Finite element analysis of crack problems for strain gradient material model

Abstract

A strain gradient material model is developed within the framework of infinitesimal deformation theory and implemented using a finite element simulation. Discussing the governing equations involving the second gradient terms, a complete form of the strain gradient material model is derived. The generalized variational principle, the so-called ‘Hu-Washizu principle’, is applied to the mixed-type finite element stiffness equation, in which the displacement, the strain, and the second gradient of displacement are variants. The stress–strain concentration is examined, and emphasis is placed on the explicit scale dependence of the objective domain. Stress relaxation behaviour near the crack tip is, in general, observed for small cracks, and the energy release rate calculated through the conventional J-integral is no longer path-independent for such scale-dependent crack problems.

Acknowledgements

This work started while the first author was at LMM, UPMC/CNRS under the support of the French government, which is acknowledged. Part of the current study is supported as grant-in-aid for scientific research (No. 14655044) by the Ministry of Education, Japan.