Limit Elastic Analysis of FG Ceramic Rotating Disk on the Basis of Effective Mechanical Properties

Abstract

Functionally graded materials are prevalent among industries for its low weight compared to alloys. Depending upon the applications ceramic-ceramic materials can be graded to obtain novel functional combination. In a disk, yielding mainly occurs at the root, hence to overcome the yielding problem, Al2O3 material was used at the root and graded with different ceramics; as ceramics possess high yield strength. In composites, many methods are available to calculate effective mechanical properties and modified rule of mixture (MROM) is one of these. In MROM only single parameter, stress to strain transfer ratio needs to be identified which in present case was calculated using inverse approach i.e. by comparing Halpin-Tsai model with MROM. Variational formulation method was employed taking radial displacement field as unknown variable. The effective Young’s modulus calculated was then verified with experimental data and good agreement was seen. Further effective yield stress distribution was quantified and plotted to compare with von-Mises stress. Limit elastic speed then obtained was used to rank different ceramic-ceramic graded combinations. Limit speed defines the performance of disk and thus it was chosen as a selection parameter.

A Dynamic Programming Setting for Functionally Graded Thick-Walled Cylinders

“…Problem 1. Given the dynamical system (11) and the boundary conditions (12) and (13), find the control function w :…”

“…The application of Pontryagin's Principle, therefore, leads to a system of six nonlinear and coupled first-order differential equations described by (11) and (23) that have to be solved by intervals for a constant value of w (in each interval either w ≡ w + or w ≡ w − ) and taking into account the six boundary conditions (12), (13) and (26).…”

“…To solve this boundary value problem (BVP), a numerical integration approach is needed. One way is to implement the so-called shooting method [24], i.e., to start from the inner radius and guess x 2 (1), p 1 (1), and p 3 (1), and then forward integrate (11) and (23) as an initial value problem (IVP) until the outer radius, where a check is made whether (12) and (26) are satisfied. If so, a solution is found, if not the initial guesses are adjusted.…”

“…A significant amount of work has been devoted to understand the thermal and mechanical behavior of FG axisymmetric bodies. In particular, the study of the stress and strain fields in FG cylinders [ 2 , 3 ], spheres [ 4 , 5 ], pressure vessels [ 6 , 7 ], circular plates [ 8 , 9 ], and rotating disks [ 10 , 11 , 12 ] has been widely investigated. In all the above-mentioned works, a generic property (e.g., Young’s modulus, mass density, or a thermal expansion coefficient) is assumed to vary along the radial coordinate according to some pre-defined functions.…”

“…Given the boundary conditions (12) and (13), the solution of system (11) is unique for any choice of the input w; hence, the cost functionals J i (i = 1, 2, 3), although defined with respect to the state variables, are, in turn, functions of w.…”