Wavelength-dependent strain gradient modelling of one-dimensional mass-spring lattices

By extending the long-wave assumption generally employed in most continua, we have proposed a strain gradient (SG) continuum model to simulate one-dimensional (1D) mass-spring chains with local/nonlocal interactions. By accounting for the periodicity of wave motion, we establish a wavelength-dependent Taylor expansion formulation of displacement. The present continuum is reduced to the existing Mindlin-type SG counterparts at the long-wave limit and remains accurate in the medium- and short-wave range. Based on the proposed model, the following fundamental topics are analyzed: (1) The selection of unit cell strongly influences the performance of the continuum. The model is found to work at its best when the periodic unit cell is composed of one single complete mass point and all half springs directly linked to it. (2) Nonlocal interactions increase local extremums in the dispersion curve, and we propose an explicit criterion to judge the workable range of the long-wave approximation, which becomes narrower with increasing the nonlocal interaction span. (3) A higher-order SG continuum does NOT necessarily enhance the ability of predicting short-wave dispersion properties unless the Taylor expansion method proposed here is adopted. The fourth-order equation of motion, hereafter called EOM, of the present SG continuum has been found to adequately capture the dispersion relation throughout the first Brillouin zone. This work sheds light on short-wave homogenizations of periodic structured medium such as mechanical metamaterials.