A numerical investigation of mixedmode crack propagation in functionally graded ZnO2NiCr (FG) beams using a regularized variational formulation is presented. The simulation results were compared with mixedmode three points bending experimented by Jin. et al. (Eng. Frac. Mech. 76 (2009) 1800 1810). Our numerical investigation shows that the numerical scheme based on hybrid phase field model captures well the crack propagation behavior including initial kinking angle, crack path and loaddisplacement relationship. The effect of elastic and fracture toughness mismatch on crack path behavior is also analyzed.
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A Numerical Study of Mixed-mode Crack Propagation in ZnO2/NiCr Functionally Graded Materials by A Hybrid Phase-Field Method
Phuc Minh Phama , Duc Hong Doanb, Tinh Quoc Buic, Nguyen Xuan Nguyend,
Nguyen Minh Dunge, Nguyen Binh Khieme, Nguyen Dinh Ducb,e
a
Faculty of Basic Sciences, University of Transport and Communications, Hanoi, Vietnam
b
Advanced Materials and Structures Lab, University of Engineering and Technology, VNU - Hanoi
c
Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Japan
d Department of Mathematics, Mechanics and Informations - Hanoi University of Science,
VNU -Hanoi
e
Department of Mechano-Informatics, Graduate school of Information Science and Technology -
The University of Tokyo
d
Infrastructure Engineering Program, Vietnam-Japan University (VJU), VNU-Hanoi, Vietnam
Abstract
A numerical investigation of mixed-mode crack propagation in functionally graded ZnO2/NiCr (FG) beams using a regularized variational formulation is presented The simulation results were compared with mixed-mode three points bending experimented by Jin et al (Eng Frac Mech 76 (2009) 1800 1810) Our numerical investigation shows that the numerical scheme based on hybrid phase field model captures well the crack propagation behavior including initial kinking angle, crack path and load-displacement relationship The effect of elastic and fracture toughness mismatch on crack path behavior is also analyzed
Key Words: Functionally graded materials; ZnO 2 /NiCr; hybrid phase field model
1 Instructions
Functionally graded ZnO2/NiCr was fabricated
with structure of laminated ZnO2/NiCr with
different composite percentage of NiCr from 0%
to 50% (Jin, 2009) Therefore, the behaviors of
crack at interface of laminated ZnO2/NiCr are
dominated fracture mode This study aims at to
investigate the behaviors and mechanisms of a
crack propagation at interface laminated
ZnO2/NiCr by a hybrid phase field model The
effects of elastic and fracture toughness
mismatch on crack path behavior is also analyzed
2 Governing equations
In terms of phase field fracture modeling, the cracks, which can be regarded as internal discontinuities with respect to the macroscopic field, are essentially represented by a phase field
variable s bounded between 0 and 1 The phase field variable s varies continuously from 1 for undamaged materials to 0 for completely
damaged materials The hybrid (isotropic-anisotropic) phase field model for brittle fracture
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is finally formulated as follows (Doan, 2016 and
Ambati, 2015):
(1) (2)
(3)
where , are the stress and strain tensors,
respectively; is the displacement field, div is
the divergence operator, and the superposed dot
represents the partial differentiation with respect
to time In Eq (2), l stands for the length scale
parameter introduced to account for the width of
the crack Gc denotes the material fracture
toughness independent of the crack speed, sis
the Laplacian of the phase field parameter, while
H introduces a strain history field of maximum
positive reference energy, 0 , obtaining in
a loading process, in order to handle the
irreversibility of the crack phase field evolution
[25]
0t 0
H ( , ) : maxxt [ , ] ( ( , ))x (4)
By only applying the phase field parameter to
the tensile part of the elastic energy density
function, 0 , we thus prohibit crack
propagation under compression, yielding:
0
1
with the elastic constant 0 ,
1
2
tr tr tr , and is the viscosity
parameter
3 Results and discussion
3.1 Crack path compared with experimental
result
We start by considering a mixed-mode experimental test of beam conducted by Jin (2009) as shown in Fig 1(a) The beam is made
of ZnO2/NiCr with the material composite of 50%ZnO2/50%NiCr The crack path and deformation are shown in Fig 1(b) Our numerical result shows that the numerical scheme based on hybrid phase field model captures well the crack propagation behavior as initial kinking angle of 10
(a) Schematic of beam and geometry parameter
(b) Deformation of beam with crack path Fig 1 Schematic of mixed-mode experimental set up and calculation crack path
3.2 Crack step-over mechanism at Compliant-to-Stiffer Interface
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Fig 2 Schematic of three points bending
experimental set up with pre-crack located on
compliant side
(a) Loading displacement: u=0.14 mm
(b) Loading displacement: u=0.15 mm
Fig 3 Crack step-over at compliant-to-stiffer
interface
Figure 2 shows a schematic of three points
bending experimental set up with pre-crack with
length of 2.2 mm located on compliant side
Compiant and stiffer side are made of
ZnO2/NiCr with the material composite of
50%ZnO2/50%NiCr and ZnO2, respectively
Interestingly, the crack is generated firstly at
stiffer side than propagates to compliant side as
shown in Fig 3 This mechanism is called crack
step-over, which occurs when a crack propagates from compliant-to-stiffer interface It
is worth noting that, although crack step-over has been predicted in Leguillon (2013), it is first time visualized by numerical simulation in this work
3.3 Crack branching at Stiffer-to-Compliant Interface
Next, we consider a case when pre-crack located
at stiffer side A calculation set up is same as in Fig 2 with only a change of location of stiffer side and compliant side Calculation results are shown in Fig 4 As crack propagates to stiffer-to-compliant interface, crack is arrested and branching at the interface
(a) Loading displacement: u=0.26 mm
(b) Loading displacement: u=0.28 mm
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(c) Loading displacement: u=0.6 mm
Fig 4 Crack branching at stiffer-to-compliant
interface
4 Conclusions
In this study, investigate the behavior and
mechanism of a crack propagation at interface
laminated ZnO2/NiCr by a hybrid phase field
model Simulation results demonstrates the
crack step-over mechanism at interface of
compliant-to-stiffer At interface of
stiffer-to-compliant interface, crack is arrested and
branching
Acknowledgement: This research is funded by
Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2015.03 and ISPS-Japan The authors are grateful for this support
References
Ambati, M et al (2015) , A phase-field model for ductile fracture at finite strains and its experimental verification, Comput Mech 57 pp 149-267
Doan, HD et al (2016), Hybrid Phase Field Simulation of Dynamic Crack Propagation in
Functionally Graded Glass-Filled Epoxy, Composites Part B 99 pp 266-276
Leguillon, D et al (2013), The strengthening effect caused by an elastic contrast part I: the bimaterial
case Int J Fract 179 pp 157 167
Jin, X et al (2009), Experimental investigation of the mixed-mode crack propagation in ZrO2/NiCr
functionally graded materials, Engineering Fracture Mechanics 76 pp 1800 1810
Trang 10437
Tinh Quoc Bui 253, 259, 265, 383
Duc Hong Doan 253, 259, 265, 383
Van Manh Hoang 84, 138, 154, 161
Nguyen Dinh Duc 253, 259, 265, 292, 302,
310, 383
Nguyen Quang Hoang 107, 114, 207 Nguyen Quang Huan 334, 393
Nguyen Van Khang 124, 167, 207
Khoa Viet Nguyen 217, 225, 233, 387 Nguyen Dinh Kien 269, 326, 334, 393, 422
Ngoc Linh Nguyen 132, 154
Trang 11438 Authors Index
Nguyen Van Long (UCE) 409
Van Long Nguyen (UET) 132
Nguyen Xuan Nguyen 265, 383
Nguyen Thi Phuong 342, 355, 429
Quang Van Nguyen 225, 233
Minh Triet Nguyen 132, 138
Ngoc Viet Nguyen 84, 132, 138, 154
Manh Thang Pham 84, 132, 138 154, 161
Vu Hoai Nam 342, 355, 429
Trang 12Steering Committee
Nguyen Huu Duc, Nguyen Viet Ha, Duong Ngoc Hai, Nguyen Hoa Thinh
Program Committee
Chairman: Nguyen Dong Anh
Vice Chairman: Nguyen Dinh Duc, Dinh Van Manh
Dao Huy Bich, Le Van Canh, Truong Huu Chi, Doan Minh Chung, Dao Van Dung, Duc Pham (UK), Nguyen Van Diep, Dimitri V Georgievskii (Russia), Duong Ngoc Hai, Nguyen Xuan Hung, Cao The Huynh, Le Xuan Huynh, Seung Chul Jung (Korea), Kazuyoshi Fushinobu (Japan), Nguyen Van Khang, Nguyen Tien Khiem, Kim Chun-Gon (Korea), Nguyen Thi Viet Lien, Nguyen Cao Menh, Nguyen Van Pho, No-Cheol Park (Korea), Dinh Van Phong, Pham Hong Phuc, Do Sanh, Suong H Van (Canada), Pham Manh Thang, Truong Tich Thien, Tran Ich Thinh, Bui Dinh Tri, Nguyen Thoi Trung, Pham Anh Tuan, Tuan D Ngo (Australia), Pham Chi Vinh, Hui-Shen Shen (China)
Organising Committee
Chairman: Nguyen Dinh Duc
Vice Chairman: Nguyen Viet Khoa, Pham Manh Thang
Dang The Ba, Tran Mau Danh, Tran Thi Thu Ha, Pham Duy Hung, Seung Chul Jung (Korea), Nguyen Ha Nam, Bui Trung Ninh, Nguyen Hoang Quan, Nguyen Phuong Thai, Pham Minh Trien
Secretariat
Chairman: Dao Nhu Mai
Vu Thi Thuy Anh, Nguyen Ngoc Linh, Phan Thi Cam Ly, Hoang Van Manh, Tran Quoc Quan, Nguyen Cao Son, Nguyen Ngoc Viet, Tran Hai Yen