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.
383 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 b Faculty of Basic Sciences, University of Transport and Communications, Hanoi, Vietnam 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; ZnO2/NiCr; hybrid phase field model 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 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 and The phase field variable s varies continuously from for undamaged materials to for completely damaged materials The hybrid (isotropicanisotropic) phase field model for brittle fracture 384 Phuc Minh Pham et al is finally formulated as follows (Doan, 2016 and Ambati, 2015): (1) (2) 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 (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, s is the Laplacian of the phase field parameter, while H introduces a strain history field of maximum positive reference energy, , obtaining in (a) Schematic of beam and geometry parameter a loading process, in order to handle the irreversibility of the crack phase field evolution [25] H ( x, t ) : max [ 0,t ] ( ( x, )) (4) By only applying the phase field parameter to the tensile part of the elastic energy density function, , we thus prohibit crack (b) Deformation of beam with crack path propagation under compression, yielding: ( ): with tr( ) the tr ( ) : (tr ( ) 2 tr( ) (5) elastic constant , is the viscosity tr ( ) ) , and parameter Results and discussion 3.1 Crack path compared with experimental result Fig Schematic of mixed-mode experimental set up and calculation crack path 3.2 Crack step-over mechanism at Compliantto-Stiffer Interface Crack Propagation in ZnO2/NiCr Functionally Graded Materials 385 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 Fig Schematic of three points bending experimental set up with pre-crack located on compliant side Next, we consider a case when pre-crack located at stiffer side A calculation set up is same as in Fig with only a change of location of stiffer side and compliant side Calculation results are shown in Fig As crack propagates to stifferto-compliant interface, crack is arrested and branching at the interface (a) Loading displacement: u=0.14 mm (a) Loading displacement: u=0.26 mm (b) Loading displacement: u=0.15 mm Fig Crack step-over at compliant-to-stiffer interface Figure 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 This mechanism is called crack (b) Loading displacement: u=0.28 mm 386 Phuc Minh Pham et al 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 (c) Loading displacement: u=0.6 mm Fig Crack branching at stiffer-to-compliant interface 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-tocompliant interface, crack is arrested and branching 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 437 Bui Manh Cuong Trong Hieu Bui Bui Thi Thuy Tinh Quoc Bui Dinh Tri Bui 334 Le Ngoc Ly 355 73 Hoai T Le 10 Phuong Hoai Le 94 253, 259, 265, 383 Le Quang Vinh 284 39 Luu Xuan Hung 199 181 167 Bui Van Tuyen 269, 393 Luu Quynh Huong Mai Van Cao 217, 387 Atsushi Mitani Younho Cho 387 Dang The Ba 20, 33 Nguyen Dong Anh Nguyen Manh Cuong 94 146, 174 284 Chien Van Dang 80 Cuong T Nguyen 10 Dang Thuy Dong 429 Nguyen T Doan 107 Dang Thi Ha 409 Nguyen Ba Duan 376 Dang Van Hieu 247 Nguyen Dinh Duc 253, 259, 265, 292, 302, Dang Viet Hung 368 310, 383 Do Thi Ngoc Anh Do Nam 50, 58 Nguyen Van Duc 20, 33 181 Nguyen Dinh Dung 207 Do Tran Thang 99 Nguyen Minh Dung 383 Thom Van Do 259 Duc Hong Doan D.T Dam Nguyen Thi Hong Hanh 253, 259, 265, 383 Nguyen Duc Hieu 326 174 Nguyen Nhu Hieu 99, 146 Duong Ngoc Hai Duong Ngoc Hao 174 Nguyen Quang Huan Duong Thi Thanh Tam 417 Nguyen V Hoi Huong T.T Duong 99 Nguyen Quang Hoang 107, 114, 207 334, 393 107 10 Nguyen Van Khang 124, 167, 207 265 Nguyen Duy Khanh 292 Ki-Jong Gil 80 Nguyen Binh Khiem 383 Tien Vinh Ha 39 Nguyen Tien Khiem 181 Shinichi Hirai 94 Nguyen Minh Khoa 342 Sohichi Hirose 253 Khoa Viet Nguyen Kazuyoshi Fushinobu Van Manh Hoang Jong-Wook Kim 84, 138, 154, 161 217, 225, 233, 387 Nguyen Dinh Kien 269, 326, 334, 393, 422 80 Nguyen Trong Kien 189 189, 194 Nguyen Trung Kien 350 Le Thi Ha 269 Nguyen Phuong Lan 44 Le Thi Hue 417 Ngoc Linh Nguyen Le Thi Bich Nam 284 Nguyen Van Loi La Duc Viet 132, 154 401 438 Nguyen Thang Long Authors Index 20 Manh Thang Pham 84, 132, 138 154, 161 Nguyen Van Long (UCE) 409 Thu T Pham 10 Van Long Nguyen (UET) 132 Trung Xuan Pham 80 Van My Nguyen 241 Hai Dang Phan 387 Nguyen Sy Nam 124 Phan Thi Tra My 194 265, 383 Van Thang Phan 241 342, 355, 429 Phung Van Ngoc 33 Yong-Bin Shin 80 Nguyen Xuan Nguyen Nguyen Thi Phuong Nguyen Hoang Quan 26, 33 Quang Van Nguyen 225, 233 Nguyen Van Quyen 114 Nguyen Quang Thai Trung Dung Tran 277 Tran Van Dung 363 Tran Mai Lien 50 Tran Huu Quoc 409 Tran Quoc Quan 302 Nguyen Cao Thang 199 Nguyen Tat Thang Van Thang Nguyen 39 Thai Son Tran 161 Nguyen Xuan Thanh 376 Tran Ich Thinh 284 Nguyen Dang Thuy 363 Tran Thi Thom 326, 393 Nguyen Thi Minh Thuy 401 Tran Anh Thu 58 73 Tin Trung Tran 80 363 Tran Minh Tu Quang Tien Nguyen Nguyen Van Toan Minh Triet Nguyen 132, 138 Duy Trong Nguyen 39 Tuan A.H Nguyen 253 Ngoc Viet Nguyen 84, 132, 138, 154 Ninh Quang Hai Pham Hoang Anh 247 Anh V Truong 417 65 Truong Quoc Chien 167 Truong Thi Thuy Dung 417 Truong Thi Phuong Vu Thi Thuy Anh 310 44 Vu Lam Dong 422 Pham Duc Chinh 350 Vu Quoc Hien 284 Pham Ngoc Chung 146 Hoa C Vu 318 Pham Hong Cong 292 Duy Khanh Vu Pham Sy Dong 401 Vu Hoai Nam Pham Thi Viet Anh 368, 376 Tran Thanh Tuan 401, 409 Vu Dinh Quang 73 342, 355, 429 Pham Thi Thu Ha 50, 58 310 Huy Hoang Pham 94 Truong V Vu 65 Pham V B Ngoc 107 Vu Duc Vuong 114 Phuc Minh Pham 383 Soukha Yakoshi 277 Steering 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 ... Loading displacement: u=0.6 mm Fig Crack branching at stiffer-to-compliant interface Conclusions In this study, investigate the behavior and mechanism of a crack propagation at interface laminated... 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... 383 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,