Luận án: Phương pháp Phase Field bậc cao cho nứt vật liệu giòn, bán giòn

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¿U8FJNBS (FSNBOZ 4VQFSWJTPST 1SPG.BHE"CEFM8BIBC 1I% (IFOU6OJWFSTJUZ 1SPG)VOH/HVZFO9VBO 1I% )P$IJ.JOI$JUZ6OJWFSTJUZPG5FDIOPMPHZ 7JFUOBN ´3HUIHFWLRQLVLPSRVVLEOHMXVWVWULYHWRGR\RXUEHVt.µ Angela Watson Acknowledgments Support and assistance were invaluable throughout my dissertation writing. I would first like to acknowledge the Flemish Government funding the project under the financial support of the VLIR-UOS TEAM Project (VN2017TEA454A103). This project could not have reached its goal without this funding. In addition, I am very grateful to the H2020 MSCA- RISE-project BESTOFRAC (734370), which has afforded me the opportunity to visit research institutions in order to further my scientific knowledge.

I would like to give my special regards to Professor Magd Abdel Wahab. It is greatly appreciated that his excellent conceptualization and advice helped lead to exceptional results in this project. In appreciation of Professor Nguyen Xuan Hung's valuable guidance and for the unconditional and enthusiastic support throughout my doctoral journey, I would like to express my deepest gratitude. I would like to express my gratitude to my brothers, Dr.

Le Thanh Cuong, Dr. Vu Huu Truong, and Dr. Tran Minh Trang, who accompanied me in completing this dissertation. To my colleagues at Laboratory Soete, I would like to say a heartfelt "thank you" for their friendship, assistance, and support.

During our group discussion, I received constructive suggestions, professional opinions, and opinions from others that helped improve my research quality. Finally, I would like to express my enormous gratitude to my wife and my family for everything that has been done for me. journey would not have been possible without their encouragement and unconditional supports. Khuong Duy Nguyen Ghent, 12 December 2021 Contents Acknowledgments.

v List of Figures. ix List of Tables. xvii Nomenclature & Abbreviations. xxvii List of Publications .2 Phase-field model .3 Heterogenous brittle failure .4 Quasi-brittle failure .1 NURBS basis functions .3 B-spline geometry refinement .4 Multi-patch problem .5 A NURBS-based finite element approach .3 Phase-field models .1 Phase-field approximation .2 A weak form of governing equations .3 A strong form of governing equations .4 Variational principles of phase-field formulations .4 Isogeometric analysis for the phase-field model .2 Scheme algorithms for a phase-field fracture.3 Phase-field formulations .5 A mathematical model of porous functionally graded materials 70 3.6 A local refinement mesh approach.7 Pre-existing crack implementation .2 Crack propagation in brittle material .2 Porous functionally graded materials.3 Crack propagation in quasi-brittle material .1 Asymmetric three-point bending concrete beam .2 L-shaped panel test.3 Mixed-mode failure of three-point bending concrete beam 149 4.

157 DISCUSSIONS AND SUGGESTIONS .3 Suggestions for Further Research. 161 List of Figures Figure 1. The breakwater construction at Ca Mau province of Vietnam. B-spline basis functions.

Quadratic B-spline curve with knot vectors Ξ = 0,0,0,1, 2,3, 4, 4,5,5,5. A comparison between NURBS and B-spline curves. A quadratic B-spline surface with two-knot spans. Variation of control points, basis functions, elements using knot insertion.

Variation of control points, basis functions using order elevation. Variation of control points, basis functions using k-refinement. Dimensions of an L-panel domain example. Coarse mesh of the L-panel: (a) single patch and (b) multi-patch descriptions.

Refinement mesh of the L-panel: (a) single patch and (b) multi- patch descriptions. Numerical integration in IGA. Two ways describe an internal crack: (a) sharp crack and (b) smeared crack. The phase-field approach of the one-dimensional fracture surface.

A flowchart for the monolithic scheme. A flowchart of the staggered scheme. Different predictions of crack propagation for the single-edge notched shear test with different formulations [9]. A material gradation of functionally graded material along the y-axis: (a) without and (b) with porosities.

Volume fraction distribution of material gradation. Material distributions of FGM without porosities along y-axis: (a) Láme's parameter, (b) shear modulus, and (c) critical energy release rate. Material distributions of FGM with and without porosities along y-axis: (a) Láme's parameter, (b) shear modulus, and (c) critical energy release rate. Graded elements for FGM.

Refinement mesh of the L-panel: (a) single patch and multi- patch descriptions with (b) global refinement and (c) local refinement. A conceptual illustration of nonconforming patches. Boundary conditions and geometry of a single edge notched specimen under (a) mode-I and (b) mode-II loading. Multi-patch of single edge notched problem under mode-I loading: (a) patch definition and (b) the cubic NURBS control points for the coarsest mesh.

A refinement mesh of single edge notched problem under mode- I loading with the effective size of h = l0 / 2. Reaction force versus displacement for different mesh. Crack propagation for single edge notched under mode-I loading with length-scale l0 = 0.0075 mm for (a) near fully separated plate, (b) fully separated plate in the case of second-order theory and (c) near fully separated plate, (d) fully separated plate in the case of fourth-order theory. Reaction force versus displacement of single edge notched problem under mode-I loading in two cases of phase-field theories.

The mesh of single edge notched under mode-II loading: (a) the coarsest mesh and patch numbering and (b) the refinement mesh. Reaction force versus displacement for a single edge notched problem under mode-II loading in two cases of phase-field theories. A crack propagation for single edge notched problem under pure shear loading with length-scale l0 = 0.0075 mm for (a) the second-order and (b) the fourth-order phase-field theories. Symmetric three point bending specimen: (a) boundary conditions and geometry, (b) coarsest mesh and patch numbering, and (c) refined mesh.

Reaction force versus displacement curves of symmetric three- point bending in two cases of phase-field theories. A crack propagation for symmetric three-point bending with the fourth-order formulation of the phase-field model at displacement u = 0. The asymmetric double notched tensile problem: (a) boundary conditions and geometry [68], (b) coarsest mesh and patch definition and (c) refined mesh. Reaction force versus displacement curves of the asymmetric double notched tensile problem in two cases of phase-field theories.

Double-crack propagation for an asymmetric double notched tensile specimen corresponds to the fourth-order phase-field theory at the (a) initial and (b) ending step. The notched plate with holes: (a) boundary conditions and geometry, and the experimentally crack patterns: (b) fractured sample and (c) the observed crack path from Ref. The mesh of notched plate with holes example: (a) the coarsest mesh and patch numbering and (b) the refined mesh. A notched plate example with holes: (a) the initial crack, (b) the propagated crack to the hole, (c) new appearance crack and (d) separated plate.

Reaction force versus displacement curves of the notched plate with hole example in two cases of phase-field theories. The mesh of asymmetric notched three-point bending problem: (a) boundary conditions and geometry, (b) the coarsest mesh and patch definition, and (c) refined mesh. Reaction force versus displacement results of the asymmetrically three-point bending test in two cases of phase-field theories. The asymmetrically three-point bending test: (a) crack path corresponds to the fourth-order formulation of the phase-field model, (b)experimentally observed crack pattern [122].

Boundary conditions and geometry of a single tension edge notched plate. The multi-patch mesh of single edge notched problem under mode-I loading: (a) the coarsest mesh, (b) the global, and (c) local refinement mesh with the effective size element h = l0 with length-scale l0 = 0. Reaction force versus displacement. Crack paths with cubic B-spline elements with the effective size h = l0 / 2 , different length-scale parameters l0 , Y1 rule of the mixture, and the power-law exponent parameter n = 1.

Reaction force versus displacement for various power-law indexes. Reaction force versus displacement for various power-law indexes with porous parameters. The experimentally fractured patterns of the three-point bending beam from Ref. A three-point bending example: (a) boundary conditions and geometry (dimension in (mm)) and (b) the refined mesh.

Elastic modulus and fracture toughness along with the height of the FGM three-point bending beam. The present numerical crack patterns of the three-point bending plate: (a) FGM, (b) non-FGM. The three-point bending plate's crack paths: (a) FGM and (b) non-FGM without porosity. The effect of porosity parameters on the crack patterns of the FGM three-point bending plate.

The crack patterns of an asymmetrically three-point bending beam with holes: (a) an experimental result from Bittencourt [122] and (b) a numerical result from Molnár [68]. An asymmetrically three-point bending beam with holes test. The numerical crack patterns of an asymmetrically three-point bending beam with holes: (a) homogeneous and (b) graded beam without porosity. Material properties of an asymmetrically three-point bending porous FG beam with holes.

The effects of porous parameters on the crack patterns of an asymmetrically three-point bending FG beam with holes. Asymmetric three-point bending concrete beam. Reaction force versus displacement curves of the concrete beam. L-shaped panel test: (a) geometry (unit of length: mm), boundary conditions, and (b) local refinement mesh.

Reaction force versus displacement curves of the L-shaped panel test. L-shaped panel test: (a) geometry (unit of length: mm), boundary conditions, and (b) local refinement mesh. The geometry and boundary conditions of mixed-mode concrete beam test (unit of length: mm). Reaction force versus CMOD curves of the mixed-mode concrete beam.

The predicted crack patterns of the mixed-mode concrete beam compared with an experimental result from Ref. 154 List of Tables Table 3-1. Set of control points of a quadratic B-spline curve. The coordination and weights of a NURBS curve.

Set of control points of a quadratic B-spline surface. Set of control points of a linear B-spline surface. Computational time and number of DOFs in the different cases of order B-spline elements. Computational time and DOF number in four cases of linear B- spline elements.

Computational time and DOF number in the different cases of higher-order B-spline elements with local refinement mesh. Material properties of NiCr and ZrO2. DOF number and computational time of the multiple meshes of B-spline elements. Damage profile of the multiple meshes of B-spline elements.

144 Nomenclature & Abbreviations FEM : Finite element method. IGA : Isogeometric analysis NURBS : Non-Uniform Rational B-splines FGMs : Functionally graded materials DOFs : Degree of freedoms CZM : Cohesive zone model 3D : Three-dimensional VUKIMS : Virtual Uncommon-Knot-Inserted Master-Slave C : Critical energy release density µ : Shear modulus λ : Lamé’s first parameter  : Phase field variable u : Displacement variables ε : Strain tensor σ : Stress tensor g ( ) : Stress degradation function c : Scaling number  ( ) : Geometric crack function l0 : Length-scale number + : History-field variable Summary The primary purpose of this PhD dissertation is to develop an efficient computational method based on a combination of a phase-field theory and a NURBS-based isogeometric approach (IGA) for crack propagation in a domain composed of brittle and quasi-brittle materials. The work presented in this dissertation is a part of a research project, namely "An innovative solution to protect Vietnamese coastal riverbanks from floods and erosion", to protect the coastline of Vietnam. The Flemish Government funds the project under the financial support of the VLIR-UOS TEAM Project (VN2017TEA454A103).

The crucial task of the project is to investigate the applications and developments of advanced numerical methods to predict the bearing capacity of these structures subjected to ocean wave loading and coastal subsidence. This dissertation contributes to developing advanced numerical methods to predict crack propagation in a concrete structure. Based on the behaviors of the materials, ceramic and glass are classified as brittle materials, while a concrete-like material is considered a quasi-brittle material. In addition, the phase-field theory recently became the most popular approach in modelling crack propagation in solid mechanics.

Its main idea is to use a scalar auxiliary variable, namely a phase-field variable, to model a discontinuous zone in a continuous domain.