×

zbMATH — the first resource for mathematics

On certain deformation classes of compressible Hencky materials. (English) Zbl 1141.74013
Summary: For the case of plane strain and in the absence of body forces an exact solution that describes the straightening of annular cylindrical sectors composed of compressible Hencky materials is obtained and, using the energy criterion, its stability is investigated. The most general class of isotropic, compressible, hyperelastic materials for which a solution of this type is possible is also determined. Under certain restrictions, an inequality that may be regarded as a universal relation is shown to hold for all materials in this class as well as for all isotropic, compressible, hyperelastic materials that satisfy the Baker-Ericksen inequality. We also establish some other inequalities, showing that certain deformations of bodies composed of Hencky materials are necessarily accompanied by an overall volume increase (which may also be viewed as universal relations for Hencky materials).

MSC:
74B20 Nonlinear elasticity
74G05 Explicit solutions of equilibrium problems in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hencky, H., Zeitschrift für Technische Physik 9 pp 215– (1928)
[2] Bonet, J., Nonlinear Continuum Mechanics for Finite Element Analysis (1997) · Zbl 0891.73001
[3] Anand, L., Transactions of the ASME, Journal of Applied Mechanics 46 pp 78– (1979) · Zbl 0405.73032 · doi:10.1115/1.3424532
[4] Bruhns, O. T., Archives of Mechanics 52 pp 489– (2000)
[5] Bruhns, O. T., J. Elasticity 66 pp 237– (2002) · Zbl 1078.74517 · doi:10.1023/A:1021959329598
[6] Bruhns, O. T., Proceedings of the Royal Society of London A pp 2207– (2001) · Zbl 1048.74505 · doi:10.1098/rspa.2001.0818
[7] Truesdell, C., Handbuch der Physik III/3 (1965)
[8] Aron, M., J. Elasticity 60 pp 165– (2000) · Zbl 1001.74012 · doi:10.1023/A:1011083214837
[9] Krawietz, A., Archive for Rational Mechanics and Analysis 58 pp 127– (1975) · Zbl 0324.73006 · doi:10.1007/BF00275784
[10] Ogden, R. W., Non-linear Elastic Deformations (1984) · Zbl 0541.73044
[11] Aron, M., International Journal of Solids and Structures 34 pp 2803– (1997) · Zbl 0942.74528 · doi:10.1016/S0020-7683(96)00216-8
[12] Murphy, J. G., J. Elasticity 60 pp 151– (2000) · Zbl 1001.74013 · doi:10.1023/A:1010843015909
[13] Carroll, M. M., Quarterly Applied Mathematics 48 pp 767– (1990) · Zbl 0716.73037 · doi:10.1090/qam/1079919
[14] Aron, M., Mathematics and Mechanics of Solids 3 pp 131– (1998) · Zbl 1001.74531 · doi:10.1177/108128659800300201
[15] Beatty, M. F., Transactions of the ASME, Journal of Applied Mechanics 53 pp 807– (1986) · Zbl 0606.73037 · doi:10.1115/1.3171862
[16] Knowles, J. K., Archive for Rational Mechanics and Analysis 63 pp 321– (1977) · Zbl 0351.73061 · doi:10.1007/BF00279991
[17] Hardy, G. H., Inequalities (1952)
[18] Baker, M., Journal of the Washington Academy of Science 44 pp 33– (1954)
[19] Ogden, R. W., Mathematical Proceedings of the Cambridge Philosophical Society 81 pp 313– (1977) · Zbl 0354.73023 · doi:10.1017/S030500410005338X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.