Conservation laws in viscoelasticity

Caviglia, Giacomo; Morro, Angelo

doi:10.1090/qam/1074965

Authors: Giacomo Caviglia and Angelo Morro
Journal: Quart. Appl. Math. 48 (1990), 503-516
MSC: Primary 73F15; Secondary 49S05, 73B99
DOI: https://doi.org/10.1090/qam/1074965
MathSciNet review: MR1074965
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The evolution equations of a linear viscoelastic solid are written in terms of the Laplace transform of the displacement field. A corresponding reformulation of the condition of vanishing divergence for vector fields is then proposed and, through a systematic procedure, an explicit representation for a very large family of such conserved vectors is derived. As an application it is shown how a suitable choice of the admissible parameters leads to specific conservation laws which involve spatial means of linear momentum, angular momentum, stress, and displacement, in terms of the known body force, and initial and boundary data. As a further application a Betti-type reciprocity relation is derived. The connection with Noether’s approach to conservation laws is also discussed.

Peter J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR 836734
F. Bampi and A. Morro, The inverse problem of the calculus of variations applied to continuum physics, J. Math. Phys. 23 (1982), no. 12, 2312–2321. MR 685699, DOI https://doi.org/10.1063/1.525322
Peter J. Olver, Conservation laws in elasticity. I. General results, Arch. Rational Mech. Anal. 85 (1984), no. 2, 111–129. MR 731281, DOI https://doi.org/10.1007/BF00281447
Dominic G. B. Edelen, Applied exterior calculus, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. MR 816136
Dominic G. B. Edelen and Izak M. Snyman, Cartan forms for multiple integral problems in the calculus of variations, J. Math. Anal. Appl. 120 (1986), no. 1, 218–239. MR 861916, DOI https://doi.org/10.1016/0022-247X%2886%2990212-X
G. Caviglia and A. Morro, Noether-type conservation laws for perfect fluid motions, J. Math. Phys. 28 (1987), no. 5, 1056–1060. MR 887023, DOI https://doi.org/10.1063/1.527546
Giacomo Caviglia, Symmetry transformations, isovectors, and conservation laws, J. Math. Phys. 27 (1986), no. 4, 972–978. MR 832778, DOI https://doi.org/10.1063/1.527117
G. Caviglia, Composite variational principles and the determination of conservation laws, J. Math. Phys. 29 (1988), no. 4, 812–816. MR 940342, DOI https://doi.org/10.1063/1.527975
Richard Arens, The conserved currents for the Maxwellian field, Comm. Math. Phys. 90 (1983), no. 4, 527–544. MR 719433

M. J. Leitman and G. M. C. Fisher, The linear theory of viscoelasticity, Encyclopedia of Physics, Vol. VI A/3, Springer, Berlin, 1973, pp. 1–123

Qing Jiang, Conservation laws in linear viscoelastodynamics, J. Elasticity 16 (1986), no. 2, 213–219. MR 849673, DOI https://doi.org/10.1007/BF00043587
G. Caviglia and A. Morro, A general approach to conservation laws in viscoelasticity, Acta Mech. 75 (1988), no. 1-4, 255–267. MR 978225, DOI https://doi.org/10.1007/BF01174639
V. I. Fushchich and A. G. Nikitin, New and old symmetries of the Maxwell and Dirac equations, Fiz. Èlementar. Chastits i Atom. Yadra 14 (1983), no. 1, 5–57 (Russian, with English summary); English transl., Soviet J. Particles and Nuclei 14 (1983), no. 1, 1–22. MR 716301

V. I. Fushchich and A. G. Nikitin, Symmetries of Maxwell’s Equations, Reidel, Dordrecht, 1987

M. E. Gurtin and Eli Sternberg, On the linear theory of viscoelasticity, Arch. Rational Mech. Anal. 11 (1962), 291–356. MR 147047, DOI https://doi.org/10.1007/BF00253942
Peter J. Olver, Conservation laws in elasticity. III. Planar linear anisotropic elastostatics, Arch. Rational Mech. Anal. 102 (1988), no. 2, 167–181. MR 943430, DOI https://doi.org/10.1007/BF00251497
Giacomo Caviglia and Angelo Morro, Conservation laws in anisotropic elastostatics, Internat. J. Engrg. Sci. 26 (1988), no. 4, 393–400. MR 944573, DOI https://doi.org/10.1016/0020-7225%2888%2990118-8
Giacomo Caviglia and Angelo Morro, Generalized symmetries and conservation laws: a Noether-type approach, Riv. Mat. Pura Appl. 4 (1989), 33–53. MR 1068885