Micromorphic Continuum Theory: Finite Element Analysis of 3D Elasticity with Applications in Beam- and Plate-Type Structures

Abstract

Due to the failure of classical elasticity to correctly model the behavior of small-scale structures as well as inhomogeneous media, a non-classical three-dimensional (3D) finite element formulation is developed on the basis of the micromorphic theory (MMT). Possessing micro-scale rotation, shear and stretch degrees of freedom (DOFs), MMT is an appropriate candidate to take the size- and microstructural-effects into consideration in mechanical problems. First, a general 3D formulation is proposed for the micromorphic solid continua which includes three stress and strain fields with 18 elastic constants. Then, the relations are written in matricized form which is advantageous for computational aims. Using the matrix-vector MMT formulation, a 3D micromorphic element with 12 DOFs (3 classical and 9 non-classical) is developed. Also, a robust scheme is used to determine the material parameters in terms of two classical constants in such a way that the positive-definiteness of the stored energy would be guaranteed. In the next step, the static deformations of micromorphic beams and plates with various kinds of edge supports are computed to reveal the efficiency of the method. The influences of length scale parameter on the bending responses of micromorphic structures with various geometrical properties are also analyzed. From comparing the results obtained from the classical and micromorphic elasticity theories, it is indicated that MMT results do not completely converge to those of the classical elasticity theory where the size does not matter. This is because of considering the micro-deformation DOFs in MMT and shows the microstructural-effects. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.