Ellipticity and Hyperbolicity Within Nonlinear Strain Gradient Elasticity: 1D Case

We discuss the hyperbolicity properties of the equations of motion for a generalised model of structures and continua such as strain gradient elasticity and their relation to the ellipticity of the equations of statics. Unlike the classical elasticity of simple materials, here hyperbolicity does not follow from ellipticity, which requires further analysis. Here we restrict ourselves to the one-dimensional (1D) case, such as an elastic bar, and demonstrate some special cases.