Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues-with examples

Applied Mechanics Reviews

Volumn 40, Issue 12, 1987, Pages 1699-1734

  Beatty, Millard F ^a  

^a UNIVERSITY OF KENTUCKY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33749072614     PISSN: 00036900     EISSN: None     Source Type: Journal    
DOI: 10.1115/1.3149545     Document Type: Article

References (137)

1. Atkin, R J, and Fox, N (1980). An introduction to the theory of elasticity, Logman Group, London, New York.
2. Antman, S S (1973). Monotonicity and invertibility conditions in one dimensional nonlinear elasticity, Nonlinear elasticity, Academic, New York, p 57-92.
3. Antman, S S (1978). A family of semi-inverse problems in nonlinear elasticity, in Contemporary developments in continuum mechanics and partial differential equations, G M de LaPenha and L A Medeiros, Eds North-Hol-land, Netherlands, p 1-24.
4. Antman, S S (1979). The eversión of thick spherical shells. Arch Rational Mech Anal 70, 113-123.
5. Animan, S (1983). Regular and singular problems for large elastic deformations of tubes, wedges and cylinders, Arch Rational Mech Anal 83, 1-52.
6. Baker, M, and Ericksen, J L (1954). Inequalities restricting the form of stress deformation relations for isotropic elastic solids and Reiner-Rivlin fluids, J Washington Acad Sci 44, 33-35; see also Foundations of elasticity theory, C Truesdell, Ed, International Science Review Series, Gordon and Breach, New York, 1965.
7. Baker, T J (1900). The frequency of transverse vibrations of a stretched India rubber cord, Philosophical Magazine 49, 347-351.
8. Ball, J M (1977a). Convexity conditions and existence theroems in nonlinear elasticity, Arch Rational Mech Anal 63, 337-403.
9. Ball, J (1977b). Constitutive inequalities and existence theorems in nonlinear elastostatics, Nonlinear analysis and mechanics, Vol I, R J Knops, Ed, Pitman, London. 187-241.
10. Ball, J M, and Schaeffer, D (1983). Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead load tractions, Proc Cambridge Philosophical Soc 94, 315-340.
11. Batra, R C (1972). On non-classical boundary conditions, Arch Rational Mech Anal 48, 163-191.
12. Batra, R C (1975). On the coincidence of the principal axes of stress and strain in isotropic elastic bodies, Lett Appl Sci 3, 435-439.
13. Batra, R C (1976). Deformation produced by a simple tensile load in an isotropic elastic body, J Elasticity 6, 109-111.
14. Beatty, M F (1965). Some static and dynamic implications of the general theory of elastic stability, Arch Rational Mech Anal 19, 167-188.
15. Beatty, M F (1967). A theory' of elastic stability for incompressible, hyperelastic bodies, hit J Solids Struct 3, 23-37.
16. Beatty, M F (1970). Stability of hyperelastic bodies subject to hydrostatic loading, hit J Nonlinear Mech 5, 367-383.
17. Beatty, M F (1977). Elastic stability of rubber bodies in compression, Finite elasticity, ASME-AMD vol 27, R S Rivlin, Ed, p 125-150.
18. Beatty, M F (1983). Finite amplitude oscillations of a simple rubber support system, Arch Rational Mech Anal 83, 195-219.
19. Beatty, M F (1984a). A lecture on some topics in nonlinear elasticity and elastic stability, IMA preprint 99, Inst for Math and Its Appl, Univ of Minnesota, MN.
20. Beatty, M F (1984b). Finite amplitude vibrations of a body supported by simple shear springs, J Appl Mech 51, 361-366.
21. Beatty, M F (1986a). Principles of engineering mechanics, Vol 1, Kinematics— The geometry of motion, Plenum, New York and London.
22. Beatty, M F (1986b). Finite amplitude vibrations of a neo-Hookean oscillator, Quart Appl Math 44, 19-34.
23. Beatty, M F (1987). A class of universal relations in isotropic elasticity theory, J Elasticity 17, 113-121.
24. Beatty, M F, and Chow, A C (1983). On the transverse vibration of a rubber string, J Elasticity 13, 317-344.
25. Beatty, M F, and Chow, A C (1984). Free vibrations of a loaded rubber string, Int J Nonlinear Mech 19, 69-81.
26. Beatty, M F, and Hayes, M A (1988). Deformation of an elastic, internally constrained material, to appear.
27. Beatty, M F, and Stalnaker, D O (1986). The Poisson function of finite elasticity, J Appl Mech 53, 807-813.
28. Bell, J F (1983). Continuum plasticity at finite strain for stress paths of arbitrary composition and direction, Arch Rational Mech Anal 84, 139-170.
29. Blatz, P J and Ko, W L (1962). Application of finite elasticity to the deformation of rubbery materials, Trans Soc Rheol 6, 223-251.
30. Bolotin, V V (1963). Nonconservative problems of the theoiy of elastic stability, Macmillan, New York.
31. Bowen, R M, and Wang, C -C (1976). Introduction to vectors and tensors, Plenum, New York and London.
32. Bryan, G H (1888). On the stability of elastic systems, Proc Cambridge Philosophical Soc 6, 199-210, 287-292.
33. Bueche, F (1960). Molecular basis for the Mullins effect, J Appl Polym Sci 4, 108-114. '
34. Carlson, D E, and Shield, R T (1982). Finite elasticity. Proceedings of the IUTAM symposium on finite elasticity, Lehigh Univ 1980, Martinus Nijhoff, The Hague, Boston, London.
35. Charrier, J M (1970). Large elastic deformations of some cord reinforced rubber shells. Rubber Chem Tech 43, 282-303.
36. Chen, P J (1976). Selected topics in wave propagation. Noordh of T, Leyden, Netherlands.
37. Chung, D T, Horgan, C O, and Abeyaratne, R, (1986). The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible elastic materials, Int J Solids Struct 22, 1557-1570.
38. Currie, P K, and Hayes, M (1982). On non-universal finite elastic deformations, Proc of the IUTAM symposium on finite elasticity, Lehigh Univ 1980, D E Carlson and R T Shield, Eds, Nijhoff, The Hague, Boston, London, p 143-150.
39. Dcmiray, H (1976). Stresses in ventricular wall, J Appl Mech 98, 194-197.
40. Dunn, J E (1983). Certain a priori inequalities and a peculiar elastic material, Quart J Mech Appl Math 36, 351-363.
41. Ericksen, J L (1953). On the propagation of waves in isotropic, incompressible, perfectly elastic materials, J Rational Mech Anal 2, 329-337.
42. Ericksen, J L (1954). Deformations possible in every isotropic, incompressible, perfectly elastic body, Z angew Math Phys 5, 466-489.
43. Ericksen, J L (1955). Deformations possible in every compressible, isotropic, perfectly elastic material, J Math Phys 34, 126-128.
44. Ericksen, J L (1977a). Special topics in elastostatics, Advances in applied mechanics, Vol 17, Academic, New York, p 189-244.
45. Ericksen, J L (1977b). Semi-inverse methods in finite elasticity theory', Finite elasticity, ASME-AMD, Vol 27, R S Rivlin, Ed, p 11-21.
46. Ericksen, J L (1985). Constitutive theory for some constrained elastic crystals, IMA Preprint Series #132, Inst for Math and its Appl, Univ of Minnesota, MN.
47. Ericksen, J L, and Toupin, R A (1956). implications of Hadamard’s conditions for elastic stability with respect to uniqueness theorems, Can J Math 8, 432-436.
48. Eringen, A C, and Suhubi, E S (1974). Elastodynamics, Vol 1 Finite motions, Academic, New York, London.
49. Fosdick, R L (1966). Remarks on Compatibility, in Modern developments in the mechanics of continua, Proc of the int conference on rheol, S Eskinazi, Ed, Academic, New York, 109-127.
50. Fosdick, R L, and Schuler, K W (1969). On Ericksen’s problem for plane deformations with uniform transverse stretch, int J Eng Sci 7, 217-233.
51. Fosdick, R L (1971). Statically possible radially symmetric deformations in isotropic incompressible elastic solids, Z Angew Math Phys 22, 590-607.
52. Fung, Y C B (1967). Elasticity of soft tissues in simple elongation. Am J Physiology 213, 1532-1544.
53. Gent, A N (1978). Rubber elasticity: Basic concepts and behavior, in Science and technology of rubber, Academic, New York, chapter 1.
54. Green, A E, and Adkins, J E (1960). Large elastic deformations, Oxford UP (Clarendon), London and New York.
55. Green, A E, and Zerna, W (1954). Theoretical elasticity, Oxford UP (Clarendon), London and New York.
56. Gurtin, M E (1972). The linear theory of elasticity, in Mechanics of solids, Vol II, Fliigge’s handbuch der physik, Vol VIa/2, Springer, Berlin, Heidelberg, New York.
57. Gurtin, M E (1981). Topics in finite elasticity. CBMS-NSF regional conference series in appl math, SIAM, Philadelphia PA.
58. Gurtin, M E (1982). On uniqueness in finite elasticity, Proc of the IUTAM symp on finite elasticity, Lehigh Univ 1980, D E Carlson and R T Shield, Eds, Nijhoff, The Hague, Boston, London.
59. Gurtin, M E, and Spector, S J (1970). On stability and uniqueness in finite elasticity, Arch Rational Mech Anal 70, 153-165.
60. Hayes, M, and Knops, R J (1966). On universal relations in elasticity theory, Z Angew Math Phys 17, 636-639.
61. Hill, R (1957). On uniqueness and stability in the theory of finite elastic strain, J Mech Phys Solids 5, 229-241.
62. Holden, J T (1968). A class of exact solutions for finite plane strain deformations of a particular elastic material, Appl Sci Res 19, 171-181.
63. Holden, J T (1971). Note on a class of controllable deformations of incompressible elastic materials, Appl Sci Res 24, 98-104.
64. Huilgol, R R (1966). A finite deformation possible in transversely isotropic materials, Z Angew Math Phys 17, 787-789.
65. Ishihara, N, Flashitsume, N, and Tatibana, M (1951). Statistical theory of rubber-like elasticity IV, J Chem Phys 19, 1508-1520.
66. James, H M, and Guth, E (1943). Theory' of the elastic properties of rubber, J Chem Phys U, 455-481.
67. Kafadar, C B (1972). On Ericksen’s problem. Arch Rational Mech Anal 47, 15-27.
68. Kao, B, and Pipkin, A C (1972). Finite buckling of fiber-reinforced columns, Acta Mech 13, 265-280.
69. Kearsley, E A (1986), Asymmetric stretching of a symmetrically loaded elastic sheet, Int J Solids Struct 22, 111-119.
70. Klingbeil, W, and Shield, R T (1966). On a class of solutions in plane finite elasticity, Z Angew Math Phys 17, 489-501.
71. Knops R J, and Wilkes, E W (1973). Theory of elastic stability, Flügge’s handbuch der physik, Vol VIa/3, Springer, Berlin, Heidelberg, New York, p 125-302.
72. Knowles, J K (1960). Large amplitude oscillations of a tube of incompressible elastic material, Quart Appl Math 18, 71-77.
73. Knowles, J K (1962). On a class of oscillations in the finite deformation theory of elasticity, J Appl Mech 29, 283-286.
74. Knowles, J K (1979). Universal states of finite anti-plane shear, Ericksen’s problem in miniature. Am Math Monthly 86, 109-113.
75. Knowles, J K, and Sternberg, E (1975). On the ellipticity of the equations of nonlinear elastostatics for a special material, J Elasticity 5, 341-361.
76. Knowles, J K, and Sternberg, E (1977). On the failure of ellipticity of the equations of finite elastostatic plane strain, Arch Rational Mech Anal 63, 321-336.
77. Krawietz, A (1975). A comprehensive inequality in finite elastic strain, Arch Rational Mech Anal 58, 127-149.
78. von Lang, V (1899). Ueber transversale tone von kaulschukfaden. Ann Phys Chem 68, 335-342.
79. Lee, S J, and Shield, R T (1980). Variational principles in finite elastostatics, Z Angew Math Phys 31, 437-472.
80. Leipholz, H (1970). Stability theory, Academic, New York, p 176-270.
81. Macsithigh, G P (1986). Energy-minimal finite deformations of a symmetrically loaded elastic sheet, Quart J Mech Appl Math 39, 111-123.
82. Marris, A W, and Shiau, J F (1970), Universal deformations in isotropic incompressible hyperelastic materials when the deformation tensor has equal proper values, Arch Rational Mech Anal 36, 135-160.
83. Marris, A W (1975). Universal deformations in incompressible isotropic elastic matericils, J Elasticity 5, 111-128.
84. Marsden, J E, and Hughes, T J R (1983). Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, N J.
85. Martin, S E, and Carlson, D E (1976). A note on Ericksen’s problem, J Elasticity 6, 105-108.
86. Muller, W C (1970), A characterization of the five known families of solutions of Ericksen’s problem, Arch Appl Mech 22, 515-522.
87. Müller, W C (1970). Some further results on the Ericksen problem for deformation with constant strain invariants, Z Angew Math Phys 21, 633-636.
88. Mullins, L (1947). Effects of stretching on the properties of rubber, J Rubber Res 16, 275-289.
89. Mullins, L, and Tobin, N R (1965). Stress softening in rubber vulcanizates. Part I, J Polym Sci 9, 2993-3009.
90. Ogden, R W (1984). Nonlinear elastic deformations, Wiley, New York.
91. Payne, A R, and Scott, J R (1960), Engineering design with rubber, Interscience, New York, chapter 6.
92. Parry, G P (1979). Corollaries of Ericksen’s theorems on the deformations possible in every isotropic hyperelastic body, Arch Appl Mech 31, 757-760.
93. Pearson, C E (1955). General theory of elastic stability. Quart Appl Math 14, 133-144.
94. Pipkin, A C (1977). Finite deformations in materials reinforced with inextensible cords. Finite elasticity, ASME-AMD Vol 27, R S Rivlin, Ed, p 91-102.
95. Pipkin, A C (1979). Stress analysis of fiber reinforced materials. Adv Appl Mech 19, 1-51.
96. Rivlin, R S (1948a). Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure homogeneous deformations. Philosophical Trans R Soc 240, 419-508.
97. Rivlin, R S (1948b). Large elastic deformations of isotropic materials, IV. Further developments of the general theory, Philosophical Trans R Soc London Ser A 241, 379-397.
98. Rivlin, R S (1949a). Large elastic deformations of isotropic materials, V. The problem of flexure, Proc R Soc London Ser A 195, 463-473.
99. Rivlin, R S (1949b). Large elastic deformations of isotropic materials, VI. Further results in the theory of torsion, shear and flexure, Philosophical Trans R Soc London Ser A 242, 173-185.
100. Rivlin, R S (1949c). A note on the torsion of an incompressible highly elastic cylinder, Proc Cambridge Philosophical Soc 45, 485-487.
101. Rivlin, R S (1973). Some restrictions on constitutive equations, Proc of the international symposium on the foundations of continuum thermodynamics, Bassaco.
102. Rivlin, R S (1974). Stability of pure homogeneous deformations of an elastic cube under dead loading. Quart Appl Math 32, 265-272.
103. Rivlin, R S, (Ed) (1977). Finite elasticity, ASME-AMD, Vol 27, ASME, New York.
104. Rivlin, R S (1982). Some thoughts on material stability, Proceedings of the IUTAM symposium on finite elasticity, Lehigh Univ, 1980, D E Carlson and R T Shield, Eds, Ntjhoff, The Hague, Boston, London, 105-122.
105. Rivlin, R S, and Saunders, D W (1951). Large elastic deformations of isotropic materials, VII. Experiments on the deformation of rubber. Philosophical Trans R Soc A243, 251-288.
106. Rogers, C, and Baker, J A (1980). The finite elastodynamics of hyperelastic thin tubes, Int J Nonlinear Mech 15, 225-233.
107. Rogers, T G, and Pipkin, A C (1971). Small deflections of fiber-reinforced beams or slabs, J Appl Mech 38, 1047-1048.
108. Sawyers, K N (1976). Stability of an elastic cube under dead load: Two equal forces, Int J Nonlinear Mech 11, 11-23.
109. Schaffers, W J (1977), Buckling in fiber-rein forced elastomers, Textile Res J 47, 502-512.
110. Sewell, M J (1967). On configuration-dependent loading, Arch Rational Mech Anal 23, 327-351.
111. Shahinpoor, M, and Nowinski, J L (1971). Exact solution to the problem of forced large amplitude radial oscillations of a thin hyperclastic tube, Int J Nonlinear Mech 6, 193-207.
112. Shield. R T (1971). Deformations possible in every compressible, isotropic, perfectly elastic material. J Elasticity 1, 91-92.
113. Shield, R T. Equilibrium solutions in finite elasticity, J Appl Mech 50, 1171-1180.
114. Singh, M, and Pipkin, A C (1965). Note on Ericksen’s problem, Z Angew Math Phys 16, 706-709.
115. Singh, M, and Pipkin, A C (1966). Controllable states of elastic dielectrics, Arch Rational Mech Anal 21, 169-210.
116. Spector, S J (1980). On uniqueness in finite elasticity with general loading, J Elasticity 10, 145-161.
117. Spector, S J (1982). On uniqueness for the traction problem in finite elasticity, J Elasticity 12, 367-383.
118. Spencer, A J M (1970). The static theory of finite elasticity, J Inst Math Appl 6, 164-200.
119. Spencer, A J M (1972). Deformations of fibre-reinforced materials, Oxford UP (Clarendon), London and New York.
120. Tabaddor, F (1987a). Rubber elasticity models for finite element analysis, Comput Struct 25.
121. Tabaddor, F (1987b). Elastic stability of rubber products, preprint.
122. Toupin, R A (1964). Theories of elasticity with couple stress. Arch Rational Mech Anal 17, 85-112.
123. Treloar, L R G (1948). Stresses and birefringence in rubber subjected to general homogeneous strain, Proc Phys Soc London 60, 135-142.
124. Treloar, L R G (1975). The physics of rubber elasticity, 3rd ed, Clarendon, Oxford.
125. Truesdell, C (1952). The mechanical foundations of elasticity and fluid dynamics, J Rational Mech and Anal 1, 125-300.
126. Truesdell, C (1952). Corrections and additions, J Rational Mech Anal 2, 593-616.
127. Truesdell, C A (1956). The main unsolved problem in finite elasticity theory, Z Angew Math Phys 36, 97-103, English translation in Foundations of elasticity theory, International Science Review Series, Gordon and Breach, New York.
128. Truesdell, C, (1962). Solutio generalis et accurata problematum quamplurimoruin de moto coporum elasticorum incomprimibilium in de-formationibus valde magnis, Arch Rational Mech Anal 11, 106-113.
129. Addendum, Arch Rational Mech Anal 12,.1963, 427-428; see also Truesdell and Noll (1965), p 61; and Wang and Truesdell (1973), p 315-341.
130. Truesdell, C (1966). The elements of continuum mechanics. Springer, New York.
131. Truesdell, C, and Noll, W (1965). The nonlinear field theories of mechanics, Fliigge’s handbuch der physik, Vol III/3, Springer, Berlin, Heidelberg, New York.
132. Truesdell, C, and Toupin, R A (1963). Static grounds for inequalities in finite strain of elastic materials, Arch Rational Mech Anal 12, 1-33.
133. Usmani, S A, and Beatty, M F (1974). On the surface instability of a highly elastic half-space, J Elasticity 4, 249-263.
134. Wang, C -C (1965). On the radial oscillations of a spherical thin shell in finite elasticity theory. Quart Appl Math 23, 270-274.
135. Wang, C -C, and Truesdell, C (1973). Introduction to rational elasticity. NoordholT I, Leyden, The Netherlands.
136. Willis, A H (1948). Instability in hollow rubber cylinders subjected to axial loads, VII International congress of applied mechanics, London, Vol 1, p 280-296.
137. Zeigler, H (1968). Principles of structural stability, Blaisdell, Waltham MA.