Sergei Khakalo

Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software

Abstract This article is devoted to isogeometric analysis of higher-order strain gradient elasticity by user element implementations within a commercial finite element software Abaqus. The sixth-order boundary value problems of four parameter second strain gradient-elastic bar and plane strain/stress models are formulated in a variational form within an H 3 Sobolev space setting. These formulations can be reduced to two parameter first strain gradient-elastic problems of H 2 variational forms. The implementations of the isogeometric C 2 - and C 1 -continuous Galerkin methods, for the second and first strain gradient elasticity, respectively, are verified by a series of benchmark problems. With the first benchmark problem, a clamped bar in static tension, the convergence properties of the method in the energy norm are shown to be optimal with respect to the NURBS order of the discretizations. For the second benchmark, a clamped bar in extensional free vibrations, the analytical frequencies are captured by the numerical results within the classical and the first strain gradient elasticity. With three examples for the plane stress/strain elasticity, the convergence properties are shown to be optimal, the stress fields of different models are compared to each other, and the differences between the eigenfrequencies and eigenmodes of the models are analyzed. The last example, the Kraus problem, analyses the stress concentration factors within the different models.

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.

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.

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.