Viacheslav Balobanov

Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models

As a first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s strain gradient elasticity theory of form II. For both models, this leads to sixth-order boundary value problems with new types of boundary conditions that are given additional attributes singly and doubly, referring to a physically relevant distinguishing feature between free and prescribed curvature, respectively. Second, the variational formulations are analyzed with rigorous mathematical tools: the existence and uniqueness of weak solutions are established by proving continuity and ellipticity of the associated symmetric bilinear forms. This guarantees optimal convergence for conforming Galerkin discretization methods. Third, the variational analysis is extended to cover two other generalized beam models: another modification of the strain gradient elasticity theory and a modified version of the couple stress theory. A model comparison reveals essential differences and similarities in the physicality of these four closely related beam models: they demonstrate essentially two different kinds of parameter-dependent stiffening behavior, where one of these kinds (possessed by three models out of four) provides results in a very good agreement with the size effects of experimental tests. Finally, numerical results for isogeometric Galerkin discretizations with B-splines confirm the theoretical stability and convergence results. Influences of the gradient and thickness parameters connected to size effects, boundary layers and dispersion relations are studied thoroughly with a series of benchmark problems for statics and free vibrations. The size-dependency of the effective Young’s modulus is demonstrated for an auxetic cellular metamaterial ruled by bending-dominated deformation of cell struts.

Isogeometric Static Analysis of Gradient-Elastic Plane Strain/Stress Problems

In the present contribution, isogeometric methods are used to analyze the statics of the plane strain and plane stress problems based on the theory of strain gradient elasticity. The adopted strain gradient elasticity models, in particular, include only one length scale parameter enriching the classical strain energy expression and resulting in fourth order partial differential equations instead of the corresponding second order ones based on the classical elasticity. The problems are discretized by an isogeometric NURBS based \( C^1\) continuous Galerkin method which is implemented as a user subroutine into a commercial software Abaqus. Computational results for benchmark problems, a square plate in tension and a Lame problem, demonstrate the applicability of the method and verify the implementation.

Modeling chemical reaction front propagation by using an isogeometric analysis

Powered by TCPDF (www.tcpdf.org) This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Morozov, Alexander; Khakalo, Sergei; Balobanov, Viacheslav; Freidin, Alexander; Müller, Wolfgang H.; Niiranen, Jarkko

Isogeometric analysis of gradient-elastic 1D and 2D problems

In the present contribution, isogeometric methods are used to analyze the statics and dynamics of rods as well as plane strain and plane stress problems based on a simplified version of the form II of Mindlin’s strain gradient elasticity theory. The adopted strain gradient elasticity models, in particular, include only two length scale parameters enriching the classical energy expressions and resulting in fourth order partial differential equations instead of the corresponding second order ones based on the classical elasticity. The problems are discretized by an isogeometric non-uniform rational B-splines (NURBS) based \( C^{p-1} \) continuous Galerkin method. Computational results for benchmark problems demonstrate the applicability of the method and verify the implementation.

Strain gradient elasticity theories in lattice structure modelling

This contributions discusses the simulation of magnetothermal effects in superconducting magnets as used in particle accelerators. An iterative coupling scheme using reduced order models between a magnetothermal partial differential model and an electrical lumped-element circuit is demonstrated. The multiphysics, multirate and multiscale problem requires a consistent formulation and framework to tackle the challenging transient effects occurring at both system and device level.Micro Abstract Shell elements for slender structures based on a Reissner-Mindlin approach struggle in pure bending problems. The stiffness of such structures is overestimated due to the transversal shear locking effect. Here, an isogeometric Reissner-Mindlin shell element is presented, which uses adjusted control meshes for the displacements and rotations in order to create a conforming interpolation of the pure bending compatibility requirement. The method is tested for standard numerical examples.Powered by TCPDF (www.tcpdf.org) This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Niiranen, Jarkko; Khakalo, Sergei; Balobanov, Viacheslav

ISOGEOMETRIC GALERKIN METHODS FOR GRADIENT-ELASTIC BARS, BEAMS, MEMBRANES AND PLATES

Isogeometric Galerkin methods are used to analyse plate and beam bending problems as well as membrane and bar models based on Mindlin’s strain gradient elasticity theory for generalized continua. The current strain gradient models include higher-order displacement gradients combined with length scale parameters enriching the strain and kinetic energies of the classical elasticity and hence resulting in higher-order partial differential equations with corresponding non-standard boundary conditions. The problems are first formulated within appropriate higher-order Sobolev space settings and then discretized by utilizing Galerkin methods with isogeometric NURBS basis functions providing appropriate higher-order continuity properties. Example benchmark problems illustrate the convergence properties of the methods.