Institute of Mathematics of the Czech Academy of Sciences

Institute of Mathematics of the Czech Academy of Sciences

Computational analysis of quasi-brittle fracture in cement-based and similar composites, supplied by various types of rod, fibre, etc. reinforcement, is crucial for the prediction of their load bearing ability and durability, but rather difficult because of the risk of initiation of zones of microscopic defects, followed by formation and propagation of a large number of macroscopic cracks. A reasonable and complete deterministic description of relevant physical processes is rarely available. Thus, due to significance of such materials in the design and construction of buildings, semi-heuristic computational models must be taken into consideration. These models generate mathematical problems, whose solvability is not transparent frequently, which limits the credibility of all results of ad hoc designed numerical simulations. In this short paper such phenomena are demonstrated on a simple model problem, covering both micro- and macro-cracking, with references to needful generalizations and more realistic computational settings.

In the present paper, an evaluation of the damage behaviour of quasi-brittle composites exposed to mechanical, thermal, and other loads is studied by means of viscoelastic and/or viscoplastic material models, applying some non-local regularisation techniques to the initiation and development of damages. The methods above are presented as a strong tool for a deeper understanding of material structures in miscellaneous engineering disciplines like civil, mechanical, and many others. Nevertheless, all of the software packages reflect certain compromises between the need for effective computational tools, with parameters obtained from inexpensive experiments, within the possibilities and the complexity of both physical and geometrical descriptions of structure deformation within processes. The article is devoted to the mathematical aspects regarding a considerably wide class of computational modelling problems, emphasising the following ones: (i) the existence and the uniqueness of solutions of engineering problems formulated in terms of the deterministic initial and boundary value problems of partial differential equations theory; (ii) the problems of convergence of computational algorithms applied to (i). Both aspects have numerous references to possible generalisations and investigations connected with open problems.

This paper is showing the results of a study focused on the occurrence of failures in heterogeneous structural materials, especially on the issue of modelling the formation and propagation of cracks. Attention is paid to the application of the finite element method to various types of materials in order to find critical parameters determining the behaviour of materials during the damage process. Application of damage mechanisms and approaches to modelling the origin of crack propagation by modifications in FEM systems are presented. Furthermore, some practical applications are tested. The main effort is devoted to composites and searching for procedures when using cohesion models for their more accurate modelling, especially near the stress concentrators, or before the crack tip. The use of the cohesive approach of elements that represent the physical nature of energy release in front of the crack front has proven to be promising not only in the direct use of cohesive elements, but also in combination with modified methods of standard finite elements. The authors of this article use their long-term experience in the given issue, which they consistently confront with developments in this area in recent years.

During the operation of structures, stress and deformation fields occur inside the materials used, which often ends in fatal damage of the entire structure. Therefore, the modelling of this damage, including the possible formation and growth of cracks, is at the forefront of numerical and applied mathematics. The finite element method (FEM) and its modification will allow us to predict the behaviour of these structural materials. Furthermore, some practical applications based on cohesive approach are tested. The main effort is devoted to composites with fibres and searching for procedures for their accurate modelling, mainly in the area where damage can be expected to occur. The use of the cohesive approach of elements that represent the physical nature of energy release in front of the crack front has proven to be promising not only in the direct use of cohesive elements, but also in combination with modified methods of standard finite elements.