Mao-Hong Yu Generalized Plasticity Mao-Hong Yu Guo-Wei Ma Hong-Fu Qiang Yong-Qiang Zhang Generalized Plasticity With 315 Figures Professor Mao-Hong Yu Xian Jiaotong University School of Civil Engineering and Mechanics 710049 Xian, People’s Republic of China E-mail: mhyu@mail.xjtu.edu.cn Co-Authors Guo-Wei Ma Nanyang Technological University, Singapore Hong-Fu Qiang Yi’an Hi-Tech Research Institute, China Yong-Qiang Zhang National University of Singapore, Singapore Library of Congress Control Number: 2005932191 ISBN-10 3-540-25127-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-25127-0 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to to prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Camera-ready by the Author and SPI, India Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN 11009092 62/3141/SPI Appreciate the beauty of the discover the truth of the universe universe; Zhuang Zi (369 –286 B.C China) I have loved the principle of beauty in all things, and if I had had time I would have made myself remembered Keats, John (1795 – 1821, UK) A thing of beauty is a joy for ever Keats, John (1795 – 1821, UK) “Beauty is truth, truth is beauty,” — that is all ye know on earth, and all ye need to know Keats, John (1795 – 1821, UK) Preface Generalized plasticity is a generalization of the unified strength theory to the theory of plasticity It is the unification of metal plasticity for Tresca materials, Huber-von Mises materials, and twin-shear materials It is also the unification of geomaterial plasticity for Mohr-Coulomb materials and generalized twin-shear materials Moreover, it leads to unification of metal plasticity and plasticity of geomaterials, in general It is a companion volume to Unified Strength Theory and Its Applications published by Springer in 2004 Generalized Plasticity is based on the lectures on the unified theory of materials and structures given by the author at the School of Civil Engineering and Mechanics, Xi’an Jiaotong University in Xi’an, China and at the Nanyang Technological University of Singapore in 1996 It is a course entitled “Generalized Plasticity” for Ph.D students at Xi’an Jiaotong University since 1993 The main contents are the unified yield function (unified strength theory) of material, the unified slip line field theory for plane strain problem, unified characteristics field theory for plane stress problem, unified characteristics field theory for spatial axisymmetric problem, limit pressure and shakedown pressure of a pressure vessel, the plastic zone analysis at a crack tip under small-scale yielding and the unified fracture criterion Several chapters in this book have been presented in conferences and published in various journals They are: Unified Strength Theory (Yu, 1991, 1992, 1994, 2002, 2004); Unified Slip Line Field Theory for Plane Strain Problem (Yu, Yang, et al., 1997, 1999); Unified Characteristics Field Theory for Plane Stress Problem (Yu and Zhang, 1998, 1999; Zhang, Hao and Yu, et al., 2003); Unified Characteristics Field Theory for Spatial Axisymmetric Problem (Yu and Li, 2001); Unified Solution for Limit Pressure of a Pressure Vessel (Wang and Fan, 1998; Zhao et al., 1999); Unified Solution for Shakedown Pressure of a Pressure Vessel (Xu and Yu, 2004, 2005); Analysis of Plastic Zone at Crack Tip (Qiang et al., 1998, 2004); Unified Fracture Criterion (Yu, Fan, Che, Yoshimine, et al., 2003, 2004; Qiang and Yu, 2004) The beauty of the unified strength theory discussed in Chap is a part of a closing lecture delivered at the International Symposium on Developments in Plasticity and Fracture: Centenary of M.T Huber Criterion, held at Cracow, Poland in 2004 The garden of the flowers of strength theory, the beauty of the Huber-von Mises criterion, and the beauty of the unified strength theory were discussed at the lecture The analytical results of the generalized plasticity are a series of results It is different from the conventional plasticity As an example, a unified solution of bearing capacity of a plane strain structure by using the unified slip line field theory is shown in Fig The conventional solution of bearing capacity of a VIII Preface structure is adapted only for one kind of material It is shown in Fig at b = This result is obtained by using the Mohr-Coulomb strength theory (inner bound or lower bound as shown in Fig 2a), Fig 2b is a special case for Į = materials It can also be obtained by using the unified strength theory with b = The unified solution includes a series of solutions and encompasses the solution of the Mohr-Coulomb strength theory as a special case It is also possible to obtain a series of new solutions for different values of parameter b and different ratios of tension and compression strength of material, i.e., Į = ıt/ıc Fig Limit loads of a plane strain structure (a) Į = ıt/ıc materials (b) Į = ıt/ıc = materials Fig Yield loci of the unified strength theory on the deviatoric plane Preface IX The unified strength theory and the unified slip line field theory for plane strain problem can be expressed in terms of another material parameter, such as friction angle ij, it is widely used in geomechanics and geotechnical engineering The unified solutions for a plane strain problem in terms of the friction ij are shown in Fig The description of the unified solutions of plane strain problems can be seen in Chap Fig Unified solutions of a plane strain problem in terms of the friction ij For the plane stress problems, the yield loci of the unified strength theory in plane stress state is shown in Fig 4, and the unified solution of bearing capacity of a plane stress structure by using the unified characteristics line field theory is shown in Fig It can be seen that a series of new results are given whereas the Mohr-Coulomb theory and the Tresca criterion can give only one result It is a special case of the unified solution by using the unified strength theory and the unified characteristics line theory with b = and b = 0, Į = (a) Į = ıt/ıc materials (b) Į = ıt/ıc =1 materials Fig Yield loci of the unified strength theory in plane stress state X Preface 1.35 α = 0.2 α = 0.5 α = 0.8 α = 1.0 1.30 1.25 1.20 q 1.15 1.10 1.05 1.00 0.95 0.0 0.2 0.4 0.6 0.8 1.0 b Fig Limit loads of a plane stress structure The analytical results are clearly illustrated to show the effects of yield criterion on plastic limit behaviors for plane strain problems, plane stress problems, axisymmetric problems, other engineering structures, the shape and size of plastic zone at crack tip, discontinuous bifurcation, and angle of shear band Generalized plasticity gives us a series of results, which can be adapted for different materials and structures The contents of the book can be divided into five parts as follows: Part One The unified strength theory, material parameters in the unified strength theory, yield surfaces, yield loci, reasonable choice of the yield criterion, and the beauty of the unified strength theory are described in Chaps and Part Two Plastic stress–strain relation and concrete plasticity, discussed in Chaps and Part Three Twin-shear slip field and the unified slip-line field theory for plane strain problems, twin-shear characteristics field and the unified characteristics line theory for plane stress problems, and unified characteristics line field theory for axisymmetric problems and high velocity penetration problem are explained in Chaps 8–12 Part Four The unified solution of plastic zone at crack tip under small-scale yielding is given Based on the unified strength theory, a unified fracture criterion, a new closed form of plastic core region model, and variation for the angle of initial crack growth versus crack inclination under different loading conditions are obtained They are described in Chaps 13 and 14 Part Five Chapter 15 is devoted to the unified solutions of limit loads and shakedown loads for pressure vessels Stress state analysis and basic behaviors of materials under complex stress are discussed in Chaps and The description of the stress state may be found in a number of books covering mechanics of materials, solid mechanics, and elasticity and plasticity Only some basic formulae and figures as well as some new ideas are Preface XI given here Brief summaries, problems, and references and bibliography are given at the end of the chapters I would like to express my gratitude for the support of the National Natural Science Foundation of China (Grants nos 59779028, 59924033, and 50078046), the Ministry of Education of China, the China Academy of Launch Vehicle Technology, the Aircraft Strength Research Institute of China, the Department of Science and Technology of Xi’an Jiaotong University, as well as the School of Civil and Environmental Engineering, Nanyang Technological University, Singapore I am indebted to many authors and colleagues, especially Dr Ma at Nangyang Technological University, Singapore, Prof Qiang at Xi’an Hi-Tech Research Institute, and Dr Zhang at National University of Singapore for going through the manuscript of this book I am also indebted to Dr Li YM, Dr Yang SY, Dr Zhao JH, Dr Wang F, Dr Li JC, Dr Song L, Dr Liu YH, Dr Fan W, Dr Zan YW, Dr Wei XY, Dr Xu SQ, Dr Yoshimine M, Dr Yang JH, Dr Zheng H, Dr Gao JP, Dr Liu FY, and Mr Liu JY and Ms Zeng WB for their research work at Xi’an Jiaotong University Thanks are due to Ms He, Ms Lei, and others for their support during the course of writing this book I would like to thank many professors from other universities and many research scientists and engineers from various institutions for their work in the research and application of the unified strength theory I would also like to acknowledge the support from all other individuals and universities, research organizations, journals, and publishers I would also like to express my sincere thanks to Dr Dieter Merkle, Editorial Director, Engineering and Editorial Department and International Engineering Department, Springer-Verlag, Germany, and his team for their excellent editorial work on this manuscript The study of generalized plasticity based on the unified strength theory is just the beginning A lot of research in generalized plasticity is still to be done Mao-Hong Yu Spring 2005 Contents Preface Notations Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Stress Space and Stress State 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Linear Elasticity Classical Plasticity Concrete Plasticity Soil Plasticity Rock Plasticity Generalized Plasticity Generalized Plasticity Based on the Unified Strength Theory References and Bibliography Elements Stress at a Point, Stress Invariants Deviatoric Stress Tensor, Deviatoric Tensor Invariants Stresses on the Oblique Plane Hexahedron, Octahedron, Dodecahedron Stress Space Stress State Parameters Summary References Basic Characteristics of Yield of Materials under Complex Stress 3.1 3.2 3.3 3.4 3.5 3.6 Introduction Strength Difference Effect (SD effect) Effect of Hydrostatic Stress Effect of Normal Stress Effect of Intermediate Principal Stress Effect of Intermediate Principal Shear-Stress 1 8 10 15 15 15 16 17 21 22 27 31 31 33 33 33 35 37 38 43 ... is the unification of metal plasticity for Tresca materials, Huber-von Mises materials, and twin-shear materials It is also the unification of geomaterial plasticity for Mohr-Coulomb materials... the tetrahedra will be in the signs attached to l, m and n The eight tetrahedra together form an octahedra as shown in Fig 2.4e, and the eight planes form the faces of this octahedron The normal... minimum principal shear stress element (when τ12≤τ23), the minimum principal shear stress τ23 and the respective normal stress σ23, as well as the maximum principal stress σ1 act on this element