Book Review
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Book Information:
Author:
Stuart S. Antman
Title:
Nonlinear problems of elasticity
Additional book information:
Appl. Math. Sci., vol. 107, Springer-Verlag,
Berlin and New York,
1995,
xviii + 750 pp.,
ISBN 0-377-94199-1,
$59.95$
Stuart S. Antman and John E. Osborn, The principle of virtual work and integral laws of motion, Arch. Rational Mech. Anal. 69 (1979), no. 3, 231–262. MR 522525, DOI 10.1007/BF00248135
John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 475169, DOI 10.1007/BF00279992
Kaushik Bhattacharya, Nikan B. Firoozye, Richard D. James, and Robert V. Kohn, Restrictions on microstructure, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 5, 843–878. MR 1303758, DOI 10.1017/S0308210500022381
Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
Philippe G. Ciarlet and Jindřich Nečas, Unilateral problems in nonlinear, three-dimensional elasticity, Arch. Rational Mech. Anal. 87 (1985), no. 4, 319–338 (English, with French summary). MR 767504, DOI 10.1007/BF00250917
P. J. Davies, Buckling and barrelling instabilities in finite elasticity, J. Elasticity 21 (1989), no. 2, 147–192. MR 1002449, DOI 10.1007/BF00040894
Penny J. Davies, Buckling and barrelling instabilities on non-linearly elastic columns, Quart. Appl. Math. 49 (1991), no. 3, 407–426. MR 1121674, DOI 10.1090/qam/1121674
8. L Euler. Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes. Bousquent, Lausanne, 1744. In Opera Omnia I, Vol. 24, 231-297.
Lawrence C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal. 95 (1986), no. 3, 227–252. MR 853966, DOI 10.1007/BF00251360
R. L. Fosdick and R. T. Shield, Small bending of a circular bar superposed on finite extension or compression, Arch. Rational Mech. Anal. 12 (1963), 223–248. MR 145737, DOI 10.1007/BF00281227
Jerrold E. Marsden and Thomas J. R. Hughes, Mathematical foundations of elasticity, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1983 original. MR 1262126
Alexander Mielke, Saint-Venant’s problem and semi-inverse solutions in nonlinear elasticity, Arch. Rational Mech. Anal. 102 (1988), no. 3, 205–229. MR 944546, DOI 10.1007/BF00281347
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
S. Müller, Tang Qi, and B. S. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 11 (1994), no. 2, 217–243 (English, with English and French summaries). MR 1267368, DOI 10.1016/S0294-1449(16)30193-7
Vladimír Šverák, Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 1-2, 185–189. MR 1149994, DOI 10.1017/S0308210500015080
Tullio Valent, Boundary value problems of finite elasticity, Springer Tracts in Natural Philosophy, vol. 31, Springer-Verlag, New York, 1988. Local theorems on existence, uniqueness, and analytic dependence on data. MR 917733, DOI 10.1007/978-1-4612-3736-5
L. M. Zubov and A. N. Rudev, On the peculiarities of the loss of stability of a non-linearly elastic rectangular beam, Prikl. Mat. Mekh. 57 (1993), no. 3, 65–83 (Russian, with Russian summary); English transl., J. Appl. Math. Mech. 57 (1993), no. 3, 469–485. MR 1249429, DOI 10.1016/0021-8928(93)90126-7
- 1.
- S S Antman and J E Osborn. The principle of virtual work and integral laws of motion. Arch. Rat. Mech. Anal., 69:231--262, 1979. MR 80d:73020
- 2.
- J M Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal., 63:337--403, 1977. MR 57:14788
- 3.
- K Bhattacharya, N B Firoozye, R D James, and R V Kohn. Restrictions on microstructure. Proc. Royal Soc. Edinburgh, 124A:843--878, 1994. MR 95i:73025
- 4.
- P G Ciarlet. Mathematical Elasticity, Vol.I: Three-Dimensional Elasticity. North-Holland, 1988. MR 89e:73001
- 5.
- P G Ciarlet and J Ne\v{c}as. Unilateral problems in nonlinear three-dimensional elasticity. Arch. Rat. Mech. Anal., 87:319--338, 1985. MR 86e:73030
- 6.
- P J Davies. Buckling and barrelling instabilities in finite elasticity. J. Elasticity, 21:147--192, 1989. MR 90m:73033
- 7.
- P J Davies. Buckling and barrelling instabilities of nonlinearly elastic columns. Quarterly Applied Maths., 49:407--426, 1991. MR 92m:73071
- 8.
- L Euler. Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes. Bousquent, Lausanne, 1744. In Opera Omnia I, Vol. 24, 231-297.
- 9.
- L C Evans. Quasiconvexity and partial regularity in the calculus of variations. Arch. Rat. Mech. Anal., 95:227--268, 1986. MR 88a:49007
- 10.
- R L Fosdick and R T Shield. Small bending of a circular bar superposed on finite extension or compression. Arch. Rat. Mech. Anal., 12:223--248, 1963. MR 26:3265
- 11.
- J E Marsden and T J R Hughes. Mathematical Foundations of Elasticity. Prentice-Hall, 1983. MR 95h:73022
- 12.
- A Mielke. Saint-Venant's problem and semi-inverse solutions in nonlinear elasticity. Arch. Rat. Mech. Anal., 102:205--229, 1988. Corrigendum ibid. 110::351-352, 1990. MR 89h:73016; MR 91f: 73009
- 13.
- C B Morrey. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math., 2:25--53, 1952. MR 14:992a
- 14.
- C B Morrey. Multiple Integrals in the Calculus of Variations. Springer, 1966. MR 34:2380
- 15.
- S Müller, T. Qi, and B S Yan. On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré,Analyse Nonlinéaire, 11:217--243, 1994. MR 95a:73025
- 16.
- V \v{S}verák. Rank-one convexity does not imply quasiconvexity. Proc. Royal Soc. Edinburgh, 120A:185--189, 1992. MR 93b:49026
- 17.
- T Valent. Boundary Value Problems of Finite Elasticity, volume 31 of Springer Tracts in Natural Philosophy. Springer-Verlag, 1988. MR 89c:73001
- 18.
- L M Zubov and A N Rudev. On the peculiarities of the loss of stability of a non-linear elastic rectangular bar. J. Appl. Maths Mechs, 57:469--485, 1993. (English translation of Prikl. Mat. Mekh., 57:65-83, 1993.). MR 94j:73036
Review Information:
Reviewer:
J. M. Ball
Affiliation:
Heriot-Watt University
Email:
J.M.Ball@ma.hw.ac.uk
Journal:
Bull. Amer. Math. Soc.
33 (1996), 269-276
DOI:
https://doi.org/10.1090/S0273-0979-96-00648-9
Review copyright:
© Copyright 1996
American Mathematical Society