Įtempimų būvio erdviškumo įtakos plokštelių su įpjova įtempimų intensyvumo koeficientui skaitinis tyrimas ; Numerical investigation of the three–dimensionality Influence on stress intensity factor in the plate with a notch

Vlad Patryshev

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Vladislav ŽARNOVSKIJ NUMERICAL INVESTIGATION OF THE THREE–DIMENSIONALITY INFLUENCE ON STRESS INTENSITY FACTOR IN THE PLATE WITH A NOTCH Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) 1190 Vilnius 2005 VILNIUS GEDIMINAS TECHNICAL UNIVERSITY Vladislav ŽARNOVSKIJ NUMERICAL INVESTIGATION OF THE THREE–DIMENSIONALITY INFLUENCE ON STRESS INTENSITY FACTOR IN THE PLATE WITH A NOTCH Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) Vilnius 2005 Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 2001–2005 Scientific Supervisor: Prof Dr Habil Rimantas KA ČIANAUSKAS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T) Consultant: Prof Dr Habil Mykolas DAUNYS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering – 09T) The Dissertation is being defended at the Council of Scientific Field of Mechanical Engineering at Vilnius Gediminas Technical University: Chairman: Prof Dr Habil Mindaugas LEONAVI ČIUS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T) Members: Prof Dr Habil Jonas BAREIŠIS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering – 09T) Prof Dr Habil Rimantas BELEVI ČIUS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical

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• Mechanical engineering
• Sif

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• Publié par: vilnius_gediminas_technical_university
• Publié le: 01 janvier 2006
• Nombre de lectures: 49

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    Vladislav ARNOVSKIJ     NUMERICAL INVESTIGATION OF THE THREEDIMENSIONALITY INFLUENCE ON STRESS INTENSITY FACTOR IN THE PLATE WITH A NOTCH     Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T)  

Vilnius

 

2005

 

1190

 

VILNIUS GEDIMINAS TECHNICAL UNIVERSITY           Vladislav ARNOVSKIJ     NUMERICAL INVESTIGATION OF THE THREEDIMENSIONALITY INFLUENCE ON STRESS INTENSITY FACTOR IN THE PLATE WITH A NOTCH     Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T)    

Vilnius

2005

Doctoral dissertation was prepared at Vilnius Gediminas Technical University in 20012005 Scientific Supervisor: Prof Dr Habil Rimantas KAČIANAUSKAS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering  09T) Consultant: Prof Dr Habil Mykolas DAUNYS(Kaunas University of Technology, Technological Sciences, Mechanical Engineering  09T) The Dissertation is being defended at the Council of Scientific Field of Mechanical Engineering at Vilnius Gediminas Technical University: Chairman: Prof Dr Habil Mindaugas LEONAVIČIUS Gediminas (Vilnius Technical University, Technological Sciences, Mechanical Engineering  09T) Members: Prof Dr Habil Jonas BAREIIS(Kaunas University of Technology, Technological Sciences, Mechanical Engineering  09T) Prof Dr Habil Rimantas BELEVIČIUS Gediminas Technical (Vilnius University, Technological Sciences, Mechanical Engineering  09T) Prof Dr Habil Mykolas DAUNYS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering  09T) Prof Dr Habil Bronislovas SPRUOGIS Gediminas Technical (Vilnius University, Technological Sciences, Mechanical Engineering  09T) Opponents: Prof Dr Habil Juozas ATKOČIŪNAS (Vilnius Gediminas Technical University, Technological Sciences, Civil Engineering  02T) Assoc Prof Dr Gintautas   DUNDULIS Energy Institute, (Lithuanian Technological Sciences, Mechanical Engineering  09T) The 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 2   p. m. on 8 December 2005. Address: Saulėtekio al. 11, LT-10223 Vilnius-40, Lithuania Tel.: [phone]; fax [phone]; e-mail [e-mail] The summary of the doctoral dissertation was distributed on 8 November 2005. A copy of the doctoral dissertation is available for review at the Library of Vilnius Gediminas Technical University (Saulėtekio al. 14, Vilnius, Lithuania). © Vladislav arnovskij, 2005

 

 

VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS         Vladislav ARNOVSKIJ     ĮTEMPIMŲBŪVIO ERDVIKUMOĮTAKOS PLOKTELIŲSUĮPJOVAĮTEMPIMŲ   INTENSYVUMO KOEFICIENTUI SKAITINIS TYRIMAS     Daktaro disertacijos santrauka Technologijos mokslai, mechanikos ininerija (09T)      

Vilnius

2005

Disertacija rengta 20012005 metais Vilniaus Gedimino technikos universitete. Mokslinis vadovas prof. habil. dr. Rimantas KAČIANAUSKAS Gedimino (Vilniaus technikos universitetas, technologijos mokslai, mechanikos ininerija  09T). Konsultantas  prof. habil. dr. Mykolas DAUNYS (Kauno technologijos universitetas, technologijos mokslai, mechanikos ininerija  09T).    Disertacija ginama Vilniaus Gedimino technikos universiteto Mechanikos ininerijos mokslo krypties taryboje:    Pirmininkas prof. habil. dr. Mindaugas LEONAVIČIUS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos ininerija  09T). Nariai: prof. habil. dr. Jonas BAREIIS(Kauno technologijos universitetas, technologijos mokslai, mechanikos ininerija  09T), prof. habil. dr. Rimantas BELEVIČIUS(Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos ininerija  09T), prof. habil. dr. Mykolas DAUNYS(Kauno technologijos universitetas, technologijos mokslai, mechanikos ininerija 09T),  prof. habil. dr. Bronislovas SPRUOGIS(Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos ininerija  09T). Oponentai: prof. habil. dr. Juozas ATKOČIŪNAS(Vilniaus Gedimino technikos universitetas, technologijos mokslai, statybos ininerija  02T), doc. dr. Gintautas DUNDULIS(Lietuvos energetikos institutas, technologijos mokslai, mechanikos ininerija  09T). Disertacija bus ginama vieame Mechanikos ininerijos mokslo krypties tarybos posė Vilniaus Gedimino technikos 8 d. gruodio val. 14dyje 2005 m. universiteto Senato posėdiųsalėje. Adresas: Saulėtekio al. 11, LT-10223 Vilnius-40, Lietuva. Tel.: [phone]; faksas [phone], el. patas [e-mail] Disertacijos santrauka isiuntinėta 2005 m. lapkričio 8 d. Disertaciją peri galimaūrėti Vilniaus Gedimino technikos universiteto bibliotekoje (Saulėtekio al. 14, Vilnius, Lietuva). VGTU leidyklos Technika 1190 mokslo literatūros knyga.  © Vladislav arnovskij, 2005

 

GENERAL CHARACTERISTIC OF THE DISSERTATION

  Research area and topicality of the work In order to determine fracture parameters of the body and fracture conditions, stress and strain fields in the crack zone should be precisely calculated, especially near the crack tip. These goals can be achieved by performing experiments which are very complicated and expensive. More efficient numerical finite element and boundary element methods yielding the reliable data faster than experiments are usually used for practical calculations. A great number of classical analytical, experimental and numerical investigations are restricted to solving two-dimensional problems by making different assumptions. The results may be uncertain when three-dimensional state is simplified to two-dimensional one. Therefore, it is very important to study the effect of this simplification, in other words, to investigate the influence of three- dimensionality.   The dissertation research belongs to the area of numerical fracture mechanics.   The main objectives To investigate the three-dimensionality influence, it is necessary: •   stress and strain state in theTo obtain numerically the three-dimensional deformed plate including plastic zone formation.   To develop a special three-dimensional FE meshing algorithm. In order to • obtain the reliable results, a proper FE mesh which would be handy for exact evaluation of the mechanical parameters at the fixed points through the plate thickness is needed. The algorithm must be applicable to 3D discretization of the plate (specimen) with the notch and compatible with the standard FE programs. •   To determine how the simulated three-dimensional stress and strain fields and fracture parameters vary through the plate thickness and to define their dependence on the plate thickness. •   To evaluate three-dimensionality of the plate state and the influence of the plate thickness on fracture parameters both at the elastic and elastic-plastic stage, i.e. to determine the conditions when the simplification of three- dimensional stress state can be justified and when the use of such simplified two-dimensional solutions is inadmissible.  

 

Research object and methods.For numerical investigation of the three- dimensionality effect, a single-edge notch bending plate similar to commonly use standard three-point bending plate with a imitated initial defect has been chosen. The plate with a straight V-shaped through-thickness notch is loaded according to the opening mode I and considered to be a homogeneous isotropic 3D body at a macro level. The proposed approach is based on the finite element simulation, while the influence of three-dimensional stress and strain fields on the plate fracture and the shape of the plastic deformation zone through the thickness of the notch front are investigated. Novelty and originality of research •   the present research lies in new approaches to evaluatingThe novelty of stress and strain state three-dimensionality and thickness influence on the deformed bodys mechanical state and fracture factors in different areas of the plate through its thickness. •   The original FE generation and computation techniques, combining 2D in- plane adaptive unstructured mesh with structured through-thickness mesh are developed and applied to solve 3D problems of fracture mechanics and investigate 3D structures with defects. •   The formation of a particular shape of the plastic zone in the 3D plate model is described with reference to classical fracture mechanics approaches. •   The finite element model based on the 3D mesh can be used for studying complicated 3D structures with different known defects. The structure of the work. The work is written in the Lithuanian language. It consists of the introduction, five chapters, conclusions and the list of 105 references. The thesis comprises 94 pages, including 54 illustrations. Introduction.The introduction presents a brief description of the research area and topicality of the thesis, as well as the main goals of investigation, research object, scientific novelty and originality of the work. Chapter 1. Problems of fracture mechanics and their analysis using FEM.   In this chapter analysis of the current situation, the main terms and problems of fracture mechanics as well as the most popular numerical methods, in particular, the finite element method (FEM) and its application to solving problems of fracture mechanics are presented. The main problems being solved in fracture mechanics are as follows: the analysis of stress-strain fields in the vicinity of the crack tip in elastic and plastic range and evaluation of proper fracture parameters.  

Most of investigations were restricted to the two limit cases, plane stress and plane strain, respectively. Strictly speaking, these 2D solutions are only applicable to plates with vanishing or infinite thickness, where the stress state can be defined as the plane stress or plane strain. Consequently, there is a significant level of empiricism in deciding whether a particular plate could be treated as "thin" or "thick". However, the state of deformation near the crack tip is always three- dimensional (3D), and therefore the meaning of the two-dimensional (2D) solutions has not been clearly understood yet. These accounts for the fact that many three-dimensional studies have been made to explain and quantify the 3D nature of fundamental two-dimensional solutions. One difficulty in the 3D analysis lies in the enormous computational expenses in addition to a complicated character of the stress and strain fields at the 3D crack front. However, fast development of computer facilities in terms of speed and capacity has provided the possibility to increase the accuracy of solutions in three-dimensional problems. Chapter 2. Three dimensional (3D) finite element (FEM) models of the plates.with a V-shaped straight through-thickness notch isThe plate considered as a three-dimensional body. The geometry of the plate is presented in Fig 1. The in-plane dimensions are taken to correspond a standard SENB three-point bending specimen, where the support span is nominally equal to four times the widthW(W =notch angle of the V-notch is20 mm). The β   (β= 60°), the notch depth isa(a= 6 mm). The thickness of the specimenBthe influence of which should be investigated, is assumed to be variable, while its relative variation is defined in the ran eB/W= 0.1÷3.0. y  U z   2.1W  60 0   4W a  1.2  W 2 1 W x . B   Fig 1.The geometry of the plate The three-point bending conditions for mode I are simulated. The external loading is given by the controlled quasi-static central displacementU. The

 

considered plate is homogeneous, isotropic and elastic (Poissons ratioν= 0.28 and Youngs modulusE=200 GPa).   To avoid human-made uncertainties in choosing the element shapes and sizes and well-known difficulties of stress field analysis in the vicinity at a singular notch tip point, the original 3D FE mesh generator and corresponding post-processing technique were developed for the purpose of modelling. The unique generator was initially developed using the ANSYS code. The 3D finite element model is created by an independent in-plane and through-thickness mesh generation technique. The developed technique involvesh-adaptive 2D- 3D finite element generator creating the unstructured triangular mesh with the refinement near the notch tip. The above generator provides automatic refinement of the mesh around singular points and contains some specific features controlling the orthogonality of the mesh to prescribed lines, which is desired in fracture considerations. The inplane mesh density was controlled along the notch front linex= 0, while the von Mises stress played the role of the mesh density indicator. The quality of the 2D discretization was evaluated by examining radial variation of stress componentσ xx vicinity of the notch and by comparing it with in the analytical solution. The through-thickness discretization comprises the subdivision of cross-section into independent layers, while the entire 3D model is generated by sweeping the fixed 2D mesh into thickness direction. Verification tests show that the numerical solution at the free boundary is highly sensitive to the through-thickness mesh density, while peak values may be obtained only by applying very fine local subdivision. The normalised final 3D FE model with 80 layers, presenting a quarter of the plate, is shown in Fig 2. It is used in further analysis.  

  Fig 2.   Quarter of the (FE) model   In summary, the suggested 3D FE model is normalised and transferred to mesh generator to model the plate of arbitrary thickness. It is implemented as pre- and postprocessor software compatible with standard FE codes.

 

Chapter 3. Investigation of the three-dimensionality of the plate. The effect of the three-dimensionality of the mechanical fields of the plate is studied mainly by considering normal out-of-plane components of stress and strain tensorσ zz  andε zz , respectively. Two independent variables stress-based − constraint factordand strain-based constraint factork,depending on Poisson's ratioν, are explored here as three-dimensionality indicators along the notch or the crack front:   d z= 1 νσ xx (z) σ zz ( +σ z) yy (z) , (1a) k(z) = 1− ν ν ε xx z ε zz + ε yy z . (1b) More generally, the constraint factors (CF) defined by eqs. (1a) and (1b) reflect how the stress and strain fields meet the plane strain or plane stress conditions and may be treated as the approximation bounds of the actual 3D fields. The limit value ofd is 0, in the case of the plane stress, and 1, for the plane strain, while the limit value ofkis 1, in the case of the plane stress, and 0, for the plane strain. Obviously, in-between values of CF 0≤d≤1 and 1≥k≥0 , respectively, indicate the 3D state as a transition state between the plane stress and plane strain. It should be noted that, in the above context, both definitions must rely on the consistency condition:    k+d= 1. (2) To capture the three-dimensional character of stress fields and the influence of plate thickness, the three dimensional plate with 8 different values of relative thickness B/W (B/W 0.1; 0.2; 0.4; 0.6; 0.8; 1.0; 1.5; 3.0) has been = considered. Generally, the CF are 3D functions and their presentation throughout the entire volume will be very complicated, tedious and redundant. The selected characteristic views of the constraint factors are presented in Figs. 3. The in-mid- plane views illustrate the existence of the plane strain zone in the vicinity of the notch tip. For a thin plate (Fig 3 a), this zone is highly concentrated and even hardly detectable by our relatively simple graphical model. The increase of the plate thickness causes the above zone to increase too, while the values of the constraint factor are approaching their limit valuesk   →0 andd   →1. The largest observable transition zone occurs for the medium-size plate (Fig 3b), while for the thick plate (Fig 3c) the constraint factors are characterised by very shallow surfaces. It should be noted that numerically obtained CF values are well suited for the notch front linexwhile in the remaining area, especially in thin and thick= 0,  

plates, irregular oscillations may occur. These results may be explained by the sensitivity of the latter to the in-plane discretization, where a desired quality of the applied FE mesh was maintained in the vicinity of the front line.  a) c)

1.0 1 0 . 0.8 0.8 0.6 0 6 0.009 . 0.4 0.008 0.007 008 0.09 40. 0.2 0.006 0 2  0.007 . 0.0 -0.006 -0.003  0.000 0.003 0.006 0. 0.0 600. 0 0 0 30 6 0.0 00 -0.006 -0.003 0.0   x,m  b) d)   y 1.0 0.8 0.6 x   0.4 0.009 0.008    0.2 0.007 0 - . 0 0 .006-0.0030.0000.0030.00 0 6 .006   z,m   Fig 3.In-plane (x-y) variation of the strain-based constraint factor k in the mid-plane (z = 0): a) thin plate (B/W= 0.1), b) medium thickness plate (B/W= 0.8), c) thick plate (B/W= 3.0), d) identification of argument area Modelling results for all plates of various thicknesses are summarised in Figs. 4. Here, the thickness variations of the constraint factors at the notch tip are plotted. The results are mapped into the unique scale represented by a relative thickness co-ordinatez/B. The CF values are calculated by using linear extrapolation technique of stresses from the neighbouring nodes to the notch tip. The character of the curves depends on the plate thickness and illustrates the transition of the plane strain zone via relative thickness to the plane stress state by approaching it at the free boundary. The character of the CF at the boundary layer is rather complicated and may be regarded as the intersection between two 3D dominant zones, namely, a transition from the plane strain to the plane stress zone and free boundary layer. This is even better illustrated in Fig 5 by checking the consistency condition defined by eq. (2).