Conservation laws in viscoelasticity

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.

**[1]**P. J. Olver,*Applications of Lie Groups to Differential Equations*, Springer, New York, 1986 MR**836734****[2]**F. Bampi and A. Morro,*The inverse problem of the calculus of variations applied to continuum physics*, J. Math. Phys,**23**, 2312-2321 (1982) MR**685699****[3]**P. J. Olver,*Conservation laws in elasticity. I. General results*, Arch. Rational Mech. Anal.**85**, 112-129 (1984) MR**731281****[4]**D. G. Edelen,*Applied Exterior Calculus*, Wiley, New York, 1985 MR**816136****[5]**D. G. B. Edelen and I. M. Snyman,*Cartan forms for multiple integral problems in the calculus of variations*, J. Math. Anal. Appl.**120**, 218-239 (1986) MR**861916****[6]**G. Caviglia and A. Morro,*Noether-type conservation laws for perfect fluid motions*, J. Math. Phys.**28**, 1056-1060 (1987) MR**887023****[7]**G. Caviglia,*Symmetry transformations, isovectors, and conservation laws*, J. Math. Phys.**27**, 972-978 (1986) MR**832778****[8]**G. Caviglia,*Composite variational principles and the determination of conservation laws*, J. Math. Phys.**29**, 812-816 (1988) MR**940342****[9]**R. Arens,*The conserved currents for the Maxwellian field*, Commun. Math. Phys.**90**, 527-544 (1983) MR**719433****[10]**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**[11]**Q. Jiang,*Conservation laws in linear viscoelastodynamics*, J. Elasticity**16**, 213-219 (1986) MR**849673****[12]**G. Caviglia and A. Morro,*A general approach to conservation laws in viscoelasticity*, Acta Mech.**75**, 255-267 (1988) MR**978225****[13]**V. I. Fushchich and A. G. Nikitin,*New and old symmetries of the Maxwell and Dirac equations*, Sov. J. Part. Nucl.**14**, 1-22 (1983) MR**716301****[14]**V. I. Fushchich and A. G. Nikitin,*Symmetries of Maxwell's Equations*, Reidel, Dordrecht, 1987**[15]**M. E. Gurtin and E. Sternberg,*On the linear theory of viscoelasticity*, Arch. Rational Mech. Anal.**11**, 291-356 (1963) MR**0147047****[16]**P. J. Olver,*Conservation Laws in Elasticity. III. Planar Linear Anisotropic Elastostatics*, Arch. Rational Mech. Anal.**102**, 167-181 (1988) MR**943430****[17]**G. Caviglia and A. Morro,*Conservation laws in anisotropic elastostatics*, Int. J. Engrg. Sci.**26**, 393-400 (1988) MR**944573****[18]**G. Caviglia and A. Morro,*Generalized symmetries and conservation laws: a Noether-type approach*, Riv. Mat. Pura Appl.**4**, 33-54 (1989) MR**1068885**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73F15,
49S05,
73B99

Retrieve articles in all journals with MSC: 73F15, 49S05, 73B99

Additional Information

DOI:
https://doi.org/10.1090/qam/1074965

Article copyright:
© Copyright 1990
American Mathematical Society