Axis-Symmetrical Solutions for n-Plies Functionally Graded Material Cylinders under Strain No-Decaying Conditions

Abstract

The proposal of the present work is to furnish a general approach to construct exact elastic solutions for FGM cylinders, made of a central core and n arbitrary cylindrical hollow homogeneous and isotropic phases. The hypothesis of axis-symmetrical boundary conditions is here assumed in order to analyze a class of elastic problems which present no-decaying of selected mechanical quantities and in particular of the axial strains ϵ 33 in the radial direction, being x 3 the axis of the laminated cylinder. To construct a robust mathematical procedure for obtaining exact elastic solutions for axis-symmetrical n-plies-Functionally Graded Material Cylinders (n-FGMCs), a theorem is first given for qualifying the space of the solutions and then their mathematical form is identified, when the object exhibits no-decaying of the axial strain. By starting from the classical Boussinesq-Somigliana-Galerkin vector and specializing it to torsionless composite cylinders characterized by no-decaying of the axial strain, a special form of the bi-harmonic Love's function χ (i) (r, x3) is finally obtained. It is then demonstrated that the differential boundary value problem (BVP) always can be translated in an equivalent linear algebraic one, first solving an in cascade one-dimensional Euler-like differential system (field equations) and then writing the boundary conditions by means an algebraic system ruled by a (6n + 4)-order square matrix P. At the end, constructive and existence theorems are formulated and proved, showing examples in comparison with literature data.