Jeong, Jena; Neff, Patrizio Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. (English) Zbl 1197.74009 Math. Mech. Solids 15, No. 1, 78-95 (2010). Summary: We investigate the weakest possible constitutive assumptions on the curvature energy in linear Cosserat models still providing for existence, uniqueness and stability. The assumed curvature energy is \(\mu L^{2}_{ c} \parallel \text{dev sym} \nabla \text{axl} \overline A\parallel ^{2}\) where axl \(\overline A\) is the axial vector of the skewsymmetric microrotation \(\overline A \in \mathfrak{so}(3)\) and dev is the orthogonal projection on the Lie-algebra \(\mathfrak{sl}(3)\) of trace free matrices. The proposed Cosserat parameter values coincide with values adopted in the experimental literature by R. S. Lakes. It is observed that unphysical stiffening for small samples is avoided in torsion and bending while size effects are still present. The number of Cosserat parameters is reduced from six to four. One Cosserat coupling parameter \(\mu _{c} > 0\) and only one length scale parameter \(L_{c} > 0\). Use is made of a new coercive inequality for conformal Killing vectorfields. An interesting point is that no (controversial) essential boundary conditions on the microrotations need to be specified; thus avoiding boundary layer effects. Since the curvature energy is the weakest possible consistent with non-negativity of the energy, it seems that the Cosserat couple modulus \(\mu _{c} > 0\) remains a material parameter independent of the sample size which is impossible for stronger curvature expressions. Cited in 49 Documents MSC: 74A35 Polar materials 74B99 Elastic materials 74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) 74G30 Uniqueness of solutions of equilibrium problems in solid mechanics Keywords:polar-materials; microstructure; parameter-identification; structured continua; solid mechanics; variational methods PDFBibTeX XMLCite \textit{J. Jeong} and \textit{P. Neff}, Math. Mech. Solids 15, No. 1, 78--95 (2010; Zbl 1197.74009) Full Text: DOI References: [1] Neff, P., The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric (2006) · Zbl 1104.74007 [2] Cosserat, E., Théorie des Corps dÉformables, Librairie Scientifique, Hermann, Paris, 1909 (Translation: Theory of Deformable Bodies, NASA TT F-11 561, 1968) [3] Eringen, A.C., International Journal of Engineering Science 2 pp 189– (1964) · Zbl 0138.21202 [4] Eringen, A.C. Theory of micropolar elasticity, in Fracture. An Advanced Treatise, Vol. II, ed. H. Liebowitz, pp. 621-729. Academic, New York , 1968. [5] Toupin, R.A., Archive for Rational Mechanics and Analysis 11 pp 385– (1962) · Zbl 0112.16805 [6] Toupin, R.A., Archive for Rational Mechanics and Analysis 17 pp 85– (1964) · Zbl 0131.22001 [7] Green, A.E., Archive for Rational Mechanics and Analysis 17 pp 113– (1964) · Zbl 0133.17604 [8] Mindlin, R.D., Archive for Rational Mechanics and Analysis 11 pp 415– (1962) · Zbl 0112.38906 [9] Schaefer, H. Das, Zeitschrift für Angewandte Mathematik Mechanik (ZAMM) 47 pp 485– (1967) · Zbl 0189.27102 [10] Truesdell, C., The non-linear field theories of mechanics (1965) [11] Eringen, A.C. and Kafadar, C.B. Polar field theories, in Continuum Physics IV: Polar and Nonlocal Field Theories, ed. A. C. Eringen, pp. 1-73, Academic, New York , 1976. [12] Eringen, A.C., Microcontinuum Field Theories (1999) · Zbl 0953.74002 [13] Capriz, G., Annali di Matematica Pura ed Applicata, Series IV 115 pp 17– (1977) · Zbl 0374.73005 [14] Capriz, G., Continua with Microstructure (1989) · Zbl 0676.73001 [15] Maugin, G.A., Philosophical Transactions of the Royal Society of London A 356 pp 1367– (1998) · Zbl 0916.73005 [16] Nowacki, W., Theory of Asymmetric Elasticity. (Polish original, 1971) (1986) [17] Iesan, D., International Journal of Engineering Science 9 pp 59– (1971) · Zbl 0218.73003 [18] Duvaut, G., Journal de Mécanique 9 pp 325– (1970) · Zbl 0312.73030 [19] Hlavacek, I., II: Mindlin’s elasticity with micro-structure and the first strain gradient. Journal of Applied Mathematics 14 pp 387– (1969) · Zbl 0195.27003 [20] Gheorghita, V., I. Archives of Mechanics 26 pp 933– (1974) · Zbl 0297.73009 [21] Gheorghita, V., Archives of Mechanics 29 pp 355– (1974) · Zbl 0297.73009 [22] Iesan, D., International Journal of Engineering Science 35 pp 1277– (1997) · Zbl 0902.73012 [23] Iesan, D., International Journal of Engineering Science 33 pp 399– (1995) · Zbl 0899.73461 [24] Iesan, D., International Journal of Engineering Science 32 pp 991– (1994) · Zbl 0924.73202 [25] Neff, P., Journal of Elasticity 87 pp 239– (2007) · Zbl 1206.74019 [26] Neff, P., Trends in Applications of Mathematics to Mechanics, STAMM Proceedings, Seeheim 2004 [27] Neff, P., Proceedings of the Royal Society of Edinburgh A 136 pp 997– (2006) · Zbl 1106.74010 [28] Gauthier, R.D., CISM Lectures [29] Lakes, R.S., Journal of Biomechanics 25 pp 1409– (1995) · Zbl 0900.73005 [30] Anderson, W.B., Journal of Materials Science 29 pp 6413– (1994) [31] Gauthier, R.D., ASME Journal of Applied Mechanics 42 pp 369– (1975) [32] Cowin, S., Zeitschrift für Angewandte Mathematik und Physik 21 pp 494– (1970) · Zbl 0198.58503 [33] DOI: 10.1007/978-3-540-45286-7 [34] Koiter, W.T., Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen B 67 pp 17– (1964) [35] Dain, S., Calculus of Variations and Partial Differential Equations 25 pp 535– (2006) · Zbl 1091.35097 [36] Hlavacek, I., I. Archive for Rational Mechanics and Analysis 36 pp 305– (1968) · Zbl 0193.39001 [37] Lakes, R.S., International Journal of Solids and Structures 22 pp 55– (1985) [38] Lakes, R.S. Experimental methods for study of Cosserat elastic solids and other generalized elastic continua, in Continuum Models for Materials with Microstructure , ed. H. B. Mühlhaus, pp. 1-25, Wiley , New York, 1995. · Zbl 0900.73005 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.