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Numerical 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