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].

References:

[1] Lazar, M., Agiasofitou, E., Po, G., Three-dimensional nonlocal anisotropic elasticity: a generalized continuum theory of Ångström-mechanics, Acta Mechanica 231, 743–781 (2020).
[2] Lazar, M., Agiasofitou, E., Böhlke, T., Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin-Mindlin anisotropic first strain gradient elasticity, Continuum Mechanics and Thermodynamics 34, 107–136 (2022).
[3] Toupin, R.A., Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11, 385–414 (1962).
[4] Mindlin, R.D., Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis 16, 51–78 (1964).
[5] Po, G., Admal, N.C., Lazar, M., The Green tensor of Mindlin's anisotropic first strain gradient elasticity, Materials Theory 3 (3), (2019).