Analytical solutions for the thick-walled cylinder problem modeled with an isotropic elastic second gradient constitutive equation

Numerical modeling of localization phenomena shows that constitutive equations with internal length scale are necessary to properly model the post-localization behavior. Moreover, these models allow an accurate description of the scale effects observed in some phenomena like micro-indentation. This paper proposes some analytical results concerning a boundary value problem in a medium with microstructure. In addition to their own usefulness, such analytical solutions can be used in benchmark exercises for the validation of numerical codes. The paper focuses on the thick-walled cylinder problem, using a general small strain isotropic elastic second gradient model. The most general isotropic elastic model involving seven different constants is used and the expression of the analytical solutions is explicitly given. The influence of the microstructure is controlled by the internal length scale parameter. The classical macrostress is no more in equilibrium with the classical forces at the boundary. Double stresses are indeed also generated by the classical boundary conditions and, as far as the microstructure effects become predominant (i.e. the internal length scale is much larger than the thickness of the cylinder), the macrostresses become negligible. This leads to solutions completely different from classical elastic ones.