The Concept of a Minimal State in Viscoelasticity: New Free Energies and Applications to PDEs
- 136 Downloads
We show here the impact on the initial-boundary value problem, and on the evolution of viscoelastic systems of the use of a new definition of state based on the stress-response (see, e.g., [48, 16, 41]). Comparisons are made between this new approach and the traditional one, which is based on the identification of histories and states. We shall refer to a stress-response definition of state as the minimal state . Materials with memory and with relaxation are discussed.
The energetics of linear viscoelastic materials is revisited and new free energies, expressed in terms of the minimal state descriptor, are derived together with the related dissipations. Furthermore, both the minimum and the maximum free energy are recast in terms of the minimal state variable and the current strain.
The initial-boundary value problem governing the motion of a linear viscoelastic body is re-stated in terms of the minimal state and the velocity field through the principle of virtual power. The advantages are (i) the elimination of the need to know the past-strain history at each point of the body, and (ii) the fact that initial and boundary data can now be prescribed on a broader space than resulting from the classical approach based on histories. These advantages are shown to lead to natural results about well-posedness and stability of the motion.
Finally, we show how the evolution of a linear viscoelastic system can be described through a strongly continuous semigroup of (linear) contraction operators on an appropriate Hilbert space. The family of all solutions of the evolutionary system, obtained by varying the initial data in such a space, is shown to have exponentially decaying energy.
Unable to display preview. Download preview PDF.
- 2.Bourdaud, G., Lanza de Cristoforis, M., Sickel, W.: Superposition operators and functions of bounded p-variation. Preprint, Institut de Mathématiques de Jussieu, Projet d'analyse fonctionelle, 2004Google Scholar
- 5.Coleman, B.D.: Thermodynamics of materials with memory. Arch. Ration. Mech. Anal. 17, 1–45 (1964)Google Scholar
- 6.Coleman, B.D., Mizel, V.J.: Norms and semi-groups in the theory of fading memory. Arch. Ration. Mech. Anal. 23, 87–123 (1967)Google Scholar
- 9.Coleman, B.D., Owen, D.R.: On thermodynamics and elastic-plastic materials. Arch. Ration. Mech. Anal. 59: 25–51 (1975)Google Scholar
- 13.Day, W.A.: The Thermodynamics of Simple Material with Fading Memory. Springer, New York, 1972Google Scholar
- 14.Day, W.A.: The thermodynamics of materials with memory. In: Materials with Memory D. Graffi ed., Liguori, Napoli (1979)Google Scholar
- 15.Del Piero, G.: The relaxed work in linear viscoelasticity. Mathematics Mech. Solids, 9, no. 2, 175–208 (2004)Google Scholar
- 19.Deseri, L.: Restrizioni a priori sulla funzione di rilassamento in viscoelasticità lineare (in Italian). Doctorate Dissertation (Advisor: Prof. G. Del Piero), The National Library of Florence, Italy 1993Google Scholar
- 21.Deseri, L., Gentili, G., Golden M.J.: Free energies and Saint-Venant's principle in linear viscoelasticity. Submitted for publicationGoogle Scholar
- 22.Deseri, L., Golden, J.M.: The minimum free energy for continuous spectrum materials. Submitted for publicationGoogle Scholar
- 23.Dill, E.H.: Simple materials with fading memory. In: Continuum Physics II. A.C. Eringen ed. Academic, New York, 1975Google Scholar
- 24.Fabrizio, M.: Existence and uniqueness results for viscoelastic materials. In: Crack and Contact Problems for Viscoelastic Bodies. G.A.C. Graham and J.R. Walton eds., Springer-Verlag, Vienna, 1995Google Scholar
- 34.Fabrizio, M., Morro, A.: Mathematical Problems in Linear Viscoelasticity. SIAM, Philadelphia, 1992Google Scholar
- 36.Gohberg, I.C., Krein, M.G.: Systems of integral equations on a half-line with kernels depending on the difference of arguments. Am. Math. Soc. Transl. Ser. 2, 14 217–287 (1960)Google Scholar
- 38.Golden, M.J., Graham, G.A.C.: Boundary Value Problems in Linear Viscoelasticy. Springer, Berlin, 1988Google Scholar
- 41.Graffi, D., Fabrizio, M.: Sulla nozione di stato per materiali viscoelastici di tipo ``rate''. Atti Acc. Lincei Rend. Fis, (8), 83, 201–208 (1989)Google Scholar
- 45.Halmos, P.R.: Finite-dimensional Vector Spaces. Springer-Verlag, New York, 1972Google Scholar
- 47.Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Groningen, 1953Google Scholar
- 49.Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Lectures Notes in Mathematics, 10, University of Maryland, 1974Google Scholar
- 50.Sneddon, I.N.: The Use of Integral Transforms. McGraw-Hill, New York, 1972Google Scholar
- 51.Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Clarendon, Oxford, 1937Google Scholar
- 52.Truesdell, C.A., Noll, W.: The Non-linear Filed Theory of Mechanics. Handbuck der Physik, III, 3, Flugge (Ed.). Springer Verlag, Berlin, 1965Google Scholar
- 53.Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, 1963Google Scholar
- 54.Widder, E.T.: The Laplace Transform. Princeton University Press, 1941Google Scholar