×

On locking materials. (English) Zbl 0594.73037

Locking materials are hyperelastic materials for which the strain tensor is constrained to stay in some convex set. The strain energy density of these materials is convex. This paper is concerned with the boundary value problem of locking materials. The boundary value problem is formulated as a dual variational problem. The author gives the dual variational principles governing the equilibrium state of an elastic- locking material. Two minimum theorems of Prager are extended and inf-sup equality is proved by means of a penalty method.
The locking limit analysis is presented and discussed. The upper and lower bound formulae for the safety factor are obtained by the simultaneous use of the two variational principles (statical and kinematical methods).
The results of convex analysis are applied to get the mathematical proof of the duality relations and of the equality inf-sup. The appendices of the paper deal with the stress concentrations in a problem of torsion and with the limited compressibility. In limited compressibility, only volume changes are constrained. These materials lead to simpler mathematical problems. The existence of a stress solution which satisfies the boundary conditions is proved for the limited compressibility material.
Reviewer: I.Ecsedi

MSC:

74R20 Anelastic fracture and damage
49J40 Variational inequalities
74S30 Other numerical methods in solid mechanics (MSC2010)
49J20 Existence theories for optimal control problems involving partial differential equations
74C99 Plastic materials, materials of stress-rate and internal-variable type
74G70 Stress concentrations, singularities in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Adams, R. S.:Sovolev Spaces, Academic Press, New York, 1975.
[2] Cyras, A.: ?Optimization Theory of Perfectly Locking Bodies?,Arch. Mech. 24 (1972) 203-210. · Zbl 0262.73011
[3] Demengel, F.: Déplacements à déformation bornée et contrainte mesures, à paraître dans les Annales de l’école normale supérieure de Pise.
[4] Demengel, F.: ?Relaxation et existence pour le problème des matériaux à blocage?, to be published inRairo. · Zbl 0579.73017
[5] Duvaut, G. and Lions, J. L.:Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. · Zbl 0298.73001
[6] Ekeland, L. and Temam, R.:Analyse Convexe et Problèmes variationnels, Dunod, Paris, 1974. · Zbl 0281.49001
[7] Lions, J. L. and Magenes, E.:Problèmes aux limites non homogènes et applications, Vol. 1 et 2, Dunod, Paris, 1968. · Zbl 0165.10801
[8] Prager, W.: ?On Ideal Locking Materials?,Trans. Soc. Rheology 1 (1957) 169-175. · Zbl 0098.37702
[9] Prager, W.: ?Unilateral Constraints in Mechanics of Continua?, Estratto dagli atti del Simposio Lagrangiano. Accademia delle Science di Torino, 1964, pp. 1-11.
[10] Prager, W.: ?Elastic Solids of Limited Compressibility?,Proc. 9th Int. Congress Appl. Mech., Vol. 5, Brussels, 1958, pp. 205-211.
[11] Rockafellar, R. T.:Convex Analysis, Princeton University Press, 1970. · Zbl 0193.18401
[12] Salencon, J.:Calcul à la Rupture et Analyse Limite, Presses de l’E.N.P.C., Paris, 1983.
[13] Strang, G.: ?A Minimax Problem in Plasticity Theory?, inFunctional Analysis Methods in Numerical Analysis, Lecture Notes in Math. No. 701, Springer-Verlag, Berlin, Heidelberg, New York, 1979, pp. 319-333.
[14] Strang, G. and Kohn, R.: ?Hencky-Prandtl nets and Constrained Michell Trusses?,Comp. Meth. Appl. Mech. Eng. 36 (1983) 207-222. · Zbl 0501.73095
[15] Temam, R.:Problèmes mathématiques en Plasticité, Dunod, Paris, 1983.
[16] Temam, R.:Navier-Stokes Equations, North-Holland, Amsterdam, 1979.
[17] Temam, R. and Strang, G.: ?Duality and Relaxation in the Variational Problems of Plasticity?,J. Mécanique,19 (1980) 493-528. · Zbl 0465.73033
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.