Advanced Structured Materials

Transcription

1 Advanced Structured Materials Volume 86 Series editors Andreas Öchsner, Esslingen, Germany Lucas F.M. da Silva, Porto, Portugal Holm Altenbach, Magdeburg, Germany

2 More information about this series at

3 Vladimir A. Kolupaev Equivalent Stress Concept for Limit State Analysis 123

4 Vladimir A. Kolupaev Fraunhofer Institute for Structural Durability and System Reliability (LBF) Darmstadt Germany ISSN ISSN (electronic) Advanced Structured Materials ISBN ISBN (ebook) Library of Congress Control Number: Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved 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, express 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. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 To my parents

6 Acknowledgements This book was written during my work as a Research Assistant at the German Institute for Polymers (now Fraunhofer LBF) in Darmstadt. I thank Prof. Matthias Rehahn for the given opportunity to apply my experience and to broaden my knowledge in applied mechanics. I was inspired by Prof. Holm Altenbach to investigate the subject of strength hypotheses. I would like to thank him sincerely for being my academic advisor, for motivation, and his considerable contribution to each topic. I wish to express special thanks to Prof. Yu Mao-Hong for his interest in this work and several valuable suggestions that helped me to improve my book. I thank Prof. Wilfried Becker for his help and constructive discussions. Furthermore, I thank Prof. Ralf Cuntze for numerous comments and particularly for his critical remarks which led to the thoroughness of this work. I would like to thank a lot Prof. Ryszard B. Pęcherski for his active interest in my topic and scientific exchange. Professor Murat N. Serazutdinov is sincerely thanked for the guidance of my first steps in mechanics and his interest in my current research. My heartfelt thanks go out to my friends and colleagues Dr. rer. nat. Alexandre Bolchoun, Dr.-Ing. Michael Drass, Dr.-Ing. Nina-Carolin Fahlbusch, Dr. Tibor Macko, Dr.-Ing. Holger Massow, and Dr. rer. nat. Michael Rudschuck for a fertile collaboration. I owe particular thanks and recognition to my colleagues Dipl.-Ing. Axel Nierbauer, M.Sc. Eddy M. Kiegelmann, and Ing. Reinhold Riesner for the excellent realization of the loading schemata and for the challenging tests on hard foams. I would like to thank my family whose loving support has made this work possible. Darmstadt, Germany August 2017 Vladimir A. Kolupaev vii

7 Contents 1 Introduction Need of Criteria Equivalent Stress Concept Invariants of Stress Tensor Lode Parameter Formulation of Criteria Methodology... 7 References Criteria of Equivalent Stress Concept Zero-Parameter Criteria Rankine Tresca Maxwell, Huber von Mises Hencky Mariotte Schmidt Ishlinsky, Burzyński Ivlev Ko Sokolovsky and Ishlinsky Ivlev Conclusion One-Parameter Criteria Mariotte-St. Venant Beltrami Mohr-Coulomb Sandel Burzyński Torre, Balandin Drucker Drucker Prager, Mirolyubov Sayir Haythornthwaite, Candland ix

8 x Contents Sdobyrev, Pisarenko Lebedev Capurso Freudenthal Gou, Cazacu Barlat Spitzig Edelman Drucker Maitra, Iyer Dodd Naruse Hoek Brown Kolupaev Two-Parameter Criteria Burzyński Yagn Schleicher Leckie Hayhurst Podgórski, Bigoni Piccolroaz Yu Criteria with More Than Two Parameters Birger Altenbach Zolochevsky Altenbach Bolchoun Kolupaev Obvious Systematization References Visualization Methods Spatial Representation of the Limit Surface p-plane Burzyński-Plane Diagram r I r II Diagram r 11 s Assessment of the Loading Points References Formulations of Classical Strength Hypotheses Normal Stress Hypothesis Tresca Hypothesis von Mises Hypothesis Schmidt Ishlinsky Hypothesis Remarks About Graphical Forms References Dimensionless Values for Comparison Basic Stress States p-plane Values Elastic Poisson s Ratio Inelastic Poisson s Ratio References

9 Contents xi 6 Visualization of Standard Criteria Strain Criterion Burzyński Yagn Criterion Linear Combinations of the Classical Hypotheses Mohr Coulomb Criterion Sdobyrev, Pisarenko Lebedev Criterion Comparison of Standard Criteria Discussion of Standard Criteria References Alternative Formulations of Standard Criteria Hoek Brown Criterion Modifications of the Pisarenko Lebedev Criterion Pisarenko Lebedev Criteria of Higher Power Reduced Cubic Criterion Leckie Hayhurst Criterion Sayir s Cone Modification of Twin-Shear Theory Simple Generalizations Discussion References Conditions and Assumptions of Strength Criteria Remarks Concerning the State of the Art of Research Need of a Generalized Criterion Applicability of the Criteria Formulation of Conditions Necessary Conditions Plausibility Assumptions References Generalized Pressure-Insensitive Criteria Yield Criteria of Trigonal Symmetry Sayir Criterion Capurso Criterion Haythornthwaite Criterion Podgórski, Bigoni Piccolroaz Criterion Modified Altenbach Zolochevsky Criterion Capurso Haythornthwaite Linear Combination Capurso Haythornthwaite Multiplicative Ansatz Radcig Criterion Cosine Ansatz of Altenbach Bolchoun Kolupaev Tri-quadratic Criterion Yield Criteria of Hexagonal Symmetry Unified Yield Criterion of Yu

10 xii Contents Bi-cubic Criterion Multiplicative Ansatz Criterion Generalized Criterion of Hexagonal Symmetry Cosine Ansatz of Even Powers Criteria of the Stress Power n ¼ Smooth and Polyhedral Criteria Non-convex Yield Criteria in the p-plane Conclusion References Generalized Pressure-Sensitive Criteria Extension of Yield Criteria Conical and Pyramidal Criteria Unified Strength Theory of Yu Criteria Recommended for Applications Modified Theory of Yu Rotationally Symmetric Criteria Geometrical-Mechanical Criterion Capurso Haythornthwaite Generalization Criterion Normalized on Uniaxial Compressive Stress Conclusion References Multi-surface Criteria Motivation Systematization of Multi-surface Criteria Criteria with C 0 -Transition Davidenkov Fridman Criterion Modifications of the Normal Stress Hypothesis Modifications of the Strain Criterion Criteria with C 1 -Transition Ad hoc Approach According to Berg Huber Criterion Kuhn Criterion Modified Criterion of Huber Combined Geometrical-Mechanical Criterion Sikora Criterion C 1 -Continuous Surface Varying in the p-plane Conclusion References Fitting of Criteria Mathematical Objective Functions Physical Objective Functions

11 Contents xiii 12.3 Geometry-Motivated Choice of Solution Requirements on the Parameters of the Criterion Fitting Features References Experiments and Equipment Discussion of the Necessary Tests Tests on Hard Foams Specimen Preparation and Density Determination Tubular Specimen Tension, Compression, and Torsion Tests Internal and External Pressure Tests Enlacement Test Biaxial Tensile Test Hydrostatic Compression Test Hydrostatic Tensile Test Definition of Failure References Applications Measured Data by Naghdi for Aluminum Alloy Measured Data by Dillenberger for Polyamide Measured Data by Münch for the Hard Foam EPP Rotationally Symmetric Criteria Geometrical-Mechanical Criterion Measured Data by DeRuntz for Syntactic Foam Geometrical-Mechanical Criterion C 1 -Combined Criterion Measurements on Polymethacrylimide Foam Experimental Results Evaluation of Measurements on PMI Foams Measured Data by Tasuji for Concrete References Summary and Outlook Summary Outlook References

12 Abbreviations ACI American Concrete Institute ACS American Chemical Society AIAA American Institute of Aeronautics and Astronautics ASCE American Society of Civil Engineers ASME American Society of Mechanical Engineers ASTM American Society for Testing and Materials BS British Standard CAE Computer-Aided Engineering CAS Computer Algebra System CISM International Centre for Mechanical Sciences DIN Deutsches Institut für Normung Diss. Dissertation DKI Deutsches Kunststoff-Institut Dt. Deutscher DTIC Defense Technical Information Center EN Europäische Norm FLTM Ford Lab Test Method GAG Gutachtergruppe GmbH Gesellschaft mit beschränkter Haftung IABSE International Association for Bridge and Structural Engineers ICEM International Conference on Experimental Mechanics ICF International Conference on Fracture IGF Industrielle Gemeinschaftsforschung Int. International J. Journal JSCE Japan Society of Civil Engineers Mat. Materials Mech. Mechanics Mitt. Mitteilung NAMRC North American Metalworking Research Conference xv

13 xvi Abbreviations Ph.D. Phys. PSS PU SFB Trans. Rep. VEB VDI Verl. ZAMM ZAMP Doctor of Philosophy Physical or Physics Plane Stress State Polyurethane Sonderforschungsbereich Transactions Report Verlag für Bauwesen Verein Deutscher Ingenieure Verlag Zeitschrift für Angewandte Mathematik und Mechanik Zeitschrift für angewandte Mathematik und Physik

14 Abstract In this book, criteria for isotropic materials based on the equivalent stress concept are discussed. These criteria are widely applied in the limit state analysis. For general use, they are formulated in the invariants of the stress tensor. At first, so-called classical hypotheses which are useful in the case of absolutely brittle or ideal ductile material behavior are presented. Different techniques of model visualization are discussed for clear evaluation. Nine tests for basic stress states are suggested and imaged in different planes together with the criteria. At second, standard criteria describing the intermediate range between the absolutely brittle and ideal ductile behavior are introduced and compared in different diagrams. Some alternative formulations of these criteria are discussed. At third, the formulation principles for the criteria are presented. In the main part, pressure-insensitive and pressure-sensitive criteria are separated. Some criteria, which contain a number of convex forms in the p-plane, are recommended for applications. In the case of partly not reliable measured data combined criteria with continuously differentiable transition can be used. Fitting methods based on mathematical, physical, and geometrical objective functions are necessary. Finally, some tests for multi-axial stress states are introduced and evaluated. Reference measurements for five materials are given. Examples demonstrate the application of different approaches in modeling certain limit behaviors. In the summary, the most important criteria of the equivalent stress concept are resumed. Possible directions for development of the equivalent stress concept are shown in the outlook. Keywords Strength hypotheses Yield criteria Limit surfaces Pressure-insensitive criteria Pressure-sensitive criteria Fitting xvii