TY - JOUR
T1 - Homogenization of shear-deformable beams and plates with periodic heterogeneity
T2 - A unified equilibrium-based approach
AU - Han, Shilei
AU - Xiao, Yanze
AU - Tian, Qiang
N1 - Publisher Copyright:
© 2024 Elsevier B.V.
PY - 2025/2/15
Y1 - 2025/2/15
N2 - This paper presents a novel equilibrium-based approach to the linear homogenization of shear-deformable beams and plates with periodic heterogeneity. The proposed approach leverages the fact that, under equilibrium, the stress resultants and sectional strains in beams and plates vary at most linearly with respect to the axial or in-plane coordinates. Consequently, the displacement fields within a representative volume element (RVE) are composed of rigid-body, constant-strain, and linear-strain deformation modes, which are proportional to the stress resultants at the center of the beam or plate. By enforcing kinematic compatibility and equilibrium conditions on the lateral surfaces of adjacent RVEs, along with energetic equivalence conditions, the local and global equilibrium equations of the RVEs are derived, leading to singular linear equations for the warping matrix and sectional compliance matrix. The proposed method accounts for all potential stiffness couplings, resulting in fully coupled 6 × 6 and 8 × 8 sectional stiffness matrices for periodic beams and plates, respectively. Notably, the approach addresses stiffness coupling between transverse shear and other deformation modes, which are overlooked in other homogenization methods for periodic structures. Additionally, an equivalent minimization formulation is introduced to determine the warping field and sectional compliance matrix, addressing the homogenization problem in a variational manner. Numerical examples demonstrate that the macro-beam and plate models, using the predicted stiffness matrices, provide accurate displacement fields and three-dimensional stress fields within the linear deformation range. The limitations of the proposed method in addressing problems with significant geometric nonlinearities are also highlighted through numerical examples.
AB - This paper presents a novel equilibrium-based approach to the linear homogenization of shear-deformable beams and plates with periodic heterogeneity. The proposed approach leverages the fact that, under equilibrium, the stress resultants and sectional strains in beams and plates vary at most linearly with respect to the axial or in-plane coordinates. Consequently, the displacement fields within a representative volume element (RVE) are composed of rigid-body, constant-strain, and linear-strain deformation modes, which are proportional to the stress resultants at the center of the beam or plate. By enforcing kinematic compatibility and equilibrium conditions on the lateral surfaces of adjacent RVEs, along with energetic equivalence conditions, the local and global equilibrium equations of the RVEs are derived, leading to singular linear equations for the warping matrix and sectional compliance matrix. The proposed method accounts for all potential stiffness couplings, resulting in fully coupled 6 × 6 and 8 × 8 sectional stiffness matrices for periodic beams and plates, respectively. Notably, the approach addresses stiffness coupling between transverse shear and other deformation modes, which are overlooked in other homogenization methods for periodic structures. Additionally, an equivalent minimization formulation is introduced to determine the warping field and sectional compliance matrix, addressing the homogenization problem in a variational manner. Numerical examples demonstrate that the macro-beam and plate models, using the predicted stiffness matrices, provide accurate displacement fields and three-dimensional stress fields within the linear deformation range. The limitations of the proposed method in addressing problems with significant geometric nonlinearities are also highlighted through numerical examples.
KW - Beams
KW - Homogenization
KW - Periodic heterogeneity
KW - Plates
KW - Shear-deformable
UR - http://www.scopus.com/inward/record.url?scp=85211077548&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2024.117620
DO - 10.1016/j.cma.2024.117620
M3 - Article
AN - SCOPUS:85211077548
SN - 0045-7825
VL - 435
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
M1 - 117620
ER -