Ghiba, Ionel-Dumitrel; Neff, Patrizio; Madeo, Angela; Placidi, Luca; Rosi, Giuseppe The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics. (English) Zbl 1338.74007 Math. Mech. Solids 20, No. 10, 1171-1197 (2015). Summary: We study well-posedness for the relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. In contrast to classical micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. Another interesting feature concerns the prescription of boundary values for the micro-distortion field: only tangential traces may be determined which are weaker than the usual strong anchoring boundary condition. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch, and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. Cited in 1 ReviewCited in 34 Documents MSC: 74A60 Micromechanical theories 35Q74 PDEs in connection with mechanics of deformable solids Keywords:micromorphic elasticity; symmetric Cauchy stresses; dynamic problem; dislocation dynamics; gradient plasticity; dislocation energy; generalized continua; microstructure; micro-elasticity; non-smooth solutions; well-posedness; Cosserat couple modulus; wave propagation PDFBibTeX XMLCite \textit{I.-D. Ghiba} et al., Math. Mech. Solids 20, No. 10, 1171--1197 (2015; Zbl 1338.74007) Full Text: DOI arXiv References: [1] Neff P, Cont Mech Therm (2013) [2] DOI: 10.1007/978-1-4612-0555-5 · Zbl 0953.74002 [3] Neff P, Trends in Applications of Mathematics to Mechanics pp 337– (2005) [4] Mariano PM, Arch Comput Meth Eng 12 pp 392– (2005) · Zbl 1152.74315 [5] Mariano PM, ESAIM: COCV 15 (2) pp 377– (2009) · Zbl 1161.74006 [6] Mariano PM, Quaderni di Matematica 20 pp 80– (2007) [7] Steigmann DJ, Int J Non-Linear Mech 47 pp 734– (2012) [8] DOI: 10.1002/zamm.200510281 · Zbl 1104.74007 [9] Jeong J, Math Mech Solids 15 (1) pp 78– (2010) · Zbl 1197.74009 [10] DOI: 10.1002/zamm.200800156 · Zbl 1157.74002 [11] DOI: 10.1007/978-1-4419-5695-8_6 · Zbl 1396.74012 [12] DOI: 10.1007/BF00248490 · Zbl 0119.40302 [13] Eringen AC, Int J Eng Sci 2 pp 189– (1964) · Zbl 0138.21202 [14] Kröner E, Fundamental Aspects of Dislocation Theory 1 pp 1054– (1970) [15] Kröner E, Proceedings of 11th international congress of applied mechanics pp 143– (1964) [16] Claus WD, Proceedings of the 12th Midwestern mechanics conference on developments in mechanics 6 pp 349– [17] Eringen AC, Fundamental Aspects of Dislocation Theory 1 pp 1023– (1970) [18] Claus WD, Int J Eng Sci 9 pp 605– (1971) · Zbl 0221.73008 [19] Chang CS, J Eng Mech–ASCE 116 (5) pp 1077– (1990) [20] Misra A, Int J Solids Struct 30 (18) pp 2547– (1993) · Zbl 0782.73009 [21] DOI: 10.1016/j.ijsolstr.2010.07.002 · Zbl 1196.74161 [22] Yang Y, Int J Solids Struct 49 (18) pp 2500– (2012) [23] Merkel A, Physical Review Letters 107 pp 225502– (2011) [24] Neff P, J Elasticity 87 pp 239– (2007) · Zbl 1206.74019 [25] DiCarlo A, Mech Res Commun 29 (6) pp 449– (2002) · Zbl 1056.74005 [26] Soós E, Int J Eng Sci 7 pp 257– (1969) · Zbl 0167.24703 [27] Hlaváček I, J Apl Mat 14 pp 387– (1969) [28] Ieşan D, Int J Eng Sci 39 pp 2051– (2001) · Zbl 1210.74022 [29] Ieşan D, Int J Eng Sci 40 pp 549– (2002) · Zbl 1211.74070 [30] Neff P, Proc Roy Soc Edinb A 136 pp 997– (2006) · Zbl 1106.74010 [31] Klawonn A, ESAIM: Math Mod Num Anal 45 pp 563– (2011) · Zbl 1268.74037 [32] Pazy A, Semigroups of Linear Operators and Applications to Partial Differential Equations (1983) · Zbl 0516.47023 [33] Vrabie I, C0-Semigroups and applications 191 (2003) [34] Neff P, C R Acad Sci Paris Ser I 349 pp 1251– (2011) · Zbl 1234.35010 [35] Neff P, Math Methods Appl Sci 35 pp 65– (2012) · Zbl 1255.35220 [36] Bauer S, Proc Appl Math Mech (2013) [37] Bauer S, C R Acad Sci Paris Ser I (2013) [38] Lankeit J, Z Angew Math Phys 64 pp 1679– (2013) · Zbl 1414.35046 [39] Galeş C, J Therm Stresses 34 pp 1241– (2011) [40] Galeş C, Eur J Mech–A/Solids 31 pp 37– (2012) · Zbl 1278.74050 [41] Grekova EF, Int J Eng Sci 43 pp 494– (2005) · Zbl 1211.74052 [42] Maugin GA, Generalized Continua from the Theory to Engineering Applications 541 pp 301– (2013) · Zbl 1279.74013 [43] DOI: 10.1007/s00161-013-0329-2 · Zbl 1341.74085 [44] DOI: 10.1177/1077546304038224 · Zbl 1078.74026 [45] DOI: 10.1007/s004190050142 · Zbl 0908.73067 [46] Maurini C, Comput Struct 84 (22) pp 1438– (2006) [47] Maurini C, J Phys 115 pp 307– (2004) [48] Porfiri M, Int J Appl Electrom 21 (2) pp 69– (2005) [49] Vidoli S, Eur J Mech–A/Solids 20 (3) pp 435– (2001) · Zbl 0988.74047 [50] Adams RA, Sobolev Spaces 65, 1. ed. (1975) [51] DOI: 10.1007/BFb0063447 · Zbl 0413.65081 [52] DOI: 10.1007/978-3-663-10649-4 [53] Eringen AC, Int J Eng Sci 2 pp 389– (1964) [54] DOI: 10.1016/0020-7225(68)90020-7 · Zbl 0159.56903 [55] Vrabie I, Differential equations. An introduction to basic concepts (2004) · Zbl 1070.34001 [56] DOI: 10.1002/zamm.201100022 · Zbl 1247.74031 [57] Placidi L, Math Mech Solids 19 (5) pp 555– (2014) · Zbl 1305.74047 [58] Rosi G, Int J Solids Struct 50 pp 1721– (2013) [59] Engheta N, Metamaterials: Physics and Engineering Explorations (2006) [60] Zouhdi S, Metamaterials and Plasmonics: Fundamentals, Modelling, Applications (2009) [61] Dell’Isola F, Arch Appl Mech 67 (4) pp 215– (1997) [62] DOI: 10.1007/BF01170371 · Zbl 0897.73003 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.