cracks caused by the residual stresses: a) plate 432; b) plate 433

1

  mm from each edge was removed by polishing. Still, the first observation from Figure III-2 is that the crack density in different layers is quite different. The surface +45layers have much higher crack density than the rest of layers; in the 90-layers the crack density is larger than in the ±45-layers, where it is practically negligible.

Figure III- 2 .

 2 Figure III-2. C a k de sit o the edge of u t eated spe i e s: a) plate 432; b)

Figure III- 3

 3 Figure III-3 Normalized thermal stress distributions in [(+45/-45)/(90/0)] 2s laminates of finite width W=4 mm: a) transverse stress ℎ ; b) in-plane shear stress ℎ .

Figure III- 4 ,

 4 is much higher than at the edge. It correlates with higher crack density in specimens from the middle of this plate; see Figure III-2a from left to right, especially in the surface layer.

[ 64 - 65 ]

 6465 showing that in mechanical fatigue the cracking starts first in layers with voids but with increasing number of cycles the difference in crack density disappears. In the 433-plate diluent was used to reduce the viscosity, the void content is very low and the micro-damage state is uniform over the plate; see Figure III-2b.

Figure III- 4 .

 4 Figure III-4. Void content measured on the specimen edges.

[ 66 ] 2 )In (III- 3 ) 8 )

 66238 . The relationship between ERR for surface and internal crack growth in thickness direction for the same stress are given in[START_REF] Berglund | Transverse Cracking and Local Delamination in[END_REF] Thus many more cracks are initiated in the surface layer of the quasi-isotropic laminate under thermal loading. When the crack becomes sufficiently large in the ply thickness direction, propagation of the initiated crack along the fibre direction in the layer becomes energetically possible. The crack initiated at certain stress may propagate or not, dependent on the available energy for propagation at this stress level. The energy approach for crack propagation along fibres is described in [68-70]. The Mode I ERR for the steady-state crack growth along fibres can be written in terms of crack opening displacements is the thickness of the damaged layer which in our case is the same for all layers, is transverse modulus of the layer. The average normalized crack opening displacement is different in surface and in internal layers. Expression (III-3) is valid for low crack densities when stress perturbations from cracks are not interacting which is also the case in this study. The following approximate functions for were presented in [20] obtained using FE parametric analysis is thickness of the adjacent layer (in our case all layers have the same thickness, = ) and = is the modulus of the adjacent layer in the Tdirection of the cracked layer. Using the same elastic constants for CF UD composite as in FE modelling we obtain = . (III-9)

Figure III- 5 .

 5 Figure III-5. Edge crack density growth in two thermal fatigue ramps (R1 (-60; RT)°C;

Figure III- 6 .

 6 Figure III-6. Crack density distribution in the layers along thickness on the edge of

Figure III- 7 .

 7 Figure III-7. Cracks density dependence on the edge distance for thermally cycled

  hours and cooled down to RT to count cracks on specimen edges. If the time at high temperature would affect the composite transverse failure properties (aging), thermal stresses at RT would introduce additional cracks. The crack density versus the number of aging hours is presented in (Figure III-8), two specimens from each plate were tested.The counting was performed 24 hours after removing specimens from the oven. The crack density was not high enough to have significant effects of crack interaction on evolution. In a not aged specimen the surface +45-layers have the highest initial crack density (FigureIII-2) which increases linearly with the aging time reaching at the end of the test approximately 1cr/mm. The crack density on the edge of the 90-layer is increasing with a higher rate comparing to the surface +45-layer reaching about 1.7cr/mm at the end of the test. The crack density in the internal ±45-layers starts to increase later, after around 400 hours of aging time and the cracking process is accelerating. Comparing the aging effect on crack development in both materials (plate 432 and plate 433) we cannot observe any clear difference in the behaviour.

Figure III- 8 .

 8 Figure III-8. Crack density on the specimen edge versus aging time: a) surface 45-

Figure III- 9 .Figure III- 3 :

 93 Figure III-9. Crack density inside a 40days aged specimen (A-433-2).

  generally leads to weight loss. An average of weight loss taken over 4 specimens from each plate is plotted versus the aging hours is Figure III-10. The weight loss did not exceed 1.4% after 40 days of aging at 288°C.

Figure III- 10 .

 10 Figure III-10. Weight loss due to aging at 288°C

Figure III- 12 . 4 . 4 .

 1244 Figure III-12. Aging and thermal cycling effects on edge crack density

  laminate and the CLT thermal shear stress ℎ = . The in-plane thermal stress distributions at the edge are presented in Figure III-3.

Figure

  III-14 shows crack density in different layers as a function on the distance from edge. There is no statistically confirmed trend with more cracks on the edge of the specimen. This conclusion holds for NA as well as for A specimens.

Figure III- 14

 14 Figure III-14 Crack density versus distance from edge: a) NA-, ε= %, b) A-432, ε= . %.

[ 75 - 14 )

 7514 76]. In [67] relationship (III-12) was used for analysis of cracking in surface and internal layers of laminates resulting in the follows relationship between crack initiation stress in internal and surface layers of the same orientation = Intralaminar cracks initiate in stress concentration regions (local fibre clustering, voids, edges) and in weakest positions where the fibre/matrix interface or the matrix is i pe fe t fo e a ple, defe ts due to i p eg atio p o le s i high fi e o te t regions). The crack initiation process is characterized by coalescence of fibre/matrix debonds leading to formation of a crack which grows in the ply thickness direction. It would be convenient to express the thickness direction growth criterion during initiation in terms of ply-scale stress state calculated using Classical Laminate theory (CLT) and the factor to reflect the difference between surface and internal layers. The initiation of a crack is a stochastic process with certain probability and with random position if the stress distribution is uniform along the layer. This is because the weakest position for the crack to come is related to local inhomogeneity with random location. The local fibre clusters of size of 3-5 fibre diameters ( ≈ .) define the critical size when the coalescence of defects may be considered as an unstable crack. In simulations of crack initiation we assume that any layer in its transverse direction can be considered as consisting of a chain of elements of this size. The elements all-together follow the two-parameter Weibull strength distribution, but the location of an element with certain strength is random. According to (III-13) a larger amount of cracks is initiated in the surface layer of the quasi-isotropic laminate than in internal layer of the same orientation. Parameter is introduced in (III-15) to reflect the favourable situation for crack initiation in the surface layer of the laminate.Performing Monte-Carlo simulation we randomly select, for example, = elements from a virtually infinite pool of elements to create a specimen of length = • = . This selection does not exactly represent the initial Weibull distribution which means that each selection is a new specimen of the same material.

  the estimated thermal stresses. Using (III-16) the experimental crack density data for 90-layers was transformed to probability of failure and (III-15) was used to estimate Weibull parameters. The values for these parameters for both composites (aged as well as not aged) are given in TableIII-3. The log-log relationship is shown in Figure III-15a. The first point on the left corresponds to crack density due to thermal stresses with rather uncertain stress value. The last three points deviate from linearity which is a sign of crack interaction. According to Figure III-15b, where the test data and the simulation using (III-16) (solid line labelled and referred as -approach) are shown, deviation starts at 9 ≈ . / . In Figure III-15b Monte-Carlo simulations with the same parameters are also shown (different symbols for shear lag a d Hashi 's odel de o st ati g oti ea le de iatio et ee the -approach which ignores crack interaction and Monte-Carlo simulations accounting for it, starting at 9 ≈ . / which is much lower than 9 ≈ . / estimated from deviation between the approach and experimental data. Of course, the result of the Monte-Carlo simulation depends on the used stress model and it is well known that Hashi 's odel o e esti ates the i te a tio of st ess pe tu atio s, he eas i the shear lag model case the result depends on the used modification (we used shear lag model which assumes existence of a resin rich region of thickness = . between layers). Nevertheless, Figure III-15b confirms the conclusions from [77] obtained from probabilistic consideration: a) crack interaction starts to affect results at rather low crack density; b) for non-uniformly distributed cracks interaction effects are noticeable at lower crack densities than for uniformly distributed cracks. Returning to results of Monte-Carlo simulations in Figure III-15 we have to conclude that the Weibull parameters obtained using (III-16) ( -approach) are rather inaccurate: the parameters inTable III-3 give good fit to used data but they are not the real Weibull parameters.

3

 3 were used to predict crack density in internal +45-layers and in the surface +45-layer of the NA-432 laminate. All internal +45-layers behave in a very similar way. The data and simulations using theapproach are presented in Figure III-16 which shows good agreement proving that a) cracking in +45-layers has the same transverse stress dependence as in 90-layers, indicating that the edge effects in 90-layers discussed previously are not significant; b) at least for low loads the effect of shear stresses in cracking simulation can be neglected. At low strains the predicted crack density in +45-layers is slightly higher tha the e pe i e tal hi h a e att i uted to the i pe fe t pa a ete determination in 90-layer and to the discussed edge effect in 90-layer.

Figure III- 15

 15 Figure III-15 Cracking in 90-layers (NA-432): a) Weibull plot of test data; b) crack density versus transverse stress 9 , -approach (4); Monte-Carlo simulation using

Figure III- 16

 16 Figure III-16 Crack density growth with strain in +45-layers of NA-432 (data and

  interior of the layer. May be the first group of cracks could be characterized by = . √ in (III-15) whereas the second group by = . An indecisive support to this is given by Figures III-16b and III-17b where the first cracks in the surface layer are close to predictions using = . √ whereas later it becomes closer to the = curve. If so, these two predictions could be used as an upper and lower bounds to the cracks density. Unfortunately, cracks interaction would have a similar effect: at high crack density cracks in surface layers are interacting much more than in internal layers.The crack density growth with strain in 90-layers of the NA-433 plate which has slightly modified composition is shown in Figure III-17a. This relationship was used to estimate Weibull parameters, see TableIII-3, using (III-15) and (III-16) as described above for NA-432 plate. These parameters were used in the -approach to predict cracking in +45-layers. The predictions, shown with solid line in Figure III-17a, are in a good agreement with cracking evolution in this layer.

Figure III- 17

 17 Figure III-17 Microcracking development in NA-433

3 ,

 3 were obtained from 90-layers cracking data. These parameters were used in the -approach to predict crack density growth in[START_REF] Chun | Predictions of Elastic Properties of Multi-axial Warp Knitted Fabric Composites[END_REF]layers, see Figure III-18 for A-432 and Figure III-19 for A-433. For internal +45-layers in both aged laminates predictions are in good agreement with test data.

Figure III- 18

 18 Figure III-18 Microcracking in A-432 composite

Figure III- 20

 20 Figure III-20 Probability of transverse failure curves for 90-layers of 432 and 433

Figure III- 21 .

 21 Figure III-21. Cracking in -45-layers of the 432 quasi-isotropic laminate (layers are

Figure III- 22

 22 Figure III-22 Cracking in -45-layers of the 433 quasi-isotropic laminate (layers are

  surrounded by +45 layer (the same marginal effect from cracks in this layer) and by 90layer where crack density increases first. Cracks in 90-layers cause strong stress concentration in the adjacent -45-layer. These cracks, often called stitch cracks due to their appearance [31-32], have been observed in mechanical as well as in thermal fatigue and also in quasi-static loading. There is no difference in cracking of NA-432 and NA-433 laminates. In aged specimens (A-432 and A-433 in Figure III-21b and Figure III-22b)) the difference in cracking between L7 layers and L3 layers is very large. Whereas in the L3

u

  ed ⁰ a d ⁰-layers are represented by flat layers with effective stiffness properties and classical laminate theory (CLT) is used to calculate the macroscopic stiff ess. The effe ti e ⁰-la e stiff ess is al ulated a al si g isolated u ed ⁰layers subjected not only to end loading, but also to surface loads. The surface loads are identified in a detailed FE-analysis and approximated by a sinus shaped function with amplitude depending on the waves parameters. The sinus shaped surface loads are then applied to a isolated u ed ⁰-layer FE-model together with end loading to al ulate the effe ti e stiff ess of the la e . Fi all , the effe ti e ⁰-layer stiffness was successfully used to calculate the macroscopic stiffness of the composite proving validity of the approach being used and showing that, without losing accuracy, elastic p ope ties i the ⁰-layers with bundle structure can be replaced by the transverse stiff ess of the ho oge ized ⁰-layer material.

o

  ple NCF o posites ith ⁰-layer waviness. The study of the effect of the local meso-st u tu e defe ts o elasti p ope ties a e o ti ued o Poisso 's atio fo example.In the second part of this thesis, the damage in carbon fibre T650 8-harness satin weave fabric composites with thermosetting polyimide resin designed for high service temperatures is studied. High thermal stresses develop after cooling down to room temperature, which lead to multiple cracking in bundles/layers of the studied quasiisotropic laminates. The composites were subjected to two ramps of thermal cycling quantifying the increase of crack density in layers. Comparison of two ramps with the same lowest temperature shows that the highest temperature in the cycle has a significant effect on thermal fatigue resistance. During thermal aging tests at 288°C the mechanical properties are degrading with time and the crack density versus aging time was measured. Aging and fatigue effects were separately analysed showing that part of the cracking in thermal cycling tests is related to material aging during the high temperature part of the cycle. Numerical edge stress analysis and fracture mechanics were used to explain observations. In a second stage, intra-bundle cracking due to tensile transverse thermal stresses was observed in the studied [(+45/-45)/(90/0)] 2s laminates after the thermal aging at 288°C for 40 days which caused many new cracks. Both aged and not aged specimens were tested in uni-axial quasi-static tension quantifying damage development. Differences and similarities in cracking in different layers were analysed using probabilistic approaches (simple non-interactive as well as Monte Carlo simulations). It is shown that cracking in off-axis layers not being in contact with the 90-layer can be predicted based on Weibull analysis of the 90-layer, whereas in layer which is in contact with the 90-layer the crack density is much higher due to local stress concentrations caused by cracks in 90-layer. The thermal treatment degraded cracking resistance in the surface layer and in the next layer whereas failure resistance of layers close to the mid-plane did not change.

3. Effective stiffness determination using isolated beam subjected to appropriate boundary conditions

	interface in the NCF composite are conducted and sinus shaped functions are
	introduced to represent the surface load distribution. This approximation is then used
	in fu the u e i al odelli g to al ulate the effe ti e odulus of the u ed ⁰-
	layer. It is demonstrated that a CLT based analytical model, in which the
	o i al/a e age thi k ess of the ⁰-la e , the effe ti e stiff ess of the u ed ⁰-
	layer and the effective stiffness of the 90⁰-layer with varying thickness are used,
	renders good accuracy.
	In this approach the curved tows/layers are replaced by straight ones with effective
	elastic properties. Isolated curved layers with appropriate boundaries and end
	conditions are suggested for effective properties determination. To identify what type
	of su fa e loads o the isolated u ed ⁰-layer will represent its behaviour in the NCF
	composite, FE-a al sis of the st ess/t a tio dist i utio s at the ⁰-la e / ⁰-layer

Table II -

 II

1.These results show good agreement between the analytical CLT approach ith the effe ti e stiff ess of the ⁰-layer and FE calculation of the laminate stiffness.

Table II -

 II

4. Fitti g o sta t

Since the c-constant is material dependent, each new material requires one FE calculation. Estimation of data in Table

II

-4 leads to conclusion that in all comparable cases this constant is roughly three times larger for CF/EP composite. This value is rather close to the longitudinal modulus ratio of these two materials in Table

II

-1 (141/46.8=3.01

Table III - 3

 III3 Weibull parameters from 90-layer cracking

	Composite m	S0 (MPa)
	432-NA	7.84	135.2
	433-NA	8.33	138.8
	432-A	6.88	134.0
	433-A	7.3	144.2

:

  Les composites avec mesostructure en faisceaux : Propriétés élastiques et endommagement Mots clés : Les composites textiles, endommagement, mécanique de rupture, rigidité, défauts de mesostructure. Les p op i t s lasti ues et la sista e à l'e do age e t des composites textiles avec mesostructure en faisceaux ont été étudiées dans cette thèse. Le premier chapitre de cette thèse traite la modélisation des propriétés élastiques des composites NCF (Non-Crimp Fabric) pour étudier l'effet des défauts de mesostructure sur la dégradation des propriétés mécaniques. Un modèle pour la mesostructure des composites NCF est réalisé pour étudier l'effet de l'ondulation sur la réduction de la rigidité. Cette dernière est do i e pa la du tio de la igidit de la ou he ⁰. La igidit effe ti e de la ou he ⁰ peut t e d te i e soit pa od lisatio d'u e seule couche ondulée soumise à un chargement réparti, qui reproduit son interaction avec les couches voisines, avec application des symétries dans les conditions aux limites, ou en utilisant une approche de la courbe maîtresse. Une expression analytique est suggérée. Cette expression permet la détermination du facteur de réduction de rigidité pour toute longueur d'onde et amplitude donnés. L'i itiatio et le d eloppement des endommagements sont présentés dans le deuxième chapitre, où les composites textiles désignés aux applications à haute température ont été étudiés dans des conditions thermiques sévères pour vérifier leur stabilité thermique et leur résistance aux endommagements thermiques. Finalement, la performance mécanique des composites destinés pour les hautes températures, est étudiée, et l'effet du vieillissement thermique a été analysé. Des modèles 3D ont été réalisés par éléments finis pour expliquer l'effet des bords sur l'évolution des fissures observées lors des essaies de traction. En outre, les différences et les similarités au niveau de fissuration dans les différentes couches sont analysés á travers des approches probabilistes (Monte Carlo, Hashin and shear lag) et aussi la mécanique de la rupture.

Résumés :

Acknowledgments ..................................................................................................... Appendix A ............................................................................................................... References ................................................................................................................. xiv Acknowledgments It is gratefully acknowledged that the work performed in the first part of this thesis is partly funded by the Swedish 5th National Aeronautical Research Program (NFFP5) through the project ReFACT, and also through the Joint European Doctoral Program in Material Science and Engineering (DocMase). The second part was financially supported by EXCELL project funded by local government of Norbotten, SWEDEN and partial funding from the strategic innovation programme LIGHTer provided by VINNOVA are also acknowledged.

Performing the FE calculation for n=10 waviness with one chosen value of the amplitude / * = . , the calculated knock-down factor is * = .

. From (II-33) * = + •( / * )  / * =0.312

Then a set of values of / is chosen, i.e. the first row in Table II- According to (III-3) the energy release rate (ERR) is 2.42 times larger in the surface layer and according to the ERR based crack propagation criterion the propagation of initiated crack in the surface layer would start at much lower thermal stresses.

Summarizing the above discussion we can state that in thermal loading: a) the number of initiated cracks in surface layer is much higher; b) the number of propagating cracks in the surface layer is much higher.

Damage state at RT in thermally cycled composites

After certain number of cycles of thermal loading, one specimen was saved and the number of cracks in this specimen was counted along 53mm on both edges. Each data point for crack density in Figure III-5 is the average on two edges of one specimen.

First, the damage state resulting from two different thermal loading ramps can be compared. For all layer orientations the R2 (-60°C to 288°C) ramp induced more damage than R1 (-60°C to RT). One can think that the upper temperature in R2 (288°C) is close to the stress free temperature, so less damage could be expected. However, data reveal that the interval between the minimum and maximum temperature, ∆ = at fixed has important effect on the cracking in thermal fatigue. It is consistent with the Paris law for growth of individual crack in cyclic loading, stating that the crack growth rate is a power function of the ERR difference

This relationship for intralaminar cracks was validated in Much larger crack density in surface layers was observed also in [START_REF] Bechel | Permeability and damage in unloaded cryogenically cycled PMCs[END_REF] where thermal cycling at cryogenic temperatures was performed. In [START_REF] Bechel | Permeability and damage in unloaded cryogenically cycled PMCs[END_REF] the crack density determined on the edge of 90-layer was also higher than in internal 45-layers.

To explain these two features we have to recall that even in fatigue there is an initiation phase and propagation phase. Certain number of cycles is required to initiate crack and a different number of cycles is required for its propagation along fibres to cover the whole width of the specimen. If ≫ the fatigue life u e of les fo full de elop e t of the a k is p opagatio go e ed, i contrary, if ≫ the cracking in fatigue is initiation governed. In surface layer, the ERR for crack growth along fibres is much higher than in internal layers and crack propagation is almost instant (initiated cracks instantly grow through the specimen).

The initiation is a stochastic process with a certain probability. In At 1% strain the total transverse stress in the 45-layer is > + =

whereas the shear stress ≈ + = , ( 9 ≈ % ). At lower loads the fraction of the shear stress is even lower. For this reason, analysing failure in follows we will neglect the shear stress contribution assuming that crack initiation is by transverse stresses. Comparing predictions with data we will show the applicability of this assumption.

The mechanical transverse stress at the edge has the following features:

 In 90-layers it is magnified by 50% comparing to the CLT value. However the stress concentration zone is very small (1mm).

 In the internal +45-layers it is by 30% lower than in the CLT zone. In the surface layer the reduction is 60%

If the crack density in an i th layer is low enough to allow for large plateau regions in stress distribution between cracks, the probability of failure as a function of transverse stress can be estimated using expression versus log and use the standard procedure to estimate Weibull parameters. The result is rather sensitive to the used data points on both extremes: data points at high crack density are affected by crack interaction whereas the stress at first crack varies a lot from specimen to specimen, requiring large amount of specimens to be reliable.

The so obtained estimates of Weibull parameters can be used in Monte-Carlo simulations over the whole crack density region and adjusted to increase the accuracy.

When the crack becomes sufficiently large in the ply thickness direction, propagation of this crack along the fibre direction in the layer may become energetically possible.

The crack initiated at certain stress may propagate or not, dependent on the available energy for propagation (ERR) at this stress level. The energy approach for crack propagation along fibres is described in The constants for layers made of the different composite materials found by fitting are given in Summary: Textile composites with bundle meso-structure have been studied in this thesis for elastic properties and damage investigations.

The first chapter of this thesis deals with elastic properties modeling for Non-crimp fabric (NCF) based composites for investigating the effect of meso-structure defects on mechanical properties degradation. The objective of the work is to formulate a model for the NCF composite mesostructure in an attempt to investigate the effect of the waviness on stiffness reduction. The cross-ply NCF composite stiffness reduction is do i ated the stiff ess edu tio of the ⁰-la e . The ⁰-layer effective stiffness can be determined either by modeling a single curved tow subjected to distributed load, to reproduce its interaction with the neighboring layers, together with symmetry boundary conditions, or using a master curve approach, where a knock down factor is introduced to characterize the stiffness reduction and analytical expression is suggested. This expression allows for determination of knock down factor for any given wavelength and amplitude of the waviness.

The damage initiation and development is presented is the second chapter, where woven fabric composites designated for high temperature application were investigated under severe thermal conditions to study their thermal stability and their resistance to thermal damage.

Finally, the mechanical performance of the composites designated to high temperature applications was studied. The effect of aging was also investigated. 3D models were realized with Finite elements in order to explain the edge effect on the evolution of the cracks observed during the tensile tests. In addition, the differences and similarities in cracking in different layers were analysed using probabilistic approaches (a simple one as ell as Mo te Ca lo si ulatio s ith Hashi 's a d also shear lag model) and fracture mechanics arguments.