Citation: Alrayes, O.; Könke, C.; Hamdia, K.M. A Numerical Study of Crack Mixed Mode Model in Concrete Material Subjected to Cyclic Loading. Materials 2023, 16, 1916. https://doi.org/10.3390/ ma16051916 Academic Editors: Michele Bacciocchi, Angelo Marcello Tarantino, Raimondo Luciano and Carmelo Majorana Received: 5 February 2023 Revised: 19 February 2023 Accepted: 22 February 2023 Published: 25 February 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). materials Article A Numerical Study of Crack Mixed Mode Model in Concrete Material Subjected to Cyclic Loading Omar Alrayes 1 , Carsten Könke 1 and Khader M. Hamdia 2, * 1 Institute of Structural Mechanics, Bauhaus Weimar University, Marienstraße 15, 99423 Weimar, Germany 2 Institute of Continuum Mechanics, Leibniz Universität Hannover, 30167 Hannover, Germany * Correspondence: hamdia@ikm.uni-hannover.de Abstract: In quasi-brittle materials such as concrete, numerical methods are frequently used to simulate the crack propagation for monotonic loading. However, further research and action are required to better understand the fracture properties under cyclic loading. For this purpose, in this study, we present numerical simulations of mixed-mode crack propagation in concrete using the scaled boundary finite element method (SBFEM). The crack propagation is developed based on a cohesive crack approach combined with the thermodynamic framework of a constitutive concrete model. For validation, two benchmark crack-mode examples are modelled under monotonic and cyclic loading conditions. The numerical results are compared against the results from available publications. Our approach revealed good consistency compared to the test measurements from the literature. The damage accumulation parameter was the most influential variable on the load- displacement results. The proposed method can provide a further investigation of crack growth propagation and damage accumulation for cyclic loading within the SBFEM framework. Keywords: mixed mode crack propagation; cohesive zone method; cyclic loading; SBFEM 1. Introduction The application of fatigue fractures is essential in analysing the performance of con- crete structures. In fracture mechanics, concrete discontinuities also have the most sig- nificant investigation in the field of engineering [1,2]. To better understand the rapid failure of concrete structures under cyclic loading, a detailed procedure of fatigue crack propagation is required. The prediction of the direction of crack propagation and orienta- tion of quasi-brittle material as concrete is essential for the robust and reliable design of concrete structures. In concrete material, modelling of crack propagation and the numerical simulation of crack growth remains an outstanding issue and a critical topic of ongoing research. Primarily, the finite element technique is mainly used to simulate the crack behaviour numerically. Still, discontinuities in material simulation cannot be fully demonstrated, since the finite element method (FEM) is based on a continuum approach. The cracks are typically mapped by areas of high strain rates when using the smeared crack approach, as in Ref. [3]. The division of the crack opening into an equivalent element length of a finite element causes the effect of smeared crack formation. This method has a drawback in that it cannot accurately reflect the actual fracture pattern because the distortion and discontinuity in the displacement field are not mapped. Alternately, discontinuities are added at the element edges in the discrete crack approach [4]. This method is affiliated with a high numerical effort since each iteration step has a continuous re-meshing process. Based on the extensions of the conventional FEM, cohesive numerical approaches in modelling crack propagation have been developed to avoid this disadvantage [5–8]. Particular crack tip components were created to reduce the mesh quality essential for crack Materials 2023, 16, 1916. https://doi.org/10.3390/ma16051916 https://www.mdpi.com/journal/materials