Variational continuum damage accumulation model in mechanics of materials


Аuthors

Mi L. 1*, Lurie S. A.2**

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. Institute of Applied Mechanics of Russian Academy of Science, IAM RAS, 32a, Leninskii av., Moscow, В-334, GSP-1, 119991, Russia

*e-mail: mishaleng@mail.ru
**e-mail: salurie@mail.ru

Abstract

This file contains an example of a new article layout for a magazine, with an updated color scheme. In traditional mechanics models, damage accumulation during the elastic stage is often neglected. However, it alters material properties and becomes significant during long-term operation, particularly in stress concentrators under static and cyclic loads. This study develops a variational model for damage accumulation in materials, which accounts for the mutual influence between stress levels and material property degradation effects. The model describes the evolution of damage and, overall, the physically nonlinear behavior of materials. The damage model is constructed by introducing a scalar damage parameter (scalar damage function), which reflects the relationship between the macroscopic degradation of mechanical properties and microscopic defect fields that physically cause damage growth. The variational model is introduced by expanding the list of arguments for the potential energy density, using the principle of local equilibrium, and assuming a linear law for the degradation of mechanical properties. As a result, a mathematical model of material deformation accounting for property degradation is formulated, including a system of governing equations and boundary conditions. The proposed model implicitly accounts for the degree of mechanical property degradation due to the stress state and, conversely, considers the influence of the material damage growth process (property degradation) on stress fields. In this work, an iterative approximate solution method, analogous to the method of elastic solutions, is proposed to solve the formulated nonlinear boundary value problem.
The damage growth model under loading is implemented in a one-dimensional test plane problem for a rod, where stress concentration is introduced through variable geometric parameters. Comparisons of solutions to specific test problems with the finite element method confirm the convergence of the proposed approach and its high accuracy. It is shown that significant degradation of the elastic modules occurs in the stress concentration zone, while the damage parameter changes nonlinearly under linearly varying external load amplitudes.


Keywords:

damage model; damage parameter; degradation model; variational method; mathematical model; iterative solution method.

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