scholarly journals Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase-field fracture by nonlocal operator method

2021 ◽  
Author(s):  
Huilong Ren ◽  
Xiaoying Zhuang ◽  
Erkan Oterkus ◽  
Hehua Zhu ◽  
Timon Rabczuk
Keyword(s):  
Thin Plate ◽  
Phase Field ◽  
Nonlocal Elasticity ◽  
Gradient Elasticity ◽  
Operator Method ◽  
Integral Form ◽  
Physical Models ◽  
Nonlocal Operator ◽  
Weighted Residual Method ◽  
Physical Problems

AbstractThe 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.

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

2021 ◽  
Vol 384 ◽  
pp. 113963
Author(s):  
Huilong Ren ◽  
Xiaoying Zhuang ◽  
Nguyen-Thoi Trung ◽  
Timon Rabczuk
Keyword(s):  
Finite Deformation ◽  
Gradient Elasticity ◽  
Operator Method ◽  
Higher Order ◽  
Nonlocal Operator

scholarly journals Implicit implementation of the nonlocal operator method: an open source code

2022 ◽  
Author(s):  
Yongzheng Zhang ◽  
Huilong Ren
Keyword(s):  
Open Source ◽  
Variational Principles ◽  
Source Code ◽  
Gradient Elasticity ◽  
Operator Method ◽  
Nonlinear Problems ◽  
Energy Functional ◽  
Benchmark Problems ◽  
Nonlocal Operator ◽  
Open Source Code

AbstractIn this paper, we present an open-source code for the first-order and higher-order nonlocal operator method (NOM) including a detailed description of the implementation. The NOM is based on so-called support, dual-support, nonlocal operators, and an operate energy functional ensuring stability. The nonlocal operator is a generalization of the conventional differential operators. Combined with the method of weighed residuals and variational principles, NOM establishes the residual and tangent stiffness matrix of operate energy functional through some simple matrix without the need of shape functions as in other classical computational methods such as FEM. NOM only requires the definition of the energy drastically simplifying its implementation. The implementation in this paper is focused on linear elastic solids for sake of conciseness through the NOM can handle more complex nonlinear problems. The NOM can be very flexible and efficient to solve partial differential equations (PDEs), it’s also quite easy for readers to use the NOM and extend it to solve other complicated physical phenomena described by one or a set of PDEs. Finally, we present some classical benchmark problems including the classical cantilever beam and plate-with-a-hole problem, and we also make an extension of this method to solve complicated problems including phase-field fracture modeling and gradient elasticity material.

On the carbon nanotube mass nanosensor by integral form of nonlocal elasticity

2019 ◽  
Vol 150 ◽  
pp. 445-457 ◽  
Author(s):  
Mahmood Fakher ◽  
Sasan Rahmanian ◽  
Shahrokh Hosseini-Hashemi
Keyword(s):  
Carbon Nanotube ◽  
Nonlocal Elasticity ◽  
Integral Form

scholarly journals Thermal buckling of functionally graded piezomagnetic micro- and nanobeams presenting the flexomagnetic effect

2021 ◽  
Author(s):  
Mohammad Malikan ◽  
Tomasz Wiczenbach ◽  
Victor A. Eremeyev
Keyword(s):  
Bifurcation Point ◽  
Heterogeneous Media ◽  
Thermal Environment ◽  
Shear Deformation Theory ◽  
Gradient Elasticity ◽  
Functionally Graded ◽  
Small Scale ◽  
Weighted Residual Method ◽  
Axial Stability ◽  
Graded Material

AbstractGalerkin weighted residual method (GWRM) is applied and implemented to address the axial stability and bifurcation point of a functionally graded piezomagnetic structure containing flexomagneticity in a thermal environment. The continuum specimen involves an exponential mass distributed in a heterogeneous media with a constant square cross section. The physical neutral plane is investigated to postulate functionally graded material (FGM) close to reality. Mathematical formulations concern the Timoshenko shear deformation theory. Small scale and atomic interactions are shaped as maintained by the nonlocal strain gradient elasticity approach. Since there is no bifurcation point for FGMs, whenever both boundary conditions are rotational and the neutral surface does not match the mid-plane, the clamp configuration is examined only. The fourth-order ordinary differential stability equations will be converted into the sets of algebraic ones utilizing the GWRM whose accuracy was proved before. After that, by simply solving the achieved polynomial constitutive relation, the parametric study can be started due to various predominant and overriding factors. It was found that the flexomagneticity is further visible if the ferric nanobeam is constructed by FGM technology. In addition to this, shear deformations are also efficacious to make the FM detectable.

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