Dynamic Pulse Buckling: Theory and Experiment

394 Pages·1987·8.094 MB·English

by  Herbert E. Lindberg, Alexander L. Florence (auth.)

Dynamic Pulse Buckling
MECHANICS OF ELASTIC STABILITY
Editors: H.H.E. Leipholz and G.LE. Oravas
H H.E. Leipholz, Theory of elasticity. 1974.
ISBN 90-286-0193-7
L. Librescu, Elastostatics and kinetics of anisotropic and heterogeneous
shell-type structures. 1975.
ISBN 90-286-0035-3
C.L. Dym, Stability theory and its application to structural mechanics.
1974.
ISBN 90-286-0094-9
K. Huseyin, Nonlinear theory of elastic stability. 1975.
ISBN 90-286-0344-1
H.H.E. Leipholz, Direct variational methods and eigenvalue problems in
engineering. 1977.
ISBN 90-286-0106-6
K. Huseyin, Vibrations and stability of multiple parameter systems. 1978.
ISBN 90-286-0136-8
H.H.E. Leipholz, Stability of elastic systems. 1980.
ISBN 90-286-0050-7
V.V. Bolotin, Random vibrations of elastic systems. 1984.
ISBN 90-247-2981-5
D. Bushnell, Computerized buckling analysis of shells. 1985.
ISBN 90-247-3099-6
L.M. Kachanov, Introduction to continuum damage mechanics. 1986.
ISBN 90-247-3319-7
H.H.E. Leipholz and M. Abdel-Rohman, Control of structures. 1986.
ISBN 90-247-3321-9
H.E. Lindberg and A.L. Florence, Dynamic pulse buckling. 1987.
ISBN 90-247-3566-1
Dynamic Pulse Buckling
Theory and Experiment
By
Herbert E. Lindberg
APTEK, Inc., San Jose, CA, USA
Alexander L. Florence
SRI International, Menlo Park, CA, USA
This work was sponsored by the Defense Nuclear Agency under RDT&E RMSS
Code B342078464 N99QAXAH30103 H2590D and published as report number
DNA 6503H, under contract number DNA 001-78-C-0287.
1987 MARTINUS NIJHOFF PUBLISHERS •
II
a member of the KLUWER ACADEMIC PUBLISHERS GROUP
DORDRECHT / BOSTON / LANCASTER
Distributors
for the United States and Canada: Kluwer Academic Publishers, P.O. Box 358,
Accord Station, Hingham, MA 02018-0358, USA
for the UK and Ireland: Kluwer Academic Publishers, MTP Press Limited,
Falcon House, Queen Square, Lancaster LAI lRN, UK
for all other countries: Kluwer Academic Publishers Group, Distribution Center,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands
Library of Congress Cataloging in Publication Data
Lindberg, Herbert E., 1930-
Dynamic pulse buckling.
(Mechanics of elastic stability ; 12)
Bibliography: p.
Includes index.
1. Buckling (Mechanics) 2. Elasticity. 3. Stability
of structures. I. Florence, Alexander L., 1928-
II. Title. III. Series.
TA656. 2. L53 1987 624.1' 6 87-14049
ISBN-13: 978-94-0 I 0-8 136-8 e-ISBN-13: 978-94-009-3657-7
001: 10.1007/978-94-009-3657-7
Copyright
© 1987 by Martinus Nijhoff Publishers, Dordrecht.
Softcover reprint of the hardcover I st edition 1987
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, mechanical,
photocopying, recording, or otherwise, without the prior written permission of
the publishers,
Martinus Nijhoff Publishers, P.O. Box 163, 3300 AD Dordrecht,
The Netherlands.
v
FOREWORD
This book originally appeared as a text prepared for the Defense Nuclear Agency
to summarize research on dynamic pulse buckling, by the authors and their colleagues
at SRI International, during the period from 1960 to 1980. The original printing of 300
copies by the DNA Press was followed shortly by a small second printing to meet the
demand by readers who heard of the book from the primary recipients. This supply was
also quickly exhausted, to researchers and practicing engineers outside the DNA
community and to academics who wanted to include the material in courses on elastic
and plastic stability of structures. Commercial publication by Martinus Nijhoff
Publishers was therefore undertaken to meet the needs of this broader community.
The objective of the book was to gather into a cohesive whole material that had
been published in reports and the open literature during the two decade period. In the
process of knitting this material together, a substantial amount of new work was done.
The book therefore contains many new results never published in the open literature.
These include the systematic development of the concepts of pulse buckling in
Chapter 2, with a simple bar as an example, and a demonstration of the relationships
between pulse buckling and the more familiar features of static buckling. In particular,
the representation of imperfections that trigger static buckling, developed a century ago,
is shown to be useful, in an appropriate random form, for pulse buckling. New
experimental results are given for equivalent imperfections in thin bars made of sheet
material typically used in thin-walled shells.
Similar development from classical static buckling theory is given in Chapter 3 for
shells under radial impulse. For convenience of the reader, the Donnell shell equations
needed for both static and dynamic analysis under both radial and axial loads are
developed from first principles. The theory for radial impulse is developed from low
impulse levels, for which instability is in auto parametric resonance, to intermediate
levels, for which higher order terms must be included in the equations of motion in order
to analyze autoparametric resonance, to high levels, at which buckling takes place during
a single inward motion in the hoop mode. This again illustrates the relationship between
more familiar types of buckling and single-pulse buckling, which is the main topic of
the monograph. As a supplement to application of the theory for buckling during
circumferential plastic flow, new experimental results and a unique constitutive relation
are given for plastic stress-strain behavior of aerospace metals.
These ideas, developed to explain and extend experimental observations for ideally
VI
impulsive loading, are then extended to radial pressure pulses with finite pulse durations.
Formulas are derived to estimate critical combinations of peak pressure and impulse
intensity that cause buckling for loads over the entire range from ideally impulsive to
step pressures of infinite duration. As for impulsive loads, extensive experimental results
are given as a guide to the theoretical development and for confirmation of the accuracy
of the theoretical results.
Chapter 4 gives advanced theory for plastic-flow buckling under radial impulse. The
first examples are of buckling in shells of various lengths, in which the axial boundary
conditions allow bi-axial plastic flow. This introduces a new buckle-resisting bending
moment, called the directional moment, which occurs because of a kinematic constraint
that forces yield points for material at various positions through the shell thickness to
move along the yield surface in such a way as to resist bending. A new theory for
viscoplastic flow buckling with directional moments is also developed. These theories are
applied to prediction of threshold buckling in the conventional sense, and also to
threshold buckling for shells thrown in at very high velocities such that the shell material
becomes a solid mass, as used in explosive pipe closures and shaped charges for oil well
pipe perforation.
In Chapter 5, buckling theory is developed for cylindrical shells under axial impact.
As in Chapter 3, relationships are demonstrated among static buckling, buckling under
long duration step loads, and pulse buckling which occurs during a single transit of the
axial stress wave up and down the shell. These relationships show that the theory for
step loading cannot be applied for impulsive loading in the manner suggested in the
literature. As for radial impulse, thicker-walled shells buckle under very high velocity
impact, which results in bi-axial plastic flow. Again, directional moments play an
important roll in this buckling. Both the elastic and plastic-flow buckling theories are
guided and confirmed by experimental results summarized in the text.
Experimental results are also given for gross buckling collapse of moderately thick
shells, and a simple kinematic energy theory is derived to predict the average collapse
force. These results demonstrate the order-of-magnitude change in time scales between
the dynamics that initiate buckles and the quasi-static energy absorption from later
stretching and folding. The last chapter gives theoretical and experimental results for
plastic flow buckling of rectangular plates.
H. E. Lindberg
A. L. Florence
March, 1987
VII
CONTENTS
FOREWORD .................................................................................... V
PREFACE. .......................... ......... ....................................................................... XIII
1. INTRODUCTION ......................................................................................... 1
1.1 FORMS OF DYNAMIC BUCKLING .................................................. 1
1.2 EXAMPLES OF DYNAMIC PULSE BUCKLING .............................. 3
2. IMPACT BUCKLING OF BARS .................................................................. 11
2.1 INTRODUCTION ................................................................................ 11
2.2 ELASTIC BUCKLING OF LONG BARS ............................................ 12
2.2.1 Equations of Motion................................................................ 13
2.2.2 Static Elastic Buckling of a Simply Supported Bar.................... 15
2.2.3 Theory of Dynamic Elastic Buckling of a Simply
Supported Bar .............................................................. "........... 19
2.2.4 Amplification Functions........................................................... 25
2.2.5 Dynamic Elastic Buckling under Eccentric Load........ .............. 27
2.2.6 Dynamic Elastic Buckling with Random Imperfections ........... 33
2.2.7 Framing Camera Observations of Dynamic Elastic Buckling... 43
2.2.8 Streak Camera Observations--Effects of the Moving
S tress Wave.............................................................................. 45
2.2.9 Experiments on Rubber Strips--Statistical Observations .......... 49
2.2.10 Buckling Thresholds in Aluminum Strips ................................ 53
2.3 DYNAMIC PLASTIC FLOW BUCKLING OF BARS ......................... 57
2.3.1 Introduction ............................................................................. 57
2.3.2 Differential Equation of Motion .............................................. 61
2.3.3 The Initially Straight Bar .......................................................... 64
2.3.4 The Nearly Straight Bar............................................................ 65
2.3.5 Comparisons of Theoretical Model and Experimental Results. 67
3. DYNAMIC PULSE BUCKLING OF RINGS
AND CYLINDRICAL SHELLS FROM RADIAL LOADS ......................... 75
VIII
3.1 INTRODUCTION ................................................................................ 75
3.2 DYNAMIC PLASTIC FLOW BUCKLING OF RINGS AND LONG
CYLINDRICAL SHELLS FROM UNIFORM RADIAL IMPULSE.... 76
3.2.1 Introduction ............................................................................. 76
3.2.2 Postulated Character of the Motion--Dynamic Flow Buckling. 78
3.2.3 Equation of Motion.................................................................. 80
3.2.4 Perfectly Circular Ring, Almost Uniform Initial
Radial Velocity......................................................................... 82
3.2.5 Strain Rate Reversal................................................................. 85
3.2.6 The Buckling Terms--Representative Numerical Cases ........... 86
3.2.7 Experimental Technique and Characteristic Results................ 93
3.2.8 Comparison of Experiment with Theory.................................. 97
3.2.9 Buckling Threshold .................................................................. 103
3.3 DYNAMIC ELASTIC BUCKLING OF RINGS AND
CYLINDRICAL SHELLS FROM UNIFORM RADIAL IMPULSE .... 104
3.3.1 Introduction ............................................................................. 104
3.3.2 Theory of Elastic Shell Motion ................................................ 105
3.3.3 Initial Growth of the Flexural Modes--The Stability
Parameter ................................................................................. 112
3.3.4 Small Initial Velocity--Autoparametric Vibrations ................... 116
3.3.5 Intermediate Initial Velocity--Onset of Pulse Buckling............ 120
3.3.6 High Initial Velocity--Pulse Buckling ....................................... 128
3.4 CRITICAL RADIAL IMPULSES FOR ELASTIC AND
PLASTIC FLOW BUCKLING OF RINGS AND LONG
CYLINDRICAL SHELLS ..................................................................... 135
3.4.1 Approach .................................................................................. 135
3.4.2 Strain Hardening in Engineering Metals .................................. 136
3.4.3 Equations of Motion ................................................................ 138
3.4.4 Plastic Flow Buckling ............................................................... 140
3.4.5 Summary of Formulas for Critical Impulse.............................. 149
3.4.6 Buckling with a Cosine Impulse Distribution........................... 152
3.4.7 Effects of Strain Rate Reversal................................................ 156
3.5 DYNAMIC PULSE BUCKLING OF CYLINDRICAL SHELLS
FROM TRANSIENT RADIAL PRESSURE ....................................... 158
3.5.1 Approach and Equations of Motion......................................... 158
3.5.2 Donnell Equations for Elastic Buckling ................................... 159
3.5.3 Fourier Series Solution--Static Buckling................................... 171
3.5.4 Critical Pressure-Impulse Curves for Dynamic Buckling ......... 178
3.5.5 Simple Formulas for Critical Curves........................................ 186
3.5.6 Experimental Results and Comparison with Theory ................ 189
IX
4. FLOW BUCKLING OF CYLINDRICAL SHELLS
FROM UNIFORM RADIAL IMPULSE....................................................... 203
4.1 PLASTIC FLOW BUCKLING WITH HARDENING AND
DIRECTIONAL MOMENTS .............................................................. . 203
4.1.1 Theory of Plastic Cylindrical Shells ........................................ .. 203
4.1.2 Effect of Shell Length on Strain Rates .................................... . 205
4.1.3 The Unperturbed Motion ........................................................ . 206
4.1.4 Axial Strain Distribution ........................................................ .. 209
4.1.5 Perturbed Motion .................................................................... . 210
4.1.6 Directional Moments .............................................................. . 212
4.1.7 Governing Equation ................................................................ . 215
4.1.8 Modal Solution ........................................................................ . 216
4.1.9 Amplification Functions .......................................................... . 218
4.1.10 Asymptotic Solutions for Terminal Motion ............................ . 220
4.1.11 Strain Hardening Moments Only ............................................ . 221
4.1.12 Directional Moments Only ...................................................... . 223
4.1.13 Directional and Hardening Moments ...................................... . 226
4.1.14 Displacement and Velocity Imperfections .............................. .. 233
4.1.15 Threshold Impulse ................................................................. .. 234
4.1.16 Comparison of Theory and Experiment.. ................................ . 236
4.2 VISCOPLASTIC FLOW BUCKLING
WITH DIRECTIONAL MOMENTS .................................................. .. 240
4.2.1 Viscoplastic Moments ............................................................ .. 241
4.2.2 Theory of Viscoplastic Cylindrical Shells ................................ . 241
4.2.3 The Unperturbed Motion ........................................................ . 242
4.2.4 Perturbed Motion .................................................................... . 244
4.2.5 Governing Equation ................................................................ . 245
4.2.6 Modal Solution ........................................................................ . 245
4.2.7 Amplification Functions ......................................................... .. 247
4.2.8 Approximate Solutions for Terminal Motion .......................... . 249
4.2.9 Preferred Modes and Threshold Impulses .............................. . 251
4.2.10 Displacement and Velocity Imperfections .............................. .. 252
4.2.11 Visco plastic and Directional Moments .................................... . 253
4.2.12 Comparison of Theory and Experiment.. ................................ . 253
4.3 CRITICAL VELOCITY FOR COLLAPSE OF CYLINDRICAL
SHELLS WITHOUT BUCKLING ........................................................ 260
4.3.1 Strain-Hardening Moments Only............................................. 262
4.3.2 Strain Rate Moments Only....................................................... 267
x
5. DYNAMIC BUCKLING OF CYLINDRICAL SHELLS
UNDER AXIAL IMPACT ............................................................................ 281
5.1 DYNAMIC BUCKLING OF CYLINDRICAL SHELLS
UNDER ELASTIC AXIAL IMPACT .................................................. 281
5.1.1 Analytical Formulation............... .............................................. 281
5.1.2 Static Buckling .......................................................................... 284
5.1.3 Amplification Functions for Dynamic Buckling ....................... 287
5.1.4 Buckling From Random Imperfections .................................... 290
5.1.5 Impact Experiments ................................................................. 293
5.1.6 Formula for Threshold Buckling .............................................. 297
5.1.7 Dynamic Buckling Under Step Loads ...................................... 299
5.2 AXIAL PLASTIC FLOW BUCKLING
OF CYLINDRICAL SHELLS .............................................................. 308
5.2.1 Introduction ............................................................................. 308
5 .2.2 Unperturbed Motion.......... .............. ........................ ................ 310
5.2.3 Perturbed Motion..................................................................... 312
5.2.4 Governing Equations.................. .......... .............. ........ ............. 314
5.2.5 Modal Solutions ....................................................................... 316
5.2.6 Amplification Functions ........................................................... 318
5.2.7 Preferred Mode and Critical Velocity Formulas ....................... 321
5.2.8 Directional and Hardening Moments....................................... 322
5.2.9 Description of Experiments ..................................................... 323
5.2.10 Comparison of Theory and Experiment................................... 325
5.2.11 Slow Buckling........................................................................... 329
5.2.12 Axial Impact of Plates .............................................................. 331
5.3 FORCES AND ENERGY ABSORPTION DURING AXIAL PLASTIC
COLLAPSE OF TUBES........................................................................ 336
5.3.1 Axial Collapse Experiments ..................................................... 336
5.3.2 Theoretical Estimates of Collapse Forces................................. 340
5.3.3 Comparison of Theory and Experiment................................... 344
6. PLASTIC FLOW BUCKLING OF RECTANGULAR PLATES .................. 349
6.1 INTRODUCTION ................................................................................ 349
6.2 PERTURBATIONAL FLEXURE ......................................................... 350
6.3 GOVERNING EQUATION .................................................................. 354
6.3.1 General Loading....................................................................... 354
6.3.2 Uniaxial Compression .............................................................. 354