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The general nonlinear theory of elasticity

A review

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Literature Cited

  1. A. Ya. Aleksandrov, M. Kh. Akhmetzyanov, G. N. Albaut, and V. N. Baryshnikov, “Polarized light investigations for large strains,” Zh. Prikl. Mekh. i Tekhn. Fiziki, No. 5 (1969).

  2. S. A. Alekseev, “Foundations of the theory of flexible shells,” in: Calculation of Three-Dimensional Structures [in Russian], No. 11, Gosstroiizdat, Moscow (1967).

    Google Scholar 

  3. R. L. Aliev, A. M. Lokoshchenko, and S. A. Shesterikov, “The large deflections of a nonlinear elastic ring under external pressure,” Vestnik MGU, Matematika i Mekhanika, No. 3 (1969).

  4. V. L. Berdicheskii, “Construction of models of solid media by means of the variational principle,” Prikl. Matem. i Mekh.,30, No. 3 (1966).

  5. V. L. Biderman, “Problems of calculating rubber elements,” in: Strength Calculations [in Russian], No. 3, GNTI, Moscow (1958).

    Google Scholar 

  6. V. L. Biderman, “The stability ofa bar of a neo-Hookean material,” Inzhenernyi Zhurnal, Mekhanika Tverdogo Tela, No. 3 (1968).

  7. V. L. Biderman and A. D. Zhukov, “Pure flexure of a rubber strip with large strains,” Izv. VUZ. Mashinostroenine, No. 1 (1969).

  8. V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  9. S. I. Vol'vich, “Stability of finite strains of a plate,” in: Stability Problems in Building Mechanics [in Russian], Gosstroiizdat, Moscow (1965).

    Google Scholar 

  10. S. I. Vol'vich and Yu. F. Fokin, “On the flexure theory of plates, with finite strains taken into account,” Proceedings of the Sixth All-Union Conference on the Theory of Shells and Plates, Baku, 1966 [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  11. A. S. Vol'mir, Stability of Elastic Systems [in Russian], Fizmatgiz, Moscow (1967).

    Google Scholar 

  12. K. Z. Galimov, “A survey of papers by Kazan investigators on the theory of elasticity and plasticity,” in: Investigations of the Theory of Plates and Shells [in Russian], No. 5, Izd. Kazanskogo Univ. (1967).

  13. I. I. Gol'denblat, Certain Problems of the Mechanics of Deformable Media [in Russian], Gostekhizdat, Moscow (1955).

    Google Scholar 

  14. I. I. Gol'denblat, Nonlinear Problems of the Theory of Elasticity [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  15. A. S. Grigor'ev, “Equilibrium of momentless shells of revolution with large strains,” Prikl. Matem. i Mekh.,25, No. 6 (1961).

  16. A. S. Grigor'ev, “Stability of momentless shells under conditions of tension,” Proceedings of the Sixth All-Union Conference on the Theory of Shells and Plates, Baku, 1966 [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  17. A. Green and J. Adkins, Large Elastic Strains and the Nonlinear Mechanics of a Solid Medium [Russian translation], Mir, Moscow (1965).

    Google Scholar 

  18. A. N. Guz', “Stability of elastic-plastic bodies,” Prikladnaya Mekhanika,5, No. 8 (1969).

  19. A. N. Guz', “On linearized problems of the theory of elasticity,” Prikladnaya Mekhanika,6, No. 2 (1970).

  20. L. K. Zarembo and V. A. Krasil'nikov, An Introduction to Nonlinear Acoustics [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  21. N. V. Zvolinskii and P. M. Riz, “On certain problems of the nonlinear theory of elasticity,” Prikl. Matem. i Mekh.,2, No. 4 (1939).

  22. K. Zener, Elasticity and Inelasticity of Metals [Russian translation], IL, Moscow (1964).

    Google Scholar 

  23. L. M. Zubov, “An approximate theory of buckling of thin plates of semilinear material with affine initial strain,” Prikl. Matem. i Mekh.,33, No. 4, (1969).

  24. D. D. Ivlev, “On setting up a theory of elasticity,” Dokl. Akad. Nauk SSSR,138, No. 6, (1961).

  25. A. A. Il'yushin, “On the plasticity postulate,” Prikl. Matem. i Mekh.,25, No. 3 (1961).

  26. A. A. Il'yushin, Plasticity [in Russian], Izd. AN SSSR, Moscow (1963).

    Google Scholar 

  27. H. Kauderer, Nonlinear Mechanics [Russian translation], IL, Moscow (1961).

    Google Scholar 

  28. S. M. Kloizner and A. S. Kosmodamianskii, “Nonlinear problems of the plane theory of elasticity for multiply connected media,” Prikladnaya Mekhanika,5, No. 8 (1969).

  29. Yu. I. Koifman, “The stress-strain state of pipes and annular discs of highly elastic, nonlinearly elastic material,” in: Dynamics and Strength of Machines [in Russian], No. 3, Izd. Khar'kovsk. Politekhn. Inst., Khar'kov (1966).

    Google Scholar 

  30. Yu. I. Koifman, “Large elastic strains of cylindrical flexure of rectangular plates,” Prikladnaya Mekhanika,4, No. 5 (1968).

  31. Yu. I. Koifman, “Plane nonlinear problems of elastic equilibrium of multiply connected bodies,” Prikladnaya Mekhanika,6, No. 2 (1970).

  32. Yu. I. Koifman and A. Sh. Langleiben, “Calculation of multilayered cylinders and discs of highly elastic polymer materials,” Mekhanika Polimerov, No. 2 (1967).

  33. Yu. I. Koifman and A. Sh. Langleiben, “An analysis of the stress state of massive multichannel chambers of high pressure,” Mekhanika Polimerov, No. 4 (1969).

  34. E. Bair (editor), Structural Properties of Plastics [Russian translation], Khimiya (1967).

  35. Cosmonautics. Little Encyclopedia [in Russian], Entsiklopediya, Moscow (1968).

  36. E. Kroner, The General Continuum Theory of Dislocations and Natural Stresses [Russian translation], Mir, Moscow (1965).

    Google Scholar 

  37. D. I. Kutilin, The Theory of Finite Strains [in Russian], GITTL, Moscow (1947).

    Google Scholar 

  38. L. D. Landau and E. M. Livshits, The Mechanics of Solid Media [in Russian], Gostekhizdat, Moscow (1953).

    Google Scholar 

  39. A. M. Lokoshchenko and S. A. Shesterikov, “A circular viscoelastic membrane subjected to uniform pressure,” Inzhenernyi Zhurnal, Mekhanika Tverdogo Tela, No. 5 (1967).

  40. A. A. Lukashev, S. V. Veremeenko, and G. G. Agas'ev, “Certain results of investigations into the nonlinearity of solids by means of ultrasound,” Summaries of Papers Read at the Fourth All-Union Symposium on Propagation of Elastic and Elastic-Plastic Waves [in Russian], Kishinev (1968).

  41. A. I. Lur'e, “Birfurcation of equilibrium of an ideally elastic body,” Prikl. Matem. i Mekh.,30, No. 4 (1966).

  42. A. I. Lur'e, “Certain problems of the nonlinear theory of elasticity,” Notes on Papers of the Third All-Union Congress on Theoretical and Applied Mechanics [in Russian], Moscow (1968).

  43. A. I. Lur'e, “A theory of elasticity for a semilinear material,” Prikl. Matem. i Mekh.,32, No. 6 (1968).

  44. N. G. Matchenko and L. A. Tolokonnikov, “Setting up a nonlinear theory of plate flexure,” Prikladnaya Mekhanika,3, No. 12 (1967).

  45. R. E. Mkrtchyan, “Problems of large elastic strains for extension, inflation and shear of rotating composite tubes,” Izv. Akad. Nauk ArmSSR, Mekhanika,22, No. 1 (1969).

  46. Kh. M. Mushtari and K. Z. Galimov, The Nonlinear Theory of Flexible Shells [in Russian], Tatknigoizdat, Kazan (1957).

    Google Scholar 

  47. W. Mason, Methods and Instruments of Ultrasonic Investigations [Russian translation], Part A, Mir, Moscow (1966).

    Google Scholar 

  48. O. V. Naumkin, Yu. P. Smirnov, and L. A. Tolokonnikov, “The possibilities of setting up a quadratic theory of elasticity,” Prikladnaya Mekhanika,3, No. 6 (1967).

  49. V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity [in Russian], GITTL, Moscow (1948).

    Google Scholar 

  50. V. V. Novozhilov, “On the connection between stresses and strains in a nonlinearly elastic medium,” Prikl. Matem. i Mekh.,15, No. 2 (1951).

  51. V. V. Novozhilov, “On the principles of processing results of static tests of isotropic materials,” Prikl. Matem. i Mekh.,15, No. 6 (1951).

  52. V. V. Novozhilov, The Theory of Elasticity [in Russian], Sudpromgiz, Moscow (1958).

    Google Scholar 

  53. L. V. Ovsyannikov, Lectures on the Theory of Group Properties of Differential Equations [in Russian], Izd. Novosibirsk. Univ., Novosibirsk (1965).

    Google Scholar 

  54. Ya. Orkish, “The equilibrium of momentless shells of revolution of materials similar to rubber,” Izv. AN SSSR, Mekhanika, No. 4 (1965).

  55. W. Prager, An Introduction to the Mechanics of Solid Media [Russian translation], IL, Moscow (1963).

    Google Scholar 

  56. Yu. N. Rabotnov, Creep of Structural Elements [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  57. A. L. Rachev, “The stability and vibrations of a long bar of rectangular cross section under finite elastic strain,” Izv. Inst. Tekhn. Mekhaniki Bolgarskoi Akad. Nauk, No. 5 (1968).

  58. Rubber — Structural Material of Modern Mechanical Engineering [in Russian], Khimiya, Moscow (1967).

  59. M. Reiner, Rheology [Russian translation], Nauka, Moscow (1965).

    Google Scholar 

  60. R. S. Rivlin, “Large elastic strains,” in: Rheology [Russian translation], IL, Moscow (1962).

    Google Scholar 

  61. G. N. Savin, Stress Distribution Close to Holes [in Russian], Naukova Dumka, Kiev (1968).

    Google Scholar 

  62. G. N. Savin and Yu. I. Koifman, “Certain problems of the plane nonlinear theory of elasticity,” Prikladnaya Mekhanika,7, No. 6 (1961).

  63. G. N. Savin and Yu. I. Koifman, “Nonlinear effects in problems of stress concentration close to holes with reinforced edges,” Prikladnaya Mekhanika,1, No. 9 (1965).

  64. G. N. Savin, A. A. Lukashev, E. M. Lysko, S. V. Veremeenko and S. M. Bozhevskaya, “Propagation of elastic waves in a solid in the case of a nonlinearly elastic model of a solid medium,” Prikladnaya Mekhanika,6, No. 2 (1970).

  65. A Collection of Papers on the Theory of Elasticity [in Russian], Izd. Tul'sk. Politekhn. Inst., Tula (1968).

  66. L. I. Sedov, An Introduction to the Mechanics of Solid Media [in Russian], Fizmatgiz, Moscow (1962).

    Google Scholar 

  67. L. I. Sedov, “On the tensor of energy impulse and on macroscopic internal interactions in a gravitational field and in material media,” Dokl. Akad. Nauk SSSR,164, No. 3 (1965).

  68. L. I. Sedov and M. E. Églit, “Construction of nonholonomic models of solid media, with finiteness of the strains and certain physicochemical effects taken into account,” Dokl. Akad. Nauk SSSR,142, No. 4 (1962).

  69. S. S. Sekoyan and N. E. Eremeev, “Measurement of elasticity constants of the third order for steel by the ultrasonic method,” Izmeritel'naya Tekhnika, No. 7 (1966).

  70. G. S. Taras'ev and L. A. Tolokonnikov, “Finite plane strains of a compressible material,” Prikladnaya Mekhanika,2, No. 1 (1966).

  71. L. A. Tolokonnikov, “On the connection between stresses and strains in the nonlinear theory of elasticity,” Prikl. Matem. i Mekh.,20, No. 3 (1956).

  72. L. A. Tolokonnikov, “Equations of the nonlinear theory of elasticity in terms of displacements,” Prikl. Matem. i Mekh.,21, No. 6 (1957).

  73. L. A. Tolokonnikov, “Finite plane strains of an incompressible material,” Prikl. Matem. i Mekh.,23, No. 1 (1959).

  74. L. A. Tolokonnikov, “A variational equation of stability of a state of equilibrium,” Scientific Works of the Tula Mining Institute [in Russian], No. 3, Gostekhizdat, Moscow (1961).

    Google Scholar 

  75. V. I. Feodos'ev, “On the modes of equilibrium of a spherical rubber shell under internal pressure,” Prikl. Matem. i Mekh.,32, No. 2 (1968).

  76. N. P. Fleishman and Yu. I. Koifman, “Large elastic strains of plane flexure of a rectangular plate subjected to moments at the ends,” Prikladnaya Mekhanika,2, No. 2 (1966).

  77. Yu. F. Fokin, “The stability of finite strains in circular plates subjected to a radial load,” in: Certain Problems of the Strength and Stability of Plane and Spatial Systems [in Russian], No. 25, Izd. Saratovsk. Politekhn. Inst. (1966).

  78. E. M. Chernykh, “The thermodynamic relationships on the surface of a strong fracture in an elastic medium with finite strains,” Dokl. Akad. Nauk SSSR,177, No. 3 (1967).

  79. E. M. Chernykh and S. G. Kotlyarova, “Disturbance propagation in a nonlinearly elastic medium with finite strains,” Summaries of Papers Read at the Fourth All-Union Symposium on Propagation of Elastic and Elastic-Plastic Waves [in Russian], Kishinev (1968).

  80. J. D. Achenbach, “Shear waves in finite elastic strain,” Quart. Appl. Math.,25, No. 3 (1967).

  81. H. A. Alexander, “A constitutive relation for rubberlike materials,” Int. J. Engng. Sci.,6, No. 9 (1968).

  82. J. R. Asay and A. H. Guenther, “Experimental determination of ultrasonic wave velocities in plastics as functions of temperature,” J. Appl. Polymer Sci.,10, No. 5 (1966);11, No. 7 (1967).

  83. J. R. Asay, P. L. Lamberson, and A. H. Guenther, “Ultrasonic technique for determining sound-velocity changes in high loss materials,” J. Acoust. Soc. Amer.,3, No. 45 (1969).

  84. J. H. Baltruconis and R. N. Vaishnav, “Finite deformations under pressurization in infinitely long, thick-walled, elastic cylinder ideally bonded to a thin elastic case,” Trans. Soc. of Rheology,9, No. 1 (1965).

  85. Z. P. Bazant, “L'instabilité d'un milieu continu et la résistance en compression,” Bull. RILEN, No. 35 (1967).

  86. M. F. Beatty, “Stability of the undistorted states of an isotropic elastic body,” Int. J. Nonlinear Mechanics,3, No.3 (1968).

  87. M. F. Beatty and D. E. Hooyk, “Some experiments on the stability of circular rubber bars under end thrust,” Int. J. Solids Struct.,4 (1968).

  88. J. E. Besseling and H. H. Voetman, “Thermoelastic effects in rubber,” Arch. Mech. Stosow.,20, No. 2 (1968).

  89. S. Bharatha, “Cylindrically symmetrical deformations of Mooney materials,” Arch. Mech. Stosow.,19, No. 6 (1967).

  90. S. Bharatha, “Simultaneous extension, inflation, and shear of cylindrical tube,” Appl. Sci. Res.19, No. 1 (1968).

  91. R. Bhimsain and G. Lakshiminarayana, “Finite bending of plates,” Proc. Nat. Acad. Sci. India, Ser. A,35, No. 2 (1965).

  92. M. P. Bieniek and M. Shinozuka, “Finite deformations of a thick-walled cylinder,” AIAA Journal,2, No. 3 (1964).

  93. M. A. Biot, Mechanics of Incremental Deformation, Wiley, New York (1965).

    Google Scholar 

  94. E. H. Bocardus, “Third-order elastic constants of Ge, MgO, and fused SiO2,” J. Appl. Phys.,36 (1965).

  95. S. F. Borg, “Thin plates with large deflection including the effect of the third-order elastic constants,” Ann. N. Y. Acad. Sci.,147, No. 6 (1968).

  96. F. Böhm, “Zur Mechanik des Luftreifens,” Diss. Techn. Hochschule, Stuttgart (1966).

  97. M. M. Caroll, “Some results of finite-amplitude elastic waves,” Acta Mech.,3, No. 2 (1967).

  98. B. Cavell, “The axially loaded homogeneous cylindrical rubber spring,” Acta Polytechn. Scand., Mech. Engn., Ser. A, No. 28 (1967).

  99. H. R. Chaudhry, “Straightening out an initially curved incompressible cuboid of highly elastic and transversely isotropic material,” J. Sci. Engng. Res.,11, No. 2, (1967).

  100. H. R. Chaudhry and A. N. Chawla, “Rotation of reinforced circular cylindrical tube,” Proc. Cambridge Phil. Soc.,64, No. 2, (1968).

  101. Seh-Ich Chou and G. D. Blomstron, “Finite deformation of a spinning elastic membrane,” AIAA Journal,7, No. 8 (1969).

  102. R. M. Christensen, “Large elastic deformation of spherical wedge,” Int. J. Nonlinear Mechanics,2, No. 3 (1967).

  103. R. F. Craine, “Spherically symmetric problems in finite thermoelastics,” Quart. J. Mech. Appl. Math.,21, No. 3 (1968).

  104. D. Crecraft, “Ultrasonic wave velocities in stressed nickel steel,” Nature,195 (1962).

  105. M. T. Crochet and P. M. Naghdi, “Small motions superimposed on large static deformations in porous media,” Acta Mech.,4, No. 4 (1967).

  106. M. T. Crochet and P. M. Naghdi, “Large deformation solutions for an elastic Cosserat surface,” Int. J. Engng. Sci.,7, No. 3 (1969).

  107. T. C. Doyle and J. L. Ericksen, “Nonlinear elasticity,” Adv. Appl. Mech.,4 (1956).

  108. A. J. Durelli, “Experimental strain and stress analysis of solid propellant rocket motor,” Mech. and Chem. Solid Propellant, Pergamon Press, New York, (1967).

    Google Scholar 

  109. A. J. Durelli, V. J. Parks, and J. L. Chen, “Stress concentrations in a rectangular plate with circular perforations along its two bonded edges and subjected to restrained shrinkage,” J. of Strain and Stress Analysis,1, No. 5 (1969).

  110. A. J. Durelli, V. J. Parks, and H. Feng, “Experimental methods of large strain analysis,” Int. J. Nonlinear Mech.,2, No. 4 (1967).

  111. B. D. Dyszczyk, “Statecznosc grubosciennej rury obciazonej cisnieniem zewnetrznym,” Pozpz. Inz.,15, No. 2 (1967).

  112. F. N. Earle, “Large amplitude oscillations of some axially symmetric elastic bodies,” Doctoral Dissertation, Cornell University (1966); Dissert. Abstrs, Ser. B,28, No. 2 (1968).

  113. A. S. Eringen, Nonlinear Theory of Continuous Media, McGraw-Hill, New York (1962).

    Google Scholar 

  114. R. J. Evans and K. S. Pister, “Constitutive equations for a class of nonlinear elastic solids,” Int. J. Solids Struct.,2 (1966).

  115. A. D. Fine and R. T. Shield, “Second-order effects in the propagations of elastic waves,” Int. J. Struct.,2 (1966).

  116. R. L. Fosdick, “Dynamically possible motions of incompressible isotropic simpler materials,” Arch. Rat. Mech. Anal.,29, No. 4 (1968).

  117. H. O. Foster, “Inflation of a plane circular membrane,” Trans. ASME, No. 3, (1967).

  118. W. L. Fourney, “Stability study of a finitely deformed elastic shell,” Doctoral Dissertation (1966); Dissert. Abstrs., Ser. B,27 (1967).

  119. W. L. Fourney and M. Stern, “Stability of a finitely deformed thin cyclindrical shell,” Int. J. Engng. Sci.,6, No. 11 (1968).

  120. B. Fung Yuang-Cheng, “Biomechanics, its scope, history and some problems of continuum mechanics in physiology,” Appl. Mech. Rev.,21, No. 1 (1968).

  121. A. N. Gent and P. B. Lindley, “The compression of bonded rubber blocks,” Proc. Inst. Mech. Engrs.,173, No. 3 (1959).

  122. A. N. Gent and R. S. Rivlin, “Experiments on the mechanics of rubber,” Proc. Phys. Soc., Ser. B.65 (1952).

  123. A. N. Gent and A. G. Thomas, “Form for the strain energy function for vulcanized rubber,” J. Polymer Sci., No. 28 (1958).

  124. A. E. Green and E. B. Spratt, “Second-order effects in the deformations of elastic bodies,” Proc. Roy. Soc., Ser. A, No. 224 (1954).

  125. A. E. Green and W. Zerna, Theoretical Elasticity, 2nd edition, Clarendon Press, Oxford-London (1968).

    Google Scholar 

  126. A. E. Green, R. J. Knops, and N. Law, “Large deformations superimposed on small deformations and stability of elastic rods,” Int. J. Solids Struct.,4, No. 5 (1968).

  127. A. S. Grigoriev, “Tensile instability in cases of large deformations,” Geophys. J. Roy. Astron. Soc.,14 (1967).

  128. G. Grioli, Mathematical Theory of Elastic Equilibrium (Recent Results), Springer Verlag, Berlin (1962).

    Google Scholar 

  129. R. A. Grot, “Relativistic theory of the propagation of wave fronts in nonlinear elastic materials,” Int. J. Engng. Sci.,6, No. 5 (1968).

  130. Ihsan Gurgoze, “The effect of rotation on the finite elastic deformations,” Istanbul Tekn. Univ. Ful., No. 19 (1966).

  131. M. M. Hall, “Studies of pure homogeneous deformations in unfilled natural and butyl rubbers,” Coll. Aeron. in Cranfield Note, Mater.,4, No. 16 (1968).

  132. L. J. Hart-Smith, “Elasticity parameters for finite deformations of rubber-like materials,” Z. Angew. Math. und Phys.,17, No. 5 (1966).

  133. L. J. Hart-Smith, “An analysis of inflatable paraboloids,” Mech. and Chem. Engng., Trans. Inst. Engrs. Austral.,3, No. 2 (1967).

  134. L. J. Hart-Smith and J. D. Grisp, “Large elastic deformations of thin rubber membranes,” Int. J. Engn. Sci.,5, No. 7 (1967).

  135. M. A. Hayes, “A remark on Hadamard materials,” Quart. J. Mech. and Appl. Math.,21, No. 2 (1968).

  136. G. S. Healy, “A finite strain theory and its applications to the plane stress response of polyurethane,” Doctoral Dissertation, Va Polytechn. Inst. (1966); Dissert. Abstrs., Ser. B,27, No. 7 (1967).

  137. G. Herrmann, “On second-order thermoelastic effects,” Z. Angew. Math. und Phys.,15, No. 4 (1964).

  138. R. Hill, “On uniqueness and stability in the theory of finite elastic strain,” J. Mech. Phys. Solids,5, No. 2 (1957).

  139. R. Hill, “On constitutive inequalities for simple materials. I,” J. Mech. Phys. Solids,16, No. 4-5 (1968).

  140. D. S. Hughes and J. L. Kelly, “Second-order elastic deformations of solids,” Phys. Rev., No. 92 (1953).

  141. R. R. Huilgol, “Bending of a cylinder in second-order elasticity,” J. Mec.,6, No. 2 (1967).

  142. R. R. Huilgol, “A class of small displacements superimposed on large deformations in finite elasticity,” Bull. Acad. Polon. Sci., Ser. Sci. Techn.,16, No. 4 (1968).

  143. R. L. Huston, “Nonlinear stress-strain equations,” Int. J. Solids Struct.,1 (1965).

  144. F. John, “On finite deformations of elastic isotropic materials,” Inst. Math. Sci. N. Y. Univ. Report IMM (1958).

  145. F. John, “Plane strain problems for a perfectly elastic material of harmonic type,” Commun. Pure and Appl. Math.,13 (1960).

  146. F. John, “Perfectly elastic bodies of harmonic type,” Proceedings of Intern. Symposium on Applications of Theory of Functions in Mechanics of Solids, Tbilisi, 1963 [in Russian], Vol. 1, Nauka, Moscow (1965).

    Google Scholar 

  147. Z. Karni and M. Reiner, “The complete tensor expressions for the Green and Almansi measures of deformations,” Z. Angew. Math. und Phys.,18, No. 1 (1967).

  148. A. D. Kerr and S. Tang, “The instability of a rectangular elastic solid,” Acta Mech.,4, No. 4 (1967).

  149. W. W. Klingbeil and R. T. Shield, “Some numerical investigations on empirical strain energy function,” Z. Angew Math. und Phys.,15, No. 3 (1964).

  150. W. W. Kingbeil and R. T. Shield, “Large deformation analysis on bonded elastic mounts,” Z. Angew. Math. und Phys.,17, No. 3 (1966).

  151. V. A. Krasilnikov and L. K. Zarembo, “Nonlinear interaction of elastic waves in solids,” IEEE Trans., Sonics and Ultrasonics,14, No. 1 (1967).

  152. W. R. Krigbaum and R. J. Roe, “Survey of the theory of rubberlike elasticity,” Rubber Chem. and Technol.,38, No. 5 (1965).

  153. A. D. Kydoniefs, “The finite inflation of an elastic toroidal membrane,” Int. J. Engng. Sci.,5, No. 6 (1967).

  154. A. D. Kydoniefs and A. M. Spencer, “Finite axisymmetric deformations of an initially cylindrical elastic membrane,” Quart. J. Mech. and Appl. Math,22 (1969).

  155. Lee Jen-Shin, “Application of finite elastic theory to the deformation of cylindrical tubes. I, II,” Doctoral Dissertation, Calif. Inst. Technol., Dissert. Abstrs., Ser. B,27, No. 5 (1966).

  156. J. W. Leonard, “Inflatable shell: pressurization phase,” J. Engng. Mech. Div. Proc. Amer. Soc. Civil Engrs.,93, No. 2 (1967).

  157. M. Levinson, “Stability of a compressed neo-Hookean rectangular parallelepiped,” J. Mech. Phys. Solids,16, No. 6 (1968).

  158. H. J. McSkimin and P. J. Andreatch, “Measurement of very small changes in the velocity of ultrasonic waves in solids,” J. Acoust. Soc. Amer.,41, No. 4 part 2 (1967).

  159. J. Meixner, “Process in simple thermodynamic materials,” Arch. Rat. Mech. Anal.,33, No. 1 (1969).

  160. F. D. Murnaghan, “Finite deformation of an elastic solid,” Amer. J. Math., No. 59, (1937).

  161. F. D. Murnaghan, Finite Deformation of an Elastic Solid, 2nd edition, New York (1967).

  162. Nono Takayuki, “On the stored energy function of a hyperelastic material: an application of the theory of Lie groups,” J. Math. Analysis and Applic.2, No. 24 (1968).

  163. J. L. Nowinski and T. T. Wu, “A nonlinear dynamic problem for a thick-walled cylinder of electrostrictive material,” Int. J. Engng. Sci.,6, No. 1 (1968).

  164. J. Orkisz, “Finite deformation of a circularly symmetric shell under membrane state of stress in some nonlinear cases,” Bull. Acad. Polon. Sci., Ser. Sci. Techn.,15, No. 1 (1967).

  165. W. A. Ortwein, “A nonlinear stress-strain relation,” Int. J. Solids Struct.,3, No. 4 (1967).

  166. Y. Partom, “The dilatational and distortional partition of the energy function of finite strain,” Is. J. Technol.,4, No. 3 (1966).

  167. S. Pauthier-Camier, “Recherches experimentales surles effects élastiques nonlinéaires,” Annal. Phys., No. 1 (1966).

  168. C. E. Pearson, “General theory of elastic stability,” Quart. Appl. Math., No. 14 (1956/57).

  169. H. J. Petroski and D. E. Carlson, “Controllable states of elastic heat conductor,” Arch. Rat. Mech. Anal.,31, No. 2 (1968).

  170. W. Philippoff, “Elasticity in steady flow,” Trans. Soc. Rheology,10, No. 1 (1966).

  171. Proceedings of the IUTAM Symp. on Second-Order Effects in Elasticity, Plasticity, and Fluid Dynamics, Haifa, 1962 (1964).

  172. M. Reiner, “Research on second-order effects in the elastic response of metals,” Tech. Found. Report., No. 70 (1965).

  173. R. S. Rivlin, “The solution of problems in second-order elasticity theory,” J. Rat. Mech. Anal.,2, No. 1 (1953).

  174. R. S. Rivlin, “Some topics in finite elasticity, structural mechanics,” Proceedings of the First Symposium on Naval Structural Mechanics, Pergamon Press, New York (1960).

    Google Scholar 

  175. R. S. Rivlin, “Fundamental tensor relations of nonlinear continuum mechanics,” Theory of Plates and Shells, Bratislava (1966).

  176. R. S. Rivlin and C. J. Topakogly, “A theorem in the theory of finite elastic deformations,” J. Rat. Mech. Anal.,3, No. 5 (1954).

  177. Sakurai Masayuki, Sato Yoshiyasi, Shimada Kojui, Nomura Akimasa and Nakagawa Takeshi, “Large elastic behavior of SBR and BR vulcanizates,” Repts. Progr. Polymer Phys. Japan,12 (1969).

  178. Satake Masao, “Elasticity law of large deformations,” Proc. 15th Japan Nat. Congr. Appl. Mech., Tokyo, 1965 (1966).

  179. D. W. Saunders, “Large Deformations in amorphous polymers,” in: Biomechanics and Related Bioengineering Topics, Pergamon Press, New York (1965).

    Google Scholar 

  180. A. Seeger and O. Buck, “Die experimentelle Ermittlung der elastischen Konstanten höherer Ordnung,” Z. Naturforsch.,15a, No. 12 (1960).

  181. C. B. Senceinig, “Instability of thick elastic solids,” Commun. Pure and Appl. Math.,17, No. 4 (1964).

  182. P. L. Sheng, “Secondary elasticity,” Chin. Assoc. Adv. Sci. (1955).

  183. A. Signorini, Questioni di Elasticita Non-linearezzatta, Rome (1960).

  184. Monahar Singh, “Small finite deformations of elastic dielectrics,” Quart. App. Math.,25, No. 3 (1967).

  185. Wargam Singh and H. R. Chaudhry, “Finite flexure of reinforced aelotropic elastic cuboid,” Arch. Mech. Stosow.,19, No. 5 (1967).

  186. I. L. Smith, “Large deformation tensile properties of elastomers, temperature dependence of Ci and C2 in the Mooney-Rivlin equation,” J. Polymer Sci., Ser. C, No. 16, part 2 (1967).

  187. M. Stern. “An approximate shell theory for unrestricted elastic deformations,” Int. J. Solids Struct.,3, No. 5 (1967).

  188. R. Stojanovich, “A geometric derivation of the nonlinear stress-strain relations for isotropic elastic solids,” Z. Angew. Math. und Mech., No. 39 (1959).

  189. F. Stopelli, “Sulla svilluppabilita in serie di potenzedi un parametro delle soluzioni dell equazioni dell elastostatica isoterma,” Ricerche Mat, No. 4 (1955).

  190. E. G. Tickner and A. H. Sacks, “A theory for the static elastic behavior of blood vessels,” Biorheology, No. 4 (1967).

  191. R. A. Toupin and B. Bernstein, “Sound waves in deformed perfectly elastic materials, acoustoelastic effect,” J. Acoust. Soc. Amer., No. 33 (1961).

  192. C. Truesdell, “The mechanical foundations of elasticity and fluid dynamics. I,” J. Rat. Mech. Anal., No. 1 (1952); “The mechanical foundations of elasticity and fluid dynamics. II,” J. Rat. Mech. Anal., No. 2 (1953).

  193. C. Truesdell, Six Lectures on Modern Natural Philosophy, Springer Verlag, New York (1966).

    Google Scholar 

  194. C. Truesdell and W. Noll, “The nonlinear field theories of mechanics,” in: Handbuch der Physik, III/3, S. Flügge (1965).

  195. K. S. Valanis and R. F. Lander, “The strain energy function of a hyperelastic material in terms of the extension ratios,” J. Appl. Phys.,38, No. 7 (1967).

  196. A. S. D. Wang, “On free oscillations of elastic incompressible bodies in finite shear,” Int. J. Engng. Sci.,7, No. 12 (1969).

  197. T. M. Ward and L. P. Pinnock, “The mechanical properties of solid polymers,” Brit. J. of Appl. Phys.,17, No. 1 (1966).

  198. Z. Weselowski, “Equations of stability of finitely deformed hollow sphere,” Bull. Acad. Polon. Sci., Techn.,15, No. 2 (1967).

  199. Z. Weselowski, “Stability of an elastic, thick-walled, spherical shell loaded by an external pressure,” Arch. Mech. Stosow.,19, No. 1 (1967).

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Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev and L'vov. Translated from Prikladnaya Mekhanika, Vol. 6, No. 12, pp. 3–26, December, 1970.

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Savin, G.N., Koifman, Y.I. The general nonlinear theory of elasticity. Soviet Applied Mechanics 6, 1275–1295 (1970). https://doi.org/10.1007/BF00888819

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