ĐẠI HỌC QUỐC GIA TP HCM TRƯỜNG ĐẠI HỌC BÁCH HOA Topic How to Determine Displacement and Stress of Cracked Plate by Extended Finite Element Method (XFEM) NATIONAL CENTRAL UNIVERSITY Good afternoon Professor, good afternoon class Today, I present my topic about for short XFEM 1 OUTLINE INTRODUCTION 1 2 BASIS THEORY 3 CALCULATION EXAMPLES 4 CONCLUSIONS 5 REFERENCES 2 This is the outline of my presentation, including 5 parts Introduction, literature review, basis theory, calculation examples and[.]
Trang 1How to Determine Displacement and Stress of Cracked Plate
by Extended Finite Element Method
(XFEM)
NATIONAL CENTRAL UNIVERSITY
Trang 31 INTRODUCTION
Cracked structure
Trang 41 INTRODUCTION
How to calculate when cracks appear in structure?
Trang 51 INTRODUCTION
Some models for simulation of a crack problem
discrete crack
quarter point nodes
enriched nodes initial crack
enriched element
smeared crack discrete crack
discrete crack
crack tip slitting line originaluncracked
element
Trang 6Enriched element
Trang 10N H
Tip enrichment node
Edge enrichment nodeStandard node
Crack
Trang 112 BASIS THEORY
Enrichment functions in XFEM [1]
+ Heaviside function of edge element:
x With ( )x level set function [1]
+ Crack tip enrichment functions:
polar coordinates at the tip of the crack
O(x0,y0) A(xA,yA)
B(xB,yB)
B1(xCr2,yCr2)
A1(xCr1,yCr1) vết nứt
cạnh phần tử
a) xác định loại phần tử b) phần tử thân vết nứt c) phần tử đỉnh vết nứt
O(x0,y0) A(xA,yA)
B(xB,yB)
B1(xCr2,yCr2)
A1(xCr1,yCr1) vết nứt
cạnh phần tử a) xác định loại phần tử b) phần tử thân vết nứt c) phần tử đỉnh vết nứt
Trang 122 BASIS THEORY
Enrichment functions in XFEM [1]
+ Crack tip enrichment functions:
polar coordinates at the tip of the crack
O(x0,y0) A(xA,yA)
B(xB,yB)
B1(xCr2,yCr2)
A1(xCr1,yCr1) vết nứt
cạnh phần tử a) xác định loại phần tử b) phần tử thân vết nứt c) phần tử đỉnh vết nứt
Trang 132 BASIS THEORY
Enrichment functions in XFEM [1]
+ Heaviside function of edge element:
x With ( )x level set function [1]
O(x0,y0) A(xA,yA)
B(xB,yB)
B1(xCr2,yCr2)
A1(xCr1,yCr1) vết nứt
cạnh phần tử
a) xác định loại phần tử b) phần tử thân vết nứt c) phần tử đỉnh vết nứt
Trang 142 BASIS THEORY
Displacement equation in XFEM for element “e”
In which, the global stiffness matrix ( ) in XFEM is defined as follows:
u: standard element, a: Edge enrichment element, b: Tip enrichment element
(1)
(2)
(3)
(4) (5)
Trang 152 BASIS THEORY
If the plate has no crack => use finite element method (FEM) to calculate
Displacement equation in XFEM for element “ e ”
Force vector in XFEM:
Stiffness Matrix in XFEM:
(3)(4)
Trang 162 BASIS THEORY
If the plate has small crack => use Extended finite element method (XFEM) to calculate
Displacement equation in XFEM for element “ e ”
Force vector in XFEM:
Stiffness Matrix in XFEM:
u: standard element, b: Tip enrichment element
Trang 172 BASIS THEORY
If the plate has large crack => use Extended finite element method (XFEM) to calculate
Displacement equation in XFEM for element “ e ”
Force vector in XFEM:
Stiffness Matrix in XFEM:
(9)
(10)
u: standard element, a: Edge enrichment element, b: Tip enrichment element
ij ij ij e
Trang 18THEORY ELEMENT METHOD ELEMENT METHOD EXTENDED FINITE EXTENDED FINITE
Calculate Displacement
and Stress plate
Analyze the results
+
Using by Matlab program
Comparing the results with previous studies
2 BASIS THEORY
The calculation procedure
Trang 193 EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3,
σ = 1000MPa, a = cracked length, homogeneous material
Trang 203 EXAMPLE
σ = 1000
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa,
Poisson ν = 0.3, σ = 1000MPa, a = cracked length
Trang 223 EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, homogeneous
material, Poisson ν = 0.3, σ = 1000MPa, a = cracked length
σ = 1000
X
Y
Comparing max displacement between
XFEM and FEM
Trang 233 EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν =
0.3, σ = 1000MPa , a = crack length = 20mm
Y
Trang 243 EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν =
0.3, σ = 1000MPa , a = crack length = 20mm
Y
Trang 253 EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν =
0.3, σ = 1000MPa , a = crack length = 20mm
Y
Trang 273 EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν =
0.3, σ = 1000MPa , homogeneous material
Comparing max displacement between
XFEM and FEM
Y
Trang 314 CONCLUSIONS
The analysis result of cracked plate by XFEM is suitable with the
previous studies
As the cracks increase, the displacement increase.
The displacement and stress depend on the position of the cracked
plate
The greater the crack length, the more the displacement changes
XFEM is easy to simulate cracks without having to re-mesh the
structure
Trang 32 Propagation analysis of the craked plate using XFEM.
Develop the problem with many crack using XFEM
Trang 33[1] Moes N, et al., “A finite element method for crack growth without remeshing,” International Journal for numerical methods in Engineering, 46,
131-150, 1999.
[2] Soheil Mohammadi, eXtended Finite Eement Method, School of Civil
Engineering University of Tehran Tehran, Iran, 2008.
[3] Sukumar, et al., “Modeling quasi-static crack growth with the extended finite element method,” Part I: Computer implementation International
Journal of Solids and Structures, 40, 7513–7537, 2003.
[4] Sukumar, et al., “Extended finite element method and fast marching method for three – dimensional fatigue crack propagation,” Engineering
Fracture Mechanics, 70, 29 – 48 2003a.
[5] Dolbow, et al., “An extended finite element method for modeling crack growth with frictional contact,” Finite Elements in Analysis and Design, 36 (3)
Trang 34LOGO THANK YOU SO MUCH!