A boundary element method for solving 2-D and 3-D static gradient elastic problems: Part I: Integral formulation Original Research Article

A boundary element formulation is developed for the static analysis of two- and three-dimensional solids and structures characterized by a linear elastic material behavior taking into account microstructural effects. The simple gradient elastic theory of Aifantis expressed in the framework of Mindlin’s general theory is used to model this material behaviour. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution for both two- and three-dimensional cases is explicitly derived and used to construct the boundary integral representation of the solution with the aid of the reciprocal integral identity especially established for the gradient elasticity considered here. It is found that for a well-posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. Explicit expressions for interior displacements and stresses in integral form are also presented. All the kernels in the integral equations are explicitly provided.