Abstract
The objective of this contribution is the computation of the Airy stress function for functionally graded beam-type structures subjected to transverse and shear loads. For simplification, the material parameters are kept constant in the axial direction and vary only in the thickness direction. The proposed method can be easily extended to material varying in the axial and thickness direction. In the first part an iterative procedure is applied for the determination of the stress function by means of Boley’s method. This method was successfully applied by Boley for two-dimensional (2D) isotropic plates under plane stress conditions in order to compute the stress distribution and the displacement field. In the second part, a shear loaded cantilever made of isotropic, functionally graded material is studied in order to verify our theory with finite element results. It is assumed that the Young’s modulus varies exponentially in the transverse direction and the Poisson ratio is constant. Stresses and displacements are analytically determined by applying our derived theory. Results are compared to a 2D finite element analysis performed with the commercial software ABAQUS. It is found that the analytical and numerical results are in perfect agreement.
Highlights
Graded materials (FGMs) are composite materials where the material properties vary throughout the body
The literature on Functionally graded materials (FGMs), where the focus is laid on the derivation of a mechanical model which is valid for arbitrary material property variations, is limited
The beam model is verified by a two-dimensional (2D) finite element calculation performed with ABAQUS where analytical and numerical results are in excellent agreement
Summary
Graded materials (FGMs) are composite materials where the material properties vary throughout the body. A higher-order shear-deformable finite beam element for an FGM sandwich structure is derived by Li [15] where higher-order strain assumptions are taken into account, but the transversal strain is neglected as in [14] Our motivation in this contribution is to provide an analytical method (under classical plane-stress assumption) to determine a beam model without any a priori assumptions on the displacement or the stress field. For a supported piezoelectric bimorph with sinusoidal mechanical and electrical actuation, the result is compared to (2D) analytical results Motivated by the latter contributions, the iterative method from Boley is extended to FGMs. Instead of the bi-potential equation, we derive a fourth-order partial differential equation for the Airy stress function from the compatibility equations. The beam model is verified by a two-dimensional (2D) finite element calculation performed with ABAQUS where analytical and numerical results are in excellent agreement
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