Finite element analysis of micromorphic and micropolar continua based on two-dimensional elasticity

Abstract

In this paper, within the framework of two-dimensional (2D) elasticity, a novel finite element formulation is proposed based on the micropolar theory (MPT) and the micromorphic theory (MMT). First, general formulations are developed for the micromorphic and micropolar continua in the context of 2D elasticity. Then, they are presented in a matrix form which is useful from the computational viewpoint. In the next step, using the matricized MPT and MMT formulations, a linear finite element approach including the effects of micro-deformation and micro-rotation degrees of freedom (DOFs) of material particles is developed, and a quadratic size-dependent element is proposed accordingly. Two test problems are solved to reveal the efficiency of the developed formulation. The influence of the length scale parameter on the bending of micromorphic and micropolar plates is illustrated in the given examples. Furthermore, comparisons are made between the results obtained from classical elasticity theory and those calculated based upon MPT and MMT.

Vibration analysis of two-dimensional micromorphic structures using quadrilateral and triangular elements

“…(2019) for 2D micromorphic plate. From Tables 15 and 16, one can observe very good agreement between the results obtained herein and those reported in Bazdid-Vahdati et al. (2019).…”

“…Also, the dimensionless displacement parameters are defined as { U 1 , U 2 } = { u 1 , u 2 } G / Fa in which u 1 , u 2 , G, F and a denote the displacements in x and y −axis directions, shear modulus, load amplitude and the side length of the plate, respectively. The comparison between the predicted results in Bazdid-Vahdati et al. (2019) and the present approach is shown in Table 15.…”

“…In the next problem, the 2D micromorphic plate is supported at the bottom edge and is subjected to a compressive load on the top edge. In Table 16, comparisons are made between the present results and those of Bazdid-Vahdati et al. (2019) for 2D micromorphic plate.…”

“…(2019). In the first problem, by considering the material constants and the geometrical data of Bazdid-Vahdati et al. (2019), a 2D square plate is modeled.…”

“…In another validation study, using the eight-node micromorphic element, the bending analysis of 2D micro-structure is performed and the obtained results are compared with the reported ones in Bazdid-Vahdati et al (2019). In the first problem, by considering the material constants and the geometrical data of Bazdid-Vahdati et al (2019) Analysis of 2D micromorphic structures the right edge. Also, the dimensionless displacement parameters are defined as {U 1 ,U 2 } 5 {u 1 ,u 2 }G/Fa in which u 1 , u 2 , G, F and a denote the displacements in x and yÀ axis directions, shear modulus, load amplitude and the side length of the plate, respectively.…”