# The J-contour integral in peridynamics via displacements

Authors
• 1 Luleå University of Technology, Division of Operation and Maintenance Engineering, Luleå, Sweden , Luleå (Sweden)
• 2 Luleå University of Technology, Division of Mechanics of Solid Materials, Luleå, Sweden , Luleå (Sweden)
Type
Published Article
Journal
International Journal of Fracture
Publisher
Springer Netherlands
Publication Date
Feb 21, 2019
Volume
216
Issue
2
Pages
173–183
Identifiers
DOI: 10.1007/s10704-019-00351-3
Source
Springer Nature
Keywords
Peridynamics is a nonlocal formulation of solid mechanics capable of unguided modelling of crack initiation, propagation and fracture. Peridynamics is based upon integral equations, thereby avoiding spatial derivatives, which are not defined at discontinuities, such as crack surfaces. Rice’s J-contour integral is a firmly established expression in classic continuum solid mechanics, used as a fracture characterizing parameter for both linear and nonlinear elastic materials. A corresponding nonlocal J-integral has previously been derived for peridynamic modelling, which is based on the calculation of a set of displacement derivatives and force interactions associated with the contour of the integral. In this paper, we present an alternative calculation of the classical linear elastic J-integral for use in peridynamics, by writing Rice’s J-integral as a function entirely of displacement derivatives. The accuracy of the proposed J-integral on displacement formulation is investigated by applying it to the exact analytical displacement solution of an infinite specimen with a central crack and comparing the exact analytical expression of its J-integral KI2/E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_I^2/E$$\end{document}. Further comparison with a well-known peridynamic crack problem shows very good agreement. The suggested method is computationally efficient and further allows testing of the accuracy of a peridynamic model as such.