Abstract
The paper reviews the existing approaches to calculating the destruction of solids. The main attention is paid to algorithms using a unified approach to the calculation of deformation both for nondestructive and for the destroyed states of the material. The thermodynamic derivation of constitutive relations for solids with elastic, viscous and plastic properties accounting possible destruction is presented. Explicit and implicit non-matrix algorithms for calculating the evolution of deformation and fracture development are presented. Implicit schemes are implemented using iterations of the conjugate gradient method, with the calculation of each iteration exactly coinciding with the calculation of the time step for two-layer explicit schemes. Therefore the solution algorithms are very simple. The results of solving typical problems of destruction of solid deformable bodies for slow (quasistatic) and fast (dynamic) deformation processes are presented. Recommendations are given for modeling the processes of destruction and ensuring the reliability of numerical solutions.
Highlights
Numerical computations for destruction processes have been conducted since computers and numerical algorithms appeared, i.e. more than 50 years
It seems that the way to get out of this difficulty for fracture mechanics as well as for gas dynamics with multiple shock waves is to use through calculation methods with trapping narrow zones of large solution gradients
The destruction was mathematically expressed in the replacement of the usual bond of stresses and strains in the framework of the theory of elasticity and plasticity by a bond describing the behavior of the destroyed material
Summary
Numerical computations for destruction processes have been conducted since computers and numerical algorithms appeared, i.e. more than 50 years. The level of studying the processes of destruction is the study of the distribution of individual cracks to predict the possibility and time to continue the safe operation of structures in cases where the mentioned defects have already been found Such problems are solved by analytical and numerical-analytical methods of brittle fracture mechanics [1]. Because of this at the time step the operators of the elastic-plasticity problems lose the property of positive definiteness, so the boundary problem becomes incorrect according to Hadamard, the numerical solutions become physically meaningless and the calculation ends abnormally In continuum mechanics this phenomenon is known as a violation of the material resistance criterion according to the Drucker [4]. The accuracy of through calculation methods for damage modeling may be significantly improved by the use of adaptive movable computational grids in order to minimize approximation errors in areas of large gradients of solutions (reviews and descriptions are in [18 – 20])
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