Book Review
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MathSciNet review:
1567869
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Book Information:
Author:
T. Valent
Title:
Boundary value problems of finite elasticity
Additional book information:
Springer-Verlag, Heidelberg, 1988, xii+192 pp., $64.00. ISBN 0-387-96550-5.
Stuart Antman, Existence of solutions of the equilibrium equations for non-linearly elastic rings and arches, Indiana Univ. Math. J. 20 (1970/71), 281–302. MR 266478, DOI 10.1512/iumj.1970.20.20025
S. S. Antman, Regular and singular problems for large elastic deformations of tubes, wedges, and cylinders, Arch. Rational Mech. Anal. 83 (1983), no. 1, 1–52. MR 695361, DOI 10.1007/BF00281086
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
J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 3-4, 315–328. MR 616782, DOI 10.1017/S030821050002014X
5. M. van Buren, On the existence and uniqueness of solutions to boundary value problems in finite elasticity, Thesis, Carnegie-Mellon University, Pittsburgh, 1968.
D. R. J. Chillingworth, J. E. Marsden, and Y. H. Wan, Symmetry and bifurcation in three-dimensional elasticity. I, Arch. Rational Mech. Anal. 80 (1982), no. 4, 295–331. MR 677564, DOI 10.1007/BF00253119
D. R. J. Chillingworth, J. E. Marsden, and Y. H. Wan, Symmetry and bifurcation in three-dimensional elasticity. II, Arch. Rational Mech. Anal. 83 (1983), no. 4, 363–395. MR 714980, DOI 10.1007/BF00963840
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
P. G. Ciarlet and P. Destuynder, A justification of the two-dimensional linear plate model, J. Mécanique 18 (1979), no. 2, 315–344 (English, with French summary). MR 533827
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
Philippe G. Ciarlet and Jindřich Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal. 97 (1987), no. 3, 171–188. MR 862546, DOI 10.1007/BF00250807
Bernard D. Coleman and Walter Noll, On the thermostatics of continuous media, Arch. Rational Mech. Anal. 4 (1959), 97–128 (1959). MR 109527, DOI 10.1007/BF00281381
R. L. Fosdick and J. Serrin, On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids, J. Elasticity 9 (1979), no. 1, 83–89. MR 528599, DOI 10.1007/BF00040982
Morton E. Gurtin, An introduction to continuum mechanics, Mathematics in Science and Engineering, vol. 158, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 636255
Hervé Le Dret, Structure of the set of equilibrated loads in nonlinear elasticity and applications to existence and nonexistence, J. Elasticity 17 (1987), no. 2, 123–141 (English, with French summary). MR 885602, DOI 10.1007/BF00043020
J. E. Marsden and T. J. R. Hughes, Topics in the mathematical foundations of elasticity, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. II, Res. Notes in Math., vol. 27, Pitman, Boston, Mass.-London, 1978, pp. 30–285. MR 576233
17. J. E. Marsden and T. J. R. Hughes, Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, 1983.
Francesco Stoppelli, Un teorema di esistenza ed unicità relativo alle equazioni dell’elastostatica isoterma per deformazioni finite, Ricerche Mat. 3 (1954), 247–267 (Italian). MR 74237
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Band III/3, Springer-Verlag, Berlin, 1965. MR 0193816
Tullio Valent, Local theorems of existence and uniqueness in finite elastostatics, Proceedings of the IUTAM Symposium on Finite Elasticity (Bethlehem, Pa., 1980) Nijhoff, The Hague, 1982, pp. 401–421. MR 676677
21. T. Valent, Pressure boundary problems infinite elasticity. Results on local existence, uniqueness and analyticity, in Proceedings, meeting on finite thermoelasticity (G. Grioli, Editor), Accademia Nazionale dei Lincei, Roma, 1986, pp. 241-264.
Tullio Valent and Giuseppe Zampieri, The differentiability of an operator connected with a class of quasilinear differential systems, Rend. Sem. Mat. Univ. Padova 57 (1977), 311–322 (1978) (Italian, with English summary). MR 526198
C. C. Wang and C. Truesdell, Introduction to rational elasticity, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics of Continua, Noordhoff International Publishing, Leyden, 1973. MR 0468442
- 1.
- S. S. Antman, Existence of solutions of the equilibrium equations for nonlinearly elastic rings and arches, Indiana Univ. Math. J. 20 (1970), 281-302. MR 0266478
- 2.
- S. S. Antman, Regular and singular problems for large elastic deformations of tubes, wedges, and cylinders, Arch. Rational. Mech. Anal. 83 (1983), 1-52. MR 695361
- 3.
- J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337-403. MR 475169
- 4.
- J. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh 88A (1981), 315-328. MR 616782
- 5.
- M. van Buren, On the existence and uniqueness of solutions to boundary value problems in finite elasticity, Thesis, Carnegie-Mellon University, Pittsburgh, 1968.
- 6.
- D. R. J. Chillingworth and J. E. Marsden, Symmetry and bifurcation in three-dimensional elasticity, Part I, Arch. Rational Mech. Anal. 80 (1982), 295-331. MR 677564
- 7.
- D. R. J. Chillingworth and J. E. Marsden, Symmetry and bifurcation in three-dimensional elasticity, Part II, Arch. Rational Mech. Anal. 83 (1983), 363-395. MR 714980
- 8.
- P. G. Ciarlet, Mathematical elasticity, Vol. I: Three-dimensional elasticity, North-Holland, Amsterdam, 1988. MR 936420
- 9.
- P. G. Ciarlet and P. Destuynder, A justification of a nonlinear model in plate theory, Comput. Methods Appl. Mech. Engrg. 17/18 (1979), 227-258. MR 533827
- 10.
- P. G. Ciarlet and J. Nečas, Unilateral problems in nonlinear, three-dimensional elasticity, Arch. Rational Mech. Anal. 87 (1985), 319-338. MR 767504
- 11.
- P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch Rational Mech. Anal. 19 (1987), 171-188. MR 862546
- 12.
- B. D. Coleman and W. Noll, On the thermostatics of continuous media, Arch. Rational Mech. Anal. 4 (1959), 97-128. MR 109527
- 13.
- R. L. Fosdick and J. Serrin, On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids, J. Elasticity 9 (1979), 83-89. MR 528599
- 14.
- M. E. Gurtin, In introduction to continuum mechanics, Academic Press, New York, 1981. MR 636255
- 15.
- H. Le Dret, Structure of the set of equilibrated loads in nonlinear elasticity and applications to existence and nonexistence, J. Elasticity 17 (1987), 123-141. MR 885602
- 16.
- J. E. Marsden and T. J. R. Hughes, Topics in the mathematical foundations of elasticity, in Nonlinear analysis and mechanics: Heriot-Watt symposium, Vol. 2, Pitman, London, 1978, pp. 30-285. MR 576233
- 17.
- J. E. Marsden and T. J. R. Hughes, Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, 1983.
- 18.
- F. Stoppelli, Un teorema di esistenza e di unicità relativo alle equazioni dell'elastostatica isoterma per deformazioni finite, Ricerche Mat. 3 (1954), 247-267. MR 74237
- 19.
- C. Truesdell and W. Noll, The non-linear field theories of mechanics, in Handbuch der Physik, Vol. III/3 (S. Flügge, Editor), Springer-Verlag, Berlin, 1965, pp. 1-602. MR 193816
- 20.
- T. Valent, Local theorems of existence and uniqueness infinite elastostatics, in Finite elasticity (D. E. Carlson and R. T. Shield, Editors), Nijhoff, The Hague, 1978, pp. 401-421. MR 676677
- 21.
- T. Valent, Pressure boundary problems infinite elasticity. Results on local existence, uniqueness and analyticity, in Proceedings, meeting on finite thermoelasticity (G. Grioli, Editor), Accademia Nazionale dei Lincei, Roma, 1986, pp. 241-264.
- 22.
- T. Valent and G. Zampieri, Sulla differenziabilità di un operatore legato a una classe di sistemi differenziali quasi-lineari, Rend. Sem. Mat. Univ. Padova 57 (1977), 311-322. MR 526198
- 23.
- C.-C. Wang and C. Truesdell, Introduction to rational elasticity, Noordhoff, Groningen, 1973. MR 468442
Review Information:
Reviewer:
Philippe G. Ciarlet
Journal:
Bull. Amer. Math. Soc.
23 (1990), 209-222
DOI:
https://doi.org/10.1090/S0273-0979-1990-15934-8