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

Full-text PDF

References | Similar Articles | Additional Information

**[1]**F. B. Hildebrand, E. Reissner, and G. B. Thomas,*Notes on the foundations of the theory of small displacements of orthotropic shells*, Tech. Notes Nat. Adv. Comm. Aeronaut.,**1949**(1949), no. 1833, 59. MR**0030886****[2]**P. M. Naghdi,*On the theory of thin elastic shells*, Quart. Appl. Math.**14**(1957), 369–380. MR**0084284**, https://doi.org/10.1090/S0033-569X-1957-84284-7**[3]**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)**[4]**J. M. Whitney, and C. T. Sun,*A refined theory for laminated anisotropic cylindrical shells*, J. Appl. Mechan.**41**, 471-476 (1974)**[5]**Eric Reissner,*Stress strain relations in the theory of thin elastic shells*, J. Math. Physics**31**(1952), 109–119. MR**0048283****[6]**C. T. Sun and J. M. Whitney,*Axisymmetric vibrations of laminated composite cylindrical shells*, J. Acoust. Soc. Am.**55**, 1238-1246 (1974)**[7]**J. A. Zukas,*Effects of transverse normal and shear strains in orthotropic shells*, AIAAJ.**12**, 1753-1755 (1974)**[8]**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**[9]**V. Manea,*Some problems of the theory of elastic flat plates*(in Roumanian), Edit. Acad. Rep. Soc. Roum., pp. 193-221 (1966)**[10]**R. B. Nelson and D. R. Lorch,*A refined theory for laminated orthotropic plates*, J. Applied Mechanics**41**, 177-183.(1974)**[11]**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)**[12]**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)**[13]**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)**[14]**L. M. Habip and I. K. Ebicoglu,*On the equations of motion of shells in the reference state*, Ingenieur Archiv.**34**, (1965)**[15]**A. W. Leissa,*Vibrations of shells*, NASA SP-288, 1973**[15']**J. R. Vinson and T. W. Chou,*Composite materials and their use in strucutres*, Chapter 7, John Wiley & Sons, New York, Toronto, 1974**[16]**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.)**[17]**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**[18]**K. Z. Galimov,*The theory of shells with transverse shear deformation effect*(in Russian), Kazanskogo Universiteta, 1977**[19]**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)**[20]**L. Librescu,*Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures*, Noordhoff, Leyden, 1975**[21]**P. M. Naghdi,*Foundations of elastic shell theory*, Progress in Solid Mechanics, Vol. IV, North-Holland, Amsterdam, 1963, pp. 1–90. MR**0163488****[22]**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)**[23]**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**, https://doi.org/10.1016/0020-7225(63)90006-5**[24]**P. M. Naghdi,*Further results in the derivation of the general equations of elastic shells*, Internat. J. Engrg. Sci.**2**(1964), 269–273 (English, with French, German, Italian, and Russian summaries). MR**0167022**, https://doi.org/10.1016/0020-7225(64)90024-2**[25]**H. Leipholz,*Theory of elasticity*, Noordhoff International Publishing, Leyden, 1974. Mechanics of Elastic Stability, No. 1; Monographs and Textbooks on Mechanics of Solids and Fluids. MR**0413649****[26]**A. L. Gol'denveizer,*Qualitative investigation of the stress state in thin elastic shells*(in Russian), Prikl. Matem. i. Mechan.**IX**, 463-478 (1945)**[27]**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****[28]**R. B. Rikards and G. A. Teters,*Stability of shells from composite materials*(in Russian), Publ. House, Zinatne-Riga, 1974, pp. 82-87.**[29]**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****[30]**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****[31]**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****[32]**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**0173390****[33]**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****[34]**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**[35]**G. Sansome,*Orthogonal functions*, Interscience, New York, 1959

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73L99

Retrieve articles in all journals with MSC: 73L99

Additional Information

DOI:
https://doi.org/10.1090/qam/652046

Article copyright:
© Copyright 1982
American Mathematical Society