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.

This article is devoted to the prediction of the service life of selected structural materials under simulated operating conditions. Special attention is paid to the so-called representative volume element, which characterizes the damage behaviour, since it includes a critical number of microdefects. The overall damage prediction is based on the energy approach, and the development of damage comes from the traction separation laws; the shape of the damage varies for different materials. The calculations were performed using the extended finite element method (XFEM), where several minor modifications were made. This method has been successfully used in many areas of engineering sciences for research, simulation, and prediction of the behaviour of structures. XFEM reformulates the continuous boundary and initial value problems into similar variational forms instead of using the classical forms of differential equations. The simulation of fracture and damage phenomena is presented for two different materials: austenitic steel with a pronounced grain structure under creep (viscous) loading conditions and cement pasta reinforced with metal fibres under conditions of predominantly static loading.

The present study utilises the extended finite element method (XFEM) to model fibre-reinforced composites, with a focus on crack initiation and propagation. Silicon nitride-based ceramics were selected as a model material; they represent a broad class of short fibre ceramics and have received a lot of attention in recent decades. Some peculiarities when using the XFEM, including its selected modifications, are discussed in response to applied external stresses, mainly in the viscoelastic range. Promising approaches are recommended, which lead to a more accurate description of these materials under operating conditions, focusing on the correct calculation of the macroscopic stress ahead of the propagating crack front. The authors draw on years of experience with the material and investigate the possible improvements and modifications to the XFEM.

Computational modelling of quasi-brittle fracture in cement-based composites needs to cover both i) the damage caused by micro-fractured zones, referring to some nonlocal strain-stress relations, respecting quite different behaviour of such composites in tension and compression, and ii) the initiation and propagation of macroscopic cracks, exploiting the cohesive zone model, handled by some modification of the finite element technique, together with the discretization in time. A fundamental issue for such model is the introduction of a traction-separation (stress-displacement) relationship. This contribution pays particular attention to the design, identification and estimation of material parameters for the traction separation law suitable for predicting the deformation behaviour of samples of materials and structures.

Computational prediction of damage in cement-based composites, as steel fibre reinforced ones, under mechanical, thermal, etc. loads, manifested as creation of micro-fractured zones, followed by potential initiation and evolution of macroscopic cracks, is a rather delicatematter, due to the necessity of bridging between micro- and macro-scales. This short paper presents a relatively simple approach, using certain extension of the finite element technique, open to possible generalizations. Such model admits proper verification of its existence andconvergence results, from the physical and mathematical formulation up to software implementation of relevant algorithms. Its practical applicability is documented on computational examples.