Dvimačių struktūrų irimo modeliavimas naudojant prisitaikančiuosius baigtinių elementų tinklus ; Simulation of 2D structures fracture using adaptive finite element meshes
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2004
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Eugeniuš Stupak SIMULATION OF 2D STRUCTURES FRACTURE USING ADAPTIVE FINITE ELEMENT MESHES Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) 1042 Vilnius “Technika” 2004Vilnius Gediminas Technical University Eugeniuš Stupak SIMULATION OF 2D STRUCTURES FRACTURE USING ADAPTIVE FINITE ELEMENT MESHES Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) Vilnius “Technika” 2004 Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 2000-2004.
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Eugeniu Stupak SIMULATION OF 2D STRUCTURES FRACTURE USING ADAPTIVE FINITE ELEMENT MESHES Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) Vilnius Technika 2004
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Vilnius Gediminas Technical University Eugeniu Stupak SIMULATION OF 2D STRUCTURES FRACTURE USING ADAPTIVE FINITE ELEMENT MESHES Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T)
Vilnius Technika 2004
Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 2000-2004. Scientific supervisor: Prof Dr Habil Rimantas KAČ Gediminas TechnicalIANAUSKAS (Vilnius University, Technological Sciences, Mechanical Engineering 09T) Consultant: Prof Dr Habil Rimantas BELEVIČ Gediminas TechnicalIUS (Vilnius University, Technological Sciences, Mechanical Engineering 09T) Dissertation is being defended at the Council of Scientific Field of Mechanical Engineering at Vilnius Gediminas Technical University: Chairman: Prof Dr Habil Mindaugas LEONAVIČIUS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering 09T) Members: Prof Dr Habil Rimantas BELEVIČ Gediminas TechnicalIUS (Vilnius University,Technological Sciences, Mechanical Engineering 09T) Prof Dr Habil Mykolas DAUNYS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering 09T) Dr Gintautas DUNDULIS (Lithuanian Energy Institute, Technological Sciences, Mechanical Engineering 09T) Prof Dr Habil Gintaris KAKLAUSKAS (Vilnius Gediminas Technical University, Technological Sciences, Civil Engineering 02T) Opponents: Prof Dr Habil Juozas ATKOČIŪNAS (Vilnius Gediminas Technical University, Technological Sciences, Civil Engineering 02T) Prof Dr Habil Rimantas BARAUSKAS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering 09T) Dissertation will be defended at the public meeting of the Council of Scientific Field of Mechanical Engineering in the Senate Hall of Vilnius Gediminas Technical University, at 1 p.m. on October 29, 2004. Address: Saulėtekio al. 11, LT-10223 Vilnius-40, Lithuania Phone: (+370 5) 2744952, fax: (+370 5) 2700112 E-mail: doktor@adm.vtu.lt The summary of the doctoral dissertation was distributed on 29thof September 2004. A copy of the doctoral dissertation is available for review at the Library of Vilnius Gediminas Technical University (Saulėtekio al. 14, Vilnius). © Vilnius Gediminas Technical University, 2004
Vilniaus Gedimino technikos universitetas Eugeniu Stupak DVIMAČIŲ STRUKTŪRŲ IRIMO MODELIAVIMAS NAUDOJANT PRISITAIKANČIUOSIUS BAIGTINIŲ ELEMENTŲ TINKLUS Daktaro disertacijos santrauka Technologijos mokslai, mechanikos ininerija (09T)
Vilnius Technika 2004
Disertacija rengta 2000-2004 metais Vilniaus Gedimino technikos universitete. Mokslinis vadovas: prof. habil. dr. Rimantas KAČIANAUSKAS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos ininerija 09T) Konsultantas: prof. habil. dr. Rimantas BELEVIČIUS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos ininerija 09T) Disertacija ginama Vilniaus Gedimino technikos universiteto Mechanikos ininerijos mokslo krypties taryboje: Pirmininkas: prof. habil. dr. Mindaugas LEONAVIČIUS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos ininerija 09T) Nariai: prof. habil. dr. Rimantas BELEVIČ Gedimino technikosIUS (Vilniaus universitetas, technologijos mokslai, mechanikos ininerija 09T) prof. habil. dr. Mykolas DAUNYS (Kauno technologijos universitetas, technologijos mokslai, mechanikos ininerija 09T) dr. Gintautas DUNDULIS (Lietuvos energetikos institutas, technologijos mokslai, mechanikos ininerija 09T) prof. habil. dr. Gintaris KAKLAUSKAS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, statybos ininerija 02T) Oponentai: prof. habil. dr. Juozas ATKOČIŪ Gedimino technikosNAS (Vilniaus universitetas, technologijos mokslai, statybos ininerija 02T) prof. habil. dr. Rimantas BARAUSKAS (Kauno technologijos universitetas, technologijos mokslai, mechanikos ininerija 09T) Disertacija bus ginama vieame Mechanikos ininerijos mokslo krypties tarybos posėdyje 2004 m. spalio mėn. 29 d. 13 val. Vilniaus Gedimino technikos universiteto Senato posėdiųsalėje. Adresas: Saulėtekio al. 11, LT-10223 Vilnius-40, Lietuva Tel. (+370 5) 2744952, faksas (+370 5) 2700112 el.patas doktor@adm.vtu.lt Disertacijos santrauka isiuntinėta 2004 m. rugsėjo mėn. 29 d. Disertaciją peri galimaūrėti Vilniaus Gedimino technikos universiteto bibliotekoje (Saulėtekio al. 14, Vilnius). VGTU leidyklos Technika 1042 mokslo literatūros knyga. © Vilniaus Gedimino technikos universitetas, 2004
DESCRIPTION OF THE DOCTORAL DISSERTATION Research Area and Topicality of Investigation Fracture as an actual problem has been considered in mechanical, civil and other fields of engineering. It should be stated, however, that even using advanced production technologies it is impossible to produce any materials without defects, which are the main sources of fracture. The problems of fracture are solved in some complex manner combining theoretical knowledge, numerical models and experimental testing. Nowadays the advanced numerical methods and computational technologies are employed in fracture mechanics. The finite element method (FEM) is the most popular among them. The most characteristic aspect in solving the problems of fracture mechanics is determination of stress-strain state in the vicinity of crack (defect) tip with higher accuracy, so there is needed rather very fine finite element mesh (FE) near the tip when applying FEM. Besides, no mathematically approved recommendations about gradation of the FE mesh and its element sizes exist. The novel FE mesh generation approaches are based on adaptive unstructured FE meshes. From the theoretical point of view, adaptive FE meshes are more accurate than simple structured FE meshes. They allow using small elements at the crack tip or in the path of its possible propagation. Fracture analysis of the engineering structures and application of adaptive FE meshes for simulation of the fracture, expanding the possibilities of numerical experiments, are actual problems not only in the field of computational fracture mechanics but, generally, in the field of mechanical engineering. The Main Objective The main objective of current investigation is development of adaptive FE meshing strategy and its application for two-dimensional linear and non-linear analysis of mechanical structures with defects of different shape, also for simulation of crack propagation. Research Object 2D elastic and elastic-plastic isotropic non-homogeneous mechanical structures are investigated. The fixed shaped notches and the cracks are
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presented in these structures. The standard experimental specimens with defects, which are used in evaluation of fracture parameters, are investigated numerically in this work. Novelty and Originality The novelty of this research work is development and application of adaptive FE strategy for solving 2D problems of fracture mechanics. The original FE generation technique based on the adaptive approach is developed for the 2D structures with defects. The stress indicator employed in the adaptive analysis is able to capture high gradients of stress in the vicinity of the defect. Performance of this technique is checked against the defects of different geometry. Finally the above-mentioned technique exposes its suitability for solving fracture problems. Approbation Results of the research were presented at the following national and international scientific conferences: • Modelling of Crack Propagation using FEM Package ABAQUS Conference// International 2001 Mechanics Kaunas, , Lithuania, 5 April, 2001. • Calculation of Bending Specimen with Various Path Cracks using Adaptive Finite Element Meshes// Xth Lithuanian Conference on Computational Mechanics, Vilnius, Lithuania, 15 March, 2002. • FE Analysis for Crack Propagation in SENB SpecimenAdaptive // 1st CEACM Conference on Computational Mechanics, 15th International Conference CMM-2003 Computer Methods in Mechanics, Gliwice/Wisla, Poland, 3-6 June, 2003. • Adaptive Finite Element Meshes to Nonlinear Problem ofApplication of SENB Specimen// XIIth Conference on Computational Lithuanian Mechanics, Vilnius, Lithuania, 9 April, 2004. • Three-dimensional Correction of Two-dimensional Fracture Criteria using a Constraint Factor// XXI International Congress of Theoretical and Applied Mechanics, Warsaw, Poland, 15-21 August, 2004.
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Structure and Volume The work is written in the Lithuanian language. It consists of the introduction, five chapters, conclusions and the list of 127 references. The thesis comprises of 87 pages, including 65 illustrations. CONTENTS OF THE DISSERTATION 1. Introduction There is presented a brief description of the research area and topicality of the thesis, goals of investigation, research object, scientific novelty and originality in this part. 2. Problems of Fracture Mechanics and Calculations using FEM There are presented analyses of the current situation, including the main terms and problems of fracture mechanics in this chapter, the most popular numerical methods, especially the finite element method (FEM) and its application for solving problems of fracture mechanics. The main problems solved in fracture mechanics are as following: correct analysis of stress-strain fields in the vicinity of the crack tip in elastic and plastic range; implementation of proper fracture parameters and simulation of crack propagation. From the pioneering work of A. Griffith in 1920, the main methods used in fracture mechanics were theoretical (analytical) methods and alternatively experimental testing. Nowadays the problems of fracture mechanics are solved in complex manner combining both above-mentioned methods with some numerical models, which are dominating since the last few decades. The finite element method (FEM) is the most popular among them, both in the world and in Lithuania. The boundary element method (BEM) allows obtaining exact solutions only for problems of the linear elastic fracture mechanics, because it uses Greens functions. This method is more convenient for solving problems of dynamic fracture, but its use for solving nonlinear problems is still complicated. Rather very fine FE mesh is needed at the vicinity of crack tip it is one of the most characteristic aspects in solving the problems of fracture mechanics. This aspect can be rather easy implemented by using the novel FE mesh generation techniques based on adaptive unstructured FE meshes. The 7
above-mentioned meshes are more accurate than simple structured FE meshes, because it is simpler to control in self-automatic manner the size of small elements at the crack tip or in the path of its propagation. However, application of adaptive technique to fracture problems is still limited, because the special algorithms are needed, and investigations improving its suitability have to be done. 3. Adaptive FE Strategy using Stress Indicator This chapter describes a proposed adaptive FE strategy. Exist some kinds of adaptivity:h−adaptivity (when more FE are used),p−adaptivity (when ranges of elements shape functions are increased in order to have finer FE mesh),r−adaptivity, etc. In this work theh−adaptivity is implemented, because its implementation into FEM programs is rather easy. The simple stress based indicator is proposed for adaptive control. It describes regions of calculation domain where: 1) stresses are maximal and FE mesh size must be decreased, 2) stresses are rather small, so FE mesh size can be increased. Proposed stress indicator can be expressed as: =/max, (1) heremax some stress (in global maximal value of normal stress, axes direction, von Mises stress, etc.). Proposed stress based adaptive indicator is verified by numerical experiment omitting strong mathematical considerations. Easy implementation into programs may be considered as a significant advantage. The flowchart of proposedh−adaptive FE analysis approach is presented in Figure 1. At the first stage FEM model is prepared. Next the numerical analysis is done using some FE code. As result we can get the stress distribution set over the calculation domain. The accuracy of solution in the vicinity of stress concentrator tip is checked: σi−σi−1≤ηtol⋅σ*, (2) hereσiandσi-1 stress in concentrator tip at theiandi-1 analysis steps;ηtol user specified tolerance value;σ * nominal stress. If accuracy is achieved, then the adaptive FE analysis is finished, else a new FE mesh is generated. 8
Information about mathematical model
Generator Analysis using FE code Analysis of results
NoConvergence achieved? Yes End Fig. 1.A flowchart ofh−adaptive FE strategy Programming realization of this algorithm is rather simple. Advancing front technique is used for FE mesh generation. Firstly a coarse initial FE mesh covering the calculation domain is generated. This mesh generator needs the next information: about background mesh, covering domain; about geometry and boundary conditions; node distribution density function. The latter function describes the distances between nodes; it valuesδm are prescribed in the fixed nodes of the background mesh. The data about generated FE mesh, which will be used as a background mesh in the next step of the analysis, is stored. Self-automatic initial FE mesh generation approach, based on advancing front technique was implemented also. It allows taking into account some specific aspects of the solution domain: rather complex geometry, loading conditions, stress gradients, etc. In such a way, semi-optimal FE mesh is generated at the first iteration of adaptive FE analysis, and smaller numbers of iterations are needed to obtain the optimum FE mesh. The efficiency of the algorithm is verified by generating FE meshes in the domains of complex geometry with stress concentrators.
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4. Adaptive FE Meshes in Linear Fracture Mechanics Many calculations of 2D linear fracture mechanics problems have been performed in order to check the accuracy of the proposedh−adaptive FE strategy. The verification analyses were performed for the tensioned plate with central crack and single edge notched bending specimen (SENB) as shown in Figure 2. The tensioned plate is characterised by Youngs modulus E and Poissons ratio= 200 GPa,= 0.2 ;b length 20 mm,= 10 mm, thickness 1 mm,= ,1000 Pab/l= 10,d/l= 0.05, SENB specimen by Youngs modulusE and Poissons ratio= 200 GPa,= 0. thickness28 ; B= 15.8 mm, time-depended vertical displacementU= 0.04 mm. a
b) U(t)
600 1,2 S80 mm = L= 90 Fig. 2.of tensioned plate (a), SENB specimen (b)Geometry Triangle finite elements with an adaptive meshing for the simulation of plane stress problems are employed. Stress analysis and adaptive meshing were performed in the elastic range; the computations reported here are generated using the FE codes ALGOR, ANSYS. 10
