3D models of Random Composite Materials

 

 

Vincent Riboulet

 

May-August 1999

 

 

 

 

CENTER FOR STRUCTURAL AND APPLIED MECHANICS

 

 

 

 

Contents

 

 

Abstract………………………………………………………………………………      3

 

The Work place………………………………………………………………………     4

The University of Virginia

The Center for structures and applied mechanics

Professor Graham’s team

 

Problem presentation………………….………………….…………………………      6

Material samples

Objectives

 

The Moving Window GMC……………………………..……………………………     10

Overview

Application

Results

Statistical description of the data

 

Finite element models, stress distribution in Random Composite………….……    15

2D model

Relation between sxx and Elastic Modulus

Effects of the mesh sizes

Effects of the Window size

3D models of layered Carbon-Carbon Composite………………………...    21

4 point Bending Test of a Beam..……………………………………   23

Slot pull test on a part of a jet brake disk…………….…………….    28

 

Conclusions……………………………………………………………………………    33

 

Acknowledgements……………………………………………………………………   34

 

References……………………………………………………………………………     35

 

Appendix………………………………………………………………………………     36

 


Photographs

4

6

6

7

7

12

12

21

22

23

28

 
 


1.      University of Virginia (the Rotunda)………………………………………..……..     4

2.      Silicon carbide fibers in titanium matrix..………………………………………….     6

3.      Graphite fiber tow in epoxy matrix …………………………………………….….     6

4.      Cellular aluminum…………………………………………………………….……     7

5.      Functionally graded material……………………………………………………….     7

6.      Micrograph of a functionally graded composite…………………………………...     12

7.      Micrograph of a graphite epoxy fiber tow………………………………………….    12

8.      Global view of the disk……………………………………………………………..     21

9.      Broken lug………………………………………………………………………….     22

10.  Micrograph of actual Carbon-Carbon composite…………………………………..     23

11.  Slot Pull test………………………………………………………………………...     28

 

 

 

 

 

 

 

Figures

8

9

11

14

17

19

23 & 25

24

26

 
 


1.      Beam with varying Young’s modulus……………………………………..………..…   8

2.      Principles of the research………………………………………………….…………..   9

3.      Moving Window GMC…………………………………………….………………...   11

4.      2D problem………………………….…………………………….……….………....   14

5.      Effects of the Mesh size……………………………………………………………...   17

6.      Effects of the Window size…………………………………………………………..   19

7.      Layered carbon-carbon composite in configuration 1………………………….

8.      4 points bending test…………………………………………………………………   24

9.      Layered carbon-carbon composite in configuration 2……………………………….   26


Abstract

 

 

 

Most models of the behavior of composite materials assume an effective homogenized set of material properties. These models fail to capture the true behavior of the wide variety of composite materials that exhibit significant inherent randomness: soil, cellular materials, concrete, particulate composites, fiber-reinforced composites. This research project is aimed at modeling the local stress in such materials.

My personal contribution is the creation of 3D finite element models using Abaqus to highlight the differences in the calculated stress values between results based on homogenized material properties and a model where each element has individual values for the material properties. These values have been calculated from a sample with the Moving Window GMC technique and are put in the Abaqus code using Fortran programs.

The specific models presented here are a layered carbon-carbon beam under 4 point bending load and a part of a carbon-carbon brake disk under slot-pull testing. These models are based on real samples provided by a leading materials manufacturer in the United States.

 

 

 

 

 

 

 

 

 

 

 

The training

 

Adviser at the University of Virginia: Lori L. Graham

Adviser at the ENS de Cachan: Olivier Allix

Training from the 3rd of May to the 12th of August

The Work Place

The University, the laboratory, the team.

The University of Virginia

 

The University of Virginia (which includes approximately 2000 faculty and a total full time student enrollment of about 17000) offers professional degrees under the schools of Architecture, Law, Medicine, Commerce, Business Administration, Education, Engineering and Applied science. The school of engineering and applied sciences is an integral part of the university community, which provides opportunities for interdisciplinary work in pursuit of the basic goals of education, research, and public service.

 

Photo1. University of Virginia: the Rotunda

The center for structural and solid mechanics

 

The center for structural and solid mechanics is a laboratory of the Civil Engineering Department. The computational room is well equipped with several new PC’s and some SUN and SGI work-stations. They provide via the network access to any computer of the engineering school, such as flemming, which was used for most of this project.

This project is funded by the National Science Foundation, in collaboration between the Civil Engineering Department of the University of Virginia and the Mechanical Engineering Department of the University of South Carolina.

Professor Graham’s Team

 

Professor Lori Graham is a full time assistant professor at the civil engineering department of the University of Virginia. Her research focuses on stochastic finite elements, an area that merges finite element techniques with the principles of probabilistic mechanics to provide such information as system reliability. Her other research interests include random heterogeneous materials, structural dynamics, and the simulation of non-Gaussian stochastic processes

 

For this project her team is composed of:

 

·           Aimee LeBlanc (Master )

·           Bligh Wollner (Master )

·           Eman Siragy ( Ph D )

·           Fernando Ferrante ( Master )

·           Jacob Agran (Undergraduate )

·           Kevin Smith (Undergraduate)

 

She also works in collaboration with Professor Sarah Baxter of the Department of Mechanical Engineering at the University of South Carolina

 

 


Problem presentation

Creating a model for composites

Material Samples

 


Many composite models assume that inclusions such as fibers are spaced at "regular" intervals. In some cases this assumption is valid; the image below shows a micrograph of Silicon Carbide fibers in a titanium matrix (from “Mechanics of Composite Materials”, by C. T. Herakovich, 1999)

 


Photo 2  Silicon Carbide fibers in a titanium matrix

 

But this is not always true. The following are images of a few samples taken from various composite materials. All of them present significant variation in the position of the fibers.

·                   

Graphite fiber tow in epoxy matrix (Kunze & Herakovich)

 


Photo 3 Graphite fiber tow in epoxy matrix


 The white areas correspond to graphite fibers and the dark areas to epoxy matrix. This image of a fiber tow shows areas with relatively few fibers (see orange-circled area on figure above), as well as clusters of many fibers (see yellow-circled area on figure above).

 

·                    Cellular Aluminum (Wadley)

 


 


Photo 4 Cellular Aluminum

 

 

 

The white areas in this image correspond to aluminum and the black areas to voids. The void spaces vary greatly in size, concentration, and shape.

·        Functionally Graded Material


 


Photo 5: functionally graded material

 

In this case, material properties change not only due to functional gradation, but also because of a fairly high degree of randomness in the fiber distribution.


                We have seen that a lot of composites don’t have a regular arrangement in their composition. It’s now interesting to have a look at the differences in the results we get from a model assuming an homogenized material properties and a model taking into account the variation in the material properties. We consider a classical beam under flexion with a varying Young’s Modulus, E(x)

 

 

E varies randomly over length (E(x))

 
 

 

 

 

 

 

 

 

 


Fig 1: Beam with varying Yong modulus

 

 

 

If we call the variation in displacement vector U, we have:

KU=P

 

Past research has shown small that scale fluctuations in E(x) do not greatly affect K and therefore U. But now if we consider stress:

s(x)=D(x)BU

 

We see that s(x) is directly impacted by fluctuations in D(x) (and therefore E(x)).

 

Moreover, stress is often more important to design than displacement. So, we need  appropriate descriptions of material properties for proper analysis of stress variability.

 

 

 

 

 

 

 

 

 

 

 

Objectives of this work:

 

We are working on composites that exhibit significant randomness in the microstructure. As we have seen, this can have a serious impact on critical behavior such as local stresses.

 

In order to calculate these local stresses, we will calculate elastic and inelastic material property fields for a given 2-Dimensional microstructure with the moving-window GMC technique. Then we will use these fields as input to finite element analysis (Abaqus). Ultimately, the goal is to calculate variability of critical response of a given structure. As we see in the graph underneath, there are 5 steps to the research described here:

1. Collection of digitized material micrographs

2. Generation of local material property fields from micrographs collected in part 1, using moving-window GMC

3. Finite Element Analysis to obtain local stress field of a given structure/ loading condition using local properties obtained in part 2

4. Statistical characterization of material property fields calculated in part 2 and subsequent Monte Carlo simulation of "new" local property fields.

5. Incorporation of simulated property fields (part 4) into finite element analysis (part 3) in order to obtain statistics of local stresses.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Fig 2 principles of the research.


Moving Window GMC

A micromechanical analysis: Generalized Method of Cells.

 

Overview

 

·        The method of cells:

 

The method of cells (Adoubi, 1991), and it’s extension, the generalized method of cells (Paley and Adoubi, 1995), are approximate analytical methods for predicting the elastic as well as inelastic response of fibrous composites. The methods can be used for two-dimensional or three-dimensional analysis.

 

The theory is not the subject here. You can find it in [1], [2] (see references). The input data are the properties of the individual components and the geometry of the material. The overall behavior of composites generated by the method of cells is displayed in term of:

 

Effective elastic moduli

Effective coefficients of thermal expansion

Effective thermal conductivities

Effective stress-strain response in the inelastic region

 

·        The generalized method of cells

 

In the generalized formulation, a repeating unit cell is subdivided into an arbitrary number of subcells. This generalization extends the modeling capability of the method of cells to include the following:

 

Modeling of variable fiber shapes

Analysis of different fiber arrays

Modeling of porosities and damage

Modeling of interfacial regions aroound inclusios, including interfacial degradation

 

·        The moving window GMC

 

In this technique, one applies a GMC model to a part of the sample (called the window) to calculate the local material properties for the point in the middle of this window. The properties of the next point are calculated by moving the window.


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Fig 3: Moving window GMC


Application of the moving GMC

 

We will now look at the results that the moving-window GMC provides for 2 different composites. We will have a more in-depth look at the results in the finite elements 2D part.

·        Functionally Graded Microstructure

 

Zone de Texte: Axial modulus                                   Transverse modulus y

Photo 6: Micrograph of a functionally graded composite

 

Blue and purple correspond to high values of the moduli, red and orange are low values. Note that fiber volume fraction increases with x coordinate. All results are based on a square window of 7.1% x 1.8%

·       

Graphite-epoxy fiber tow

Photo 7: Micrograph of a graphite epoxy fiber tow                        Local Tranverse Elastic Modulus


Results from Moving-Window GMC

 

As we have seen, the moving window GMC offers an interesting representation of the local material properties. Currently there is work underway to identify techniques of choosing appropriate window size: Cross-validation, Relation to underlying correlation distance, Effect on Stress Response. Another important step is to simulate new material property fields, based on a statistical description of the original local material properties

 

 

Generating a Statistical Description of Data

 

Based on the material property fields generated, Monte Carlo Simulations can be performed that will provide new fields with the same statistics as the original field. In order to perform such simulations, two important pieces of information must be estimated:

·        The Probability Density Function defining the distribution of values at any given point

·        The correlation function defining the correlation between two points at various distances apart

For more information on this part, look at the website: http://scooter.ce.virginia.edu/~jandk/home 

 

Such Monte Carlo simulations will be used as input to finite element analyses, providing statistics on the maximum stress for a number of samples of the same material.


Finite element analysis

 Stress Distribution in Random Composite

2D Finite Element Analysis

 

This work is a part of the Ph.D. of Eman Siragy, which has not been published yet. I would like to thank her and Professor Graham for allowing me to use it. The goal here is to have an idea of the size of the mesh and the size of the windows in the moving window GMC in order to get accurate results. The problem is a 2D square under uniform traction. The material is a composite with random placed circular fibers.

 

 

 

 


Fig 4 : 2D problem

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Transverse Elastic Modulus Ezz

 
                                                               

Relation between sxx

and Elastic Modulus

 

 

sxx, max=1.37 MPa

 
 

 

 

 

 

 

 

 

 

 

 

 


We see here that there is only approximately 60% correlation between Exx and sxx. This shows us that the configuration is as important as the local elastic modulus in calculating local stresses.


1.              Effects of the mesh sizes

 

·               sxx for various mesh sizes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


The mesh size has strong impact if the mesh is too coarse.


·        Maximum and Minimum Stress vs Mesh for 5%x5% Window Size

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Fig5: Effects of the mesh size.

 

 

 

We see that the results for stress converge between 32x32 mesh and 64x64 mesh.

 

We will now try to change the window size of underlying material property fields in order to know the effects of the Window size.


2.  Effects of the Window size

 

·         sxx for 2.5%x2.5% window size and 5%x5% window size

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


There is a difference of over 20% in results after changing the window size in the moving-window GMC analysis. We can conclude that window size has a large impact on local stress results

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

·        “Exact” Results for sxx

 

 

 

To get rid of the effects of the mesh, we now use a 200x200 mesh as a basis of comparison for the result with different window and mesh size. The stress pattern is similar to those of other meshes

 

With this mesh, we have:

Maximum stress in fiber elements= 3.59 MPa; Maximum stress in matrix elements=1.90 MPa

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Fig 5:  Effects of the window size

 

 

To compare the effects of the size of the window, we have to compare the yellow and red line (Sxx ) and the purple and blue (Syy ). We see that the results are equal for the very coarse mesh and that they begin to differ when the mesh becomes finer. The difference, especially for Syy becomes important.


·        Ratio of local stress sxx to local yield stress sYxx

 

In this example, Matrix is assumed to have a yield stress of 300.5 Mpa and Fiber assumed to have an infinite yield stress

 

 

 

 

 

 

 

Transverse Yield Stress sYxx:

 
 

 

 

 

 

 

 

 

 

 

 

 

 


We see that areas of high elastic moduli tend to have higher corresponding local yield stresses.

 

Let’s Calculate the maximum ratio of local stress sxx to local yield stress sYxx , for the 64x64 element mesh, when pressure load p=100 MPa:

 

5%x5% window:             36.7% of yield

2.5%x2.5% window:       42.2% of yield

 

200x200 mesh:               63.0% of yield

 

 

Difference has decreased somewhat from 62% to 41% (for the 5%x5% windowing) and 51% to 33% (for the 2.5%x2.5% windowing). These results are interesting, as they suggest that consideration of stress alone may not be appropriate without accounting for some local maximum stress, such as yield stress. Clearly, this idea merits further study

 

 

 

Conclusions on the 2D models

 

 

 

•In calculating local stresses, configuration is as important as local elastic moduli or local fiber volume fraction

•For a given field of material properties, the finite element results converged with increasing mesh - smaller windowing scheme called for finer meshing to achieve convergence

•Consideration of local stress variation relative to a consistent benchmark (e.g., yield stress from the same GMC windows) yields better results, may be more rational

•Nature of sample (fibers placed at purely random locations) leads to short correlation distance of underlying material - worst-case scenario?

Technique may be applied to real materials, in 2D

 

 

 

 

 

 

 

 

 

 

3D Examples

 

The goal is now to apply the technique we have applied on the 2D square to real 3D samples. A carbon-carbon jet brake disk supplied by one of the leading materials manufacturer in the United States, which is working with the team, is used for this analysis.


Photo 8 : Global view of the disk

As we can see there is a problem with fracture at the lug.



Photo 9 : broken lug

 


Our goal is to apply our technique to see the effect on the local stress at the base of the lug, in order to learn if the difference in local stress can explain the rupture.

The manufacturer of the disk gave us information on 2 tests they had performed. The first one is a test of a 4-layer beam of this material, and the second one is a slot pull test on a part of the disk.

 

 

Layered Carbon-Carbon Composite

 

 

 

 

 

 

 


Fig 7: layered composite in

configuration 1

 

 

 

 

 

 

 


The beam is made of 4 layers of carbon carbon composite wich are in 0 90 direction. 2 models will be made, depending on the direction of the bottom fibers.

 

To begin we used this model of the fibers:

 

 


Micrograph Assumed in first Finite Element Models

 

Then we used data from a real sample. The values given here are the one we got with the real material. Values of the first model are available in the appendix.

 

 


Photo 10 : Micrograph of actual Carbon-Carbon Composite - 1 layer

 

 

The Material is assumed not to vary in fiber direction


Loading: 4-point Bending Test

 

 

 

 

 

 

 

 

 

 

 

 


Fig 8 : 4 points bending test

This test was made in order to get an idea of the flexural strength. It was performed per ASTM D970. The specimen is 0.25 inches thick, 0.80 inches wide, and 5.0 inches in length. The outer span (support) is 4.2 inches while the inner span (load point ) is 1.4 inches. They found a typical structural strength of 15.0 Ksi.

We compare 2 models. One assumed that the composite is homogeneous. Each element has the average values of the material properties. The second model takes the values of the moving window GMC. To do that, a Fortran program reads the values and then makes all the calculation to give to each element is own values. The output is a text file which is used as input for abaqus.

The materials properties given by the Moving GMC are given for a rectangle of 1layer * 0.80. There are 1800 * 140 points For each different mesh an average value is calculated on the values of the point inside an element to give to this element is value. The elements that are next in the fiber direction get the same value.

 


Beam with bottom fibers in direction y

 

 

 

 

 

 

 

 

 


Fig 7 : Layered composite in

configuration 1

 

Using varying microstructure as input:

 
 

 


Fibers in bottom layer follow y-direction

 

The values of sxx are:

 

·        Homogeneous materials:

sxxmax=137 MPa

sxxmin=-177 MPa

 

·        Varying microstructure:

sxxmax=156 MPa

sxxmin=-191 MPa

 

 

 

 

 


Beam with bottom fibers in direction x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Fig 9 : Layered composite in

configuration 2

 

 

 

 

 

 

 

 

 

 

 

 


Fibers in bottom layer

 follow y-direction

 

The values of sxx are:

 

·        Homogeneous materials:

sxxmax=112 MPa

sxxmin=-201 Mpa

 

·        Varying microstructure:

sxxmax=130 MPa

sxxmin=-201 MPa


The results show an interesting difference in the values of sxx. So we decided to make more models of the beam in order to make some comparisons with different meshes as we have done in 2D. The beam has 4 layers in the thickness so at least 4 elements in the direction of the thickness. We have to keep some proportions for the shape of the elements, the computational time of the first model was pretty low (about 30minutes). So we decide to make models with more fine meshes. 6 models were created:

·        100*6*8=4800 elements

·        125*20*12=30000

·        150*24*16=57600

·        175*28*16=78400

·        200*32*20=128000

·        225*36*28=226800

 

All this models have been created together with a special enumeration to allow the use of a unique fortran program, which takes in account all the possibilities. You can find all the (big) listing of theses models in the appendix.

 

The problem I encountered is that the computer ran out of memory. I was using Flemming.seas.virginia.edu, an IBM RS6000 with 1 Go of RAM. With the help of a system engineer, we logged the Abaqus job and discovered that it was trying to write a file which was bigger than 1Go, which is the default limit size of a file under the Unix system. So we set the limit to 2 Go, which is the maximum allocated by the OS, but then we encounter a lack of disk space in my account. We finally decided to switch to the IBM SP, which is a big multiprocessor RS600 (24 processors in parallel ) and which is used for big and long running jobs. As I was the first in the lab to use it, I went through some troubles to configure everything in my Unix account. I finally succeeded but to discover that the SP suffers the same limitations as Flemming in the maximum size of a file, so my jobs didn’t worked. You can find the HTML 4 tutorial I have written on the use of the SP in the appendix.

Theses computational problems were very frustrating as I spent a lot of time on my models (especially the huge unique fortran program). It really shows the limitations of the numerical problems in 3D. Hopefully, one will be able to use my work when a system engineer will have fixed all the problems.

 

 

 

 

 

 

 

 

Slot-pull specimen

 

 

 



This test was made in order to know the resistance of the slot. The disk is cut in the radial direction so that there is only one slot remaining. Then there is a load on this slot of about 6 Kips (1000 lbs).

 


Zone de Texte: Pict 11 :The geometry of the disk is really complicated and was difficult to model under abaqus without the help of a pre processor (which automatically creates the mesh). Basically, two meshes were created, one for the inside of the disk and the other one for the friction part. Then I  “tied” the corresponding nodes of the 2 models. The other problem is that I used cylindrical coordinates for the disk but that the fixation, were the disk is bond is not radial.

This time, I also made 2 models for the homogenized values but as the differences were not important, I made one model with a varying microstructure.

 


Disk with bottom fibers in radial direction

 

 

 


 

 

 

 

 


srr

sqq

szz

Mises

Min

-5.28E2

-3.83E3

-9.44E1

1.63

Max

1.57E03

3.83E03

3.57E2

4.29E3


 

 


As expected, the elements which are in the area where the crack appeared are the one with the more stress.



Disk with bottom fibers in ortho-radial direction

 

 

 

 

 


srr

sqq

szz

Mises

Min

-2.39E3

-2.86E3

-3.07E2

3.59

Max

1.88E3

3.6E3

3.27E3

3.52E3

 

 

 

The values are a bit different from the other case


Disk with varying microstructure

 


 

 

 


srr

sqq

szz

Mises

Min

-2.67E3

-7.11E3

-9.53E2

1.32

Max

2.24E3

7.73E3

9.9E2

6.73E3

 


 

 

 


General comparaison:

 

 

 

srr

sqq

szz

Mises

Homogeneous dir R

Min

-5.28E2

-3.83E3

-9.44E1

1.63

Max

1.57E03

3.83E03

3.57E2

4.29E3

Homogeneous dir q

Min

-2.39E3

-2.86E3

-3.07E2

3.59

Max

1.88E3

3.6E3

3.27E3

3.52E3

Varying microstructure

Min

-2.67E3

-7.11E3

-9.53E2

1.32

Max

2.24E3

7.73E3

9.9E2

6.73E3

 

We see here that the results with the varying microstructure are not so different from the one we get assuming a homogeneous composite. This can be related to the fact that the mesh of the model is pretty coarse, so the material value for each element of the model is an average of a lot of points, minimizing the variation of the values.

 

 

 

 

 


Conclusions

 

 

The application of the moving-Window GMC technique to model the behavior of composite materials (layered carbon-carbon composite with randomly placed circular fibers) proved to be a very interesting technique.

 

The extended study of a 2D square under tension load showed us that:

-window size has a large impact on local stress results.

-the results are equal for very coarse meshes and that they begin to differ when the mesh becomes finer.

-consideration of stress alone may not be appropriate without accounting for some local maximum stress, such as yield stress. (this idea merits further study)

 

The beam and slot pull test were the first 3D models generated using this technique. What took the most time was generating the mesh "by hand". A preprocessor could have been much faster but would not have give us the same control other the enumeration, which proved to be very helpful while writing the Fortran programs (alloting to each element its material properties). I tried to make an exhaustive study in 3D as it has been made in 2D, but faced a lot of numeric problems ( memory limitations ). A major future direction of research is to try to overcome theses problems in order to get an idea of the importance of window and mesh size.

 


Acknowledgments

 

I would like to thank the following persons:

 

In France

Olivier Allix, my adviser at the ENS de Cachan for this training, to have accepted to be my adviser and to have followed me.

Sebastien Le-Loch, who has put me in contact with the UVA team and has given me a lot of wise advises.

 

In the USA

All the staff of the international center, to have given to me a friendly place to stay.

Cindy Sites, secretary of the department, for her help in all the administrative approach, and to have loan to me a so useful bike for the summer.

Jim Damberg, to have allowed me to use all his tools, and for is help.

Brad Sayler, computer system engineers, for his help in setting up a working station.

E H. Carruth and Jeffrey L. Chisholm , ITC staff, system engineer on the Abaqus station, for all their help to try to fix all the computer related problems.

Carlt T. Herakovich, emeritus professor, to have allowed me to contact Professor Graham, and for his discussions in French!

Furman W. Barton, Civil engineering department, for his help with the Abaqus license and all the Abaqus documentation

Eman Siragy, Ph.D. Candidate, for all her precious advises with Abaqus.

 

And most of all

Lori L. Graham, supervisor of the training, to let me free and to trust me for the choices of my research ways, for her welcome and all she made for me before and during this 3 months.

 

 

References

 

[1] C.T. Herakovich,

Mechanics of Fibrous Composites, 1997

[2] M.Paley and J.Aboudi

Micromechanical analysis of composites by the generalized method of cell model,

Mechanic of Materials, 1992

[3]All the Abaqus documentation, especially the example manual.

[4]Installation guide for Linux, all the ITC on-line documentation about Unix.