Introduce students to the possibilities of numerical methods application to problems of fracture mechanics. Introducing students to the application of finite element method in the analysis of nonlinear problems. Understanding and studying the problems of coupled external loads on welded structures. The development of an independent and practical work using licensed software.

By attending this course the student will master advanced application of finite element method, especially in the field of welding and welded structures. The importance of the application of computational fracture mechanics to structures when there are already noted one or more of the initial cracks. Students are trained to use computational methods to determine whether the stress fields on the constructions will lead to further growth of the crack, and whether crack will be stable or unstable, and based on that can determine the remaining life of the structure. Theoretical considerations, computational exercises and work with the licensed software, will allow students to synergize the previously acquired knowledge of mathematics, mechanics, structures integrity and mechanical materials, and apply this knowledge in engineering practice.

Elastic and elastic-plastic fracture mechanics. Fracture mechanics parameters. Stress intensity factor, crack tip opening, J integral. Application of fracture mechanics in structural integrity. Solving nonlinear problems using the FEM; types of nonlinearities, a review. Introduction to nonlinearity of the materials, the basics of the theory of plasticity. Presentation of the different criteria of plastic flow of materials in the FEM. The influence of building up the material. The influence of material anisotropy. The problem of heterogeneous materials - application on the welded joints. Problems of the material porosity. Viscoplasticity. Algorithms for solving nonlinear problems; incremental - iterative procedures. Nonlinearity of geometry; analysis of large deformations. Viscoelasticity. Nonlinear boundary conditions: solution for contact problems using formulation of FEM.Application of FEM in fracture mechanics and failure. Singular FE. Calculations of J-integrals in the FEM. Crack growth, techniques of node release. Determination of stress intensity factors using numerical methods. Adaptive finite element meshes and their application in the analysis of stress concentration. Numerical analysis in the local approach. The extended finite element method.

Determination of fracture mechanics parameters in elastic and elastic-plastic field. Experimental, numerical and analytical methods.Application of various algorithms in solving nonlinear problems; the accuracy and convergence of the solutions. Examples of FEM formulation of nonlinearities of geometry. Developments of FEM contact models. FEM formulation of dynamic and impact loadings. Post-processing. Techniques of introducing residual stresses - application on different welding procedures. FEM solutions in assessing fracture integrity of the weld. Examples of calculating J-integral for welded joints. Numerical determination of stress intensity factors in the real structure. Numerical simulation of crack propagation using XFEM.

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[1] Written lessons from lectures (handouts) [2] Kojic M., Computational Prrocedures in Inelastic Analysis of Solids and Structures, Kragujevac, 1997. [3] Sekulović M., The finite element method, Građevinska knjiga, Beograd, 1988. [4] A. Sedmak, Use of the fracture mechanics on the structure integrity assessment, Faculty of Mechanical Engineering, Belgrade, 2003. [5] G Jovicic., Zivkovic M., S Vulović., Computational fracture mechanics and fatigue, Faculty of Mechanical Engineering, Kragujevac, 2011. [6] Markо. P. Rakin, Local access to a ductile fracture of metallic materials. TMF, Belgrade, 2009. [7] S. Sedmak, A. Sedmak, Experimental and numerical methods of fracture mechanics in structural integrity assesment, TMF, Belgrade, 2000.

Total assigned hours: 65

New material: 45
Elaboration and examples (recapitulation): 5

Auditory exercises: 0
Laboratory exercises: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 0
Review and grading of seminar papers: 10
Review and grading of the project: 0
Test: 0
Test: 0
Final exam: 5

Activity during lectures: 10
Test/test: 20
Laboratory practice: 20
Calculation tasks: 0
Seminar paper: 20
Project: 0
Final exam: 30
Requirement for taking the exam (required number of points): 40

Kojic M., Computational Prrocedures in Inelastic Analysis of Solids and Structures, Kragujevac, 1997.; Reddy J. N.,An Introduction to the Finite Element Method, McGraw-Hill: New York, 2005.; Mohammadi S., Extended finite element method for fracture analysis of structure, Blackwell Publishing Ltd., Oxford, UK, 2008.; O. C. Zienkiewicz, R. L. Taylor and J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Sixth Edition, 2005; T.L. Anderson, Fracture Mechanics: Fundamentals and Applications 3rd ed. CRC Press, London, 2005.