Table Of ContentNumerical simulation
of mechanical behaviour
of composite materials
Sergio Oller
Lecture Notes
on Numerical Methods
in Engineering and Sciences
Numerical Simulation of
Mechanical Behavior of
Composite Materials
Lecture Notes on Numerical Methods in
Engineering and Sciences
Aims and Scope of the Series
This series publishes text books on topics of general interest in the field of computational
engineering sciences.
The books will focus on subjects in which numerical methods play a fundamental role for
solving problems in engineering and applied sciences. Advances in finite element, finite
volume, finite differences, discrete and particle methods and their applications are examples
of the topics covered by the series.
The main intended audience is the first year graduate student. Some books define the
current state of a field to a highly specialised readership; others are accessible to final year
undergraduates, but essentially the emphasis is on accessibility and clarity.
The books will be also useful for practising engineers and scientists interested in state of
the art information on the theory and application of numerical methods.
Series Editor
Eugenio Oñate
International Center for Numerical Methods in Engineering (CIMNE)
School of Civil Engineering, Technical University of Catalonia (UPC), Barcelona, Spain
Editorial Board
Francisco Chinesta,Ecole Nationale Supérieure d'Arts et Métiers, Paris, France
Charbel Farhat, Stanford University, Stanford, USA
Carlos Felippa, University of Colorado at Boulder, Colorado, USA
Antonio Huerta,Technical University of Catalonia (UPC), Barcelona, Spain
Thomas J.R. Hughes, The University of Texas at Austin, Austin, USA
Sergio R. Idelsohn,CIMNE-ICREA, Barcelona, Spain
Pierre Ladeveze,ENS de Cachan-LMT-Cachan, France
Wing Kam Liu, Northwestern University, Evanston, USA
Xavier Oliver, Technical University of Catalonia (UPC), Barcelona, Spain
Manolis Papadrakakis, National Technical University of Athens, Greece
Jacques Périaux, CIMNE-UPC Barcelona, Spain & Univ. of Jyväskylä, Finland
Bernhard Schrefler,Università degli Studi di Padova, Padova, Italy
Genki Yagawa,Tokyo University, Tokyo, Japan
Mingwu Yuan,Peking University, China
Titles:
1. E. Oñate, Structural Analysis with the Finite Element Method.
Linear Statics. Volume 1. Basis and Solids, 2009
2. K. Wiśniewski, Finite Rotation Shells. Basic Equations and
Finite Elements for Reissner Kinematics, 2010
3. E. Oñate, Structural Analysis with the Finite Element Method.
Linear Statics. Volume 2.Beams, Plates and Shells, 2013
4. E.W.V. Chaves. Notes on Continuum Mechanics. 2013
5. S. Oller. Numerical Simulation of Mechanical Behavior of Composite Materials, 2014
6. S. Oller. Nonlinear Dynamics of Structures, 2014
Numerical Simulation of
Mechanical Behavior of
Composite Materials
Sergio Oller
Centre Internacional de Mètodes Numèrics en Enginyeria(CIMNE)
School of Civil Engineering
Universitat Politècnica de Catalunya (UPC)
Barcelona, Spain
ISBN: 978-3-319-04932-8(HB)
ISBN:978-3-319-04933-5(e-book)
Depósito legal: B-2223-2014
A C.I.P. Catalogue record for this book is available from the Library of Congress
Lecture Notes Series Manager:Mª Jesús Samper,CIMNE, Barcelona, Spain
Cover page: Pallí Disseny i Comunicació, www.pallidisseny.com
Printed by: Artes Gráficas Torres S.L.
Huelva9, 08940Cornellà de Llobregat (Barcelona), España
www.agraficastorres.es
Printed on elemental chlorine-free paper
Numerical Simulation of Mechanical Behavior of Composite Materials
Sergio Oller
First edition, 2014
(cid:164)International Center for Numerical Methods in Engineering (CIMNE), 2014
Gran Capitán s/n, 08034 Barcelona, Spain
www.cimne.com
No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form
or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise,
without written permission from the Publisher, with the exception of any material supplied
specifically for the purpose of being entered and executed on a computer system, for exclusive
use by the purchaser of the work.
This work is dedicated to my wife and son, and also to all my loved ones
PPPrrreeefffaaaccceee
This is a summary of the research work on "composite materials” carried out by the
author since 1990. It reflects my personal work with other researchers and also with senior
engineering and PhD students whose thesis I have advised on this research line1.
The subjects included in the present work are oriented mainly towards the constitutive
modeling and structural evaluation of structures built with composite materials. Many of
these subjects were and are still today original. Therefore, they do not represent a simple
continuity of the research line trends of the past.
All the concepts presented here are a summary of a more extensive work carried out by
the author and other collaborators whose texts can be found in the references mentioned
in each case. It is worthwhile mentioning that all these subjects are still under development
and require the additional participation and dedication of more people. This summary
shows the subjects are not at the same conceptual and technical level. Some ideas are more
developed than others, however, in general, all of them show that the numerical-
mechanical study of composite materials is a very promising area for both, research work
and industrial potential, and consequently to improve their use for the best interest of
society.
This research work started in 1990 and has progressed on two different research lines:
the “Mixing Theory” and its modifications and the “Homogenization Theory” and its
comprehensive study. Both lines have given way to a significant scientific production.
1 PhD theses involved in this book:
- Luccioni B. (1993). Formulación de un Modelo Constitutivo para el Tratamiento de Materiales
Ortótropos. Universidad Nacional de Tucumán - Laboratorio de Estructuras Tucumán, Argentina.
Advisors: S. Oller, R. Danesi.
- Car E. (2000). Modelo Constitutivo para el Estudio del Comportamiento Mecánico de los Materiales
Compuestos. Universidad Politécnica de Cataluña - Departamento de Resistencia de Materiales y
Estructuras en la Ingeniería – Barcelona, Spain. Advisors: S. Oller, E. Oñate.
- Zalamea F. (2001). Modelización de Compuestos Mediante la Teoría de Homogeneización. Universidad
Politécnica de Cataluña - Departamento de Resistencia de Materiales y Estructuras en la Ingeniería –
Barcelona, Spain. Advisors: J. Miquel, S. Oller.
- Nallim L. (2002). Mecánica de placas anisótropas: Un enfoque variacional. Universidad Nacional de Salta -
Facultad de Ingeniería – Salta, Argentina. Advisors: R. O. Grossi, S. Oller.
- Rastellini F. (2006). Modelización Numérica de la No-linealidad Constitutiva de Laminados Compuestos.
Universidad Politécnica de Cataluña - Departamento de Resistencia de Materiales y Estructuras en la
Ingeniería – Barcelona, Spain. Advisors: S. Oller, E. Oñate.
- Martínez X. (2008). Micro-Mechanical Study of Composites by Using the Serial/Parallel Mixing Theory.
Departamento de Resistencia de Materiales y Estructuras en la Ingeniería – Barcelona, Spain. Advisor: S.
Oller.
- Molina M. (2012). Simulación de los materiales compuestos como refuerzo en estructuras de hormigón
armado. Departamento de Resistencia de Materiales y Estructuras en la Ingeniería - Barcelona, Spain.
Advisor s: S. Oller, A. Barbat.
- Badillo H. (2012). Numerical modelling based on the multiscale homogenization theory. Application in
composite materials and structures. Departamento de Resistencia de Materiales y Estructuras en la
Ingeniería – Barcelona, Spain. Advisor: S. Oller.
VI MECHANICALBEHAVIOR OF COMPOSITES
The original objective of the research project was and is still today to design composite
materials from their own components basic properties, without handling the composite as
a single material. This “concern” has permanently pushed forward the research
development and has opened the path towards the “material design” suitable for each
structure. The subject’s complexity and difficulty have demanded the intensive
implementation of many techniques previously developed by the author in other research
lines such as plasticity, anisotropy, introduction of rigid-body movements using the
constitutive equation, local instability through a constitutive formulation, large strains,
delamination problems, composites fatigue, etc. Simultaneously, a thorough data base and
parallelization intensive computational work has been carried out. Moreover, a finite
element code (cid:16)PLCd2(cid:16) has been developed and used as a “development and testing tool”
to implement these formulations. All this technology has been transferred to the COMET3
program, which is a finite element code developed at CIMNE for more general purposes.
Currently, research on composites is still under way and the temperature and humidity
effects on the behavior have been included for a more comprehensive study of the matrix-
fiber sliding phenomena. We are also trying to take these formulations to the structural
finite elements, such as shells and membranes.
This work has been possible thanks to the institutional support of CIMNE (International
Center for Numerical method in Engineering), which has financially supported this book
since its first edition in Spanish in 2003, and later in its English edition. In this latter task
many people have participated, and particularly I thank Ms. Hamdy Briceño and Prof.
Miguel Cerrolaza for their careful translation and revision of this text. I also thank all my
students who have contributed to the correction of the text during the eleven years that
this book has been used as a syllabus of "the Nonlinear Dynamics course" in the
Department of Strength of Materials, of the Technical University of Catalonia, Spain.
I hope these notes will contribute to a better understanding of the non-linear dynamics
and encourage the reader to study this subject in greater depth.
Sergio Oller
Barcelona, March 2014
2 PLCd-Manual. Non-linear thermomachanic finite element code oriented to PhD student education.
Code developed at CIMNE, 1991-to present (https://web.cimne.upc.edu/users/plcd/).
3 Cervera M., Agelet C. and Chiumenti M. (2002). COMET: Coupled Mechanical and Thermal Analysis. Data
input manual. Technical Report CIMNE Nº IT-308. Barcelona, Spain.
CCCOOONNNTTTEEENNNTTTSSS
1 INTRODUCTION 1-1
1.1 Use of composite materials 1-1
1.1.1 The use of composite materials in the automobile industry 1-1
1.1.2 The use of composite materials in the aeronautic industry 1-2
1.1.3 Composite materials in the naval industry 1-3
1.1.4 The use of composite materials in Civil Engineering 1-4
1.2 Composite properties 1-5
1.3 Classification of composite materials 1-7
1.3.1 Classification by topology 1-7
1.3.2 Classification by components 1-7
1.3.3 Classification by structure 1-8
2 MECHANICAL ANISOTROPY 2-1
2.1 Introduction 2-1
2.2 Generalities on the anisotropic formulation 2-2
2.3 Yield function and plastic potential for isotropic materials 2-5
2.4 General explicit definition of the isotropic yield criterion in the
referential configuration 2-6
2.5 General explicit definition of the orthotropic yield criterion in the
referential configuration 2-8
2.6 General implicit definition of the orthotropic criterion in the
referential configuration 2-10
2.6.1 Stress space transformation 2-11
2.6.2 Strain space transformation 2-13
2.7 General definition of the stress space transformation tensor 2-15
2.8 Numerical calculation of the adjusting tensor matrix form 2-17
2.9 Mises-Hill orthotropic criterion verification by the space mapping
theory 2-19
2.10 Anisotropy in the updated configuration 2-20
2.10.1 Transformation of the stress space 2-21
2.10.2 Transformation of the strain space 2-22
2.11 Plastic flow rule. Evolution law of internal variables 2-24
2.11.1 Referential configuration 2-24
