2024 - Sustainable Industrial Processing Summit
SIPS 2024 Volume 11. Rowlands Intl. Symp / Mathematics
| Editors: | F. Kongoli, A. Bountis, M. Johnson, S. Karam, L. Kauffman, P. Mandell, M. Mikalajunas, W. Miller, G. Ord, R.M. Santilli, E. Suhir, E. Trell, T. Vougiouklis |
| Publisher: | Flogen Star OUTREACH |
| Publication Year: | 2024 |
| Pages: | 372 pages |
| ISBN: | 978-1-998384-24-2 (CD) |
| ISSN: | 2291-1227 (Metals and Materials Processing in a Clean Environment Series) |
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TOUPIN-MINDLIN FIRST STRAIN GRADIENT ELASTICITY FOR CUBIC AND ISOTROPIC MATERIALS AT SMALL SCALES
Markus Lazar1; Eleni Agiasofitou1;1KARLSRUHE INSTITUTE OF TECHNOLOGY (KIT), Karlsruhe, Germany;
Type of Paper: Regular
Id Paper: 106
Topic: 38
Abstract:
Nonlocal elasticity and strain gradient elasticity theories are challenging generalized continuum theories to model crystals at small scales like the Ångström-scale (see,e.g., [1,2]), where classical elasticity is not valid and leads to unphysical singularities. The theory of first strain gradient elasticity in its modern form dates back to Toupin [3] and Mindlin [4]. 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 [2]. 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. The 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. The numerical values of the obtained quantities are computed for some representative cubic materials using an interatomic potential (MEAM) [2, 5]. Moreover, the isotropy conditions of strain gradient elasticity are given and discussed. A generalization of the Voigt average towards the sixth-rank constitutive tensor of the gradient-elastic constants is given to determine the 5 isotropic gradient-elastic constants [2].