Raymond D. Mindlin R. D. Mindlin and ~rrlied Mechanics A collection of studies in the development of Applied Mechanics Dedicated to Professor Raymond D. Mindlin by his former students Edited by GEORGE HERRMANN PERGAMON PRESS INC. New York • Toronto • Oxford • Sydney PERGAMON PRESS INC. Maxwell House, Fairview Park, Elmsford, N.Y. 10523 PERGAMON OF CANADA LTD. 207 Queen's Quay West, Toronto 117, Ontario PERGAMON PRESS LTD. Headington Hill Hall, Oxford PERGAMON PRESS (AUST.) PTY. LTD. Rushcutters Bay, Sydney, N.S.W. Copyright © Pergamon Press, Inc. Library of Congress Cataloging in Publication Data Main entry under title: R. D. Mindlin and applied mechanics. CONTENTS: Herrmann, G. Preface.—List of publica- tions of R. D. Mindlin (p. vii) —Drucker, D. C., Duffy, J., and Fox, E. A. Remarks of the past, present, and likely future of photoelasticity.—Goodman, L. E. Development of the three-dimensional theory of elastic- ity. [etc.] 1. Mechanics, Applied—Addresses, essays, lectures. 2. Mindlin, Raymond David, 1906- —Bibliography. I. Mindlin, Raymond David, 1906- II. Herrmann, George, 1921- ed. TA350.3. R2 1974 620.1 73-22346 ISBN 0-08-017710-7 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, photo- copying, recording or otherwise, without prior permission of Pergamon Press Inc. PREFACE Tkts VOLUME is dedicated to Professor Raymond D. Mindlin of Columbia University in recognition of his achievements as a teacher of numerous students and as a most influential scholar in Applied Mechanics. The past several decades have witnessed rapid progress in Applied Me- chanics, a branch of Applied Science and Engineering. Much of this progress was made in the area of Mechanics of Solids, and here many of the most signifi- cant advances bear the stamp and style of Mindlin's own contributions, which began to appear some forty years ago. This influence grew through the activities of Mindlin's students and others associated with him. Since Mindlin's research work has appeared in many different publications, it seemed desirable to collect the results and to assemble them between the covers of a single volume. This could have been accomplished, for example, by publishing a representative collection of reprints. Another possible course of action, the one followed in the present volume, consisted of preparing accounts of the development of specific areas of Mechanics of Solids to which Mindlin has contributed most. Eight such areas were identified and are represented by the eight articles of this volume, all of which were written by authors who had been Mindlin's students. It fell upon the undersigned, who considers him- self a de facto former student, to coordinate this project. Each of the eight articles attempts to present, in a unified fashion, the substance of Mindlin's contributions, on the one hand, and the general develop- ment of a field, on the other hand. Thus each of the articles is an up-to-date and self-contained review and indicates the relevance and significance of Mindlin's own work. Hence, this volume should be regarded also as a tribute to R. D. Mindlin by his former students. When one examines the evolution of Applied Mechanics during the last several decades, one is led to realize that the work of R. D. Mindlin possesses a distinctive style and character. It is not only a matter of what results are ultimately achieved, but also why and how. Above all, Mindlin's work is moti- vated by a concern for physical reality. His analytical studies always begin with an intense desire to explain and interpret, in mathematical terms, observed but poorly understood physical phenomena. It might suffice to mention, as examples, problems of elastic contact and of granular media related to the behavior of the carbon microphone and the problem areas of generalized elastic continua and anisotropic plates, related to the behavior of quartz crystals as frequency standards, filters, delay lines and transducers. His continued efforts to bridge the gap between continuum and lattice theories aided decisively in 111 viii PREFACE explaining certain observed anomalies. Mindlin's experimental studies in photoelasticity are characterized by attention and keen appreciation both of practical and theoretical considerations. In planning the contents of the reviews, it became apparent that not all areas of Mindlin's work could be readily covered in this volume. Mention should be made in particular of Mindlin's contributions to the dynamics of package cushioning, which became a classical study and which has had much influence on the design of supports and containers for the protection of equip- ment from mechanical shock and vibration. Further, his work on the interaction of elastic shells with a surrounding fluid helped to set the stage for numerous later investigations on dynamic response of submerged structures. In this Preface we have omitted accounts of Mindlin's dedicated activities in several professional societies and organizations, his work as a consultant to various Government agencies, and his many years of service and loyalty to Columbia University. In recognition of these, numerous awards and honors have been bestowed upon him, of which we mention here only the Presidential Medal for Merit, the Von Karman Medal of the American Society of Civil Engineers, and the Timoshenko Medal of the American Society of Mechanical Engineers. What his students surely remember best is what and how he taught them in the classroom and after lectures, as well as during discussions of re- search projects. It is the memory of those years which will live on in this volume. GEORGE HERRMANN Stanford University List of Publications of R. D. Mindlint 1. A Reflection Polariscope for Photoelastic Analysis, Review of Scientific Instruments 5, 224-228 (1934) . 2. Welded Joints Studied with New Type Polariscope, Engineering News-Record, 621- 622 (1934) . 3. Contribution au probleme d'equilibre de l'elasticite d'un solide indefini limite par un plan, Coinpies Rendus 201, 536 (1935) . 4. Force at a Point in the Interior of a Semi-Infinite Solid, Physics, 7, 195-202 (1936) . 5. Note on the Galerkin and Papkovitch Stress Functions, Bulletin of the American lathe- matical Society 42, 373-376 (1936) . 6. Pressure Distribution on Retaining Walls, Section J--Earth Pressure, Proceedings of the International Conference on Soil Mechanics, Cambridge, Mass., Vol. III (1937). 7. On the Distribution of Stress around a Pile, Section E—Stress Distribution in Soils, Proceedings of the International Conference on Soil Mechanics, Cambridge, Mass., Vol. III (1937). 8. Analysis of Doubly Refracting Materials with Circularly and Elliptically Polarized Light, Journal of the Optical Society of America 27, 288-291 (1937) . 9. Stress Systems in Circular Disk under Radial Forces, Journal of Applied Mechanics 4, A115-118 (1937) . 10. Distortion of the Photoelastic Fringe Pattern in an Optically Unbalanced Polariscope, Journal of Applied Mechanics 4, A170-172 (1937) . 11. The Equiangular Strain-Rosette, Civil Engineering 8, 546-547 (1938) . 12. Gravitational Stresses in Bi-polar Coordinates, Proceedings of the Fifth International Congress of Applied Mechanics, Cambridge, Mass., pp. 112-116 (1938) . 13. Stresses in a Heavy Disk Suspended from an Eccentric Peg, Journal of Applied Physics 9, 714-717 (1938) . 14. Stresses in an Eccentrically Rotating Disk, London, Edinburgh and Dublin Philosophical Magazine, Ser. 7, 26, 713-719 (1938) . 15. Stress Distribution around a Tunnel, Proc. Amer. Soc. C.E. 619-642 (1939) . 16. A Review of the Photoelastic Method of Stress Analysis, Journal of Applied Physics 10, 222-241, 273-294 (1939) . 17. An Extension of the Photoelastic Method of Stress Measurement of Plates in Bending, Journal of Applied Mechanics 8, A187-189 (1941) . 18. Stress Analysis by Three-Dimensional Photoelastic Methods (with D. C. Drucker) , Journal of Applied Physics 11, 724-732 (1940) . 19. Optical Aspects of Three-Dimensional Photoelasticity, Journal of the Franklin Institute 233, 349-363 (1942) . 20. The Present Status of Three-Dimensional Photoelasticity, Civil Engineering 12, 255-258 (1942). 21. Dynamics of Package Cushioning, Bell System Technical Journal 24, 353-461 (1945) . t These publications are referred to in text by the letter M before a number in brackets, e.g., [12]. ix x LIST OF PUBLICATIONS 22. Response of Damped Elastic Systems to a Pulse Acceleration, 6th International Congress of Applied Mechanics, Paris, France, September 1946. 23. Analogy between Multiply-Connected Slabs and Slices, Quarterly of Applied Mathematics 4, 279-290 (1946) . 24. Activities of the Society for Experimental Stress Analysis, International Technical Con- gress, Paris, France, September 1946, and Proceedings of the Society for Experimental Stress Analysis, 4 (2) , ix—xi (1947) . 25. Stress Distribution around a Hole near the Edge of a Plate under Tension, Proceedings of the Society for Experimental Stress Analysis 5 (2) , 56-68 (1948). 26. Response of Damped Elastic Systems to Transient Disturbances (with W. F. Stubner and H. L. Cooper) , Proceedings of the Society for Experimental Stress Analysis 5(2), 69-87 (1948). 27. The Optical Equations of Three-Dimensional Photoelasticity (with L. E. Goodman), Journal of Applied Physics 20, 89-95 (1949) . Also, Proceedings of the Seventh Inter- national Congress f or Applied Mechanics, London, 1948, p. 353. 28. A Mathematical Theory of Photo-Viscoelasticity, Journal of Applied Physics 20, 206-216 (1949). 29. Compliance of Elastic Bodies in Contact, Journal of Applied Mechanics 16, 259-268 (1949). 30. Thermo-Elastic Stress around a Cylindrical Inclusion of Elliptic Cross-Section (with H. L. Cooper) , Journal of Applied Mechanics 17, 265-268 (1950) . 31. Beam Vibrations with Time-Dependent Boundary Conditions (with L. E. Goodman), Journal of Applied Mechanics 17, 377-380 (1950) . 32. Analogies (with M. G. Salvadori), Handbook of Experimental Stress Analysis, Wiley, New York, 1950, pp. 700-827. 33. Nuclei of Strain in the Semi-Infinite Solid (with David H. Cheng), Journal of Applied Physics 21, 926-930 (1950) . 34. Thermo-Elastic Stress in the Semi-Infinite Solid (with David H. Cheng) , Journal of Applied Physics 21, 931-933 (1950) . 35. Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates, Journal of Applied Mechanics 18, 31-38 (1951) . 36. Thickness-Shear and Flexural Vibrations of Crystal Plates, Journal of Applied Physics 22,316-323 (1951). 37. Development of the Mathematical Theory of Three-Dimensional Photoelasticity, Ap- plied Mechanics Reviews 4, 537-539 (1951) . 38. Forced Thickness-Shear and Flexural Vibrations of Piezoelectric Crystal Plates, Journal of Applied Physics 23, 83-88 (1952) . 39. A One-Dimensional Theory of Compressional Waves in an Elastic Rod (with G. Herr- mann) , Proceedings of the First U.S. National Congress of Applied Mechanics, pp. 187-191 (1951). 40. Effects of an Oscillating Tangential Force on the Contact Surfaces of Elastic Spheres (with W. P. Mason, T. F. Osmer and H. Deresiewicz) , Proceedings of the First National Congress of Applied Mechanics, pp. 203-208 (1951) . 41. Response of an Elastic Cylindrical Shell to a Transverse, Step Shock-Wave (with H. H. Bleich) , Journal of Applied Mechanics, 20, 189-195 (1953) . 42. Elastic Spheres in Contact under Varying Oblique Forces (with H. Deresiewicz) , Journal of Applied Mechanics, 20, 327-344 (l.953). LIST OF PUBLICATIONS xi 43. Thickness-Shear and Flexural Vibrations of Contoured Crystal Plates (with M. Forray) , Journal of Applied Physics 25, 12-20 (1954) . 44. Thickness-Shear Vibrations of Piezoelectric Crystal Plates with Incomplete Electrodes (with H. Deresiewicz) , Journal of Applied Physics 25, 21-24 (1954) . 45. Suppression of Overtones of Thickness-Shear and Flexural Vibrations of Crystal Plates (with H. Deresiewicz) , Journal of Applied Physics 25, 25-27 (1954) . 46. Force at a Point in the Interior of a Semi-Infinite Solid, Proceedings of the First Mid- western Conference on Solid Mechanics, 56-59 (1955) . 47. Thickness-Shear and Flexural Vibrations of a Circular Disk (with H. Deresiewicz) , Journal of Applied Physics 25, 1329-1332 (1954) . 48. Axially Symmetric Flexural Vibrations of a Circular Disk (with H. Deresiewicz) , Journal of Applied Mechanics 22, 86-88 (1955) . 49. Mechanics of Granular Media, Proceedings of the Second U.S. National Congress of Applied Mechanics, 13-20 (1954) . 50. Timoshenko's Shear Coefficient for Flexural Vibrations of Beams (with H. Deresiewicz), Proceedings of the Second U.S. National Congress of Applied Mechanics, pp. 175-178 (1954) . 51. Mathematical Theory of Three-Dimensional Photoelasticity, Colloque de Photoelasticite et Photoplasticite, Union Internationale de Mecanique Theoretique et Appliquee, Bruxel- les, 1954, p. 47. 52. Thickness-Shear and Flexural Vibrations of Rectangular Crystal Plates (with H. Dere- siewicz) , Journal of Applied Physics 26, 1435-1442 (1955) . 53. Flexural Vibrations of Rectangular Plates (with A. Schacknow and H. Deresiewicz) , Journal of Applied Mechanics 23, 430-436 (1956) . 54. High-Frequency Extensional Vibrations of Plates (with T. R. Kane), Journal of Applied Mechanics 23, 277-283 (1956) . 55. Simple Modes of Vibration of Crystals, Journal of Applied Physics 27, 1462-1466 (1956) . 56. Waves on the Surface of a Crystal (with H. Deresiewicz), Journal of Applied Physics 28, 669-671 (1957) . 57. Velocities of Long Waves in Plates (with D. C. Gazis), Journal of Applied Mechanics, 24, 541-546 (1957) . 58. Stress-Strain Relations and Vibrations of a Granular Medium (with J. Duffy) , Journal of Applied Mechanics 24, 585-593 (1957) . 59. Vibrations of a Monoclinic Crystal Plate (with E. G. Newman) , Journal of the Acoustical Society of America 29, 1206-1218 (1957) . 60. Vibrations of an Infinite, Elastic Plate at Its Cut-Off Frequencies, Proceedings of the Third U.S. National Congress of Applied Mechanics, pp. 225-226 (1958) . 61. Extensional Vibrations of Elastic Plates (with M. A. Medick), Journal of Applied Mechanics 26, 561-569 (1959) . 62. Axially Symmetric Waves in Elastic Rods (with H. D. McNiven) , Journal of Applied Mechanics 27, 145-151 (1960) . 63. Vibrations and Waves in Elastic Bars of Rectangular Cross Section (with E. A. Fox), Journal of Applied Mechanics 27, 152-158 (1960) . 64. Dispersion of Flexural Waves in Elastic, Circular Cylinder (with U. H. Pao) , Journal of Applied Mechanics 27, 513-520 (1960) . 65. Extensional Vibrations and Waves in a Circular Disk aad a Semi-Infinite Plate (with D. C. Gazis) , Journal of Applied Mechanics 27, 541-545 (1960) . xii LIST OF PUBLICATIONS 66. Waves and Vibrations in Isotropic, Elastic Plates, Structural Mechanics, Pergamon Press, New York, 1960, pp. 199-232. 67. On the Equations of Extensional Motion of Crystal Plates (with H. L. Cooper) , Quart. A rrRl. Math. 19, 111-118 (1961) . 68. On the Equations of Motion of Piezoelectric Crystals, Problems of Continuum Mechanics, Soc. for Ind. and Appl. Math., Philadelphia, 1961, pp. 282-290. Muskhelishvili An- niversary Volume. 69. High Frequency Vibrations of Crystal Plates, Quart. A p pl. Math. 19, 51-61 (1961) . 70. Forced Vibrations of Piezoelectric Crystal Plates (with H. F. Tiersten) , Quart. App!. Math. 20, 107-119 (1962) . 71. Vibrations of an Infinite, Monoclinic Crystal Plate at High Frequencies and Long Wave Lengths (with R. K. Kaul) , Journal of the Acoustical Society of America 34, 1895-1901 (1962) . 72. Frequency Spectrum of a Monoclinic Crystal Plate (with R. K. Kaul) , Journal of the Acoustical Society of America 34, 1902-1910 (1962) . 73. Extensional Vibrations of Thin Quartz Disks (with A. G. Lubowe) , Journal of the Acoustical Society of America 34, 1911-1918 (1962). 74. Dispersion of Axially Symmetric Waves in Elastic Rods (with M. Onoe and H. D. Mc- liven) , Journal of Applied Mechanics 29, 729-734 (1962) . 75. Strong Resonances of AT-Cut Quartz Plates (with D. C. Gazis) , Proc. Fourth U.S National Congress of Applied Mechanics, pp. 305-310 (1962) . 76. Effects of Couple-stresses in Linear Elasticity (with H. F. Tiersten) , Archive f or Rational Mechanics and Analysis 11, 415-448 (1962) . 77. Influence of Couple-stresses on Stress Concentrations, Experimental Mechanics 3, 1-7 (1963) . 78. Complex Representation of Displacements and Stresses in Plane Strain with Couple- stresses, Proceedings of the International Symposium on Applications of the Theory of Functions in Continuum Mechanics, Tbilisi, USSR, 1963, pp. 256-259. 79. Extensional Waves Along the Edge of a Plate (with J. J. McCoy) , Journal of Applied Mechanics 30, 75-78 (1963) . 80. High Frequency Vibrations of Plated Crystal Plates, Progress in Applied Mechanics, Macmillan, New York, 1963, pp. 73-84. Prager Anniversary Volume. 81. Micro-structure in Linear Elasticity, Archive for Rational Mechanics and Analysis 16, 51-78 (1964). 82. On the Equations of Elastic Materials with Micro-structure, International Journal of Solids and Structures 1, 73-78 (1965) . 83. Thickness-twist Vibrations of an Infinite, Monoclinic, Crystal Plate, International Jour- nal of Solids and Structures, 1, 141-145 (1965) . 84. Stress Functions for a Cosserat Continuum, International Journal of Solids and Struc- tures 1, 265-271 (1965) . 85. Second Gradient of Strain and Surface-tension in Linear Elasticity, International Journal of Solids and Structures, 1, 417-438 (1965) . 86. Thickness-shear and Flexural Vibrations of Partially Plated, Crystal Plates (with P. C. U. Lee) , International Journal of Solids and Structures 2, 125-139 (1966) . 87. Bechmann's Number for Harmonic Overtones of Thickness-Twist Vibrations of Rotated- Y-Cut Quartz Plates, Journal of the Acoustical Society of America 41, 969-973 (1967). 88. Anharmonic, Thickness-twist Overtones of Thickness-shear and Flexural Vibrations of LIST OF PUBLICATIONS xiii Rectangular, AT-cut, Quartz Plates (with W. J. Spencer), Journal of the Acoustical Society of America 42, 1268-1277 (1967) . 89. On First Strain-gradient Theories in Linear Elasticity (with N. N. Eshel) , International Journal of Solids and Structures 4, 109-124 (1968) . 90. Polarization Gradient in Elastic Dielectrics, International Journal of Solids and Struc- tures 4, 637-642 (1968) . 91. Optimum Sizes and Shapes of Electrodes for Quartz Resonators, Journal of the Acoustical Society of America 43, 1329-1331. 92. Face-shear Waves in Rotated-V-cut Quartz Plates, The Joseph Mann Memorial Volume, University of Toronto Press, 1968, pp. 143-145. 93. Theories of Elastic Continua and Crystal Lattice Theories, Mechanics of Generalized Continua, Springer-Verlag, 1968, pp. 312-320. 94. Lattice and Continuum Theories of Simple Modes of Vibration in Cubic Crystal Plates and Bars Symposium on Dynamics of Structural Solids, American Society of Mechanical Engineers, 1968, pp. 2-9. 95. Note on Vibrations of Prisms and Polygonal Plates, The L. I. Sedov Anniversary Volume, USSR Academy of Sciences and Society for Industrial and Applied Mathematics, Phila- delphia, 1969. 96. Crystal Lattice Theory of Torsion of a Rectangular Bar of Simple Cubic Structure, The B. Seth Anniversary Volume, Indian Journal of Mechanics and Mathematics, 1969. 97. Continuum and Lattice Theories of Influence of Electro-Mechanical Coupling on Capaci- tance of Thin Dielectric Films, International Journal of Solids and Structures 5, 1197- 1208 (1969) . 98. Lattice Theory of Shear Modes of Vibration and Torsional Equilibrium of Simple-Cubic Crystal Plates and Bars, Internationa lJournal of Solids and Structures 6, 725-738 (1970) . 99. Polarization Gradient in Elastic Dielectric, International Centre for Mechanical Sciences, Udine, Italy, 1970. 100. Thickness-Twist Vibrations of a Quartz Strip, International Journal of Solids and Struc- tures 7(1), 1-4 (1971). 101. Forced Electromechanical Vibrations of a Sodium Chloride Plate. (Published as Item 105 below.) 102. Acoustical and Optical Activity in Alpha Quartz (with R. A. Toupin) International Journal of Solids and Structures 7(9), 1219-1228 (1971) . 103. A Continuum Theory of a Diatomic, Elastic Dielectric, International Journal of Solids and Structures 8(3), 369-384 (1972) . 104. Coupled Elastic and Electromagnetic Fields in a Diatomic, Dielectric Continuum International Journal of Solids and Structures 8(4), 401-408 (1972) . 105. Electromechanical Oscillations of Centro-Symmetric Crystal Plates with Cubic Lattice, Applied Mathematics and Mechanics (PMM) 35 (3) , 446-450 (1971) . 106. Galerkin Stress Functions for Non-Local Theories of Elasticity, Galerkin Anniversary Volume, edited by A. V. Ishlinski, Moscow, USSR (1972) . 107. High Frequency Vibrations of Piezoelectric Crystal Plates, International Journal of Solids and Structures 8, 895-906 (1972) . 108. Equations of High Frequency Vibrations of Thermopiezoelectric Crystal Plates, Inter- national Journal of Solids and Structures, forthcoming. 109. Elasticity, Piezoelectricity and Crystal Lattice Dynamics, Journal of Elasticity 2, 217 (1972).