Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems

The fourth-order boundary value problems of one parameter gradient-elastic bar and plane strain/stress models are formulated in a variational form within an H2 Sobolev space setting. For both problems, the existence and uniqueness of the solution is established by proving the continuity and coercivity of the associated symmetric bilinear form. For completeness, the full sets of boundary conditions of the problems are derived and, in particular, the new types of boundary conditions featured by the gradient-elastic models are given the additional attributes singly and doubly. By utilizing the continuity and coercivity of the continuous problems, corresponding error estimates are formulated for conforming Galerkin formulations. Finally, numerical results, with isogeometric Cp−1-continuous discretizations for NURBS basis functions of order p≥2, confirm the theoretical results and illustrate the essentials of both static and vibration problems.