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].

In this work, a nonlocal elasticity model of Klein-Gordon type, characterized by nonlocality in space and time, is developed for the investigation of wave propagation in isotropic elastic media [1, 2]. Nonlocal elasticity theory having a close link to the underlying microstructure has the advantage to capture effects at small scales [3]. Specifically, nonlocal elasticity is valid down to the Ångström-scale and it can be considered as a generalized continuum theory of Ångström-mechanics as it has been shown in [4]. For the first time in the framework of nonlocal elasticity theory, the proposed nonlocal elasticity model of Klein-Gordon type possessing one characteristic internal time scale parameter in addition to the characteristic internal length scale parameter describes spatial and temporal nonlocal effects at small scales. 

The dispersion relations of the considered isotropic nonlocal model of Klein-Gordon type are analytically determined predicting in addition to the acoustic modes (low-frequency modes), optic modes (high-frequency modes) as well as frequency band-gaps between the acoustic and optic modes. The ranges of the frequency band-gaps for longitudinal and transverse waves are determined. Moreover, the phase and group velocities are calculated for the acoustic and optic branches of longitudinal and transverse waves showing that all four modes exhibit normal dispersion with positive group velocity. 

The proposed nonlocal model of Klein-Gordon type possessing only 4 constitutive parameters (2 elastic constants, 1 length scale and 1 time scale) provides an appropriate framework for the modelling of accurate frequency band-gaps and overall physically realistic dispersive wave propagation.