Lectures Notes on Advanced Structured Materials 2 (Advanced Structured Materials, 203) 3031490428, 9783031490422

Citation preview

Advanced Structured Materials

Holm Altenbach · Leonhard Hitzler · Michael Johlitz · Markus Merkel · Andreas Öchsner   Editors

Lectures Notes on Advanced Structured Materials 2

Advanced Structured Materials Volume 203

Series Editors Andreas Öchsner, Faculty of Mechanical and Systems Engineering, Esslingen University of Applied Sciences, Esslingen, Germany Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Porto, Portugal Holm Altenbach , Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

Common engineering materials are reaching their limits in many applications, and new developments are required to meet the increasing demands on engineering materials. The performance of materials can be improved by combining different materials to achieve better properties than with a single constituent, or by shaping the material or constituents into a specific structure. The interaction between material and structure can occur at different length scales, such as the micro, meso, or macro scale, and offers potential applications in very different fields. This book series addresses the fundamental relationships between materials and their structure on overall properties (e.g., mechanical, thermal, chemical, electrical, or magnetic properties, etc.). Experimental data and procedures are presented, as well as methods for modeling structures and materials using numerical and analytical approaches. In addition, the series shows how these materials engineering and design processes are implemented and how new technologies can be used to optimize materials and processes. Advanced Structured Materials is indexed in Google Scholar and Scopus.

Holm Altenbach · Leonhard Hitzler · Michael Johlitz · Markus Merkel · Andreas Öchsner Editors

Lectures Notes on Advanced Structured Materials 2

Editors Holm Altenbach Institute of Mechanics Otto-von-Guericke-Universität Magdeburg Magdeburg, Germany

Leonhard Hitzler Department of Materials Engineering Technische Universität München Munich, Bayern, Germany

Michael Johlitz Institut für Mechanik Universität der Bundeswehr München Neubiberg, Germany

Markus Merkel Institute for Virtual Product Development Aalen University of Applied Sciences Aalen, Baden-Württemberg, Germany

Andreas Öchsner Faculty of Mechanical and Systems Engineering Esslingen University of Applied Sciences Esslingen am Neckar, Baden-Württemberg Germany

ISSN 1869-8433 ISSN 1869-8441 (electronic) Advanced Structured Materials ISBN 978-3-031-49042-2 ISBN 978-3-031-49043-9 (eBook) https://doi.org/10.1007/978-3-031-49043-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The postgraduate seminar series on advanced structured materials is designed to facilitate teaching and informal discussion in a supportive and friendly environment. The seminar provides a forum for postgraduate students to present their research results and train their presentation and discussion skills. Furthermore, it allows for an extensive discussion of current research being conducted in the wider area of advanced structured materials. Doing so, it builds a wider postgraduate community and offers networking opportunities for early career researchers. In addition to focused lectures, the seminar provides specialized teaching/overview lectures from experienced senior academics. The 2023 Postgraduate Seminar entitled “Advanced Structured Materials: Development—Manufacturing—Characterization—Applications” was held from 20 to 24 February 2023 in Barcelona. The presented postgraduate lectures had a strong focus on polymer mechanics, composite materials, and additive manufacturing. Magdeburg, Germany Munich, Germany Neubiberg, Germany Aalen, Germany Esslingen am Neckar, Germany

Holm Altenbach Leonhard Hitzler Michael Johlitz Markus Merkel Andreas Öchsner

v

Contents

Overview Lectures Continuum Mechanics – Material Independent and Dependent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holm Altenbach 1 Some Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Important Stages in the Development of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Continuum Mechanics – Three-Dimensional Case . . . . . . . . . . . . . . . . . . . 2.1 Material Independent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Material Dependent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Some Comments Concerning Tensor Calculus . . . . . . . . . . . . . . . . . 3 One-Dimensional and Two-Dimensional Continuum Mechanics . . . . . . . 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Behavior of Classical Sandwich Beams . . . . . . . . . . . . . . . . . . . Andreas Öchsner 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Mechanical Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bending Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Tensile/Compressive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Shear Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Technical Sandwich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Bending Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Tensile/Compressive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Shear Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Bending Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 4 11 12 14 17 19 20 21 25 25 26 26 31 32 34 35 41 42 45 54

vii

viii

Contents

Classical Solidification Structures in Single-Step Metal Additive Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leonhard Hitzler 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Solidification Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Knowledge from Classical Processes . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Single-Step AM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additive Manufacturing as a Key Driver in the Mobility of Tomorrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirk Schuhmann and Markus Merkel 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Additive Manufacturing for Mobility Applications . . . . . . . . . . . . . . . . . . . 2.1 Additive Manufacturing of Plastic Components . . . . . . . . . . . . . . . . 2.2 Example of a Hybrid Material Made of Plastic and Metal (Copper and Steel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Practical Example of a Metallic Additive Component . . . . . . . . . . . . . . . . . 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 59 59 62 67 68 71 71 73 73 75 76 79 79 80

Postgraduate Lectures Experimental Analysis of Strain and Thermal Behaviour on 3D Printed Flexible Auxetic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Berta Pi Savall, Seyed Morteza Seyedpour, and Tim Ricken 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.1 Unit Cell Geometrical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.2 Preparation of Samples and Test Setup . . . . . . . . . . . . . . . . . . . . . . . 91 2.3 Post-processing of Obtained Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.4 Determination of Dynamic Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . 93 3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.1 Strain and Temperature Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.2 Determination of Dynamic Poisson’s Ratio ν . . . . . . . . . . . . . . . . . . 98 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Crack Evaluation of Ozone-Aged Elastomers . . . . . . . . . . . . . . . . . . . . . . . . Caroline Treib, Michael Johlitz, and Alexander Lion 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Material and Artificial Ageing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Artificial Ageing Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 104 105 105 105

Contents

3 Detection of Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Cracks on the Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Crack Depths in the Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Kinetic Approach for Evolution of Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selective Laser Sintered Pipe Specimen Under Torsional Load: Experimental Investigations and Material Modeling . . . . . . . . . . . . . . . . . . Dominik Hahne, Alexander Lion, and Michael Johlitz 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 SLS Production Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Torsion Specimen Design and Experimental Setup . . . . . . . . . . . . . 3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Kinematics of Pipe Specimen Under Torsional Load . . . . . . . . . . . . 3.2 Influence of the Building Direction on the Stress–Strain Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Influence of the Rate Dependency on the Stress–Strain Curve . . . . 3.4 Step Tests and Cyclic Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Material Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Three-Parameter Model of Viscoelasticity . . . . . . . . . . . . . . . . . . . . . 4.2 Relaxation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Parameter Identification from Relaxation Test . . . . . . . . . . . . . . . . . 4.4 Application of Parameter Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Importance of Heat Conduction and Global Temperature Fields in the Laser Powder Bed Fusion Process . . . . . . . . . . . . . . . . . . . . . . . Johannes Rottler, Till K. Tetzlaff, Alexander Lion, Kristin Paetzold-Byhain, and Michael Johlitz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Material and Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Samples and Test Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

109 109 111 116 118 119 120 123 123 124 124 125 128 128 130 131 131 134 134 135 136 137 139 140 141

142 143 143 144 145 148 149

Investigation of the Anisotropic Mechanical Behaviour of Short Carbon Fibre-Reinforced Polyamide 6 Fabricated via Fused Filament Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Julian Klingenbeck, Alexander Lion, and Michael Johlitz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

x

Contents

2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Process-Related Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Material-Related Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Anisotropy of 3D Printed Samples Under Tensile Load . . . . . . . . . 3 Investigation in Fibre Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Investigation of Interface Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of the Fatigue Behaviour of Laser Powder-Bed Fused AlSi10Mg Specimens Using Surface and Heat Treatments . . . . . . . Enes Sert, Janina Köppel, Marcel Butze, Fabian Brockmann, Markus Merkel, and Andreas Öchsner 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Process Description of Sample Production . . . . . . . . . . . . . . . . . . . . 2.2 Overview of Post Treatments and Examinations . . . . . . . . . . . . . . . . 2.3 Process Description Machining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Process Description T5 Heat Treatment . . . . . . . . . . . . . . . . . . . . . . . 2.5 Process Description Laser Polishing . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Process Description Shot Peening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Process Description Slide Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Fatigue Strength Test and Their Evaluation Methodology . . . . . . . 2.9 Crack Origin Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Wöhler Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fracture Mechanism Based on SEM Images . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sorption of Fuels in Additively Manufactured Thermoplastic Polyurethanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yvonne Breitmoser, Alexander Lion, Michael Johlitz, Sebastian Eibl, and Tobias Förster 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Experimental Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 AM Material and Aviation Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 3D Printing of Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Chemical Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Testing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Chemical Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 156 156 159 160 161 163 164 165 167 169 171

172 173 173 174 174 175 175 176 177 178 179 179 179 180 185 188 191

192 193 193 193 194 195 196 196

Contents

3.2 Behavior in Contact with Immersion Fluid . . . . . . . . . . . . . . . . . . . . 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distortion Compensation of Thin-Walled Parts by Pre-Deformation in Powder Bed Fusion with Laser Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefan Brenner and Vesna Nedeljkovic-Groha 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 FE Model and Artifact Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Refined Super Layer Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Non-Uniform Compensation Approach . . . . . . . . . . . . . . . . . . . . . . . 2.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Validation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Investigation of the Thermophysical Properties of Porous Steel Components Made by Selective Laser Melting . . . . . . . . . . . . . . . . . . . . . . . . Jens Kortsch-Banzhaf, Garvin Schultheiß, and Markus Merkel 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Density Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Transient Plane Source Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of the Heating Behavior and the Strength of Hot Gas Welded Polyamides with 3D Contours Using an Immersing Nozzle System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Schmid, Simon J. Heienbrock, Dennis Mayer, Dennis F. Weißer, and Matthias H. Deckert 1 The Hot Gas Welding Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Nozzle Systems for Hot Gas Welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hot Gas Welding System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Top Nozzle System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Types of Tested Plate Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Tested Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Influence of the Heating Time on the Weld Strength Depending on the Inclination Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

196 200 201 202 205 206 207 207 208 209 211 212 212 214 217 218 221 221 222 222 224 225 226 227 228

229

230 230 232 232 233 234 235 236

xii

Contents

4.1 Plate Specimen Without Inclination Angle . . . . . . . . . . . . . . . . . . . . 4.2 Plate Specimen with 15° Inclination Angle . . . . . . . . . . . . . . . . . . . . 4.3 Plate Specimen with 30° Inclination Angle . . . . . . . . . . . . . . . . . . . . 4.4 Plate Specimen with 45° Inclination Angle . . . . . . . . . . . . . . . . . . . . 4.5 Plate Specimen with 60° Inclination Angle . . . . . . . . . . . . . . . . . . . . 5 Discussion of the Influence of the Inclination Angle on the Weld and Component Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Investigation of the Inclination Angle on a PA6-GF30 . . . . . . . . . . 5.2 Investigation of the Plate Specimen Shape on a PA6-GF30 . . . . . . 5.3 Investigation of the Inclination Angle on a PA66-GF35 . . . . . . . . . 5.4 Investigation of the Plate Specimen Shape on a PA66-GF35 . . . . . 5.5 Investigation of the Fracture Pattern of the Plate Specimen . . . . . . 6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation of Lattice Structures with Johnson–Cook Material and Damage Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Julius Cronau and Florian Engstler 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Lattice Structure Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Mechanical Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Drop Tower Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Johnson–Cook Material Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Johnson–Cook Failure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Validation of the Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 BCC Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Vin Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Residual Stress, Surface Roughness, and Porosity on Fatigue Life of PBF-LB AlSi10Mg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lea Strauß and Günther Löwisch 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Manufacturing Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Material Testing Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Modeling the Correction to the Stress Amplitude . . . . . . . . . . . . . . .

236 238 238 240 241 242 244 244 246 246 247 249 250 253 253 256 256 257 257 257 258 259 259 260 261 262 262 262 267 267 269 270 271 275 275 277 277 278 280

Contents

3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Effect of Heat Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Effect of Printing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Modeling the Influence Factors to Predict Fatigue Life . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastoplastic Characterization of a Two-Component Epoxy-Based Structural Adhesive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Ascher, Ralf Späth, and Michael Johlitz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Static Tensile Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Torsion Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Static Tensile Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Torsion Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation of Infrared-Welded Short Fiber-Reinforced Thermoplastic Parts Based on Mori–Tanaka Homogenization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lucas Schraa, Niklas Hoenen, Kai Uhlig, Karina Gevers, Paul Töws, Volker Schöppner, Julia Decker, and Markus Stommel 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Aim of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Experimental Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Manufacturing of the Infrared-Welded Components . . . . . . . . . . . . 3.2 Determination of the Matrix Properties . . . . . . . . . . . . . . . . . . . . . . . 3.3 Quantification of the Local Fiber Fraction by TGA . . . . . . . . . . . . . 4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Reconstruction of the Fiber Orientation . . . . . . . . . . . . . . . . . . . . . . . 4.3 Adapted Mori–Tanaka Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Validation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Application to a Component Simulation . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

282 282 283 284 287 289 291 292 293 293 295 298 298 300 303 305 305

307

308 310 310 311 311 312 314 314 319 322 324 324 325 328 329

Analysis of Heterogeneous Ageing of HNBR O-Rings . . . . . . . . . . . . . . . . . 331 Maha Zaghdoudi, Anja Kömmling, Matthias Jaunich, and Dietmar Wolff 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 2 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

xiv

Contents

2.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Ftir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Compression Stress Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 IRHD Microhardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Characterisation and Modelling of the NR-BR Blend Under Harmonic Excitation Considering Self-heating Effect . . . . . . . . . . . Ondrej Farkas, Tomas Gejgus, Bruno Musil, and Michael Johlitz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Material Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Hyperelastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Thermo-viscoelastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Modelling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Material Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Constitutive and Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Governing Equations and Numerical Implementation . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

333 334 334 334 335 340 346 346 349 350 351 352 353 354 356 356 357 359 360 363 364

Thermal Analysis of Old Climbing Ropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Philippe du Maire, Max Friebertshäuser, Matthias H. Deckert, Michael Johlitz, and Andreas Öchsner 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Dynamic Differential Scanning Calorimetry . . . . . . . . . . . . . . . . . . . 2.2 Thermogravimetric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367

Notch Effect Under Cyclic Loading of Electrical Steel Strips . . . . . . . . . . . Patrick Schwarz, Peter Häfele, and Konstantin Naumenko 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Models to Account for Micro- and Macro-Notches Under Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Material Investigated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Specimen Preparation and Edges Validation . . . . . . . . . . . . . . . . . . . 2.3 Mechanical Properties of NO30-15 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Stress-Controlled Fatigue Tests on NO30-15 . . . . . . . . . . . . . . . . . .

377

368 369 370 371 371 375 375

378 379 381 381 382 386 387

Contents

2.5 2.6 2.7

Strain-Controlled Fatigue Tests on NO30-15 . . . . . . . . . . . . . . . . . . Discussion of the Quasi-Static Test Results . . . . . . . . . . . . . . . . . . . . Discussion of Stress-Controlled Tests with Unnotched Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Comparison of the Test Results with Methods for the Computational Evaluation of the Surface Influence . . . . . . . 2.9 Discussion of Test Results with Notched Specimens . . . . . . . . . . . . 2.10 Application of the Local Concept to the Test Results of NO30-15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Ways to Numerically Implement the Two-Time-Scale Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katharina Knape and Holm Altenbach 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Elastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Inelastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Application of the Two-Time-Scale Approach . . . . . . . . . . . . . . . . . 3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Backward Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stress-Update Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Return-Mapping Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of Site Calibration Set for Prefixed Wall Transmitters of Environmental Monitoring Systems in Regulated Industries . . . . . . . . . Mirna Osama and Ahmed Y. Shash 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Developments to be Done to the Calibration Set . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

390 392 393 394 395 396 398 399 401 401 402 403 403 404 405 405 406 407 408 410 411 413 413 415 418 420 420

Overview Lectures

Continuum Mechanics – Material Independent and Dependent Equations Holm Altenbach

Abstract Continuum mechanics in its rational formulation is a central element within mechanics. The basic concepts can be easily transferred to many special cases, with kinematics, stress models, balance equations, and constitutive laws being the most important elements of the theories. It also applies here that statics is the special case in Newton’s sense. Selected historical stages are briefly described below. The three-dimensional continuum theory is then briefly discussed. Finally, a note on the one- and two-dimensional special cases is given. The chapter is written from the point of view of a solid-state mechanics. However, numerous equations can easily be transferred to fluids.

1 Some Historical Remarks In this section, some basic references are introduced for further reading. There are much more sources, but they are often devoted to special items. In addition, some interesting stages in the history of continuum mechanics are discussed.

1.1 References Today there is an extensive literature on the history of mechanics, some of which are also written as autobiographies or biographies. Possibly the first comprehensive account of parts of solid mechanics is the two-volume edition by Todhunter,1 Todhunter (.∗ 23 November 1820, Rye, Sussex, England, .†1 March 1884, Cambridge, Cambridgeshire, England) was an English mathematician and well known for books on the history of sciences. He was elected as a Fellow of the Royal Society. 1 Isaac

H. Altenbach (B) Lehrstuhl für Technische Mechanik, Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke-Universität, Magdeburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. Altenbach et al. (eds.), Lectures Notes on Advanced Structured Materials 2, Advanced Structured Materials 203, https://doi.org/10.1007/978-3-031-49043-9_1

3

4

H. Altenbach

edited and completed by Pearson2 [69, 70]. Additional information on the history of mechanics one can get from [52, 53, 68, 72] among others. There are also some interesting Russian sources published in the last two–three decades [38, 62, 66, 67].

1.2 Important Stages in the Development of Continuum Mechanics Mechanics is one of the oldest sciences with a history of more than 2000 years. The great scholar Archimedes of Syracuse3 combined scientific knowledge, for example, mechanics, with those of mathematics and technology or applied them there. He formulated the first laws of mechanics, e.g., for the lever, in the language of mathematics and used them to build technical devices. The earliest remaining writings regarding levers date from the third-century BC and were provided by Archimedes “Give me a place to stand on, and I shall move the earth”. This is a remark of Archimedes who formally stated the correct mathematical principle of levers. One can easily verify that considering the forces of a person and the mass of the earth and the resulting weight force, the length of the lever assumes an unrealistic value. However, it is assumed that in ancient Egypt, constructors used the lever to move and uplift obelisks weighting more than 100 tons. The force applied (at end points of the lever) is proportional to the ratio of the length of the lever arm measured between the fulcrum (pivoting point) and application point of the force applied at each end of the lever. Mathematically, this is equivalent to expression for the moment .

M = Fd

(1)

with . F as the force and .d as the length. Archimedes himself developed the scientific basics of statics for statically determined systems from the law of the lever. Unfortunately, the definition of moment by force multiplied by lever length is only one variant. Beginning in the seventeenth century, this knowledge was gradually implemented in mechanics. It was already known in ancient Egypt that liquids have different viscosities and that they depend on external conditions such as temperature. The constitutive behavior of deformable solids came into the focus of scientists in the seventeenth century. In 1660, Robert Hooke4 discovered the first constitutive law of elasticity Karl Pearson (born as Carl Pearson, .∗ 27 March 1857, Islington, London, England, .†27 April 1936, Coldharbour, Surrey, England) was an English mathematician and biostatistician. He established the mathematical statistics and was elected as a Fellow of the Royal Society and a Fellow of the Royal Society of Edinburgh. He was a staunch supporter of eugenics. 3 About .∗ 287–.†212 BC, Syracus, Sicily. 4 .∗ 18 July 1635, Freshwater, Isle of Wight, England, .†3 March 1703, London, England, English polymath being a scientist, natural philosopher, and architect. He was one of the first scientists discovering microorganisms with a help of a compound microscope built himself. He was a member of the Royal Society and since 1662 its curator of experiments. 2

Continuum Mechanics – Material Independent and Dependent Equations

5

Fig. 1 Visualization of Hooke’s law

for deformable solids. He found a linear relationship between tension and extension in an elastic spring. He published this law in an anagram “ceiiinosssttuv” and the solution “Ut tensio, sic vis”5 was given in Latin by him in 1678. The visualization of Hooke’s law is given in Fig. 1. Hooke’s law was one of the first constitutive laws taking into account the materials’ properties. At the same time, one can say that this was the beginning of the rheological modeling since the behavior of a linear-elastic spring (Fig. 2) is similar to Hooke’s law. For systems that obey Hooke’s law, the extension produced is directly proportional to the load .

F = −k∆x

(2)

with the force . F, the extension .∆x, and the spring stiffness .k. The last one is specific for each spring and depends on the properties of the spring material and the spring geometry. Note that the modern formulation of Hooke’s law is σ = Eε

.

(3)

(with .σ as the stress, .ε as the strain and . E as Young’s modulus) was not known in Hooke’s time since stress and strain were introduced later by Cauchy and . E was established later by Young.6 The end of the seventeenth and eighteenth centuries was the period of intense development of mechanics, as scholars such as Sir Isaac Newton7 , members of 5

English translation “As the extension, so the force”. The concept was developed in 1727 by Leonhard Euler and the first experiments that used the concept of Young’s modulus in its current form were performed by the Italian Giordano Riccati (.∗ 25 February 1709, Castelfranco, Veneto, .†20 July 1790, Treviso, Veneto, contributions to music, architecture, mechanics, and mathematics.) in 1782 – predating Young’s work by 25 years. 7 .∗ 4 January 1643, Woolsthorpe-by-Colsterworth, Lincolnshire, England, .†31 March 1727, Kensington, Middlesex, Great Britain, English mathematician, physicist, astronomer, alchemist, theologian, 12.th President of the Royal Society. 6

6

H. Altenbach

Fig. 2 Linear-elastic spring as rheological model

the Bernoulli family, and Leonhard Euler8 among others established significant foundations of mechanics. Many of these scientists were mathematicians, since in the spirit of da Vinci’s9 phrase, “Mechanics is the paradise of mathematics”, for many mathematicians the application of their findings lay in mechanics. Since that time (and this view still exists today), mechanics is often referred to as Newtonian mechanics. In Newton’s famous book Principia Mathematica his axioms (sometimes named laws) were firstly published in 1687: Newton’s first law There exists a set of inertial reference frames relative to which all particles with no net force acting on them will move without change in their velocity. This law is often simplified as “A body persists its state of rest or of uniform motion unless acted upon by an external unbalanced force.” Newton’s first law is often referred to as the law of inertia. Newton’s second law Observed from an inertial reference frame, the net force on a particle is proportional to the time rate of change of its linear momentum: .

F =

d(mvv ) . dt

(4)

This law is often stated as “Force equals mass times acceleration” .

F = maa .

(5)

8 .∗ 15 April 1707, Basel, Swiss Confederacy,.†18 September 1783, Saint Petersburg, Russian Empire,

mathematician, physicist, astronomer, geographer, logician, and engineer working in Switzerland, Prussia, and in the Russian Empire. 9 Leonardo di ser Piero da Vinci (.∗ 15 April 1452, possibly Anchiano, Vinci, Republic of Florence,.†2 May 1519 (aged 67) Clos Lucé, Amboise, Kingdom of France), Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect.

Continuum Mechanics – Material Independent and Dependent Equations

7

The net force on an object is equal to the mass of the object multiplied by its acceleration. Newton’s third law Whenever a particle A exerts a force on another particle B, B simultaneously exerts a force on A with the same magnitude in the opposite direction. The strong form of the law further postulates that these two forces act along the same line. This law is often simplified into the sentence “To every action there is an equal and opposite reaction.” This law is often referred to as law of reciprocal actions. Newton’s fourth law Forces acting on a mass point or rigid body can be added as vectors. This law is often referred to as law of superposition. It is obvious that the Newtonian mechanics based on these axioms is valid only for point masses and it is surprising that Newton “ignored” independent rotations since Jacob I Bernoulli10 established the postulate of the angular momentum independent from the linear momentum. The Euler–Bernoulli beam bending theory is based on independent deflections and rotations (forces and moments). Today in some textbooks of continuum mechanics the balance of momentum and the balance of moment of momentum are named Euler’s laws of motion (which are the extension of Newton’s law of motion). They were formulated approximately 50 years after Newton and at first applied to rigid bodies. A precise discussion w.r.t. the independence of these laws in continuum mechanics can be found in Truesdell [71]. In the eighteenth century, many investigations in mechanics were performed by French and British scholars. For example, Augustin-Louis Cauchy11 defined stress as a measure of the average amount of force . F exerted per unit area . A. It is a measure of the intensity of the total internal forces acting within a body across imaginary internal surfaces, as a reaction to external applied forces and body forces. It was introduced into the theory of elasticity by Cauchy around 1822. In addition, he introduced a lemma (now named Cauchy’s Lemma or Cauchy’s axiom) σn = n · σ

.

(6)

which is the relation between the stress vector .σ n on the surface with the normal .n and the stress tensor .σ . Cauchy got the formulation from the equilibrium of forces on an infinitesimal tetrahedron (the derivation can be found among others in Naumenko and Altenbach [56]). The meaning that the lemma is based on the linear mapping of two vector spaces with the help of a second rank tensor is a misunderstanding since in Cauchy’s time vector spaces were unknown. In addition, the mapping expression σn = σ · n

.

10 .∗ 6

(7)

January 1655, Basel, Switzerland, .†16 August 1705, Basel, Switzerland, mathematician with contributions to mechanics. 11 .∗ 21 August 1789, Paris, France, .†3 May 1857, Sceaux, France, French mathematician, engineer, and physicist with pioneering contributions to continuum mechanics.

8

H. Altenbach

is equivalent only in the case if the stress tensor is a symmetric tensor (.σ = σ T ). With other words, for micro-polar continua Eq. (6) is valid and a similar equation for the non-symmetric moment stress tensor should be established (see Eremeyev et al. [34]). With the goal of developing a complete theory for the behavior of isotropic linearelastic bodies, the issue of material parameters was also important. One of such constitutive parameters was introduced by Thomas Young12 . Young’s modulus can be defined. Definition 1 (Young’s modulus) The elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance’s tendency to be deformed elastically (i.e., non-permanently) when a force is applied to it. Today, the elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region: .

E≡

σ stress = strain ε

(8)

If the stress–strain curve is linear, the modulus is constant, otherwise in the nonlinear elastic range the instantaneous elastic modulus can be defined. Young’s modulus (. E) describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. In addition to the elastic modulus, other moduli were introduced: • The shear modulus .G (or the second Lamé parameter .μ) describes an object’s tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces; it is defined as shear stress over shear strain .

G=

τ tan γ

(9)

with the shear stress .τ and the shear angle .γ . In the case of small .γ , it holds that values .tan γ ≈ γ . The shear modulus is also a part of the derivation of viscosity. • The bulk modulus (. K ) describes volumetric elasticity, or the tendency of an object’s volume to deform when under pressure; it is defined as volumetric stress over volumetric strain (. p is the hydrostatic, .εvol is volumetric strain .

K =−

p εvol

,

(10)

and is the inverse of compressibility. The bulk modulus is partly an extension of Young’s modulus to three dimensions.

12 .∗ 13 June 1773, Milverton, Somerset, England,.†10 May 1829, London, England, British polymath

with contributions to the fields of light, solid mechanics, physiology, language, musical harmony, and Egyptology.

Continuum Mechanics – Material Independent and Dependent Equations

9

• The first Lamé parameter .λ is defined as λ=

.

Eν . (1 + ν)(1 − 2ν)

(11)

Remark 1 The introduced elastic parameters and Poisson’s ratio are not independent. Assuming isotropic material behavior two of them are independent, for example, E .K = (12) 3(1 − 2v) is valid, if . E and .ν are the basic parameters. In engineering calculations . E and Poisson’s ratio .ν are often used, in rheological modeling .G and . K are preferred. In addition, in the mathematical theory of elasticity Lamé’s parameters are used. Further relations between the parameters are presented, for example, in Altenbach [3]. Remark 2 In many books, you will find the term elastic constants instead of elastic parameters. Since the introduced quantities are not constant but, for example, temperature dependent, the term material parameter is to be preferred. Remark 3 The value ranges of the elastic parameters can often be determined intuitively. Thus . E > 0 is valid and 0 as well as negative values can be excluded. For Poisson’s ratio, the experimentally measured value range .0 < ν < 0.5 was given for many years. .ν = 0.5 corresponds to the incompressibility condition. However, .−1 < ν < 0.5 can be derived from the positive definiteness of the strain energy, so that the group of auxetic materials can also be included. Siméon- Denis Poisson13 presented the equations of the three-dimensional isotropic linear-elastic behavior. There was a long discussion concerning the question one or two constitutive parameters. In some papers, there was the suggestion that the now so-called Poisson’s ratio is a fixed value for all materials. In the modern invariant form, the isotropic linear-elastic behavior can be presented as σ = α tr(εε )II + βεε + γ ε T

.

(13)

with three parameters. But if the stress tensor is symmetrical the number of parameters is reduced to two T .σ = σ ⇒ σ = 2μεε + λtrεε I (14) with the Lamé14 parameters .μ and .λ. Both can be expressed by μ=G=

.

13 .∗ 21

E , 2(1 + ν)

λ=

Eν (1 + ν)(1 − 2ν)

(15)

June 1781, Pithiviers, Kingdom of France (present-day Loiret), .†25 April 1840, Sceaux, Hauts-de-Seine, Kingdom of France, French mathematician and physicist. 14 Gabriel Lamé (.∗ 22 July 1795, Tours, .†May 1870, Paris), French mathematician.

10

H. Altenbach

It is easy to show the often in rheology presented two constitutive equations σ = 3K trεε , trσ

.

σ = 2μdevεε devσ

(16)

are equivalent since the bulk modulus can be estimated by .

K =λ=

2μ 3

(17)

and the operators tr and dev are defined for a second rank tensor . A as A = A · I, trA

.

1 A. A = A − trA devA 3

(18)

At the end of the nineteenth century, several variants of continuum mechanics were established. Among them theories for linear-elastic solid and for some fluids (for example, the Navier–Stokes equations). In addition, some advanced rheological models were established. The last one partly is discussed in [4, 10]. The late nineteenth and the twentieth centuries are characterized by a formulation of various branches of continuum mechanics. One of the reasons was the establishment of tensor calculus, for example, by Josiah Willard Gibbs15 . This powerful tool allows to present the basics of continuum mechanics in a very elegant form – invariant or index-free or with indices. François Cosserat16 and EugéneMaurice- Pierre Cosserat17 developed a new theory for deformable bodies based on the balances of momentum and moment of momentum [30]. The extended theory was without applications (the Cosserat brothers did presented constitutive equations and only in the 50.th of the twentieth-century Ericksen18 and Truesdell19 [36], Aero20 & Kuvshinskii [1], Palmov21 [60] and many others brought back the ideas of 15 .∗ 11

February 1839, New Haven, Connecticut, U.S., .†28 April 1903, New Haven, Connecticut, U.S., American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. 16 .∗ 26 November 1852, Douai, France, .†22 March 1914, France, French engineer and mathematician. 17 .∗ 4 March 1866, Amiens, France, .†31 May 1931, Toulouse, France, French mathematician and astronomer. 18 Jerald LaVerne Ericksen (.∗ 20 December 1924, Portland, Oregon, USA, .†11 June 2021, Minisota, USA), American mathematician specializing in continuum mechanics. 19 Clifford Ambrose Truesdell III (.∗ 18 February 1919, Los Angeles, USA, .†14 January 2000, Baltimore, USA), mathematician, natural philosopher, and historian of science [9]. 20 Eron Lyuttovich Aero (.∗ 14 May 1934, Naryshkino, Penza district, Soviet Union, .†11 July 2016, St. Petersburg, Russia), Soviet/Russian scientist, who made an outstanding contribution to the development of the mechanics of generalized continua such as Cosserat continuum and liquid crystals [35]. 21 Vladimir Aleksandrovich Palmov (.∗ 7 July 1934, Batumi, Soviet Union, .†15 October 2018), St. Petersburg, Russian Federation, Soviet/Russian scientist in the field of theoretical and applied mechanics [6, 7].

Continuum Mechanics – Material Independent and Dependent Equations

11

the Cosserat theory. Albert Einstein22 had made several contributions to mechanics: tensor calculus (for example, summation convention) and the development of the relativistic mechanics. An additional motivation came from David Hilbert23 , who was one of the most influential mathematicians. In one of his famous papers he published 23 problems in mathematics, unsolved at the time. The 6.th problem was: mathematical treatment of the axioms of physics: • axiomatic treatment of probability with limit theorems for foundation of statistical physics; • the rigorous theory of limiting processes “which lead from the atomistic view to the laws of motion of continua.” This was the motivation for many scholars to develop new theories, but till now there are only particular solutions.

2 Continuum Mechanics – Three-Dimensional Case From the history of continuum mechanics one can conclude: differential or integral equations can thus be employed in solving problems in continuum mechanics. Some of these equations are specific to the materials being investigated and are called constitutive equations, while others capture fundamental physical laws, such as conservation of mass (continuity equation), the conservation of momentum (equations of motion and equilibrium), and conservation of energy (first law of thermodynamics)24 . Definition 2 (Continuum) The continuum is an infinite number of material points with properties which changes continuously (distributed uniformly throughout, and completely fills the space it occupies). The basic definition is not limited by • • • • •

the dimension of the space, the degrees of freedom of material points, the validity for solids and fluids, the type of external loadings (mechanical, electrical, …), and the scale (macroscopic, microscopic, etc.).

In many cases, the following assumptions make the solution of problems easier: 22 .∗ 14

March 1879, Ulm, Kingdom of Württemberg, German Empire, .†18 April 1955, .†18 April 1955, Princeton, New Jersey, US, German-born theoretical physicist. 23 .∗ 23 January 1862, Königsberg or Wehlau, Prussia, .†14 February 1943, Göttingen, Germany, German mathematician. 24 The different interpretations of conservation laws and balance equations will be mentioned here, but not discussed.

12

• • • • • •

H. Altenbach

space: .R3 , reference point 0; time: .t0 ≤ t < ∞; body: .B with volume .V and surface . A; mass; homogeneity; isotropy.

2.1 Material Independent Equations Continuum mechanics deals with physical quantities, which are independent of any particular coordinate system in which they are observed. These physical quantities are then represented by tensors, which are mathematical objects that are independent of coordinate system (invariant representation). These tensors can be expressed in coordinate systems for computational convenience. The modern representation is based, for example, on [3, 50] • Material independent equations (as usual as integral relations): – – – – –

Balance of mass. Balance of momentum. Balance of moment of momentum. Balance of energy. Balance of entropy.

Remark 4 The balances are more general in comparison with conservation laws [3]. But there are other statements of the balance equations: instead of two terms on the right-hand side responsible for the changes of the balance properties one introduces three terms [43]. Remark 5 The balance of mass in solid mechanics is often used as a conservation law. Remark 6 The balance of entropy can be an equality or inequality. Remark 7 There are arguments in Müller [54] that conservation laws are more general. • Material dependent equations (constitutive equations). – Constitutive equations. – Evolution equations.

Continuum Mechanics – Material Independent and Dependent Equations

13

Remark 8 The constitutive equations are the individual response of the material on loads (in the general sense – mechanical, thermal, electrical, magnetic, etc.). A excellent definition of “constitutive equations” is presented, for example, in [46] or in [4]. • Conditions for solving systems of partial differential equations in solid mechanics: – Boundary Conditions in Solid Mechanics Kinematic conditions – in this case the displacements .u are prescribed and one gets .u = u 0 . Static conditions – in this case the stresses .σ are prescribed and one gets .σ = σ 0 . Assuming Hookey’s law, the stresses are functions of the strains and since the strains are related to the first derivatives of the displacements one gets conditions of first derivative type. Mixed conditions – there are two meanings: the first one means that the sum of both the functions itself and the first derivative is prescribed and the second one means that on one part the boundary kinematic and on another part the static conditions are prescribed.

Remark 9 The boundary conditions are partly named Dirichlet25 (or first-type) boundary condition, Neumann26 (or second-type) boundary condition, and Robin27 (or third-type) boundary condition. Remark 10 ln solid mechanics, dimensionally reduced models such as beams or plates are often used. These are described by ordinary or partial differential equations of higher than second order. Thus, boundary conditions with higher than the first derivative have to be introduced. Examples are boundary conditions for the bending moments or the shear forces. • Initial Conditions in Solid Mechanics – Displacements .u are prescribed, e.g., for the initial moment .t0 : .u = u (t0 ). – Velocities .u˙ are prescribed, e.g., for the initial moment .t0 : .u˙ = u˙ (t0 ). • Jump conditions If the fields represented in the theory are not smooth enough (for example, in shock wave problems) one should introduce jump conditions. Details are discussed, e.g., in [18, 29, 40, 55, 73]. Johann Peter Gustav Lejeune Dirichlet (.∗ 13 February 1805, Düren, French Empire, .†5 May 1859, Göttingen, Kingdom of Hanover) was a German mathematician. 26 Carl Gottfried Neumann (.∗ 7 May 1832, Königsberg, Prussia, .∗ 27 March 1925, Leipzig, Germany) was a German mathematician. 27 Victor Gustave Robin (.∗ 17 May 1855, Paris, France, .†20 November 1897, Paris, France) was a French mathematical analyst and applied mathematician. 25

14

H. Altenbach

2.2 Material Dependent Equations The formulation of the material dependent equations is the main challenge in continuum mechanics. They express the specific (individual) response of the given material on arbitrary load. That means that an universal constitutive equation cannot be established. The main modeling principles are related to Altenbach [3] • the inductive approach which means from the simplest to more complex models (or bottom up); • the deductive approach which means from the general frame to special cases (or top down); and • the rheological modeling. An additional problem is the identification of the constitutive equations which is based on [3, 41, 42] • • • •

experimental observations, mathematical analysis, theory of symmetry (Curie28 –Neumann’s29 principle), and constraints.

All these ideas can be found in the many monographs, for example, [41, 43, 46, 55]. In addition, the formulation of constitutive equations is widely presented in books and papers devoted to rheology [4, 39, 63, 64]. In addition, an alternative concept of development constitutive equation (based on experimental observations) is suggested in Lemaitre and Chaboche [48]. Here we make the focus on contributions of the Leningrad (now St. Petersburg) School of Mechanics in the Leningrad Polytech Institute (now Peter the Great St. Petersburg Polytechnic State University). For example, in the monograph of Anatolii Isakovich Lurie30 [50]31 the basic concepts of continuum mechanics were presented. From the contents one gets an overview of the basic ideas of continuum mechanics. Here is the contents of the book: I. Basic concepts of continuum mechanics 1. Stress tensor. 2. Deformation of a continuum. II. Governing equations of the linear theory of elasticity 3. The constitutive law in the linear theory of elasticity. Pierre Curie (.∗ 15 May 1859, Paris, France, .∗ 19 April 1906, Paris, France) was a French physicist, a pioneer in crystallography, magnetism, piezoelectricity, and radioactivity. 29 Franz Ernst Neumann (.∗ 11 September 1798, Joachimsthal, Holy Roman Empire, .†23 May 1895, Königsberg, German Empire) was a German mineralogist and physicist. 30 .∗ 6 July 1901, Mogilew, Russian Empire, .†12 February 1980, Leningrad, Soviet Union, Soviet scientist in the field of theoretical and applied mechanics and control processes. 31 The Russian edition was published in 1970. 28

Continuum Mechanics – Material Independent and Dependent Equations

15

4. Governing relationships in the linear theory of elasticity. III. Special problems of the linear theory of elasticity 5. Three-dimensional problems in the theory of elasticity. 6. Saint-Venant’s problem. 7. The plane problem of the theory of elasticity. IV. Basic relationships in the nonlinear theory of elasticity 8. Constitutive laws for nonlinear elastic bodies. 9. Problems and methods of the nonlinear theory of elasticity. V. Appendices A. B. C. D. E. F.

Basics tensor algebra. Main operations of tensor analysis. Orthogonal curvilinear coordinates. Tensor algebra in curvilinear basis. Operations of tensor analysis in curvilinear coordinates. Some information on spherical and ellipsoidal functions.

It is obvious that the concept was strongly connected with mathematics. However, mechanics first, after that the choice of appropriate mathematics was the next challenge. In addition, Lurie published books and articles only when he had fully investigated the problems. Therefore, it is not surprising that his monograph [50] focused on “linear elasticity”. In 1980 (the year of his death) followed the “Nonlinear Elasticity Theory”, which was also translated [49]. Lurie’s work continued with his pupils. First and foremost, Palmov and Pavel Andreevich Zhilin32 should be mentioned here [59, 61, 77]. The above-mentioned four items (experimental observations, mathematical analysis, theory of symmetry, and introduction of constraints) should be discussed more in detail. Below briefly some comments concerning these items are given. Experimental investigations accompanied always the mathematical description of the material behavior. This applies to both variants of the material equations: the constitutive and the evolution equations. There are different approaches for the constitutive equations. In practice, algebraic, differential, and integral equations are often used. For complex material models, more general approaches such as functionals also occur. The evolution equation describes, for example, the development over time of processes in the material such as plasticity, creep, and damage. Thus, first-order differential equations with respect to time are dominant here. However, in both cases, we have parameters or parameter functions in the equations, which should be established experimentally. Note that such experiments can be performed 32 .∗ 8

February 1942, Velikiy Ustyug, Vologda region, Soviet Union, .∗ 4 December 2005, St Petersburg, Russian Federation, Soviet/Russian scientist in the field of theoretical mechanics [11, 21, 22].

16

H. Altenbach

in the laboratory (real tests) and no more and more with the help of computers (virtual tests). The verification of the established constitutive and evolution equation can be realized only by real tests. The mathematical analysis is important in both the inductive and deductive formulation of constitutive equations. The inductive approach means that the starting point is the simplest model and the extension (generalization) is given step by step. For example, assuming elastic behavior, Hooke’s law σ = Eε

.

(19)

with the scalars .σ (stress), . E (Young’s modulus), and .ε (strain in the stress direction) can be introduced. This is a linear algebraic function. A similar linear function can be presented for the shear behavior τ = Gγ

.

(20)

with the scalars.τ (shear stress),.G (shear modulus) and.γ (shear strain). The extension to the three-dimensional state is based on the assumption that the scalars should be substituted by tensors, but the mathematical character of Eqs. (19) and (20). From the mathematics and mechanics, it follows that the scalars .σ and .ε should be substituted by their counterparts .σ and .ε . Both are second rank tensors and a linear relation between two second rank tensors can be given with the help of a fourth rank tensor σ = (4) E · ε .

.

(4) E

(21)

E is the fourth rank elasticity tensor and .· is the scalar product. It is obvious that nonlinear elastic behavior can be presented by extending Eq. (21) by a series as shown, for example, in Altenbach [3]. In the case of deductive approaches, the starting point is functionals. They can be used by the axioms of the material’s theory like equipresence, determinism, causality, or consistency [3, 41]. Finally, the general equations will be simplified step by step. The Curie–Neumann principle, or principle of symmetry, states that, if a crystal is invariant with respect to certain symmetry operations, any of its physical properties must also be invariant with respect to the same symmetry operations, or otherwise stated, the symmetry operations of any physical property of a crystal must include the symmetry operations of the point group of the crystal [31, 58]. With other words any symmetry of a cause must appear in its effect, while the effect may possess symmetry that is not symmetry of the cause. In our case, the principle can be applied not only to crystals, but to arbitrary materials. Since the principle has a clear mathematical formulation it can be applied for simplification of the constitutive equations. The last item w.r.t. the simplification of the constitutive and evolution equations is the introduction of geometric constraints like the incompressibility condition. Such constraints result in reduction of the degrees of freedom of material points or infinitesimal small bodies. The assumption of the plane stress or plane strain states is also a constraint. The application of geometric constraints generally simplifies the

.

Continuum Mechanics – Material Independent and Dependent Equations

17

problem and thus allows special solutions. But at the same time it leads to mathematical contradictions which cannot be overcome. For example, the plane stress state is associated with a reduction from six to three stresses. However, it should be noted that four strains are preserved and thus contradictions in the constitutive equations occur. These can be partially eliminated, or the error can be reduced. For example, in the mechanics of composites for unidirectional laminates the quality of the solutions will be better by introducing reduced stiffnesses [5].

2.3 Some Comments Concerning Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in continuum mechanics and the derivation of the governing equations for applied problems. Two possibilities exist for the representation of the tensors and the tensorial equations: • the direct (symbolic, coordinate-free) notation and • the index (component) notation. The direct notation operates with scalars, vectors, and tensors as physical objects defined in multi-dimensional spaces. In continuum mechanics, we limit ourselves to the three-dimensional case. For special applications (for example, plates and shells) two-dimensional spaces can be applied. A vector (first rank tensor) .a is considered as a directed line segment rather than a triple of numbers (coordinates). Definition 3 (Vector) A vector in the three-dimensional Euclidean33 space is defined as a directed line segment with specified scalar-valued magnitude and direction. • The magnitude (the length) of a vector .a is denoted by .|aa |. • Two vectors .a and .b are equal if they have the same direction and the same magnitude. • The zero vector .0 has a magnitude equal to zero. In continuum mechanics, two types of vectors can be introduced: • The vectors of the first type are directed line segments. These vectors are associated with translations in the three-dimensional space and are named polar vectors. Examples for polar vectors include the force, the displacement, the velocity, the acceleration, the momentum, etc. • The second type is used to characterize spinor motions by an axial vector. It is related to quantities, i.e., the moment, the angular velocity, the angular momentum, etc.

33

Around 300 BC, ancient Greek mathematician active as a geometer and logician.

18

H. Altenbach

Fig. 3 Spin vector and its representation: a Spin vector, b Axial vector in the right-screw oriented reference frame, c Axial vector in the left-screw oriented reference frame [56]

Fig. 4 Mirror transformation of a polar and a spinor (axial) vector

The geometrical interpretation of spin vectors is given in Fig. 3. In Fig. 4, the mirror transformation of a polar and an axial vector is shown. For the given spin vector .a ∗ , the directed line segment .a is introduced according to the following rules: • The vector .a is placed on the axis of the spin vector. • The magnitude of .a is equal to the magnitude of .a ∗ . • The vector .a is directed according to the right-handed screw or the left-handed screw. The selection of one of the two cases in the third item corresponds to the convention of orientation of the reference frame [76] (it should be not confused with the rightor left-handed triples of vectors or coordinate systems). Definition 4 (Polar Vector) The directed line segment is called a polar vector if it does not change by changing the orientation of the reference frame. Definition 5 (Axial Vector) The vector is called to be axial if it changes the sign by changing the orientation of the reference frame. Remark 11 (Extension of the definition of polar and axial vectors) • The definitions are valid for tensors of any rank. • The axial tensors are widely used

Continuum Mechanics – Material Independent and Dependent Equations

– – – –

19

in the rigid body dynamics; in the theories of rods, plates, and shells; in the asymmetric theory of elasticity; as well as in dynamics of micro-polar media.

• When dealing with polar and axial objects it should be remembered that they have different physical meanings. Therefore, a sum of a polar and an axial tensors of the same rank has no sense. Higher rank tensors are necessary for the rational description of the basics of continuum mechanics. Definition 6 (Second rank tensor) A second rank tensor . A is any finite sum of ordered vector pairs .A = a ⊗ b + . . . + c ⊗ d . (22) Definition 7 (Dyad) One ordered pair of vectors is called the dyad. The symbol .⊗ is called the dyadic (tensor) product of two vectors. A single dyad or a sum of two dyads are special cases of the second rank tensor. Higher rank tensors can be introduced in a similar way. Scalars are zeroth rank tensors. Definition 8 (Scalar) Physical properties completely defined by real numbers, which are not depending on the choice of the coordinate system, are named scalars. Examples are the mass, the volume, the energy, the temperature, etc. Since scalars are real numbers all elementary algebraic operations can be applied. Note that scalars may have dimensions. Only scalars with the same dimension can be summed up. Multiplication and division can be performed also in the case that the scalars have different dimensions. The direct notation is coordinate-free and does not need an introduction of any preferred coordinate system. The scalars, vectors, and tensors are handled as invariant (independent from the choice of the coordinate system) quantities. The basics of modern vector and tensor calculus were established by Gibbs and published by Wilson [74]. Modern representation is given, for example, in [33, 47, 50, 76]. Solving problems sometimes the use of the vector-matrix representation is helpful and more efficient. The basics are given, for example, in [26, 37, 78, 79]. A comparative presentation of tensor calculus and vector-matrix calculus is given, among others, in Rubin et al. [65].

3 One-Dimensional and Two-Dimensional Continuum Mechanics Since the 50s of the last century more and more two- and one-dimensional continuum models were developed. The starting point was the book of the Cosserat brothers [30]. In this book, it was assumed that a material point of the continuum has six degrees of

20

H. Altenbach

freedom (instead of three in the case of the classical Cauchy continuum). Ericksen and Truesdell came back to this idea applying the assumption of six degrees of freedom to deformable surfaces and lines. In a paper of Zhilin [75], a theory for deformable directed surfaces was developed and applied to homogeneous in the cross-section shells. Later the theory was extended to inhomogeneous plates [24] and viscoelastic plates [2]. On the success of this approach is reported in [17, 25] among others. At the same time, it should be noted that this approach allows the description of new problems like surface effects [16, 19, 20], foams [14, 15], and functionally grade materials [12, 13]. In a similar way, deformable line theories can be introduced so strong as the continuum mechanics. The so-called direct approach considers a rod as a deformable line. The basic assumption is that every cross section behaves like a rigid body in the sense that translations and cross-section rotations are basic degrees of freedom for every material point of the line. The mechanical interactions between two neighboring cross sections are (normal and/or shear) forces and (bending and/or twisting) moments. Basic balance equations of continuum mechanics are applied directly to the deformable line. Examples including the formulation of constitutive equations are given in [8, 27, 28]. The identification of the constitutive parameters is also given. In both cases, the one- and two-dimensional theories can be used for checking the correctness of classical theories of beams plates or shells. A last example of an one-dimensional rod theory is presented in Naumenko and Altenbach [56]. It was assumed that the rod is subjected to tensile (compressive) loading only. To describe the positions of cross sections of the rod in the reference configuration, the vector . R = Xii with the coordinate . X and the unit vector .i is introduced. The corresponding position in the actual configuration is defined by the vector .r = xii with the coordinate . x. The kinematics can be presented by comparing both configurations. The presented theory is interesting from the didactic point of view: the whole continuum mechanics including the formulation of constitutive equations is given for straight rods. Even such terms like the deformation gradient can be introduced. It is obvious that in this theory each material point has only one translational degree of freedom which means that Cosserat-type theories cannot be established on the base of this model.

4 Concluding Remarks The continuum mechanics allows • a unique formalism for presentation of all governing equations – for solids and fluids, – for various dimensions, – for various scales.

Continuum Mechanics – Material Independent and Dependent Equations

21

• a description of complex material behavior by constitutive and evolution equations. Further challenges are • • • •

better estimation of the material parameters, including higher gradients of strains and stresses [23, 32, 51, 57], development of more efficient numerical treatments, and better establishment of equivalent stresses [44, 45].

Acknowledgements This contribution is based partly on two books [3, 56]. The last one was written together with Prof. Konstantin Naumenko. The author would like to thank him for his essential contributions in the section.

References 1. Aero, E.L., Kuvshinskii, E.V.: Fundamental equations of the theory of elastic media with rotationally interacting particles. Sov. Phys. Solid State 2(7), 1272–1281 (1961) 2. Altenbach, H.: Determination of the reduced properties of multilayer viscoelastic sheets. Mech. Comp. Mater. 24(1), 52–59 (1988). https://doi.org/10.1007/BF00611335 3. Altenbach, H.: Kontinuumsmechanik - Eine elementare Einführung in die materialunabhängigen und materialabhängigen Gleichungen, 4th edn. Springer (2018). https://doi.org/10.1007/ 978-3-662-57504-8 4. Altenbach, H.: Rheological modeling – historical remarks and actual trends in solid mechanics. In: Altenbach, H., Kaplunov, J., Lu, H., Nakada, M. (eds.) Advances in Mechanics of TimeDependent Materials. Advanced Structured Materials, vol. 188. Springer, Cham (2023). https:// doi.org/10.1007/978-3-031-22401-0_1 5. Altenbach, H., Altenbach, J., Kissing, W.: Mechanics of Composite Structural Elements, 2nd ed. edn. Springer (2018). https://doi.org/10.1007/978-981-10-8935-0 6. Altenbach, H., Belyaev, A.: Palmov, Vladimir Alexandrovich. In: Altenbach, H., Öchsner , A. (eds.) Encyclopedia of Continuum Mechanics, pp. 1973–1974. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-55771-6_354 7. Altenbach, H., Belyaev, A., Eremeyev, V.: On the occasion of 75th birthday of Professor Vladimir A. Palmov. ZAMM – J. App. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 89(4), 241 (2009). https://doi.org/10.1002/zamm.200989444 8. Altenbach, H., Bîrsan, M., Eremeyev, V.A.: Cosserat-type rods. In: Altenbach, H., Eremeyev, V.A. (eds.) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences, vol. 541, pp. 178–248. Springer Vienna, Wien (2013). https://doi.org/10.1007/978-3-7091-1371-4_4 9. Altenbach, H., Bruhns, O.T.: Truesdell, Clifford Ambrose III. In: Altenbach, H., Öchsner, A.(eds.) Encyclopedia of Continuum Mechanics, pp. 2573–2574. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-55771-6_129 10. Altenbach, H., Eisenträger, J.: Rheological modeling in solid mechanics from the beginning up to now. Lect. Notes TICMI 22, 13–29 (2021) 11. Altenbach, H., Eremeyev, V., Indeitsev, D., Ivanova, E., Krivtsov, A.: On the contributions of Pavel Andreevich Zhilin to mechanics. Technische Mechanik – Eur. J. Eng. Mech. 29(2), 115–134 (2009) 12. Altenbach, H., Eremeyev, V.A.: Direct approach-based analysis of plates composed of functionally graded materials. Arch. Appl. Mech. 78(10), 775–794 (2008). https://doi.org/10.1007/ s00419-007-0192-3

22

H. Altenbach

13. Altenbach, H., Eremeyev, V.A.: On the analysis of viscoelastic plates made of functionally graded materials. ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 88(5), 332–341 (2008). https://doi.org/10.1002/zamm.200800001 14. Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM – Zeitschrift für Angewandte Mathematik und Mechanik 89(4), 242–256 (2009). https://doi.org/10.1002/ zamm.200800207 15. Altenbach, H., Eremeyev, V.A.: Thin-walled structures made of foams. In: Altenbach, H., Öchsner, A. (eds.) Cellular and Porous Materials in Structures and Processes. CISM International Centre for Mechanical Sciences, vol. 521, pp. 167–242. Springer, Wien (2010). https://doi.org/ 10.1007/978-3-7091-0297-8_4 16. Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011). https://doi.org/10.1016/j.ijengsci.2011.03.011 17. Altenbach, H., Eremeyev, V.A.: Cosserat-type shells. In: Altenbach, H., Eremeyev, V.A. (eds.) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences, vol. 541, pp. 131–178. Springer, Wien (2013). DOI https://doi.org/ 10.1007/978-3-7091-1371-4_3 18. Altenbach, H., Eremeyev, V.A., Lebedev, L.P., Rendón, L.A.: Acceleration waves and ellipticity in thermoelastic micropolar media. Arch. Appl. Mech. 80(3), 217–227 (2010). https://doi.org/ 10.1007/s00419-009-0314-1 19. Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale. Int. J. Eng. Sci. 59, 83–89 (2012). https://doi.org/ 10.1016/j.ijengsci.2012.03.004 20. Altenbach, H., Eremeyev, V.A., Morozov, N.F.: On the influence of residual surface stresses on the properties of structures at the nanoscale. In: Altenbach, H., Morozov, N.F. (eds.) Surface Effects in Solid Mechanics, Advanced Structured Materials, vol. 30, pp. 21–32. Springer, Berlin, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35783-1_2 21. Altenbach, H., Indeitsev, D., Ivanova, E., Krivtsov, A.: In memory of Pavel Andreevich Zhilin (1942–2005). ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 87(2), 79–80 (2007). https://doi.org/10.1002/zamm.200790000 22. Altenbach, H., Ivanova, E.A.: Zhilin, Pavel Andreevich. In: Altenbach, H., Öchsner, A. (eds.) Encyclopedia of Continuum Mechanics, pp. 2792–2795. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-55771-6_147 23. Altenbach, H., Müller, W.H., Abali, B. (eds.): Higher Gradient Materials and Related Generalized Continua. Advanced Structured Materials, vol. 120. Springer (2019). https://doi.org/10. 1007/978-3-030-30406-5 24. Altenbach, H., Zhilin, P.: Eine allgemeine Theorie elastischer einfacher Schalen (in Russian). Adv. Mech. 11(4), 107–148 (1988) 25. Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. App. Mech. 80(1), 73–92 (2010). https:// doi.org/10.1007/s00419-009-0365-3 26. Bellmann, R.: Introduction to Matrix Analysis. Classics in Applied Mathematics, vol. 19, 2nd edn. SIAM, Philadelphia (1997) 27. Bîrsan, M., Altenbach, H.: On the dynamical theory of thermoelastic simple shells. ZAMM – Zeitschrift für Angewandte Mathematik und Mechanik 91(6), 443–457 (2011). https://doi.org/10.1002/zamm.201000057 28. Birsan, M., Altenbach, H.: On the Cosserat model for thin rods made of thermoelastic materials with voids. Discrete Continuous Dyn. Syst. – Ser. S 6(6), 1473–1485 (2013). https://doi.org/ 10.3934/dcdss.2013.6.1473 29. Casey, J.: On the derivation of jump conditions in continuum mechanics. Int. J. Struct. Changes Solids 3(2), 61–84 (2011) 30. Cosserat, E., Cosserat, F.: Théorie des corps déformables. A. Hermann et Fils, Paris (1909) 31. Curie, P.: Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et d’un champ magnétique. Journal de Physique Théorique et Appliquée 3(1), 393–415 (1894). https://doi.org/10.1051/jphystap:018940030039300

Continuum Mechanics – Material Independent and Dependent Equations

23

32. Eremeyev, V.A., Altenbach, H.: On the direct approach in the theory of second gradient plates. In: Altenbach, H., Mikhasev, G.I. (eds.) Shell and Membrane Theories in Mechanics and Biology. Advanced Structured Materials, vol. 45, pp. 147–154. Springer, Cham (2015). https:// doi.org/10.1007/978-3-319-02535-3_8 33. Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, Singapore (2018). https://doi.org/10.1142/10959 34. Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. SpringerBriefs in Applied Sciences and Technology – Continuum Mechanics. Springer, Berlin, Heidelberg (2013). https://doi.org/10.1007/978-3-642-28353-6 35. Eremeyev, V.A., Porubov, A.V.: Aero, Eron Lyuttovich. In: Altenbach, H., Öchsner, A. (eds.) Encyclopedia of Continuum Mechanics, pp. 33–35. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-55771-6_27 36. Ericksen, J.L., Truesdell:, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1(1), 295–323 (1958). https://doi.org/10.1007/BF00298012 37. Faddejew, D.K., Faddejewa, W.N.: Numerische Methoden der linearen Algebra. Deutscher Verlag der Wissenschaften, Berlin (1964) 38. Filin, A.P.: Essays on Mechanical Scientists (in Russian). Strategiya (2007) 39. Giesekus, H.: Phänomenologische Rheologie: eine Einführung. Springer, Berlin (1994). https:// doi.org/10.1007/978-3-642-57953-0 40. Green, A.E., Naghdi, P.M.: On the derivation of discontinuity conditions in continuum mechanics. Int. J. Eng. Sci. 2(6), 621–624 (1965). https://doi.org/10.1016/0020-7225(65)90040-6 41. Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin (2002) 42. Haupt, P.: Konzepte der Materialtheorie. Technische Mechanik - European Journal of Engineering Mechanics 16(1), 13–22 (2019) 43. Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling – Continuum Mechanics, Dimensional Analysis, Turbulence. Springer, Berlin, Heidelberg (2004). https://doi.org/10. 1007/978-3-662-06402-3 44. Kolupaev, V.A.: Equivalent Stress Concept for Limit State Analysis. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73049-3 45. Kolupaev, V.A., Yu, M.H., Altenbach, H.: Fitting of the strength hypotheses. Acta Mechanica 227(6), 1533–1556 (2016). https://doi.org/10.1007/s00707-016-1566-9 46. Krawietz, A.: Materialtheorie - mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer, Berlin, Heidelberg, New York and Tokyo (1986). https:// doi.org/10.1007/978-3-642-82512-5 47. Lagally, M.: Vorlesungen über Vektorrechnung. Geest & Portig, Leipzig (1962) 48. Lemaitre, J., Chaboche, J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1994). https://doi.org/10.1017/CBO9781139167970 49. Lurie, A.I.: Non-Linear Theory of Elasticity, North Holland Series in Applied Mathematics and Mechanics, vol. 36. North Holland, Amsterdam (1990) 50. Lurie, A.I.: Theory of Elasticity. Foundations of Engineering Mechanics. Springer, BerlinHeidelberg (2005). https://doi.org/10.1007/978-3-540-26455-2 51. Lurie, S., Volkov-Bogorodskii, D., Altenbach, H., Belov, P., Nazarenko, L.: Coupled problems of gradient thermoelasticity for periodic structures. Arch. Appl. Mech. 93(1), 23–39 (2023). https://doi.org/10.1007/s00419-022-02197-z 52. Maugin, G.A.: Continuum Mechanics Through the Eighteenth and Nineteenth Centuries – Historical Perspectives from John Bernoulli (1727) to Ernst Hellinger (1914). Solid Mechanics and Its Applications, vol. 214. Springer, Dordrecht (2013). https://doi.org/10.1007/978-3-31905374-5 53. Maugin, G.A.: Continuum Mechanics Through the Twentieth Century – A Concise Historical Perspective, Solid Mechanics and Its Applications, vol. 196. Springer, Dordrecht (2013). https:// doi.org/10.1007/978-94-007-6353-1 54. Müller, W.H., Rickert, W., Vilchevskaya, E.N.: Thence the moment of momentum. ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 100(5), e202000,117 (2020). https://doi.org/10.1002/zamm.202000117

24

H. Altenbach

55. Müller, I.: Thermodynamik: die Grundlagen der Materialtheorie. Bertelsmann Universitätsverlag (1973) 56. Naumenko, K., Altenbach, H.: Modeling High Temperature Materials Behavior for Structural Analysis – Part I: Continuum Mechanics Foundations and Constitutive Models. Advanced Structured Materials, vol. 28. Springer (2016). https://doi.org/10.1007/978-3-319-31629-1 57. Nazarenko, L., Glüge, R., Altenbach, H.: Inverse Hooke’s law and complementary strain energy in coupled strain gradient elasticity. ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 101(9), e202100,005 (2021). https://doi.org/10.1002/ zamm.202100005 58. Neumann, F.E.: Vorlesungen über die Theorie der Elastizität der festen Körper und des Lichtäthers (edited by O. E. Meyer). B. G. Teubner-Verlag, Leipzig (1885) 59. Palmov, V.: Vibrations of Elasto-Plastic Bodies. Springer, Berlin, Heidelberg (1998). https:// doi.org/10.1007/978-3-540-69636-0 60. Pal’mov, V.A.: Fundamental equations of the theory of asymmetric elasticity. J. Appl. Math. Mech. 28(3), 496–505 (1964). https://doi.org/10.1016/0021-8928(64)90092-9 61. Palmov, V.A.: Nonlinear Mechanics of Deformable Bodies (in Russian). Polytechnic University Publisher, St. Petersburg (2014) 62. Pavilainen, G.V., Kuteeva, G.A., Polyakhova, E.N., Rudakova, T.V., Sabaneev, V.S., Tikhonov, A.A.: Essays on the History of Mechanics and Physics (in Russian). VVM, St. Petersburg (2020) 63. Reiner, M.: Deformation, Strain and Flow: An Elementary Introduction to Rheology. H.K. Lewis, London (1960) 64. Reiner, M.: Rheologie in elementarer Darstellung. Hanser, München (1967) 65. Rubin, D., Lai, W.M., Krempl, E.: Introduction to Continuum Mechanics. ButterworthHeinemann (2010). https://doi.org/10.1016/B978-0-7506-8560-3.X0001-1 66. Sinkevich, G.I., Nazarov, A.I. (eds.): Mathematical St. Petersburg – History, Science, Sights (in Russian), 2nd edn. Educational Projects, St. Petersburg (2018) 67. Smol’nikov, B.A.: Mechanics in the History of Science and Society (in Russian). Institute for Computer Research, Moscow, Izhevsk (2013) 68. Timoshenko, S.P.: History of Strength of Materials. Dover, New York (1983) 69. Todhunter, I., Pearson, K.: A History of the Theory of Elasticity and of the Strength of Materials from Galilei to the present time, vol. 2: Saint-Venant to Lord Kelvin. Dover, New York (1986) 70. Todhunter, I., Pearson, K.: A History of the Theory of Elasticity and of the Strength of Materials from Galilei to the present time, vol. 1: Galilei to Saint-Venant 1639–1850. Alpha Editions, New Delhi (2020) 71. Truesdell, C.: Die Entwicklung des Drallsatzes. ZAMM – J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 44(4–5), 149–158 (1964). https://doi.org/10.1002/ zamm.19640440402 72. Truesdell, C.: Essays in the History of Mechanics. Springer, Berlin (1968) 73. Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin, Heidelberg (1997). https://doi.org/10.1007/978-3-662-03389-0 74. Wilson, E.B.: Vector Analysis. Founded upon the Lectures of G.W. Gibbs. Yale University Press, New Haven (1901) 75. Zhilin, P.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976). https://doi.org/10.1016/0020-7683(76)90010-X 76. Zhilin, P.A.: Vectors and Second Rank Tensors in Three-Dimensional Space (in Russian, Ivanova, E.A., Altenbach, H., Vilchevskaya, E.N., Gavrilov, S.N., Grekova, E.F., Krivtsov, A.M. (eds.)). Nestor, St. Petersburg (2001) 77. Zhilin, P.A.: Rational Continuum Mechanics (in Russian, Ivanova, E.A., Altenbach, H., Vilchevskaya, E.N., Gavrilov, S.N., Grekova, E.F., Krivtsov, A.M. (eds.)) Polytechnic University Publisher, St. Petersburg (2012) 78. Zurmühl, R., Falk, S.: Matrizen und ihre Anwendungen, vol. 2. Springer, Berlin (1986) 79. Zurmühl, R., Falk, S.: Matrizen und ihre Anwendungen, vol. 1. Springer, Berlin (1997)

Mechanical Behavior of Classical Sandwich Beams Andreas Öchsner

Abstract The quasi-static mechanical behavior of classical sandwich beams is investigated in this chapter. The investigated load cases range from bending, over tension/compression to shear loading. The derived general expressions for the stress distributions are simplified for the special case of a so-called technical sandwich, i.e., a symmetrical configuration with thin face sheets and a soft core.

1 Introduction Lightweight design plays a central role in transportation (e.g., in the aerospace industry or in automotive engineering), since a reduction in weight is directly reflected in a reduction in fuel costs. A typical design principle in the context of lightweight structures is to compose different materials, many times in the form of layers, to so-called sandwich structures, see Fig. 1 for some typical examples. The idea is to combine the different advantages of the single constituents and to achieve overall properties, which are superior than the ones of the single constituents. The face sheets of such sandwich structures are normally made from classical dense or composite materials whereas the core might be a foam, honeycomb or cellular structure [8, 18]. The technical literature on the subject of sandwich structures is summarized in Table 1. This chapter is purely focused on one dimensional, i.e., rod- and beam-like sandwich structures and thus offers a new didactic approach to conveying the basic ideas of these structural members. The restriction to one-dimensional elements allows a relatively simple representation using equations that are easy for students to understand. Thus, there is a focus on basic design concepts and the application of the fundamentals of applied mechanics and not on complicated mathematical derivations or algorithms. Anyone who has mastered these basics can also familiarize themselves

A. Öchsner (B) Faculty of Mechanical and Systems Engineering, Esslingen University, 73728 Esslingen, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. Altenbach et al. (eds.), Lectures Notes on Advanced Structured Materials 2, Advanced Structured Materials 203, https://doi.org/10.1007/978-3-031-49043-9_2

25

26

A. Öchsner

Fig. 1 Examples of sandwich plates with metallic hollow sphere core: a carbon-fiber reinforced face sheets; b aluminum face sheets. Adapted from [19], reprinted with permission from Springer Nature publishers

relatively easily with more complicated subject areas of composite structures, such as plane two-dimensional structures.

2 Basic Mechanical Load Cases In the case of lightweight design, many times an attempt is made to achieve an optimal lightweight potential through a suitable choice and combination of different materials and their distribution in the cross section. Typical representatives of this come from the class of composite materials in their most general definition [2, 3], with sandwich elements being considered in more detail below, see [1, 25]. A typical sandwich element is schematically shown in Fig. 2. Here, two top layers (the so-called face sheets), which often consist of the same material, are connected by a core and thus kept at a defined distance. It is also noted that the top layers and the core usually have different tasks, see Hertel [10]. In the following, three different load cases are examined in more detail, see Fig. 3. For simplification, the beam theory according to Euler-Bernoulli and the bar theory (see [21]) are used as a basis. Thus, a generalized, i.e., a combination of beam and bar member, composite beam is considered.

2.1 Bending Load The average bending stiffness .EI y for such a composite, under the assumption of a perfect connection between the core and the face sheets, results from [13, 20] as follows:

Mechanical Behavior of Classical Sandwich Beams

27

Table 1 Selected books/reports on sandwich structures. The year refers to the first edition Year Author/Editor Title References 1948 1950

C. Libove, S.B. Batdorf N. J. Hoff

1966

F. J. Plantema

1969

H. G. Allen

1974 1975

K. Stamm, H. Witte L. Librescu

1995

D. Zenkert

1997

J. N. Reddy

1997

D. Zenkert

1998 1999

A. Vautrin J. R. Vinson

2001 2003

J. M. Davies L. P. Kollár, G. S. Springer J. R. Vinson

2005

2021

I. M. Daniel, E.E. Gdoutos, Y. D. S. Rajapakse T. J. Hause

2022

W. Ma, R. Elkin

2022

S. Krishnasamy, C. Muthukumar, S.M.K. Thiagamani, S.M. Rangappa, S. Siengchin

2009

A General Small-Deflection Theory for Flat Sandwich Plates Bending and Buckling of Rectangular Sandwich Plates Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates, and Shells Analysis and Design of Structural Sandwich Panels Sandwichkonstruktionen: Berechnung, Fertigung, Ausführung (in German) Elastostatics and Kinetics of Anistropic and Heterogeneous Shell-Type Structures An Introduction to Sandwich Construction Mechanics of Laminated Composite Plates and Shells: Theory and Analysis The Handbook of Sandwich Construction Mechanics of Sandwich Structures The Behavior of Sandwich Structures of Isotropic and Composite Materials Lightweight Sandwich Construction Mechanics of Composite Structures

[15] [11] [23]

[1] [25] [16]

[29] [24] [30] [26] [27] [7] [12]

Plate and Panel Structures of Isotropic, [28] Composite and Piezoelectric Materials, Including Sandwich Construction Major Accomplishments in Composite [6] Materials and Sandwich Structures Sandwich Structures: Theory and Responses Sandwich Structural Composites: Theory and Practice Sandwich Composites: Fabrication and characterization

[9] [17] [14]

28

A. Öchsner

Fig. 2 Structure of a sandwich element: 1, 3: top layer (skin, face), 2: core layer Fig. 3 Investigated load cases: a bending, b tensile or compressive loading, and c shear loading

3

EI y =

3

Ek

.

k =1

1 b 12

∆h k

3

+ b∆h k z ck

2

=

E k I yk , k =1

(1)

Mechanical Behavior of Classical Sandwich Beams

29

Fig. 4 Course of the a normal stress component .σx and b strain component .εx under bending load

where .z ck is the vertical distance (i.e., in the .z-direction) of the center of area of the partial body.k to the total center of area. The layer thickness.∆h can also be calculated using .∆h k = z k−1 − z k . In Eq. (1), the total second moment of area was calculated using the fraction with regard to the center of area of the partial body and the additional contribution from the parallel-axis theorem (‘Steiner’ fraction). Thus, from Eq. (1) for 1 bh 3 would result. a homogeneous element of width.b and height.h,.EI y = E I y = E 12 Special cases of Eq. (1) are reported later (see Table 3). In the case of a transverse strain constraint of the cover layers (e.g., by the core or a significant dimension in the . y-direction, i.e., .b ≈ L (i.e., a plate), where . L is the length of the beam along the .x-axis), instead of . E the modified modulus of elasticity . E → E/(1 − ν 2 ) can be used, see Stamm and Witte [25]. The stress distribution in the sandwich can be described using the following modified approach (see [13, 20]): σ

. x,k

(z) =

My E k EI y

× z.

(2)

According to Fig. 4a, there are discontinuities in stress at the transition from one material to the next in the case of different stiffnesses .(E k ). In contrast to this, however, the strain is assumed without discontinuities, i.e., a linear progression through the origin of the coordinate system, see Fig. 4b. The derivation of Eq. (2) is briefly presented below. Assuming that each layer .k has the same curvature .κ, the stress in each layer can be given as follows: σ

. x,k

= −E k z

d2 u z (x) = +E k zκ. dx 2

(3)

The internal bending moment results from integration over the stress distribution using:

30

A. Öchsner

Fig. 5 General configuration of a cantilever sandwich beam

.

My =

zσx dA = A

.

=

zσx,1 dA1 + A1

A2

z 2 E 1 κdA1 + A1

.

= ⎝E 1

.

=

A1

E 1 I y1

+

z 2 E 3 κdA3 A3

z 2 dA1 + E 2 +

E 3 I y3

(4)

(5)

⎞ z 2 dA3 ⎠ κ

z 2 dA2 + E 1 A2

E 2 I y2

zσx,3 dA3 A3

z 2 E 2 κdA2 + A2

⎛

zσx,2 dA2 +

(6)

A3

κ = EI y κ.

(7)

Using Eq. (3), the stress relationship (2) results from the last equation. In the following, different sandwich structures with different material and geometry combinations are compared in relation to the lightweight potential. This can be done by calculating the so-called lightweight index . M, i.e., the ratio between the externally applied force and the dead weight of the structure, see [13, 21] for details. The basic configuration for numerous examples, i.e., cantilever beam with end shear force, is as shown in Fig. 5. A homogeneous aluminum beam now serves as reference (see Fig. 6, configuration (1)). By using a foam core and then reducing the thickness of the cover layer (with the same outer dimensions of the cross section), an increase in the lightweight potential can be achieved (see Fig. 6, configurations (2) and (3)). A configuration in which the core layer has been completely removed is given as a limiting case (see Fig. 6, configuration (4)). It should be noted here that the calculation of the lightweight index was carried out with a limit value as the maximum external load. An alternative design concept can be realized by using a carbon-fiber reinforced plastic (CFRP) (characteristic values for CFRP according to Klein [13]: . = 1.50 × 10−6 kg/mm3 , . E = 120000 MPa, . R t = 1700 MPa; unidirectional layer (UD) with fiber volume fraction of .φ F = 0.55). For case (4) according to Fig. 6 this now results in . MCFRP = 4077.

Mechanical Behavior of Classical Sandwich Beams

31

Fig. 6 Lightweight index for different configurations of sandwich elements (. L = 100 mm,.b = h = 10 mm) and load criterion for bending load case .(F0 = −200 N ∧ σmax < Rp0.2 ). Characteristic kg t values of the Al foam according to Klein [13]: . = 0.4 × 10−6 mm 3 , . E = 2500 MPa, . Rp0.2 = c 4 MPa, . Rp0.2 = 6 MPa

Fig. 7 Course of the a normal stress component .σx and b strain component .εx under tensile load

2.2 Tensile/Compressive Load In the following, the load case of tension or compression (see Fig. 3b) is considered in more detail, although instabilities such as buckling or wrinkling are not yet taken into account here. Under the influence of an external axial force, the stress and strain distributions shown in Fig. 7 result. It is important to assume that all layers are perfectly connected and that the strain in each individual layer is equal to the total strain, see Fig. 7b. The internal normal force results from integration over the stress distribution using:

32

A. Öchsner

.

Nx =

σx dA = A

σx,1 dA1 + A1

σx,2 dA2 + A2

σx,3 dA3 A3

.

= E 1 εx ∆h b + E 2 εx ∆h b + E 3 εx ∆h b

.

= E 1 ∆h b + E 2 ∆h b + E 3 ∆h b εx = EAεx .

1

2

1

(8)

3

2

3

(9) (10)

The average tensile stiffness is thus: 3

EA =

E k ∆h k b.

.

(11)

k =1

Substituting in Eq. (10) the total strain using Hooke’s law for a layer and considering that the internal normal force is equal to the external force . F0 , i.e., .

N x = F0 = EA

σk , Ek

(12)

the relationship for the stress in the .k-th layer is obtained as: σ

. x,k

F0 E k

=

EA

.

(13)

2.3 Shear Load In the following, the shear load case (see Fig. 3c) is considered in more detail. In general, the shear stress distribution can be calculated according to Öchsner [22] by integration from the normal stress distribution. For the top layers (see layers 1 and 3 in Fig. 2) this results, as an example, for 2 2 layer 3 .( ∆h2 ≤ z ≤ ∆h2 + ∆h 3 ) in: ∆h 2 +∆h 3 2

τ

. zx,3

(z) =

dσx,3 (x) ' dz + c3 . dx

(14)

z

With the stress gradient

.

d dσx,3 (x) = dx dx

M y (x)E 3 EI y

the shear stress distribution results in:

z

=

E 3 Q z (x) E 3 z dM y (x) = z EI y dx EI y

(15)

Mechanical Behavior of Classical Sandwich Beams ∆h 2 +∆h 3 2

E Q z (x) 3

τ

. zx,3

(z) =

z ' dz ' + c3 =

EI y

33

E 3 Q z (x) 2EI y

z

⎡

∆h 2 ⎣ + ∆h 3 2

⎤

2

− z 2 ⎦ + c3 . (16)

The constant of integration .c3 can be determined using the condition that no shear stresses occur at the free edge, i.e., .τzx,3 (∆h 2 /2 + ∆h 3 ) = 0, as .c3 = 0. This finally gives the shear stress distribution in layer 3: τ

. zx,3

(z) =

E 3 Q z (x) 2EI y

or in layer 1: τ

. zx,1

(z) =

E 1 Q z (x) 2EI y

⎡

⎡

⎤

2

∆h 2 ⎣ + ∆h 3 2

− z2⎦ ,

∆h 2 ⎣ + ∆h 1 2

⎤

2

− z2⎦ .

For the core layer (see layer 2 in Fig. 2) we get .(− ∆h2 ≤ z ≤ 2

∆h 2 2

τ

. zx,2

(z) =

(17)

(18)

∆h 2 ): 2

dσx,2 (x) ' dz + c2 . dx

(19)

z

With the stress gradient .

dσx,2 (x) E 2 Q z (x) z = dx EI y

(20)

the shear stress distribution results in:

τ

. zx,2

(z) =

E 2 Q z (x) EI y

∆h 2 2

z ' dz ' + c2 =

E 2 Q z (x)

z

2EI y

⎡

∆h 2 ⎣ 2

2

⎤ − z 2 ⎦ + c2 .

(21)

The integration constant .c2 results from the transition condition for the stress .τzx between layers 2 and 3, i.e., .τzx,2 (z = ∆h 2 /2) = τzx,3 (z = ∆h 2 /2), to E Q z (x) 3

c =

. 2

2EI y

⎡ ⎣

∆h + ∆h 3 2 2

2

−

∆h 2

2

2

⎤ ⎦=

E 3 Q z (x) 2EI y

(∆h 2 + ∆h 3 )∆h 3 . (22)

34

A. Öchsner

Fig. 8 Course of the a shear stress component .τx z and the b shear strain .γx z under shear load

Thus, the shear stress distribution in the core layer (layer ‘2’) results in: ⎡

⎛

∆h 2 ⎣E 2 ⎝ .τzx,2 (z) = 2 2EI y Q z (x)

2

⎞

⎤

− z 2 ⎠ + E 3 ∆h 2 + ∆h 3 ∆h 3 ⎦ .

(23)

The distribution of the shear stress over the height of the sandwich element is shown in Fig. 8a. Different parabolas form the distribution here. The distribution of the shear strain shows, because .γzx (z) = Gτzxzx with different shear moduli in layers 2 and 3 (= 1), discontinuities, i.e., steps at the layer transitions, see Fig. 8b.

3 Technical Sandwich For the further explanations, a more technical configuration of a sandwich is considered, see Fig. 9. The face sheets of the symmetrical structure are generally significantly thinner than the core layer. Furthermore, the core layer is usually softer than the face sheets.1 In addition, homogeneous and isotropic materials that deform linearly and elastically are assumed for all layers. As tasks of the face sheets in a technical sandwich, the following functions can be cited: • They absorb practically the entire bending moment. • They absorb practically the entire axial tensile or compressive load. The core, on the other hand, has the following tasks: 1

A composite element with .∆h C < ∆h F , G C < G F is called an anti-sandwich, see [4, 5].

Mechanical Behavior of Classical Sandwich Beams

35

Fig. 9 Technical sandwich element: symmetrical structure with thin face sheets (F) and soft core (C)

• It practically absorbs the entire lateral force (shear). • Fixing, supporting, and stabilizing the face sheets in their mutual position (i.e., vertical spacing of the face sheets, avoidance of the face sheets sliding relative to each other, ensuring the evenness—as a stabilizing elastic bedding—of the face sheets to avoid buckling/wrinkling). In many cases, the tensile and shear-resistant assemblage between the face sheets and the core is created by adhesive bonding.2 However, self-forming adhesion after the foaming of the core, welding, nailing, screwing or dowelling can also be found in technical applications, see Stamm and Witte [25].

3.1 Bending Load For this case, according to Eq. (1), the average bending stiffness for the configuration according to Fig. 9 can be set as follows: EI y = 2E F

.

1 1 b(∆h F )3 + b∆h F (z cF )2 + E C b(∆h C )3 12 12

E F b(∆h F )3 E F b∆h F (∆h C + ∆h F )2 E C b(∆h C )3 + + 6 2 12 E F b(∆h F )3 E F b∆h F (h c )2 E C b(∆h C )3 = + + . 6 2 12

=

EI y,F

2

EI y,FSt

(24)

EI y,C

The thickness of the adhesive layer is usually neglected when considering the different layers.

36

A. Öchsner

Fig. 10 Normal stress distribution due to bending load: a exact, b soft core, (c) soft core and thin face sheets

The average bending stiffness is therefore made up of three parts: Part .EI y,F , which describes the bending stiffness with regard to the partial center of area of the face sheet, part .EI y,FSt , which describes the Steiner part (i.e., the additional part from the parallel-axis theorem) of the face sheets with regard to the overall center of area and the part .EI y,C , which relates to the bending stiffness of the core of the overall center of area.3.,4 C C according to This results in stress distribution in the core . − ∆h2 ≤ z ≤ ∆h2 Eq. (2) M y (x)E C .σ x,C (z) = (25) × z, F F 3 F E b∆h F (h c )2 E C b(∆h C )3 E b(∆h ) + + 6 2 12 or in the upper face sheet . − ∆h2 − ∆h F ≤ z ≤ − ∆h2 C

.

σ

. x,F

(z) =

C

∆h C 2

≤z≤

∆h C 2

+ ∆h F

or in the lower face sheet

: M y (x)E F

E F b(∆h F )3 E F b∆h F (h c )2 E C b(∆h C )3 + + 6 2 12

× z.

(26)

The graphical representation of these two curves is shown in Fig. 10a. For certain geometry or material pairings, the bending stiffness can be further simplified according to Eq. (24). First, let us examine for what ratio .h c /∆h F is the bending stiffness of the face sheets .(EI y,F ) neglectable compared to the Steiner fraction .(EI y,FSt ). If a limit value of 1% is set, the result from Eq. (24) is

3

In the case of the core, this is identical to the partial center of area of the core. The symbol “. ” stands for a part in relation to a center of area. The abbreviation “St” for a “Steiner fraction” (resulting from the parallel-axis theorem).

4

Mechanical Behavior of Classical Sandwich Beams

.

.

E F b(∆h F )3 6 E F b∆h F (h c )2 2 F 2

⇔

.

⇒

37

≤ 0.01,

(27)

(∆h ) ≤ 0.01, 3(h c )2

(28)

hc ≥ ∆h F

1 = 5.77. 0.03

(29)

If one also considers the relationship .h c = ∆h C + ∆h F in the last equation, the following limit value condition results: .

∆h C ≥ 4.77. ∆h F

(30)

Thus, for thin face sheets, i.e., .∆h F 100 ∆h C ∆h F

(34)

holds. Thus, the classification of sandwich structures shown in Table 2 can also be made in relation to the thickness ratio of core to face sheet.5 Next, it is examined for which condition the bending stiffness of the core .(EI y,C ) can be neglected compared to the bending stiffness of the Steiner fraction .(EI y,FSt ). If a limit value of 1% is also set here, the result from Eq. (24) is 5

In the following, however, no distinction is made between ‘thin’ and ‘very thin’ face sheets.

38

A. Öchsner

Table 2 Classification of sandwich structures with regard to the thickness ratio of core to face sheet, see Allen [1] Description

Thickness ratio

Thick face sheets

.

Thin face sheets

.100

Very thin face sheets

.

∆h C < 4.77 ∆h F ≥

∆h C ≥ 4.77 ∆h F

∆h C > 100 ∆h F

E C b(∆h C )3 12 . F E b∆h F (h c )2 2 6E F ∆h F (h c )2 . E C (∆h C )3

≤ 0.01,

(35)

≥ 100.

(36)

Thus, for soft cores,6 i.e., . E C