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Keywords:
boundary value problems; linear elasticity; law of interaction; principle of virtual displacements; principal of minimum potential energy
boundary value problems; linear elasticity; law of interaction; principle of virtual displacements; principal of minimum potential energy
Summary:
The equilibrium state of a deformable body under the action of body forces is described by the well known conditions of equilibrium, the straindisplacement relations, the constitutive law of the linear theory and the boundary conditions. The authors discuss in detail the boundary conditions. The starting point is the general relation between the vectors of stress and displacement on the boundary which can be expressed in terms of a subgradient relation. It is shown that this relation includes as special cases all known classical, bilateral and unilateral boundary conditions. Further, the principle of virtual displacements and the principle of minimum of the potential energy are established and it is shown that these principles are equivalent to the original boundary condition problem.
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References:
[1] A. Brondsted, R. T. Rockafellar: On the subdifferentiability of convex functions. Proc. Amer. Math. Soc., 16 (1965), 605-611. DOI 10.1090/S0002-9939-1965-0178103-8 | MR 0178103
[2] G. Duvaut J. L. Lions: Les inéquations en mécanique et en physique. Dunod, Paris 1972. MR 0464857
[3] G. Fichera: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, Memorie (Cl. Sci. fis., mat. e nat.), serie 8, vol. 7 (1964), 91-140. MR 0178631 | Zbl 0146.21204
[4] G. Fichera: Boundary value problems of elasticity with unilateral constraints. In: Handbuch der Physik (Herausg.: S. Flügge), Band VI a/2, Springer, 1972.
[5] H. Gajewski K. Gröger, K. Zacharias: Nichtlineare Operatorgleichurgen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin 1974. MR 0636412
[6] I. Hlaváček: Variational principles in the linear theory of elasticity for general boundary conditions. Apl. Mat., 12 (1967), 425 - 447. MR 0231575
[7] I. Hlaváček, J. Nečas: On inequalities of Korn's type. I: Boundary value problems for elliptic systems of partial differential equations. II: Applications to linear elasticity. Arch. Rat. Mech. Anal.. 36 (1970), 305-311, 312-334.
[8] A. D. loffe, V. M. Tikhomirov: The theory of extremum problems. (Russian). Moscow, 1974.
[9] F. Lené: Sur les matériaux élastiques à énergie de déformation non quadratique. J. Méc., 13 (1974), 499-534. MR 0375890
[10] J. J. Moreau: Fonctionelles convexes. College de France, 1966- 1967. MR 0390443
[11] J. J. Moreau: On unilateral constraints, friction and plasticity. In: New variational techniques in mathematical physics. C. I. M. E., Ed. Cremonese, Roma 1974, 173-322. MR 0513445
[12] J. J. Moreau: La convexité en statique. In: Analyse convexe et ses applications (ed. by J. P. Aubin), Lecture Notes Econ. and Math. Systems, No. 102 (1974), 141 - 167. Zbl 0302.70001
[13] J. J. Moreau: La notion de sur-potentiel et les liasions unilaterales en élastostatique. 12th Intern. Congr. Appl. Mech.
[14] B. Nayroles: Quelques applications variationnelles de la théorie des functions duales à la mécanique de solides. J. Méc., 10 (1971), 263-289. MR 0280053
[15] B. Nayroles: Duality and convexity in solid equilibrium problems. Laboratoire Méc. et d'Acoustique, C. N. R. S., Marseille 1974.
[16] B. Nayroles: Point de vue algebrique. Convexité et integrandes convexes en mécanique des solides. In: New variational techniques in mathematical physics. C. T. M. E., Ed. Cremonese, Roma 1974, 325-404.
[17] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague 1967. MR 0227584
[18] R. T. Rockafellar: Extension of Fenchel's duality theorem for convex functions. Duke Math. J., 33 (1966), 81-90. DOI 10.1215/S0012-7094-66-03312-6 | MR 0187062 | Zbl 0138.09301
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