Non-linear analysis of FGM composites by finite block method in cylindrical coordinates

• Higher order differential matrices can be easily obtained by first order differential matrix.
• Structures are divided into few blocks and mapping technique is applied to each blocks.
• All kinds of boundary problems with functionally graded materials can be solved.

In this paper, the Finite Block Method (FBM) is developed to solve two-dimensional general non-linear partial differential equations in the cylindrical/polar coordinate systems. The first order one-dimensional differential matrix, by using the Lagrange series with uniformly distributed nodes is established and the higher order derivative matrices for multi-dimensional problems are obtained afterwards in the normalised space. By introducing the mapping technique, a block of quadratic type in the cylindrical/polar coordinate systems is transformed into the normalised space with eight seeds. The differential matrices in the physical domain for linear and non-linear problems are then determined in the normalised transformed system. Several examples including the static/dynamic and linear/non-linear heat transfer, the elasticity and the plate bending problems in the polar and cylindrical coordinates are given and comparisons are made with the analytical solutions, such as the Finite Element Method (FEM) and Finite Difference Method (FDM), to demonstrate the degree of accuracy and the convergence of the FBM.