Yu. N. Deryugin, D. A. Shishkanov

National Research Mordovia State University (Saransk, Russian Federation)

Abstract. The article investigates the nonlocal method of peridynamics, which makes it possible to simulate the brittle fracture of a solid body without using spatial derivatives. The basic motion equation of a particle with a given volume is written in integral form. A model combining the key features of continuum mechanics and of the nonlocal method is considered. To determine the forces of pair interaction, the dependence of the Cauchy stress tensor on the rate-of strain tensor was used. This formulation correctly describes the behavior of the material during damage and allows to get rid of the limitations inherent to simple bond-based model and ordinary state-based model. The maximum value of the tensile stress is used as a criterion of fracture, which describes the process of nucleation and evolution of damage. To test the implemented model, tasks in a two-dimensional formulation were used. Using the example of the elastic problem about uniaxial tension of a thin rod, the convergence of the numerical solution is shown with a decrease of interaction horizon and an increase of particles number. The second task demonstrates the capabilities of the implemented model to describe the nucleation and evolution of a crack under uniaxial load on a plate with an initial horizontal defect.

Key Words: peridynamic, nonlocal interactions, interaction horizon, bond, fracture criterion, strain gradient tensor

For citation: Yu. N. Deryugin, D. A. Shishkanov. Continuum Model of Peridynamics for Brittle Fracture Problems. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 26:2(2024), 157–174. DOI: https://doi.org/10.15507/2079-6900.26.202402.157-174

Information about the authors:

Yuriy N. Deryugin, Dr.Sci. (Phys.-Math.), Professor of the Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ORCID: https://orcid.org/0000-0002-3955-775X, dyn1947@yandex.ru

Dmitry A. Shishkanov, Postgraduate Student of the Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ORCID: https://orcid.org/0000-0002-3063-4798, dima.shishkanov.96@mail.ru