×

Serrin integrals and second order problems of plasticity. (English) Zbl 0718.73039

Summary: We use the modern tools of the duality principles and the calculus of variations to formulate, analyse and solve a class of plasticity problems involving second order partial derivatives. The Serrin-type integrals can most appropriately facilitate the existence statements for the extrema from either side of the duality relation in a larger class of BV functions, and interpret the solutions with possible discontinuities on sets of measure zero. The exact solutions of a beam and numerical solutions of a circular plate are presented to demonstrate the theoretical conclusions.

MSC:

74R20 Anelastic fracture and damage
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
49Q20 Variational problems in a geometric measure-theoretic setting
49J22 Optimal control problems with integral equations (existence) (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambrosio, Functionals with linear growth defined on vector valued BV functions (1988)
[2] Temam, Mathematical Problems in Plasticity (1983)
[3] Giaquinta, Comment. Math. Univ. Carolin 20 pp 143– (1979)
[4] DOI: 10.1090/S0002-9947-1961-0138018-9 · doi:10.1090/S0002-9947-1961-0138018-9
[5] DOI: 10.1007/BF02415194 · Zbl 0082.26702 · doi:10.1007/BF02415194
[6] Ekeland, Analyse Convexe et Problèmes Variationnels (1974) · Zbl 0281.49001
[7] DOI: 10.1007/BF01766143 · Zbl 0668.49018 · doi:10.1007/BF01766143
[8] DOI: 10.1007/BF00276912 · Zbl 0618.49004 · doi:10.1007/BF00276912
[9] Cesari, Ann. Scuola Norm. Sup. Pisa 15 pp 219– (1988)
[10] Cesari, Ann. Scuola Norm. Sup. Pisa 5 pp 299– (1936)
[11] DOI: 10.1215/S0012-7094-64-03115-1 · Zbl 0123.09804 · doi:10.1215/S0012-7094-64-03115-1
[12] Adams, Sobolev Spaces (1975)
[13] Temam, Navier Stokes Equations (1977)
[14] Vol’pert, Analysis in classes of discontinuous functions and equations of mathematical physics (1985)
[15] Massari, Minimal surfaces of codimension one (1984) · Zbl 0565.49030
[16] Giusti, Minimal surfaces and functions of bounded variation (1984) · Zbl 0545.49018 · doi:10.1007/978-1-4684-9486-0
[17] Federer, Geometric measure theory (1969) · Zbl 0176.00801
[18] Buttazzo, Pitman Research Notes in Mathematics pp 16– (1989)
[19] DOI: 10.1007/BF00936648 · Zbl 0156.12503 · doi:10.1007/BF00936648
[20] DOI: 10.1115/1.3153659 · Zbl 0465.73021 · doi:10.1115/1.3153659
[21] DOI: 10.1115/1.3153658 · Zbl 0463.73111 · doi:10.1115/1.3153658
[22] DOI: 10.1016/0045-7825(81)90001-3 · Zbl 0466.73123 · doi:10.1016/0045-7825(81)90001-3
[23] DOI: 10.1016/0045-7825(82)90123-2 · Zbl 0478.73022 · doi:10.1016/0045-7825(82)90123-2
[24] DOI: 10.1007/BF01175720 · Zbl 0628.73038 · doi:10.1007/BF01175720
[25] DOI: 10.1007/BF01301259 · Zbl 0435.49016 · doi:10.1007/BF01301259
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.