Table Of ContentSTUDIES 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.
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Distributors for the United States and Canada:
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ISBN 0-444-42169-6 (Vol. 6)
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© Elsevier Science Publishers B.V., 1984
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or
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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
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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
