First strain gradient elasticity solution for nanotube-reinforced matrix problem

In the current paper, a rigorous proof of an important theorem which has been used frequently for derivation of field equations of first gradient elasticity is given for the first time. After that, due to the wide use of nanoparticles as reinforcements to different types of matrices, the nanotube-reinforced matrix problem is investigated in cylindrical coordinates. Then, using the most general model for an isotropic gradient elastic material, the displacement formulation is employed to solve the governing equations of the nanotube-reinforced matrix problem. For this purpose, the generalized perfect interface conditions for the nonhomogeneous representative volume element (RVE) are introduced and used to derive the solution. Numerical results reveal that as the matrix characteristic length parameter becomes larger in comparison to that of nanotube, the difference between the results of the classical theory and the strain gradient theory will increase and classical theory cannot accurately predict the mechanical response of the RVE; In addition, increasing the nanotube’s volume fraction results in reduction of the maximum compressive stress and a rise in the overall stiffness of RVE.