Form II of Mindlin's second strain gradient theory of elasticity with a simplification: For materials and structures from nano- to macro-scales

Abstract

The fundamental equations for Form II of Mindlin's second strain gradient elasticity theory for isotropic materials are first derived. A corresponding simplified formulation is then proposed, with six and two higher-order material parameters for the strain and kinetic energy, respectively. This simplified model is still capable of accounting for free surface effects and surface tension arising in second strain gradient continua. Within the simplified model, at first, surface tension effects appearing in nano-scale solids near free boundaries are analyzed. Next, a thin strip under tension and shear is considered and closed-form solutions are provided for analyzing the free surface effects. Expressions for effective Poisson's ratio and effective shear modulus are proposed and found to be size-dependent. Most importantly, for each model problem a stability analysis is accomplished disallowing non-physical solutions (befallen but not exclusively disputed in a recent Form I article). Finally, a triangular macro-scale lattice structure of trusses is shown, as a mechanical metamaterial, to behave as a second strain gradient continuum. In particular, it is shown that initial stresses prescribed on boundaries can be associated to one of the higher-order material parameters, modulus of cohesion, giving rise to surface tension. For completeness, a numerical free vibration eigenvalue analysis is accomplished for both a fine-scale lattice model and the corresponding second-order continuum via standard and isogeometric finite element simulations, respectively, completing the calibration procedure for the higher-order material parameters. The eigenvalue analysis confirms the necessity of the second velocity gradient terms in the kinetic energy density.

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

“…In addition, we need to investigate how assemblies of nano‐grains may resemble or be different from the assemblies, which we have considered here. In this context again, we have a ready algorithm and interesting results, ie, those presented in Khakalo and Niiranen …”

Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: Applications to sandwich beams and auxetics

“…For instance, [15,28,29] consider static bending tests at micrometer-and nanometer-scales, whereas the micro-inertia length scale parameter has been assessed in [30] by experiments on torsional vibrations fo fine-grained materials, and atomistic representations of elastic moduli tensors for different materials have been provided in [31]. Altogether, literature on applications of non-classical continuum models to real materials, structures and systems is still very limited; some examples can be found in [32,33,34,35,36,37,38,39]. Regarding numerical methods and numerical analysis for the models of generalized continuum mechanics, literature is limited as well, although there are some successful endeavours including rigorous studies on solvability and convergence (for recent overviews and examples, see [40,41,25,42,26,43,44,45]).…”

The macroscopic behavior of pantographic sheets depends mainly on their microstructure: experimental evidence and qualitative analysis of damage in metallic specimens

“…It is remarkable that this substructure results in a material with an exotic macroscopic behavior which can be subjected to large deformation while remaining in an elastic regime [1][2][3][4]. Notably, the design of the microstructure has been conceived so that its deformation energy depends on the second gradient of the displacement [5][6][7][8][9][10][11][12]. Morphologically, the substructure can be described as a double array of mutually orthogonal beams, also called fibers, interconnected by elastic cylinders at intersection points, also called pivots [13,14].…”