×

Regularization and existence of solutions in the equilibrium problem for an elastoplastic plate. (English. Russian original) Zbl 0902.35032

Sib. Math. J. 39, No. 3, 582-593 (1998); translation from Sib. Mat. Zh. 39, No. 3, 670-682 (1998).
The author proves an existence theorem for the equilibrium problem for an elastoplastic plate which guarantees validity of all boundary conditions. The case in which domains have smooth boundaries is considered as well as the case of domains with cuts. The proof is based on a special combination of the regularization method and penalty method. Namely, elliptic regularization is applied for solving the penalized boundary value problem that approximates the original elastoplastic problem. An existence theorem for the original problem is obtained by passing to the limit over the regularization and penalty parameters. The author notes that the proposed way of regularization turns out to be rather effective and useful for analysis of other elastoplastic problems. The essence of the method consists in the fact that the regularization of penalized equations is accompanied by special regularization conditions on the boundary.

MSC:

35J45 Systems of elliptic equations, general (MSC2000)
35A35 Theoretical approximation in context of PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. Johnson, ”Existence theorems for plasticity problems,” J. Math. Pures Appl. (9),55, 431–444 (1976). · Zbl 0351.73049
[2] P. M. Suquet, ”Evolution problems for a class of dissipative materials,” Quart. Appl. Math.,38, No. 4, 391–414 (1981). · Zbl 0501.73030
[3] R. Temam, Problémes Mathématiques en Plasticité, Gauther-Villars, Paris (1983).
[4] G. Anzellotti and M. Giaquinta, ”On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium,” J. Math. Pures Appl. (9),61, No. 3, 219–244 (1982). · Zbl 0467.73044
[5] A. M. Khludnev, ”Existence of solutions in a quasistatic problem of elastoplastic deformation of shells,” Sibirsk. Mat. Zh.,25, No. 5, 168–176 (1984). · Zbl 0597.73065
[6] A. M. Khludnev, ”Variational inequalities in contact plastic problems,” Differentsial’nye Uravneniya,24, No. 9, 1622–1628 (1988). · Zbl 0675.35044
[7] F. Demengel, ”Problemes variationnels en plasticite parfaite des plaques,” Numer. Funct. Anal. Optim.,6, No. 1, 73–119 (1983). · Zbl 0554.73030 · doi:10.1080/01630568308816155
[8] A. M. Khludnev and K.-H. Hoffmann, ”A variational inequality in a contact elastoplastic problem for a bar,” Adv. Math. Sci. Appl.,1, No. 1, 127–136 (1992). · Zbl 0747.73041
[9] A. M. Khludnev, ”A contact problem for a beam under plasticity and creep conditions,” Sibirsk. Mat. Zh.,34, No. 2, 173–179 (1993). · Zbl 0833.73047
[10] A. M. Khludnev, ”Contact viscoelastoplastic problem for a beam,” in: Free Boundary Problems in Continuum Mechanics, Birkhäuser-Verlag, Basel, Boston, and Berlin, 1992, pp. 159–166. (Internat. Ser. Numer. Math.,106). · Zbl 0819.35153
[11] A. M. Khludnev and J. Sokolowski, Modelling and Control in Solid Mechanics, Birkhäuser-Verlag, Basel, Boston, and Berlin (1997). · Zbl 0865.73003
[12] M. I. Erkhov, The Theory of Ideal Plastic Bodies and Constructions [in Russian], Nauka, Moscow (1978).
[13] J.-L. Lions, Some Methods for Solving Nonlinear Boundary Value Problems [Russian translation], Mir, Moscow (1972).
[14] O. A. Oleînik, V. A. Kondrat’ev, and I. Kopachek, ”On asymptotic properties of solutions to the biharmonic equation,” Differentisial’nye Uravneniya,17, No. 10, 1886–1899 (1981). · Zbl 0487.35034
[15] V. A. Kondrat’ev, I. Kopachek, and O. A. Oleînik, ”On behavior of weak solutions to secondorder elliptic equations and the system of elasticity in a neighborhood about a boundary point,” Trudy Sem. Petrovsk.,8, 135–152 (1982).
[16] A. M. Khludnev, ”A contact problem for a shallow shell with a crack,” Prikl. Mat. Mekh.,59, No. 2, 318–326 (1995). · Zbl 0888.73058
[17] A. M. Khludnev, ”On a contact problem for a plate having a crack,” Control Cybernet.,24, No. 3, 349–361 (1995). · Zbl 0844.73073
[18] N. F. Morozov, Mathematical Problems of the Theory of Cracks [in Russian], Nauka, Moscow (1984). · Zbl 0566.73079
[19] P. Grisvard, ”Singularities in boundary value problems,” Res. Notes Appl. Math.,22, 1–198 (1992). · Zbl 0766.35001
[20] S. Nicaise, ”About the Lamé system in a polygonal or a polyhedral domain and a coupled problem between the Lamé system and the plate equation. I. Regularity of the solutions,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4),19, No. 3, 327–361 (1992). · Zbl 0782.73041
[21] S. A. Nazarov and B. A. Plamenevskiî, Elliptic Problems in Domains with Piecewise Smooth Boundary [in Russian], Nauka Moscow (1991).
[22] R. Duduchava and W. Wendland, ”The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems,” Integral Equations Operator Theory,23, 294–335, (1995). · Zbl 1126.35368 · doi:10.1007/BF01198487
[23] E. P. Stephan, ”Boundary integral equations for mixed boundary value problems inR 3,” Math. Nachr.,134, 21–53 (1987). · Zbl 0649.35022 · doi:10.1002/mana.19871340103
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.