Investigation of initial‐boundary value problems of gradient elasticity in the realm of implicit second order evolution equations

Abstract

The initial‐boundary value problems encountered in the regime of the dynamic theory of gradient elasticity are characterized by several crucial and intrinsic issues. One of the most important characteristics is that the involving differential equation is of fourth order and mainly that the second order time derivative – appearing in it – is not given explicitly but in the contrary is incorporated implicitly in mixed‐type differential terms (via spatial‐time derivatives). The aim of the present work is the investigation of the emerged initial‐boundary value problems of gradient elasticity in one spatial dimension and the establishment of the suitable functional setting assuring existence and uniqueness of weak (or strong) solutions. Furthermore, the spectral analysis of the gradient elastic operator is investigated and compared with the well‐known results of classical elasticity. Copyright © 2016 John Wiley & Sons, Ltd.