TY  - JOUR
A1  - Ren, Huilong
A1  - Zhuang, Xiaoying
A1  - Oterkus, Erkan
A1  - Zhu, Hehua
A1  - Rabczuk, Timon
T1  - Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase-field fracture by nonlocal operator method
JF  - Engineering with Computers
N2  - The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture method. The nonlocal governing equations are expressed as an integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate.
KW  - Bruchmechanik
KW  - Elastizität
KW  - Peridynamik
KW  - energy form
KW  - weak form
KW  - peridynamics
KW  - variational principle
KW  - explicit time integration
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20211207-45388
UR  - https://link.springer.com/article/10.1007/s00366-021-01502-8
VL  - 2021
SP  - 1
EP  - 22
ER  - 
TY  - JOUR
A1  - Legatiuk, Dmitrii
A1  - Weisz-Patrault, Daniel
T1  - Coupling of Complex Function Theory and Finite Element Method for Crack Propagation Through Energetic Formulation: Conformal Mapping Approach and Reduction to a Riemann–Hilbert Problem
JF  - Computational Methods and Function Theory
N2  - In this paper we present a theoretical background for a coupled analytical–numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical–numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of complex function theory and couple it continuously with the finite element solution in the region far from the singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated through the minimization of the total energy composed of the potential energy and the dissipated energy based on the energy release rate. Within this setting, an analytical solution of a mixed boundary value problem based on complex analysis and conformal mapping techniques is presented in a circular region containing an arbitrary crack path. More precisely, the linear elastic problem is transformed into a Riemann–Hilbert problem in the unit disk for holomorphic functions. Utilising advantages of the analytical solution in the region near the crack tip, the total energy could be evaluated within short computation times for various crack kink angles and lengths leading to a potentially efficient way of computing the minimization procedure. To this end, the paper presents a general strategy of the new coupled approach for crack propagation modelling. Additionally, we also discuss obstacles in the way of practical realisation of this strategy.
KW  - Angewandte Mathematik
KW  - Finite-Elemente-Methode
KW  - Rissausbreitung
KW  - Modellierung
KW  - Bruchmechanik
KW  - fracture mechanics
KW  - crack propagation
KW  - coupling
KW  - energetic approach
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20210805-44763
UR  - https://link.springer.com/article/10.1007/s40315-021-00403-7
VL  - 2021
SP  - 1
EP  - 23
PB  - Springer
CY  - Heidelberg
ER  -