STUDIES IN APPLIED MECHANICS 1. Mechanics and Strength of Materials (Skalmierski) 2. Nonlinear Differential Equations (Fufcik and Kufner) 3. Mathematical Theory of Elastic and Elastico-Plastic Bodies An Introduction (NeCas and Hlaväöek) 4. Variational, Incremental and Energy Methods in Solid Mechanics and Shell Theory (Mason) 5. Mechanics of Structured Media, Parts A and B (Selvadurai, Editor) 6. Mechanics of Material Behavior (Dvorak and Shield, Editors) 7. Mechanics of Granular Materials: New Models and Constitutive Relations (Jenkins and Satake, Editors) STUDIES IN APPLIED MECHANICS 6 M e c h a n i cs of Material Behavior The Daniel C. Drucker Anniversary Voiunne Edited by George J. Dvorak Department of Civil Engineering, University of Utah, Salt Lake City, Utah, U.S.A. and Richard T. Shield Department of Theoretical and Applied Mechanics, University of Illinois at Urbana- Champaign, Urbana, Illinois, U.S.A. ELSEVIER Amsterdam — Oxford — New York — Tokyo 1984 ELSEVIER SCIENCE PUBLISHERS B.V. Molenwerf 1 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, NY 10017, U.S.A. ISBN 0-444-42169-6 (Vol. 6) ISBN 0-444-41758-3 (Series) © Elsevier Science Publishers B.V., 1984 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or other­ wise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., P.O. Box 330,1000 AH Amsterdam, The Netherlands Pages 231—238 were written as part of a U.S. Government sponsored research contract, and therefore "approved for public release; distribution unlimited". Printed in The Netherlands Preface Daniel C. Drucker became sixty-five years old on June 3, 1983. This book is a collection of papers by friends and colleagues in grateful and deep appreciation of Dan's outstanding and permanent contributions to the mechanics of material behavior, particularly to the theory of plasticity and its application to the design of engineering structures and components. Most of the papers in this Anniversary Volume were presented at a Sym­ posium on the Mechanics of Material Behavior which was held at the University of Illinois at Urbana-Champaign on June 6 and 7,1983. Support for the Symposium was provided by the National Science Foundation, the Office of Naval Research and the U.S. Army Research Office. We thank the contributors for their ready acceptance of the invitation to participate. We regret that limitations on the timing and size of the Volume did not allow all who wished to do so to contribute. G.J. DVORAK R.T. SHIELD Contributors Bernard Budiansky H. Kolsky Harvard University Brown University Cambridge, Massachusetts Providence, Rhode Island W.F. Chen F.A. Leckie Purdue University University of Illinois at Urbana-Champaign West Lafayette, Indiana Urbana, Illinois W.J. Drugan E.H. Lee University of Wisconsin Rensselaer Polytechnic Institute Madison, Wisconsin Troy, New York G.J. Dvorak T.H. Lin University of Utah University of California at Los Angeles Salt Lake City, Utah Los Angeles, California J. Duffy A. Litewka Brown University Technical University Providence, Rhode Island Posnan, Poland P.D. Griffin G. Maier University of Cape Town Politecnico di Milano Rondebosch, South Africa Milano, Italy W.K. Ho J.B. Martin University of California at Los Angeles University of Cape Town Los Angeles, California Rondebosch, South Africa Philip G. Hodge, Jr. Robert M. McMeeking University of Minnesota University of Illinois at Urbana-Champaign Minneapolis, Minnesota Urbana, Illinois W. Johnson J.M. Mosquera University Engineering Department Brown University Cambridge, England Providence, Rhode Island L.M. Kachanov S. Murthy Boston University Yale University Boston, Massachusetts New Haven, Connecticut P.M. Naghdi J. Lyell Sanders Jr. University of California at Berkeley Harvard University Berkeley, California Cambridge, Massachusetts A. Nappi A. Sawczuk Politecnico di Milano Academie Polonaise des Sciences, Warszawa, Poland Milano, Italy Universite Aix-Marseille, Marseille, France E,T. Onat R.T. Shield Yale University University of Illinois at Urbana-Champaign New Haven, Connecticut Urbana, Illinois A. Phillips P.S. Symonds Yale University Brown University New Haven, Connecticut Providence, Rhode Island J.L. Raphanel S.S. Wang Brown University University of Illinois at Urbana-Champaign Providence, Rhode Island Urbana, Illinois J.R. Rice C.J. Wung Harvard University University of Utah Cambridge, Massachusetts Salt Lake City, Utah Daniel C. Drucker Daniel Charles Drucker was born in New York City on June 3, 1918. His father, Moses Abraham Drucker, was a Civil Engineer and Dan fol­ lowed his example by studying Civil Engineering at Columbia University, receiving the B.S. degree in 1938. Dan stayed at Columbia for graduate study and in August 1939 he married Ann Bodin, his sweetheart of four years. His Ph.D. research was on three-dimensional photoelastic methods for stress analysis under the supervision of Professor R.D. Mindlin. The research led to his first publication, jointly with Mindlin, in 1940, the year in which he obtained the Ph.D. degree. The Druckers spent the next three years in Ithaca where Dan was an Instructor in Engineering at Cornell University. Their son Robert David was born in Ithaca in December, 1942. In the summer of 1943 they moved to Chicago so that Dan could assume his next position as Super­ visor for Solid Mechanics at the Armour Research Foundation of the Illinois Institute of Technology, a position he held for two years (1943— 45). After a year in the Air Force, Drucker returned to the Illinois Insti­ tute of Technology as an Assistant Professor of Mechanics. One year later Dan accepted an appointment at Brown University as Associate Professor of Applied Mathematics and of Engineering to begin a long and fruitful association (1947—68). The Druckers' daughter Mady (Miriam) was born in Providence in August 1948. It was at Brown University that Drucker made the major part of his outstanding contributions to the theory of plasticity and to its application to the design of engineering structures and components. The Brown group in solid mechanics was founded by William Prager and through his fine leadership the group became internationally known for its pioneering work in plasticity. Drucker contributed in full measure and while at Brown published over ninety papers, some in collaboration with colleagues in the Division of Applied Mathematics and in the Division of Engineering. The papers made important contributions to all of the areas involved in the successful development of a realistic and useful theory of plasticity. About fifteen papers dealt with the foundations of plasticity, particularly the theoretical development of plastic stress—strain relations, but the approach was balanced by eleven papers on the experimental evaluation of theoretical models for strain hardening. Drucker introduced the con­ cept of material stability, now known as Drucker's Stability Postulate, which provided a unified approach for the derivation of stress -^strain relations for the plastic behavior of metals. He was a co-author of the now 2 Daniel C. Drucker 3 classical paper in which the limit load was clearly defined and the theorems of limit analysis were established. The theorems led directly to limit design — a technique to predict the load-carrying capacity of engineering structures and components such as bridges, pressure vessels, machine parts, and so on. Following the publication of this work there was an immediate explosion of applications of the theorems to problems which formerly were beyond the scope of engineering practice. The special merit of limit analysis lies in the fact that engineers can make practical and safe decisions on the design of complex load bearing components on the basis of relatively simple calculations. Drucker was the first to show how limit analysis could be used in the design of cylindrical shells, and he later applied it effectively to the design of pressure vessels, particularly thin-walled pressure vessels with an A.S. M.E. Standard Torispherical Head. Vessels of this type are widely used and failures, some of them serious, were occurring. With the theorems of limit analysis, calculations were performed which showed that for some typical designs the pressure exceeded the limit value, so that failure was likely and revision of the A.S.M.E. Code was needed. The success of this work played a strong role in the development of the A.S.M.E. Code for Nuclear Pressure Vessels, which bases strength estimates on limit and shakedown concepts. Over 30 of Drucker's papers written at Brown cover important engineering applications of plasticity theory; besides structural mechanics and pressure vessel design, the topics include soil mechanics, metal working and metal cutting. In his later years at Brown, Drucker became active in a field sometimes known as micromechanics, which attempts to bridge the gap between the material scientist who studies material behavior at the atomic level and the engineer who works with real materials modeled by theories of continuum mechanics. Drucker made significant progress in determining the effect of particle size and shape of hard inclusions on the mechanical properties of certain high strength alloys. Among the phenomena which could be predicted in quantitative terms were dispersion and precipitation hardening of metals. A particularly remarkable prediction was that the flow stress of the ductile matrix of structural metals is determined pri­ marily by microstructural features not visible in the optical microscope; the visible particles previously assumed to be the cause of hardening could not be the agents of this behavior, and particles smaller than a micron must exist to induce the hardening which occurs. Subsequently, refined metallographic observations showed such particles exist. A related application of limit analysis techniques on the microscale was made by Drucker in the area of composite materials. By calculating upper bounds on limit loads of fibrous and particulate systems, he showed that finite hardening rates may not be attained for certain loading direc­ tions in these materials in the absence of matrix hardening. The result is useful in the development of plasticity theories for composite media. 4 Drucker's first Ph.D. student at Brown was Franz Edelman who received his degree in Applied Mathematics in 1950. Other students in Applied Mathematics were Frederick Stockton (Sc.M. 1949, Ph.D. 1953), William G. Brady (Sc.M. 1953), John Zickel (Ph.D. 1953), John Lyell Sanders, Jr. (Ph.D. 1954), WiUiam Bruce Woodward (ScM. 1958), and Jack L. Dais (Ph.D. 1967). Students in the Division of Engineering were Vincent William Howard (Sc.M. 1952), Norman C. Small (Ph.D. 1960), Halit Halil Demir (Sc.M. 1962, Ph.D. 1963), Steven Charles Batterman (Ph.D. 1964), David Rubin (Sc.M. 1965, Ph.D. 1967), Andrew Clennel Palmer (Ph.D. 1965), Wai-Fah Chen (Ph.D. 1966), David Christopher McDermott (Sc.M. 1967, Ph.D. 1969), William R. Powell (Sc.M. 1967), Thomas Wesley Butler (Ph.D. 1969), and George J. Dvorak (Ph.D 1969). In 1968 Drucker assumed his present position as Dean of the College of Engineering at the University of Illinois at Urbana-Champaign. The considerable duties of the position and his other professional commit­ ments leave little free time, but he continues to publish original research. He supervised the research of Luc Palgen who received the Ph.D. degree in Theoretical and Applied Mechanics in 1981. Drucker's contributions to engineering education other than direct instruction began in earnest when he became Chairman of the Division of Engineering at Brown University in 1953. The Division is presently recognized as having one of the best groups in materials engineering and solid mechanics in the country and this is primarily due to Drucker's leadership and example during the six years as Chairman. He contributed to education in a broader way through his chairmanship of the Physical Sciences Council of Brown during 1961—63. His remarkable abilities as an educator and an administrator made him eminently qualified to assume his present position as Dean of one of the leading Colleges of Engineering in the U.S.A. Since joining the University of Illinois in 1968 he has maintained the College's high standing; his leadership was particularly effective during the period of low engineering enrollments and reduced budgets of the 1970's. His efforts for quality in engineering education have gone beyond the State level and he is recognized as a spokesman on the country's need for highly qualified engineers in an increasingly technological society. Drucker was one of the first to draw attention to the difficulties facing schools of engineering after the mid 1970's in recruiting faculty to educate the engineers to meet industry's demands. His position as a leader in engineering education has been acknowledged by invitations to serve on national committees and to testify before Congress, and by his election to President (1981—1982) of the American Society of Engineering Education. Drucker participates actively in several professional societies. He has been most active in the American Society for Mechanical Engineers. For twelve years, 1956 to 1968, he was Technical Editor of the A.S.M.E. Journal of Applied Mechanics, the premier journal in the world for