Finite element analysis of flexible functionally graded beams with variable Poisson’s ratio

Abstract

Purpose The purpose of this paper is to deal with large deformation analysis of plane beams composed of functionally graded (FG) elastic material with a variable Poisson’s ratio. Design/methodology/approach The material is assumed to be linear elastic, with a Poisson’s ratio varying according to a power law along the thickness direction. The finite element used is a plane beam of any-order of approximation along the axis, and with four transverse enrichment schemes, which can describe constant, linear, quadratic and cubic variation of the strain along the thickness direction. Regarding the constitutive law, five materials are adopted: two homogeneous limiting cases, and three intermediate FG cases. The effect of both finite element kinematics and distribution of Poisson’s ratio on the mechanical response of a cantilever is investigated. Findings In accordance with the scientific literature, the second scheme, in which the transverse strain is linearly variable, is sufficient for homogeneous long (or thin) beams under bending. However, for FG short (or moderate thick) beams, the third scheme, in which the transverse strain variation is quadratic, is needed for a reliable strain or stress distribution. Originality/value In the scientific literature, there are several studies regarding nonlinear analysis of functionally graded materials (FGMs) via finite elements, analysis of FGMs with constant Poisson’s ratio, and geometrically linear problems with gradually variable Poisson’s ratio. However, very few deal with finite element analysis of flexible beams with gradually variable Poisson’s ratio. In the present study, a reliable formulation for such beams is presented.

A new 2D beam finite element for nonlinear elastic analysis including warping and shear effects

“…The approximation for the external points, which are out of the beam axis, follows the beam theory regarding the cross-sectional deformation. As performed in [23], the external points are mapped from the beam axis (1) or (2) and from a vector that defines the direction (or the slope) of the cross section, resulting in the following complete approximations:…”

“…This constitutive modeling is valid for beams when the transverse normal stress is negligible. Although the stress-free conditions should be satisfied at the top and bottom surfaces of the beam, in the case of moderately high bending curvatures, the transverse normal stress may achieve considerable levels along the cross section, as reported in [23]. In this case, the uniaxial constitutive model cannot be employed to describe the correct material behavior.…”

“…An alternative beam finite element has been proposed in the work of [23], in which the TBT is employed together with a transverse strain enrichment for the large deformation analysis of FG elastic beams with variable Poisson's ratio. This enrichment scheme has been initially proposed by [24] for nonlinear shell analysis in order to overcome the Poisson locking, which is resultant from the combination between the Reissner's hypothesis and the complete 3D constitutive relation.…”

“…According to [24], this additional parameter provides a locking-free behavior in the context of nonlinear analysis of triangular shell elements. In the work of [23], it is shown that the transverse enrichment is needed for a reliable strain or stress distribution of FG short (or moderately thick) beams. Another important finding of that work is the moderate levels of elastic strains achieved, considering the highest values at the end of the simulations performed.…”

“…However, one should remember that some metals and elastomers may exhibit moderately large levels of elastic (or reversible) strains. Moreover, an innovative feature in [23] is the use of the plane stress version of the linear Saint-Venant-Kirchhoff (SVK) constitutive model. In most of papers, the linear uniaxial model is combined with the linear shear model.…”