Dynamic Analysis Of Second Strain Gradient Elasticity Through A Wave Finite Element Approach

In this article, the Second Strain Gradient (SSG) theory proposed by Mindlin is used within a Wave Finite Element Method (WFEM) framework for dynamic analysis of one-dimensional Euler?Bernoulli bending beam and torsional bar. Firstly, strong forms of continuum models including governing equations and boundary conditions for bending and torsion cases, respectively, are derived using Hamilton?s principle. New ?non-local? Lattice Spring Models (LSM) are expounded, giving unified description of the SSG models for bending and torsion. These LSM can be regarded as a discrete micro-structural description of SSG continuum models and the resulting dynamic equations are transformed using Fourier series. Weak forms for both bending and torsion are established based on SSG theory. Subsequently, the WFEM is used to formulate the spectral problem and compute wave dispersion characteristics from one-dimensional unit-cell structures. Finally, dispersion relations and forced responses for bending and torsion in micro-sized structures are calculated by SSG and Classical Theory (CT), and some useful conclusions are discussed.