A nonlocal operator method for finite deformation higher-order gradient elasticity

authored by: Huilong Ren, Xiaoying Zhuang, Nguyen Thoi Trung, Timon Rabczuk
Abstract: We present a general finite deformation higher-order gradient elasticity theory. The governing equations of the higher-order gradient solid along with boundary conditions of various orders are derived from a variational principle using integration by parts on the surface. The objectivity of the energy functional is achieved by carefully selecting the invariants under rigid-body transformation. The third-order gradient solid theory includes more than 10.000 material parameters. However, under certain simplifications, the material parameters can be greatly reduced; down to 3. With this simplified formulation, we develop a nonlocal operator method and apply it to several numerical examples. The numerical analysis shows that the high gradient solid theory exhibits a stiffer response compared to a ’conventional’ hyperelastic solid. The numerical tests also demonstrate the capability of the nonlocal operator method in solving higher-order physical problems.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Bauhaus-Universität Weimar
Tongji University
Ton Duc Thang University
Type: Article
Journal: Computer Methods in Applied Mechanics and Engineering
Volume: 384
ISSN: 0045-7825
Publication date: 01.10.2021
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Mechanics of Materials, Mechanical Engineering, General Physics and Astronomy, Computer Science Applications
Electronic version(s): https://doi.org/10.1016/j.cma.2021.113963 (Access: Closed)
https://hal.science/hal-03127040v2 (Access: Open)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{0e34c3114f634c14b8a4eaaf2ce1c7c5,
title = "A nonlocal operator method for finite deformation higher-order gradient elasticity",
abstract = "We present a general finite deformation higher-order gradient elasticity theory. The governing equations of the higher-order gradient solid along with boundary conditions of various orders are derived from a variational principle using integration by parts on the surface. The objectivity of the energy functional is achieved by carefully selecting the invariants under rigid-body transformation. The third-order gradient solid theory includes more than 10.000 material parameters. However, under certain simplifications, the material parameters can be greatly reduced; down to 3. With this simplified formulation, we develop a nonlocal operator method and apply it to several numerical examples. The numerical analysis shows that the high gradient solid theory exhibits a stiffer response compared to a {\textquoteright}conventional{\textquoteright} hyperelastic solid. The numerical tests also demonstrate the capability of the nonlocal operator method in solving higher-order physical problems.",
keywords = "Finite strain, Invariant, Nonlocal operator method, Second/third-gradient strain, Variational principle",
author = "Huilong Ren and Xiaoying Zhuang and Trung, {Nguyen Thoi} and Timon Rabczuk",
year = "2021",
month = oct,
day = "1",
doi = "10.1016/j.cma.2021.113963",
language = "English",
volume = "384",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier",
}