Numerical/analytical solutions to the elastic response of arbitrarily functionally graded polar orthotropic rotating discs

Abstract

A substantial elastic analysis of uniform rotating discs made of radially functionally graded (FG) polar orthotropic materials is managed with both analytical and numerical methods by imposing possible boundary conditions and frequently used material grading rules such as a simple power and an exponential patterns. The complementary functions method (CFM) is originally chosen as a numerical technique to solve the governing equation having variable coefficients. Before applying CFM on the current two-point boundary value problem, the governing equation derived by the transformed onaxis in-plane stiffness terms is transformed into the initial value problem. To show the effectiveness of the method, as an additional study, some closed-form formulas are obtained to cover the rotating uniform discs made of functionally simple power-law graded polar orthotropic materials under the same constraints. It was shown that both analytical and numerical results display a perfect harmony. An extensive parametric study which considers physically or hypothetically exist different material types, several inhomogeneity indexes, anisotropy degrees varying in a wide range of 0.3-5, three types of boundary conditions, is conducted with the help of both analytical formulas and numerical solutions. To the best of the author's knowledge, although especially the anisotropy effects have been worked through for ordinary polar orthotropic materials, investigations on the anisotropy degrees on the elastic behaviour of discs made of an advanced material having varying properties along the desired directions such as a FG anisotropic material is still so limited and requires vast knowledge. Since different transformed stiffness terms through the radial direction may be developed by considering different fibre orientations, present results comprising anisotropy effects may also be interpreted as the results of such discs made of specially orthotropic materials such as cross-ply or balanced symmetrical laminates. It is mainly concluded that the use of a radially FG anisotropic material having an anisotropy degree less than the unit proposes admissible circumferential stresses along with the effective properties which continuously radially increase from the inner surface towards the outer under centrifugal forces. On the other hand, anisotropy degrees greater than the unit with increasing material properties from the inner surface to the outer mitigates the radial stresses. Negative inhomogeneity indexes with smaller anisotropy degrees are preferable to get smaller radial displacements.

Influence of the Replacement of the Actual Plastic Orthotropy with Various Approximations of Normal Anisotropy on Residual Stresses and Strains in a Thin Disk Subjected to External Pressure

“…The solution for Yield Criterion (3) is valid for Cases (i) and (ii). The input data for the solution for Yield Criterion (6) have been derived from Equations (7) and (8) for three n-values (Table 1)…”

“…For example, the residual stresses and, hence, the elastic springing are very sensitive to plastic anisotropy [3,4]. The effect of plastic anisotropy on the solution behavior for thin rotating disks has been revealed in [5][6][7]. However, for simplifying theoretical calculations, the real orthotropic yield criterion is often replaced with a transversely isotropic yield criterion (i.e., it is assumed that the material properties are independent of the direction within the transverse plane).…”

A Theory of Autofrettage for Open-Ended, Polar Orthotropic Cylinders

“…In the case of circular discs and cylinders, a common type of anisotropy is polar orthotropy. In particular, the effect of plastic anisotropy on stress and strain fields in rotating discs has been studied in [34][35][36][37][38][39], using different material models and boundary conditions. Various boundary value problems for orthotropic cylinders have been solved in [40][41][42][43][44].…”

Description of Residual Stress and Strain Fields in FGM Hollow Disc Subject to External Pressure

“…There is a vast amount of literature on functionally graded rotating discs. The elastic response of an arbitrary functionally graded polar orthotropic disc has been investigated in [9]. Another purely elastic solution has been given in [10], using the finite difference method.…”