Strain measures and compatibility equations in the linear high-order shell theories
Authors:
M. Brull and L. Librescu
Journal:
Quart. Appl. Math. 40 (1982), 15-25
MSC:
Primary 73L99
DOI:
https://doi.org/10.1090/qam/652046
MathSciNet review:
652046
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- P. M. Naghdi, On the theory of thin elastic shells, Quart. Appl. Math. 14 (1957), 369–380. MR 84284, DOI https://doi.org/10.1090/S0033-569X-1957-84284-7
J. E. Stoneking and A. P. Boresi, A theory for free vibration of orthotropic shells of revolution, Nucl. Eng. and Des. 17, 271–285 (1970)
J. M. Whitney, and C. T. Sun, A refined theory for laminated anisotropic cylindrical shells, J. Appl. Mechan. 41, 471–476 (1974)
- Eric Reissner, Stress strain relations in the theory of thin elastic shells, J. Math. Physics 31 (1952), 109–119. MR 0048283
C. T. Sun and J. M. Whitney, Axisymmetric vibrations of laminated composite cylindrical shells, J. Acoust. Soc. Am. 55, 1238–1246 (1974)
J. A. Zukas, Effects of transverse normal and shear strains in orthotropic shells, AIAAJ. 12, 1753–1755 (1974)
W. H. Drysdale and A. R. Zak, Structural problems in thick shells, in Thin-shell structures: theory, experiment, design, pp. 453–463, Eds. Y. C. Fung and E. E. Sechler, Prentice-Hall Inc., Englewood Cliffs, 1974
V. Manea, Some problems of the theory of elastic flat plates (in Roumanian), Edit. Acad. Rep. Soc. Roum., pp. 193–221 (1966)
R. B. Nelson and D. R. Lorch, A refined theory for laminated orthotropic plates, J. Applied Mechanics 41, 177–183.(1974)
K. H. Lo, R. M. Christensen, and E. M. Wu, A high-order theory of plate deformation: Part 1: Homogeneous plates, J. Appl. Mech. 44, 4, 663–668 (1977)
K. H. Lo, R. M. Christensen, and E. M. Wu, Stress solution determination for high order plate theory, Int. J. Solid Structures 14, 655–662 (1978)
L. Ia. Ainola, Non-linear theory of Timoshenko type in elastic shells (in Russian), Izv. A. N. Est. SSR., Ser. Fiz.-Matem. i. Techn. 14, 337–344 (1965)
L. M. Habip and I. K. Ebicoglu, On the equations of motion of shells in the reference state, Ingenieur Archiv. 34, (1965)
A. W. Leissa, Vibrations of shells, NASA SP-288, 1973
J. R. Vinson and T. W. Chou, Composite materials and their use in strucutres, Chapter 7, John Wiley & Sons, New York, Toronto, 1974
L. Librescu, Improved linear theory of elastic anistropic multilayered shells (in Russian), Mekhanika Polimerov, Part I, No. 6, 1038–1050, Nov. - Dec. 1975 and Part II, No. 1, 100–109, Jan. - Feb. 1976 (English Translat. by Plenum Publ. Corp.)
L. Librescu, Non-linear theory of elastic anisotropic, multilayered shells (in Russian), in Selected topics in applied mechanics, ed. L. I. Sedov. pp. 453–466, Nauka, Moskow, 1974
K. Z. Galimov, The theory of shells with transverse shear deformation effect (in Russian), Kazanskogo Universiteta, 1977
P. M. Nagdhi, The theory of shells and plates, in Handbuch der Physik VI a/1 (ed. S.Flügge), Springer, Berlin, Heidelberg, New York, pp. 425–640 (1972)
L. Librescu, Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures, Noordhoff, Leyden, 1975
- P. M. Naghdi, Foundations of elastic shell theory, Progress in Solid Mechanics, Vol. IV, North-Holland, Amsterdam, 1963, pp. 1–90. MR 0163488
L. Librescu, A physically nonlinear theory of elastic shells and plates, the Love-Kirchoff hypothesis being eliminated, Rev. Roum. Sci. Tech. - Mec. Appl., 15, 1263–1284 (1970)
- P. M. Naghdi, A new derivation of the general equations of elastic shells, Internat. J. Engrg. Sci. 1 (1963), 509–522 (English, with French, German, Italian, and Russian summaries). MR 0162420, DOI https://doi.org/10.1016/0020-7225%2863%2990006-5
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A. L. Gol’denveizer, Qualitative investigation of the stress state in thin elastic shells (in Russian), Prikl. Matem. i. Mechan. IX, 463–478 (1945)
- Eric Reissner, A note on stress functions and compatibility equations in shell theory, Topics in Applied Mechanics, Elsevier, Amsterdam, 1965, pp. 23–32. MR 0191178
R. B. Rikards and G. A. Teters, Stability of shells from composite materials (in Russian), Publ. House, Zinatne-Riga, 1974, pp. 82–87.
- W. T. Koiter, A consistent first approximation in the general theory of thin elastic shells, Proc. Sympos. Thin Elastic Shells (Delft, 1959) North-Holland, Amsterdam, 1960, pp. 12–33. MR 0142241
- W. T. Koiter, A consistent first approximation in the general theory of thin elastic shells, Proc. Sympos. Thin Elastic Shells (Delft, 1959) North-Holland, Amsterdam, 1960, pp. 12–33. MR 0142241
- B. Budiansky and J. L. Sanders Jr., On the “best” first-order linear shell theory, Progress in Applied Mechanics, Macmillan, New York, 1963, pp. 129–140. MR 0158595
- P. M. Naghdi, On a variational theorem in elasticity and its application to shell theory, Trans. ASME Ser. E. J. Appl. Mech. 31 (1964), 647–653. MR 173390
- I. N. Vekua, Theory of thin shallow shells of variable thickness, Akad. Nauk Gruzin. SSR Trudy Tbiliss. Mat. Inst. Razmadze 30 (1965), 3–103 (Russian, with Georgian summary). MR 0197007
N. K. Galimov, On the applicability of Legendre polynomials in the substantiation of the theory of sandwich plates and shells (in Russian) in Issledovania po teorii plastin i obolocek (ed. by K. Z. Galimov), X, 371–385, Univ. of Kazan, Kazan, 1973
G. Sansome, Orthogonal functions, Interscience, New York, 1959
F. B. Hildebrand, E. Reissner, and G. B. Thomas, Notes on the foundations of the theory of small displacements of orthotropic shells, NACA-TN-1633, Mar. 1949
P. M. Naghdi, On the theory of thin elastic shells, Quart. Appl. Math. 14, 369–380. (1957)
J. E. Stoneking and A. P. Boresi, A theory for free vibration of orthotropic shells of revolution, Nucl. Eng. and Des. 17, 271–285 (1970)
J. M. Whitney, and C. T. Sun, A refined theory for laminated anisotropic cylindrical shells, J. Appl. Mechan. 41, 471–476 (1974)
E. Reissner, Stress-strain relations in the theory of thin elastic shells, J. Math, and Physics 31, 109–119 (1952)
C. T. Sun and J. M. Whitney, Axisymmetric vibrations of laminated composite cylindrical shells, J. Acoust. Soc. Am. 55, 1238–1246 (1974)
J. A. Zukas, Effects of transverse normal and shear strains in orthotropic shells, AIAAJ. 12, 1753–1755 (1974)
W. H. Drysdale and A. R. Zak, Structural problems in thick shells, in Thin-shell structures: theory, experiment, design, pp. 453–463, Eds. Y. C. Fung and E. E. Sechler, Prentice-Hall Inc., Englewood Cliffs, 1974
V. Manea, Some problems of the theory of elastic flat plates (in Roumanian), Edit. Acad. Rep. Soc. Roum., pp. 193–221 (1966)
R. B. Nelson and D. R. Lorch, A refined theory for laminated orthotropic plates, J. Applied Mechanics 41, 177–183.(1974)
K. H. Lo, R. M. Christensen, and E. M. Wu, A high-order theory of plate deformation: Part 1: Homogeneous plates, J. Appl. Mech. 44, 4, 663–668 (1977)
K. H. Lo, R. M. Christensen, and E. M. Wu, Stress solution determination for high order plate theory, Int. J. Solid Structures 14, 655–662 (1978)
L. Ia. Ainola, Non-linear theory of Timoshenko type in elastic shells (in Russian), Izv. A. N. Est. SSR., Ser. Fiz.-Matem. i. Techn. 14, 337–344 (1965)
L. M. Habip and I. K. Ebicoglu, On the equations of motion of shells in the reference state, Ingenieur Archiv. 34, (1965)
A. W. Leissa, Vibrations of shells, NASA SP-288, 1973
J. R. Vinson and T. W. Chou, Composite materials and their use in strucutres, Chapter 7, John Wiley & Sons, New York, Toronto, 1974
L. Librescu, Improved linear theory of elastic anistropic multilayered shells (in Russian), Mekhanika Polimerov, Part I, No. 6, 1038–1050, Nov. - Dec. 1975 and Part II, No. 1, 100–109, Jan. - Feb. 1976 (English Translat. by Plenum Publ. Corp.)
L. Librescu, Non-linear theory of elastic anisotropic, multilayered shells (in Russian), in Selected topics in applied mechanics, ed. L. I. Sedov. pp. 453–466, Nauka, Moskow, 1974
K. Z. Galimov, The theory of shells with transverse shear deformation effect (in Russian), Kazanskogo Universiteta, 1977
P. M. Nagdhi, The theory of shells and plates, in Handbuch der Physik VI a/1 (ed. S.Flügge), Springer, Berlin, Heidelberg, New York, pp. 425–640 (1972)
L. Librescu, Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures, Noordhoff, Leyden, 1975
P. M. Naghdi, Foundations of elastic shell theory, in Progr. Solid Mech., ed. I. N. Sneddon and R. Hill, 4, 1 (1963)
L. Librescu, A physically nonlinear theory of elastic shells and plates, the Love-Kirchoff hypothesis being eliminated, Rev. Roum. Sci. Tech. - Mec. Appl., 15, 1263–1284 (1970)
P. M. Naghdi, A new derivation of the general equations of elastic shells, Int. J. Eng. Sci. 1, 509–522 (1963)
P. M. Naghdi, Further results in the derivation of the general equations of elastic shells, Int. J. Eng. Sci. 2, 269–273 (1964)
H. Leipholz, Theory of elasticity, Noordhoff, Leyden, 1974
A. L. Gol’denveizer, Qualitative investigation of the stress state in thin elastic shells (in Russian), Prikl. Matem. i. Mechan. IX, 463–478 (1945)
E. Reissner, A note on stress functions and compatibility equations in shell theory, in Topics in applied mechanics, eds. D. Abir, F. Ollendorf, and M. Reiner, Elsevier, Amsterdam, 1965, pp. 23–32
R. B. Rikards and G. A. Teters, Stability of shells from composite materials (in Russian), Publ. House, Zinatne-Riga, 1974, pp. 82–87.
W. T. Koiter, A consistent first approximation in the general theory of thin elastic shells, Part I: foundations and linear theory, Dept. Laboratorium voor Toegepaste Mechanica der Technische Hogeschool, August 5th, 1959
W, T. Koiter, A consistent first approximation in the general theory of thin elastic shells, Proc. IUTAM symposium on the theory of thin shells (Ed. W. T. Koiter), North-Holland, Amsterdam, 1960
B. Budiansky and J. L. Sanders, Jr., On the “best” first order linear shell theory, in The Prager anniversary volume, 129–140, Macmillian, 1963
P. M. Naghdi, On a variational theorem in elasticity and its application to shell theory, J. Appl. Mechan. 31, 647–653 (1964)
I. N. Vekua, The theory of thin shallow shells of variable thickness (in Russian), The Mathematical Inst. of Tbilisi, Metzniereba, 1965
N. K. Galimov, On the applicability of Legendre polynomials in the substantiation of the theory of sandwich plates and shells (in Russian) in Issledovania po teorii plastin i obolocek (ed. by K. Z. Galimov), X, 371–385, Univ. of Kazan, Kazan, 1973
G. Sansome, Orthogonal functions, Interscience, New York, 1959
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© Copyright 1982
American Mathematical Society
