Determination of length scale parameters of strain gradient continuum theory for crystalline solids using a computational quantum mechanical model based on density functional theory

Abstract

Although the classical continuum theory is advantageous in finding solutions to a wide range of engineering problems, it cannot describe some phenomena such as dispersion of acoustic waves, effects of surfaces and interfaces on the mechanical behavior of small-scale structures, and microstructure contribution in special materials. Owing to this fact, several enhanced continuum theories have evolved in the literature. However, the difficulty in determination of the length scale parameters that appear in the governing equations of such theories hampers their widespread use in practice. To date, except for a very limited number of materials, there is no known experimental procedure for the identification of these parameters. In this research, the internal length scales for an augmented continuum theory, i.e., Mindlin's strain gradient theory, have been theoretically determined for some crystalline materials with cubic structure that are of engineering interest, using ab initio DFT. According to the values obtained for these parameters, it can be perceived that the strain gradient theory is a valuable tool for capturing the size effects at even the smallest scales comparable to the dimensions of a unit cell of a crystal lattice.