Mechanics of the Solid State 9781487575199

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MECHANICS OF THE SOLID STATE

The contributions comprising this volume were dedicated to Professor Joseph Marin on the occasion of his sixtieth birthday in 1965. Professor Marin died during preparation of this volume, on August 21, 1966, in Monterey, California.

Professor Joseph Marin a .A.SC. and B.A. (British Columbia)

M.s. (Illinois) (Michigan)

PH.D.

·

MECHANICS OF THE SOLID STATE

Edited by F.P.J. Rimrott and J. Schwaighofer University of Toronto

UNIVERSITY OF TORONTO PRESS

Copyright, Canada, 1968, by University of Toronto Press Printed in Great Britain Reprinted in 2018 ISBN 978-1-4875-8517-4 (paper)

Contents

Contributors Professor Joseph Marin Creep Tensile Instability in Pressurized Shells of Revolution

Vil

J. A. SAUER

ix

CHARLES W . BERT

3

On the Experimental Stress Analysis of Fracture Models w. L. BINGHAM 14 Elastic Waves in Projectiles N. DAVIDS, B. P. GUPTA, and H. R. MINNICH 23 The Transient Response of Targets Subjected to Hypervelocity Impacts JOHN T. FRASIER 36 Mechanics of Viscoelasticity JOSEPH H. FAUPEL 49 Mathematical Inference from Mechanical Equivalence M. HETENYI 103 Creep in Anisotropic Stress Analysis c. c. HSIAO and s. R. MOGHE 109 Cascade Arrangement in Spherical Pressure Vessel Design for Nuclear Power L. w. HU and J.C. SCHUTZLER 116 Reactors Thin Rods and Shells Viewed as Cosserat Continua WALTER JAUNZEMIS 129 Face-Shear Waves in Rotated Y-Cut Quartz Plates R. D. MINDLIN 143 Design Relations for the Low-Cycle Notch Effect and a Chart of Conditions for R. E. PETERSON 146 Crack Propagation Theory of Streamlined Dies for Drawing and Extrusion o. RICHMOND 154 The Hyperbolic Sine Creep Law in Engineering Practice F. P. J. RIMROTT 168 A Derivation of the Stress-Strain Relationship of Composite Bars EDWARD SAIBEL and I. G. TADJBAKHSH 181 Mechanical Relaxation in Polymers J . A. SAUER 192 Stresses at Delaminations J. SCHWAIGHOFER, C. s. LEE, and H.F. MICROYS 207 Flow and Fracture of Viscoelastic Materials M. G. SHARMA 218 Non-uniform Plastic Fields in Pure Compression P. s. THEOCARIS 240 Effect of Surface Environment on Mode of Crack Formation in Fatigue M. A. WILKOV 259 Publications of Joseph Marin 271

Contributors

CHARLES W. BERT, PH.D., University of Oklahoma, Norman, Oklahoma, U.S.A. WM. L. BINGHAM, PH.D., North Carolina State University, Raleigh, North Carolina, U.S.A. NORMAN DAVIDS, PH.D., Pennsylvania State University, University Park, Pennsylvania, U.S.A. J. H. FAUPEL, PH.D., E. I. du Pont Company, Newark, Delaware, U .S.A. JOHN T. FRASIER, PH.D., Aberdeen Proving Ground, Maryland, U .S.A. B. P. GUPTA, M.S., University of Roorkee, Roorkee, India M. HETENYI, PH.D., DR. TECH. SCI. H.c., Stanford University, Stanford, California, U.S.A. C. C. HSIAO, PH.D., University of Minnesota, Minneapolis, Minnesota, U .S.A. L. W. Hu, PH.D., Pennsylvania State University, University Park, Pennsylvania, U.S.A. W. JAUNZEMIS, PH.D., Pennsylvania State University, University Park, Pennsylvania, U.S.A. C. S. LEE, M.A.SC., Morrison, Hershfield, Millman and Huggins Ltd. Toronto, Ontario H. F. MICROYS, M.A.SC., University of Toronto, Toronto, Ontario R. D. MINDLIN, PH.D., Columbia University, New York, New York, U.S.A. H. R. MINNICH, M.S., Babcock and Wilcox Company, Palm Beach, Florida, U.S.A. S. R. MoGHE, PH.D., University of Minnesota, Minneapolis, Minnesota, U.S.A. R. E. PETERSON, M.S., Westinghouse Electric Corporation, Pittsburgh, Pennsylvania, U.S.A. 0. RICHMOND, PH.D., U.S. Steel Corporation, Monroeville, Pennsylvania, U.S.A. F. P. J. RIMROTI, PH.D., DR.-ING., University of Toronto, Toronto, Ontario E. A. SAIBEL, PH.D., Rensselaer Institute of Technology, Troy, New York, U.S.A. J. A. SAUER, PH.D., Rutgers-The State University, New Brunswick, New Jersey, U.S.A. JOSEPH SCHWAIGHOFER, PH.D., DR.TECHN., University of Toronto, Toronto, Ontario J.C. SCHUTZLER, M.s., Douglas Aircraft Company, Santa Monica, California, U.S.A. M . G. SHARMA, PH.D., Pennsylvania State University, University Park, Pennsylvania, U .S.A. I. G. TADJBAKHSH, PH.D., IBM Corporation, Yorktown Heights, New York, U .S.A. P. S. THEOCARIS, D.sc., D.A.SC., Athens National Technical University, Athens, Greece M. A. WILKOV, PH.D., University of Texas, Austin, Texas, U.S.A.

Professor Joseph Marin

NO ENGINEER, scientist, scholar, or educator of note is apt to achieve success in his chosen field without the possession of certain innate qualities. Prominent among these are motivation, ingenuity, capability, and persistence. All of these qualities Professor Joseph Marin possessed to a high degree. In addition, he had a seemingly inexhaustible supply of energy, a deep concern for the welfare and advancement of his chosen field, and a warm and hospitable manner. It is not surprising therefore that, by the time of his untimely death on August 21, 1966, at the age of 61 years, Professor Marin had become one of the outstanding contributors to the field of mechanics. Joseph Marin received his education at the University of British Columbia, where he obtained a B.A.SC. and, later, a B.A. degree; at the University of Illinois, where he was awarded the M.s. degree; and at the University of Michigan, where he earned his PH.D. In the lives of all of us, there is usually one person, frequently one of our teachers, who does more than any other to guide our future careers and to inspire us to undertake challenging and difficult tasks. In the career of Joseph Marin, this all-important contact was Professor Stephen Timoshenko, a contributor to many aspects of engineering mechanics and an outstanding and inspirational teacher. It was young Joseph Marin's good fortune to have Professor Timoshenko both as a teacher and as a research supervisor. In view of this close association, it is perhaps not surprising that Professor Marin, like his famous teacher, also developed a life-long devotion to the field of theoretical and applied mechanics and to the application of his science to problems of stress analysis, vibrations, and engineering design. Professor Marin's contributions to the field of mechanics are many, significant, and extensive. He was a teacher, an author of technical books, a creative contributor to many phases of mechanics, and a supervisor of numerous graduate students. He also contributed in other ways to the advancement of his profession. He gave service to various technical societies in his field, through the organization and presentation of seminars, special lectures, and summer institutes, by developing a strong Department of Engineering Mechanics at one of the large state universities in the United States, and by acting as consultant to both industrial and governmental groups seeking technical aid. It is difficult to select, out of more than three decades of important contributions, certain specific ones for citation at this time. Yet it is only right and proper that Professor Marin's important contributions to the field of combined stresses and

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Professor Joseph Marin

theories of failure be given special mention. His interest in this field developed during his doctoral study and remained steadfast for over 30 years. Many of his more than 150 technical papers dealt with some aspect of this subject. His first book, published in 1940 by the Rutgers University Press and entitled Working Stresses, was concerned with the establishment of design stresses under combined stress conditions. Discussions of the behavior of materials under combined stresses, together with recent experimental and theoretical contributions, were included in several of Professor Marin's review articles as well as in his various textbooks. Many of his graduate students, as a result of his interest and inspiration, have also elected to carry on research investigations in this field. A significant contribution to the study of the behavior of materials under combined stresses was made by Professor Marin through his imaginative development of new types of experimental apparatus. With these facilities, it has become possible to carry out experimental studies of materials under conditions of stress not readily susceptible to investigation. Data so obtained were used by Professor Marin, and his students, to develop various engineering theories of material behavior. Over the years, more than 20 new types of testing facilities, together with the appropriate instrumentation for stress and strain recording, were developed by Dr. Marin. This development alone constitutes an outstanding accomplishment and one that is a credit to Professor Marin's ingenuity and his unfailing interest. Throughout his career, Professor Marin was a staunch advocate of the simultaneous consideration by engineers and designers of the mechanical properties aspect of the behavior of materials as well as the appropriate theoretical analysis of the system under consideration. He himself, as well as some of his former students, made important contributions to this field of the mechanical behavior of materials. Many of Dr. Marin's technical papers have been concerned with various aspects of the mechanical behavior of engineering materials. His life-long interest in this subject is also attested by the various books that he wrote. These include his second book, Mechanical Properties of Materials and Design, published in 1942; his fourth book, Engineering Materials, Their Mechanical Properties and Applications, published in 1952; and his most recent book, Mechanical Behavior of Engineering Materials, published in 1962. These books provide a thorough account of the experimental basis of mechanical properties as well as an extensive discussion of the various engineering theories that have been developed for the consideration of materials subject to simple and combined stresses, to static and fatigue loading, to elastic and plastic behavior, and to short-time and long-time deformations. Professor Marin's books have been a welcome addition to an important and significant field. They have provided engineers and designers with valuable recent information, together with the mechanics for utilizing these data. They have also served both graduate and undergraduate students as textbooks and reference books. In addition to carrying on research investigations, developing unique equipment, end writing technical books, Professor Joseph Marin also served his profession as a teacher. For over 30 years, and at three different universities, Rutgers, the Illinois

Professor Joseph Marin

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Institute of Technology and, for the last 23 years, at the Pennsylvania State University, he taught hundreds of undergraduate and graduate students of engineering. At one time or another, Professor Marin taught practically all of the courses that constitute either an undergraduate or graduate program in the field of engineering mechanics as well as many other courses that are usually given under civil engineering or mechanical engineering auspices. To every one of these courses, Dr. Marin brought the same attention to thoroughness and detail that marked all of his contributions. His lectures were clear, concise, and well prepared. In many courses, he supplemented the existing text with a separate set of lecture notes and problems which he himself prepared and distributed. These notes, in several instances, later became the basis for future textbooks. One of his books was written specifically for undergraduate students. This is the book called Strength of Materials, published originally in 1948 and in an enlarged and revised version in 1954. During his long teaching career, Dr. Marin supervised the theses of some 60 graduate students and his influence has extended to an uncounted number of others. Many of these students are now themselves making a significant contribution to the field of mechanics either through teaching or research or by practice of their profession in industry. Professor Marin's accomplishments in the field of mechanics are not limited to those of an individual teacher, author, and research investigator. For the 12 years from 1952 to 1964 he was the chairman of the Engineering Mechanics Department at the Pennsylvania State University. During this period, he made a significant impact on the advancement of both undergraduate and graduate education in the field of mechanics and materials. Under his direction, an undergraduate major in engineering mechanics was established, and this has since flourished. The graduate program was also enlarged and enriched and many new faculty members added. As a result, the Engineering Mechanics Department has become a strong department with interests and capabilities in many areas. Through his efforts and those of his colleagues, this department has successfully carried on numerous research programs. During his academic career, Professor Marin alone served as principal investigator on some 40 different research projects sponsored by government or industry. He also aided many other faculty members, especially the younger ones, in the planning of their experiments, in the analysis and interpretation of their data, in the procurement of needed facilities, and in the preparation of research proposals, technical reports, and articles. Professor Marin's concern in the welfare of individuals was not limited to students and future engineers. During his years as chairman of the Engineering Mechanics Department at the Pennsylvania State University, he probably did as much as, if not more than, any other chairman in the country to make the resources of the university available to the practicing engineer. He himself organized and offered special Saturday morning courses designed to bring recent information, and the latest methods of analysis, directly to the stress analysts and design engineers of industry. He also

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Professor Joseph Marin

frequently gave seminars at industrial plants and prepared special sets of notes for the benefit of engineers whose formal academic education had ended years before. Another important contribution was the establishment of special summer courses in particular areas of mechanics. A number of such courses were given every year. These would usually run for a week or more and as many as four or five seminars on different aspects of engineering mechanics might be held in a given summer. The lecturers included many prominent outside speakers as well as faculty members from the university. Hundreds of engineers from industry have attended these special seminars and have greatly benefited from them. All remember fondly Professor Marin's enthusiasm, his warm hospitality, and his great interest in bringing to their attention the most recent advances and the latest research results. Throughout his academic career, Professor Marin also served as a practicing professional engineer. He held a Professional Engineer's License from the State of Pennsylvania for many years. He served as a consultant to industry on dozens of specific technical problems. To these problems, he brought not only a well-trained mind, knowledgeable in a wide variety of areas of mechanics, but an ability to perceive the essence of the problem, to simplify it, and to reduce it to manageable proportions. Dr. Marin was as persistent in his attack on these industrial and design problems as he was on his own research investigations and in both areas this persistence led to notable achievements. Over the years Professor Marin served as a part-time consultant to over 40 industrial and governmental agencies. For the Curtiss-Wright Corporation alone, for whom he acted as consultant from 1942 until 1958, Dr. Marin wrote 95 technical reports and memoranda. His work contributed to the advancement of the design of high-powered aircraft propellers and other aerospace components, and a number of his analyses became standard design procedure. In several instances new concepts of material behavior were uncovered. An appreciable amount of Professor Marin's time and effort was also devoted to the advancement of professional societies in his field of interest. Professor Marin was a strong advocate of the benefits to be derived in bringing together a wide spectrum of engineers and scientists with different backgrounds but with common interests and common problems. He gladly served on some 30 different national technical committees. He helped to organize numerous society meetings and seminars, and willingly accepted editorial and administrative tasks. Only a person with his indefatigable interest and energy could have continued to take on so many extra tasks in addition to those associated with his teaching, writing, research, and administration. Because the field of mechanics is broad and of necessity interdisciplinary and because Professor Marin himself possessed an interest in all phases of mechanics, his participation in engineering societies was extensive. At the time of his death he was an active member of the American Society of Civil Engineers, the American Society of Mechanical Engineers, the American Society for Testing and Materials, the Society for Experimental Stress Analysis, the American Society of Metals, the Rheology Society, and many others. Many high honors came to Professor Joseph Marin. In 1949, he was the winner of

Professor Joseph Marin

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the George Westinghouse Award as an Outstanding Teacher. In 1952-1953, he held a Fulbright Professorship at the Norwegian Institute of Technology. In 1956, he received a John Simon Guggenheim Fellowship Award. He was once President of the Society for Experimental Stress Analysis. In 1960, he was made an honorary editor of the International Journal of Mechanical Sciences. He was also an honorary editor of the International Journal on Education in Mechanical Engineering. In 1962, he presented a series of invited lectures to various Academies of Science in the U.S.S.R. In the autumn of 1965 Professor Marin embarked on a new career at the U.S. Naval Post Graduate School at Monterey in California, which was ended through his untimely death on August 21, 1966. Professor Marin's influence will be indirectly felt for generations through the activities and contributions of his former students, who are located all over the United States as well as in many other countries. They would be the first to acknowledge that the inspiration for their own efforts has been in great measure a result of the example set by their beloved mentor, Professor Joseph Marin.

J. A.

SAUER

MECHANICS OF THE SOLID STATE

Creep Tensile Instability in Pressurized Shells of Revolution

CHARLES W. BERT

THE TIME-DEPENDENT or creep deformation in a simple uniaxial creep test under constant loading can be considered to consist of three distinct stages: (I) the primary or transient stage, (2) the secondary or steady-state creep-rate stage, and (3) the tertiary or increasing creep-rate stage, which ends in creep rupture. The first published explanation for the tertiary stage in terms of a tensile instability phenomenon is due to Andrade in 1910 [I].* If in the beginning of the secondary stage the creep rate is uniform, as more and more deformation takes place, the crosssectional area decreases as a result of the Poisson contraction effect. Since the load is constant, the decrease in cross-sectional area causes an increase in true stress and finally results in an instability. Andrade checked this explanation by devising an ingenious tensile specimen shaped in the form of a hyperboloid of revolution in which the decrease in cross-sectional area due to deformation was exactly balanced by a corresponding decrease in applied load. Therefore, the true stress in the specimen remained uniform throughout the test. In these tests, Andrade detected no tertiary stage of creep, thus lending support to his hypothesis as to the cause of the tertiary stage. It was not until 1953 that the creep tensile instability phenomenont was put on a mathematical basis by Hoff [2]. He showed that in a uniaxial tension creep test, the creep strain increases without bound when the time t reaches the value

(I)

where Suo is the initial steady creep rate and n is the exponent in the Norton-Bailey relation.

* Numbers in brackets refer to the list of references. t The term "creep tensile instability" is used to denote

the mechanism usually responsible for short-time creep rupture. This is an entirely different mechanism than that of long-time creep rupture, which is usually due to intergranular failure. The terminology suggested here appears to be consistent with the nomenclature "plastic tensile instability" widely accepted to denote failure due to ductile necking under a static load. B

Charles W. Bert

4

(2) where eu is the creep rate, Band n are material constants, and uu is the true stress. Thus, eq. (I) provides an upper bound on the rupture time. It is interesting to note that Monkman and Grant in 1956 [3] proposed eq. (1) on the basis of an empirical correlation of experimental results, without reference to Hoff's theoretical analysis. Apparently the only analyses of creep tensile instability of thin-walled shells are those of Finnie [4] in 1959, Rattinger and Padlog [5] in 1961, and Shoemaker [6]* in 1965. The first two considered only long circular cylinders with closed ends and subject to internal pressure. Shoemaker considered an arbitrary shell of revolution subject to axisymmetric centrifugal loading only, so that at the shell "equator" (largest circular cross-section), the state of stress is strictly uniaxial. It is noted that there is no general analysis of creep tensile instability of arbitrary shells of revolution under internal pressure. In contrast, the analogous problem of plastic tensile instability for pressurized shells of revolution was analyzed recently [7]. The purpose of this article is to present a simple analysis of the conditions under which creep tensile instability occurs in nearly arbitrary thin-walled shells of revolution subject to internal pressure. BASES

The hypotheses on which the analysis is based are as follows: 1. The shell is thin-walled, i.e., the wall thickness is very small compared to the smallest principal radius of curvature; however, the wall thickness may be either uniform or smoothly varying. 2. The only restriction on the meridional geometry is that it must be continuous; of course, the cross-sectional geometry is assumed to be circular. 3. The shell has no resistance to bending, so that a pure membrane state of stress exists. 4. Elastic strains, time-independent plastic strains, and primary creep strains are all assumed to be small compared to the secondary/tertiary creep strains, so that only the latter need be considered. 5. The material volume is unchanged by creep deformation. 6. The material is assumed to be isotropic and homogeneous in regard to its creep properties. 7. Any arbitrary multiaxial-stress creep relation applies. 8. The Norton-Bailey power-law relation between creep rate and true stress, eq. (2), is applicable. 9. The only loading considered is internal pressure uniform throughout and independent of time. 10. The only failure mechanism considered is creep tensile instability.

* This paper was not available to the author until after the present analysis was completed.

Creep Tensile Instability

5

Hypothesis 1 in effect defines a thin shell and permits the stresses in the thickness direction to be neglected. Hypothesis 2 is much less restrictive than in the only other analyses in this area, those of [4], [5], and [6]. Hypothesis 3 has been made in analyses [4], [5], and [6] as well as in Gemma and Warfield's creep analysis [8]. Furthermore, it has been made in all analyses of the analogous problem of plastic tensile instability. This hypothesis becomes the more realistic the larger is the deformation at which failure occurs, because internally pressurized shells tend to seek an equilibrium shape in which only membrane stresses exist. However, for elastic-range stresses, certain cases have been investigated in which bending stresses as well as membrane stresses must exist, as in shallow ellipsoids [9] and toroids of all types [10]. Similar investigations for large inelastic strains are not known to the author; however, membrane theory should be a reasonable first-order approximation. Perhaps Hypothesis 4 needs some elaboration. Most analyses of creep neglect elastic strain and time-independent plastic strain. However, in a recent uniaxial analysis, Carlson [11] considered the latter, which reduces the time to rupture. As in all previous analyses of creep tensile instability, the primary creep is neglected here. Thus, the only deformation considered is that which occurs in the secondary and tertiary stages of creep, which by the nature of this analysis (as well as those of Hoff, Finnie, and Carlson) is considered as one continuously varying stage rather than as two distinct stages. This hypothesis is consistent with Hypothesis 3 and, as stated by Finnie, restricts the analysis to short-time creep-rupture tests of shells of ductile materials which show a rather large failure strain. Hypotheses 5 and 6 have been used in all previous creep tensile instability analyses, and they appear to be reasonable from an engineering standpoint. Hypothesis 5 is consistent with Hypothesis 4, since the total strain can be considered to be composed of an elastic component having a lateral-contraction ratio (Poisson's) of 0.25 to 0.35 for most structural alloys and an inelastic (time-independent plastic and creep) component with a lateral-contraction ratio of 1/2. In recent experiments [12], the lateralcontraction ratio increased throughout the creep test and approached a value of 1/2 very closely before rupture occurred. Furthermore, a recent analysis [13] of the analogous problem in plastic tensile instability has shown that the variation in Poisson's ratio is a relatively small effect. In regard to Hypothesis 6, although actual materials are heterogeneous on a microscopic scale, they are usually considered to be homogeneous on a macroscopic basis. Some materials exhibit macroscopic behavior which is isotropic in the elastic range and inelastically anisotropic [14]. However, most materials of engineering interest behave essentially isotropically under creep conditions. Hypothesis 7 is the most general that could be used. It includes the two most popular multiaxial-stress relations : the von Mises (octahedral shear stress) and Tresca (maximum shear stress) relations. Hypothesis 8 is the most widely used empirical creep relation and thus is used here. However, other relationships among creep, stress, and time could be used in a similar fashion.

Charles W. Bert

6

Hypothesis 9 is of importance in all vessels pressurized by a sustained gas pressure, including gas-storage tanks, pressurized aircraft cabins, and solid rocket cases. The effect of loading varying with time has been considered by Singer [I 5] for the case of uniaxial loading. Hypothesis IO is an obvious one requiring no explanation. It needs to be re-emphasized that the restriction to small deformations used in linear-elastic and elastoplastic shell theory is not used here. In fact, as stated before, the larger the deformation at failure, the more accurate is the present analysis (provided the shell is not yielded immediately upon loading). Thus, the analysis is primarily limited to large deformations associated with ductile materials loaded to high effective stress levels below the yield strength of the material at the temperature concerned. ANALYSIS

Within the stated hypotheses, equilibrium at any point on the shell is satisfied by the following expressions for the true stresses in the meridional and hoop directions, respectively:

= pr8/2h,

(3)

= (pr9/h)(I - ½P),

(4)