On the principle of virtual power in continuum mechanics

Abstract

The principle of virtual power and its equivalence to the Euler laws expressing the global balance of momentum and moment of momentum is demonstrated. The demonstration includes bodies subjected to jump discontinuities (shocks) and external and internal constraints.

Keywords

Constraints jump discontinuities principle of virtual power

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References

Euler

De motu . Novi Commentarii Academiæ Scientarium Petropolitanæ 1775.

Cauchy

. Récherches sur l’équilibre et le movement intérieur des corps solides ou fluids, élastiques ou non élastiques. Bull Soc Philomath 1823; 9.

Cauchy

. De la pression ou tension dans un corps solide. Ex Math 1827; 2.

Gurtin

Mizel

Williams

. A note on Cauchy’s stress theorem. J Math Anal Appl 1968; 22: 398–401.

Gurtin

Martins

. Cauchy’s theorem in classical physics. Arch Ration Mech Anal 1976; 60: 306–324.

Antman

Osborn

. The principal of virtual work and integral laws of motion. Arch Ration Mech Anal 1979; 69: 231–262.

Germain

. La métode des puissances virtuelles en mécanique des milieu continus. Premiere partie. Théorie du second gradient. J Mécanique 1973; 12: 235–274.

Germain

. The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J Appl Math 1973; 25: 556–575.

Banfi

Marzocchi

Museti

. On the principle of virtual powers in continuum mechanics. Ricerche di Matematica 2006; 55: 299–310.

10.

Epstein

Segev

. Differentiable manifolds and the principle of virtual work in continuum mechanics. J Math Phys 1980; 21: 1243–1245.

11.

Antman

Marlow

. Material constraints, Lagrange multipliers and compatibility. Applications to rod and shell theories. Arch Ration Mech Anal 1991; 116: 257–299.

12.

Truesdell

. A first course in rational continuum mechanics. San Diego: Academic Press Inc., 1991.

13.

Nečas

. Les méthodes directes en théorie des equations elliptiques. Paris: Masson et Cie, Editeurs, Prague: Academia Editeurs, 1967.

14.

Banfi

Fabrizio

. Sul concetto di sottocorpo nella meccanica dei continui. Rend Accad Naz Lincei 1979; 66: 136–143.

15.

Gurtin

Williams

Ziemer

. Geometric measure theory and the axioms of continuum thermodynamics. Arch Ration Mech Anal 1986; 92: 1–22.

16.

Truesdell

Toupin

. The classical field theories. Encyclopedia of physics, Vol. III/1. Berlin, Gottingen, Heidelberg: Springer-Verlag, 1960.

17.

Lidström

. Moving regions in Euclidean space and Reynolds’ transport theorem. Math Mech Solid 2011; 16: 366–380.

18.

Truesdell

Noll

. The non-linear field theories of mechanics. Encyclopedia of physics, Vol. III/3. Berlin, Heidelberg, New York: Springer-Verlag, 1965.

19.

Dunford

Schwartz

. Linear operators. I. General theory. New York – London: Interscience Publishers, Inc., 1958.