×

zbMATH — the first resource for mathematics

Laminates and microstructure. (English) Zbl 0779.73050
Summary: This paper deals with the mathematical characterization of microstructure in elastic solids. We formulate our ideas in terms of rank-one convexity and identify the set of probability measures for which Jensen’s inequality for this type of functions holds. This is the set of laminates. We also introduce generalized convex hulls of sets of matrices and investigate their structure.

MSC:
74A60 Micromechanical theories
74M25 Micromechanics of solids
74E30 Composite and mixture properties
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1016/0020-7683(86)90030-2 · Zbl 0595.73001 · doi:10.1016/0020-7683(86)90030-2
[2] DOI: 10.1007/BF00278240 · Zbl 0597.73006 · doi:10.1007/BF00278240
[3] Ericksen, Phase Transformations and Material Instabilities in Solids pp 61– (1984) · doi:10.1016/B978-0-12-309770-5.50008-4
[4] Ericksen, Systems of Nonlinear Partial Differential Equations pp 71– (1983) · doi:10.1007/978-94-009-7189-9_3
[5] Ericksen, IUTAM Symp. Finite Elasticity pp 167– (1981) · doi:10.1007/978-94-009-7538-5_9
[6] DOI: 10.1137/0523001 · Zbl 0757.49014 · doi:10.1137/0523001
[7] DOI: 10.1007/BF00258233 · Zbl 0429.73007 · doi:10.1007/BF00258233
[8] DOI: 10.1007/BF00250733 · Zbl 0439.20031 · doi:10.1007/BF00250733
[9] DOI: 10.1007/BF00375279 · Zbl 0754.49020 · doi:10.1007/BF00375279
[10] DOI: 10.1016/0956-7151(91)90023-T · doi:10.1016/0956-7151(91)90023-T
[11] Tartar, Systems of Nonlinear Partial Diff. Eq. C111 pp 263– (1982)
[12] Ball, J. Math. Pures et Appl. 69 pp 241– (1990)
[13] DOI: 10.1007/BF00281246 · Zbl 0629.49020 · doi:10.1007/BF00281246
[14] Milton, J. Mech. Phys. Solids (1992)
[15] Tartar, Microstructure and phase transitions (1991)
[16] Dacorogna, Direct Methods in the Calculus of Variations (1989) · Zbl 0703.49001 · doi:10.1007/978-3-642-51440-1
[17] DOI: 10.1007/BF00251759 · Zbl 0673.73012 · doi:10.1007/BF00251759
[18] Sverak, Proc. R. Soc. 120A pp 185– (1992)
[19] DOI: 10.1007/BFb0024945 · doi:10.1007/BFb0024945
[20] Dacorogna, J. Math. Pures Appl. 64 pp 403– (1985)
[21] DOI: 10.1007/BF01135336 · Zbl 0825.73029 · doi:10.1007/BF01135336
[22] Firoozye, Microstructure and Phase Transitions (1992)
[23] Milton, Homogeneization of Effective Moduli of Materials and Media (1986)
[24] Lurie, Proc. Symp. Material Instabilities in Continuum Mechanics pp 257– (1988)
[25] Kohn, Proc. Seventh Army Conf. on Appl. Math, and Computing (1989)
[26] DOI: 10.1007/BF01597353 · Zbl 0019.35203 · doi:10.1007/BF01597353
[27] Kinderlehrer, Proc. Symp. Material Instabilities in Continuum Mechanics pp 217– (1988)
[28] Sverak, Proc. R. Soc. 114A pp 237– (1990)
[29] Serre, J. de Math. Pures et Appl. 62 pp 177– (1983)
[30] DOI: 10.1007/BFb0024935 · doi:10.1007/BFb0024935
[31] DOI: 10.1007/BF01442177 · Zbl 0567.49007 · doi:10.1007/BF01442177
[32] James, Proc. Symp. Material Instabilities in Continuum Mechanics pp 175– (1988)
[33] Kohn, Comm. Pure Appl. Math. 34 pp 113– (1987)
[34] Gurtin, Arch. Rat. Mech. Anal. 96 pp 243– (1986)
[35] Fonseca, J. Math. Pures Appl. 67 pp 175– (1988)
[36] Dacorogna, Proc. Symp. Material Instabilities in Continuum Mechanics pp 77– (1988)
[37] DOI: 10.1007/BF00250808 · Zbl 0611.73023 · doi:10.1007/BF00250808
[38] DOI: 10.1007/BF00251425 · Zbl 0697.73005 · doi:10.1007/BF00251425
[39] Nicolaides, Proc. Recent Adv. Adaptive Sensory Materials and their Appl. (1992)
[40] Ericksen, Metastability and Incompletely Posed Problems, IMA Vol. Math. Appl. 3 pp 77– (1987) · doi:10.1007/978-1-4613-8704-6_6
[41] DOI: 10.1137/0728018 · Zbl 0725.65067 · doi:10.1137/0728018
[42] DOI: 10.1007/BFb0024934 · doi:10.1007/BFb0024934
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.