×

Nonconvex energy functions. Hemivariational inequalities and substationarity principles. (English) Zbl 0538.73018

The purpose of the paper is the derivation of certain variational principles for material laws and boundary conditions resulting from nonconvex, nondifferentiable potentials. Two recently defined concepts of generalized gradient of Clarke [F. H. Clarke, Trans. Am. Math. Soc. 205, 247-262 (1975; Zbl 0307.26012)] and derivative container of Warga [J. Warga, Calc. Var. Control Theory, Proc. Symp. Math. Res. Cent., Madison 1975, 13-46 (1976; Zbl 0355.26004)] are employed. Hemivariational inequalities corresponding to discussed problems are derived.
Reviewer: W.Barański

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74A20 Theory of constitutive functions in solid mechanics
49S05 Variational principles of physics
49J40 Variational inequalities
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Moreau, J. J.: La notion de sur-potentiel et les liaisons unilat?rales en ?lastostatique. C. R. Acad. Sc.267, 954-957 (1968). · Zbl 0172.49802
[2] Panagiotopoulos, P. D.: Dynamic and incremental variational inequality principles, differential inclusions and their application to coexistant phases problems. Acta Mechanica40, 85-107 (1981). · Zbl 0471.49006
[3] Duvaut, G., Lions, J. L.: Les in?quations en m?canique et en physique. Paris: Dunod 1972. · Zbl 0298.73001
[4] Hamel, G.: Theoretische Mechanik. Berlin-Heidelberg-New York: Springer 1967. · Zbl 0149.42002
[5] Lanczos, C.: The variational priciples of mechanics. University of Toronto Press 1966. · Zbl 0151.43504
[6] Fichera, G.: Boundary value problems in elasticity with unilateral constraints. (Encyclopedia of Physics VI a/2.) Berlin-Heidelberg-New York: Springer 1972.
[7] Panagiotopoulos, P. D.: A nonlinear programming approach to the unilateral contact and friction boundary value problem in the theory of elasticity. Ing. Archiv.44, 421-432 (1975). · Zbl 0332.73018
[8] Ilyushin, A. A.: On the increment of plastic deformation and yield function (in Russian). Prikl. Math. Mech.24, 663-666 (1960).
[9] Olszak, W., Mr?z, Z., Perzyna, P.: Recent trends in the developments of the theory of plasticity. Pergamon Press and PWN-Polish Scientific Publishers 1963.
[10] Zhukov, A.: Plastic deformations of isotropic metals in combined loading (in Russian). Izv. Akad. Nauk. SSSR OTN12, 72-87 (1956).
[11] Palmer, A. C., Maier, G., Drucker, D. C.: Normality relations and convexity of yield surfaces for unstable materials of structural elements. Trans. ASME24, 464-470 (1967).
[12] Maier, G.: On elastic-plastic structures with associated stress-strain relations allowing for work softening. Meccanica2, 55-66 (1967). · Zbl 0222.73045
[13] Green, A. E., Naghdi, P. M.: A general theory of an elastic-plastic continuum. Arch. Rat. Mech. Anal.18, 251-281 (1965). · Zbl 0133.17701
[14] Salen?on, I., Trist?n-Lopez, Agustin: Analyse de la stabilit? des talus en sols coh?rents anisotropes. C. R. Acad. Sci.B290, 493-496 (1980).
[15] Drucker, C. D.: A more fundamental approach to plastic stress-strain relations, 487-491. Proc. First. U.S. National Congress on Applied Mechanics, Chicago, 1951. New York: 1952.
[16] Panagiotopoulos, P. D.: Superpotentials in the sense of Clarke and in the sense of Warga and applications. ZAMM62, (Heft 4/5) (1982). · Zbl 0514.73011
[17] Panagiotopoulos, P. D.: Superpotentials in the sense of F.H. Clarke and applications. Mech. Res. Comm.8, 335-340 (1981). · Zbl 0497.73020
[18] Clarke, F. H.: Generalized gradients and applications. Trans. A.M.S.205, 247-262 (1975). · Zbl 0307.26012
[19] Rockafellar, R. T.: Generalized directional derivatives and subgradients of non-convex functionals. Can. J. Math.32, 257-280 (1980). · Zbl 0447.49009
[20] Rockafellar, R. T.: La th?orie des sous-gradients et ses applications ? l’optimisation. Fonctions convexes et non-convexes. Les presses de l’Universit? de Montr?al 1979. · Zbl 0421.90045
[21] Warga, J.: Necessary conditions without differentiability assumptions in optimal control. J. dif. Equations18, 41-62 (1975). · Zbl 0297.49023
[22] Crouzeix, J. P.: Conjugacy in quasiconvex analysis. In: Lectures Notes in Economics, Vol. 144 (Proc. Conf. 1976 on Convex Analysis). Berlin-Heidelberg-New York: Springer 1977. · Zbl 0362.90096
[23] Toland, J. F.: A duality principle for non-convex optimization and the calculus of variations. Arch. Rat. Mech.71, 41-61 (1979). · Zbl 0411.49012
[24] Robertson, A. P., Robertson, W.: Topologische Vectorr?ume. Mannheim: Bibliographisches Institut 1967.
[25] Moreau, J. J.: On unilateral constraints, friction and plasticity. In: New Variational Techniques in Mathematical Physics. Roma: C.I.M.E. Edizioni Cremonese 1974.
[26] Panagiotopoulos, P. D.: Ungleichungsprobleme in der Mechanik. Habilitationsthesis. RWTH Aachen 1977.
[27] Br?zis, H.: Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert. Amsterdam: North-Holland/American Elsevier 1973.
[28] Aubin, J. P., Clarke, F. H.: Shadow prices and duality for a class of optimal control problems. SIAM J. Control and Optimization17, 567-586 (1979). · Zbl 0439.49018
[29] Germain, P.: Cours de m?canique des milieux continus. Paris: Masson 1973. · Zbl 0254.73001
[30] Tonti, E.: A systematic approach to the search for variational principles. In: Variational methods in engineering. Southampton: Univ. Press 1973. · Zbl 0304.49034
[31] Panagiotopoulos, P. D.: A systematic approach to the search for variational principles for bilateral and unilateral problems. ZAMM57, T246-T247 (1977).
[32] Grootenboer, H. J., Leijten, S. F. C. H., Blauuwendraad, J.: Numerical models for reinforced concrete structures in plane stress. Heron26, 1-83 (1981).
[33] Germain, P.: The role of thermodynamics in continuum mechanics. In: Foundations of continuum thermodynamics (Domingos, J., Nina, M., Whitelaw, J., eds.). Macmillan 1974.
[34] Ziegler, H.: Some extremum principles in irreversible thermodynamics with applications to continuum mechanics. Progress in Solid Mechanics4, 96-193 (1963).
[35] Mandel, J.: Plasticit? classique et viscoplasticit? (CISM, Vol. 97). Wien-New York: Springer 1971.
[36] Mandel, J.: Thermodynamics and plasticity. In: Foundations of continuum thermodynamics (Domingos, J., Nina, M., Whitelaw, J., eds). Macmillan 1974.
[37] Halphen, B., Son, N.Q.: Sur les mat?riaux standards g?n?ralis?s. J. de M?canique14, 39-63 (1975). · Zbl 0308.73017
[38] Warga, J.: Derivate containers, inverse functions and controllability. In: Calculus of variations and control theory (Russell, D. L., ed.). Academic Press 1978.
[39] Rieder, G., Heise, V., Pahnke, V., Antes, H., Glahn, H., Kompi?, V.: Berechnung von elastischen Spannungen in beliebig krummlinig berandeten Scheiben und Platten. Opladen: Westdeutscher Verlag 1976.
[40] Heise, U.: Applications of the singularity method for the formulation of plane elastostatical boundary value problems as integral equations. Acta Mechanica31, 33-69 (1978). · Zbl 0394.73021
[41] Glashoff, K., Sprekels, J.: An application of Glicksberg’s theorem to set-valued integral equations arising in the theory of thermostats. SIAM J. Math. Anal.12, 447-486 (1981). · Zbl 0472.45004
[42] Phan van Chuong: Solutions continues ? droite d’une ?quation int?grale multivoque. Sem. d’Anal. Convexe Montpellier 1979 Expos? No. 3.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.