×

Symmetric-hyperbolic equations of motion for a hyperelastic material. (English) Zbl 1180.35345

Summary: We offer an alternate derivation for the symmetric-hyperbolic formulation of the equations of motion for a hyperelastic material with polyconvex stored energy. The derivation makes it clear that the expanded system is equivalent, for weak solutions, to the original system. We consider motions with variable as well as constant temperature. In addition, we present equivalent Eulerian equations of motion, which are also symmetric-hyperbolic.

MSC:

35L60 First-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
58A10 Differential forms in global analysis
74B20 Nonlinear elasticity
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/BF00279992 · Zbl 0368.73040 · doi:10.1007/BF00279992
[2] Ciarlet P. G., Studies in Mathematics and its Applications 20, in: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity (1988) · doi:10.1016/S0168-2024(08)70055-9
[3] DOI: 10.1007/BF00281381 · Zbl 0231.73003 · doi:10.1007/BF00281381
[4] DOI: 10.1007/3-540-29089-3 · doi:10.1007/3-540-29089-3
[5] DOI: 10.1007/BF00280911 · Zbl 0614.35057 · doi:10.1007/BF00280911
[6] Dafermos C. M., Arch. Ration. Mech. Anal. 87 pp 267–
[7] DOI: 10.1007/s002050100137 · Zbl 0985.74024 · doi:10.1007/s002050100137
[8] Federer H., Die Grundlehren der Mathematischen Wissenschaften 153, in: Geometric Measure Theory (1969)
[9] DOI: 10.1073/pnas.68.8.1686 · Zbl 0229.35061 · doi:10.1073/pnas.68.8.1686
[10] Godunov S. K., Dokl. Akad. Nauk SSSR 139 pp 521–
[11] DOI: 10.1137/0718021 · Zbl 0467.65038 · doi:10.1137/0718021
[12] Hughes T. J. R., Arch. Ration. Mech. Anal. 63 pp 273–
[13] P. Lax, Contributions to Nonlinear Functional Analysis, ed. E. Zarantonello (Academic Press, New York, 1971) pp. 603–634. · doi:10.1016/B978-0-12-775850-3.50018-2
[14] DOI: 10.1007/978-1-4612-1116-7 · doi:10.1007/978-1-4612-1116-7
[15] Marsden J. E., Mathematical Foundations of Elasticity (1983) · Zbl 0545.73031
[16] DOI: 10.1016/0196-8858(88)90025-5 · Zbl 0663.73012 · doi:10.1016/0196-8858(88)90025-5
[17] Qin T., J. Elasticity 50 pp 245–
[18] Trangenstein J. A., Commun. Pure Appl. Math. pp 41–
[19] DOI: 10.1007/978-3-642-46015-9 · doi:10.1007/978-3-642-46015-9
[20] DOI: 10.1016/0022-0396(87)90188-4 · Zbl 0647.76049 · doi:10.1016/0022-0396(87)90188-4
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.