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Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity

Lazar, Markus 1; Agiasofitou, Eleni 1; Böhlke, Thomas ORCID iD icon 1
1 Institut für Technische Mechanik (ITM), Karlsruher Institut für Technologie (KIT)

Abstract:

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, 3+11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and 1+6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. ... mehr


Verlagsausgabe §
DOI: 10.5445/IR/1000137752
Veröffentlicht am 22.09.2021
Originalveröffentlichung
DOI: 10.1007/s00161-021-01050-y
Scopus
Zitationen: 21
Web of Science
Zitationen: 20
Dimensions
Zitationen: 27
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Technische Mechanik (ITM)
Publikationstyp Zeitschriftenaufsatz
Publikationsjahr 2022
Sprache Englisch
Identifikator ISSN: 0935-1175, 1432-0959
KITopen-ID: 1000137752
Erschienen in Continuum Mechanics and Thermodynamics
Verlag Springer
Band 34
Seiten 107-136
Nachgewiesen in Dimensions
Scopus
Web of Science
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