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

Sergei Khakalo*, Jarkko Niiranen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

56 Citations (Scopus)


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 H3 Sobolev space setting. These formulations can be reduced to two parameter first strain gradient-elastic problems of H2 variational forms. The implementations of the isogeometric C2- and C1-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.

Original languageEnglish
Pages (from-to)154-169
Number of pages16
JournalComputer-Aided Design
Publication statusPublished - 1 Jan 2017
MoE publication typeA1 Journal article-refereed


  • Abaqus
  • Eigenanalysis
  • First strain gradient elasticity
  • Isogeometric analysis
  • Second strain gradient elasticity
  • User elements


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