A new fracture criterion for peridynamic and dual-horizon peridynamics
Jinhai ZHAO1, Hesheng TANG1,2(), Songtao XUE1
1. Research Institute of Structural Engineering and Disaster Reduction, Tongji University, Shanghai 200092, China 2. State Key Laboratory of Disaster Prevention in Civil Engineering, Tongji University, Shanghai 200092, China
A new fracture criterion based on the crack opening displacement for peridynamic (PD) and dual-horizon peridynamics (DH-PD) is proposed. When the relative deformation of the PD bond between the particles reaches the critical crack tip opening displacement of the fracture mechanics, we assume that the bond force vanishes. A new damage rule similar to the local damage rule in conventional PD is introduced to simulate fracture. The new formulation is developed for a linear elastic solid though the extension to nonlinear materials is straightforward. The performance of the new fracture criterion is demonstrated by four examples, i.e. a bilateral crack problem, double parallel crack, monoclinic crack and the double inclined crack. The results are compared to experimental data and the results obtained by other computational methods.
Amiri F, Millan D, Arroyo M, Silani M, Rabczuk T. Fourth order phase-field model for local max-ent approximants applied to crack propagation. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 254–275 https://doi.org/10.1016/j.cma.2016.02.011
2
Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109 https://doi.org/10.1016/j.tafmec.2013.12.002
3
Areias P, Rabczuk T, de Sá J C. A novel two-stage discrete crack method based on the screened Poisson equation and local mesh re_nement. Computational Mechanics, 2016, 58(6): 1003–1018 https://doi.org/10.1007/s00466-016-1328-5
4
Thai T Q, Rabczuk T, Bazilevs Y, Meschke G. A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 584–604 https://doi.org/10.1016/j.cma.2016.02.031
5
Budarapu P, Gracie R, Bordas S, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148 https://doi.org/10.1007/s00466-013-0952-6
6
Budarapu P R, Gracie R, Yang S W, Zhuang X, Rabczuk T. Efficient Coarse Graining in Multiscale Modeling of Fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143 https://doi.org/10.1016/j.tafmec.2013.12.004
7
Silani M, Talebi H, Hamouda A S, Rabczuk T. Nonlocal damage modelling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23 https://doi.org/10.1016/j.jocs.2015.11.007
8
Talebi H, Silani M, Rabczuk T. Concurrent Multiscale Modelling of Three Dimensional Crack and Dislocation Propagation. Advances in Engineering Software, 2015, 80: 82–92 https://doi.org/10.1016/j.advengsoft.2014.09.016
9
Silani M, Ziaei-Rad S, Talebi H, Rabczuk T. A Semi-Concurrent Multiscale Approach for Modeling Damage in Nanocomposites. Theoretical and Applied Fracture Mechanics, 2014, 74: 30–38 https://doi.org/10.1016/j.tafmec.2014.06.009
10
Talebi H, Silani M, Bordas S, Kerfriden P, Rabczuk T. A Computational Library for Multiscale Modelling of Material Failure. Computational Mechanics, 2014, 53(5): 1047–1071 https://doi.org/10.1007/s00466-013-0948-2
11
Talebi H, Silani M, Bordas S P A, Kerfriden P, Rabczuk T. Molecular Dynamics/XFEM Coupling by a Three-Dimensional Extended Bridging Domain with Applications to Dynamic Brittle Fracture. International Journal for Multiscale Computational Engineering, 2013, 11(6): 527–541 https://doi.org/10.1615/IntJMultCompEng.2013005838
12
Areias P, Rabczuk T, Msekh M. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 322–350 https://doi.org/10.1016/j.cma.2016.01.020
13
Areias P, Msekh M A, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143 https://doi.org/10.1016/j.engfracmech.2015.10.042
14
Areias P M A, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63 https://doi.org/10.1016/j.tafmec.2014.06.006
15
Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137 https://doi.org/10.1016/j.engfracmech.2013.06.006
16
Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947 https://doi.org/10.1007/s00466-013-0855-6
17
Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotation. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122 https://doi.org/10.1002/nme.4477
18
Dolbow J E. An extended finite element method with discontinuous enrichment for applied mechanics. Northwestern university, 1999.
19
Fries T P, Belytschko T. The extended/generalized finite element method: an overview of the method and its applications. International Journal for Numerical Methods in Engineering, 2010, 84(3): 253–304
20
Ghorashi S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for Fracture Analysis of Orthotropic Media. Computers & Structures, 2015, 147: 138–146 https://doi.org/10.1016/j.compstruc.2014.09.017
21
Nguyen-Thanh N, Valizadeh N, Nguyen M N, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291 https://doi.org/10.1016/j.cma.2014.08.025
22
Shi G H. Numerical manifold method and discontinuous deformation analysis. 1997.
23
Cai Y, Zhuang X, Zhu H. A generalized and efficient method for finite cover generation in the numerical manifold method. International Journal of Computational Methods, 2013, 10(05): 1350028 https://doi.org/10.1142/S021987621350028X
24
Belytschko T, Lu Y Y, Gu L. Element‐free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256 https://doi.org/10.1002/nme.1620370205
25
Liu W K, Jun S, Zhang Y F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 1995, 20(8‐9): 1081–1106 https://doi.org/10.1002/fld.1650200824
26
Li S, Liu W K. Meshfree and particle methods and their applications. Applied Mechanics Reviews, 2002, 55(1): 1–34 https://doi.org/10.1115/1.1431547
27
Nguyen V P, Rabczuk T, Bordas S, Duflot M. Meshless methods: a review and computer implementation aspects. Mathematics and Computers in Simulation, 2008, 79(3): 763–813 https://doi.org/10.1016/j.matcom.2008.01.003
28
Zhuang X, Augarde C E, Mathisen K M. Fracture modeling using meshless methods and level sets in 3D: framework and modeling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998 https://doi.org/10.1002/nme.4365
29
Amiri F, Anitescu C, Arroyo M, Bordas S, Rabczuk T. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 2014, 53(1): 45–57 https://doi.org/10.1007/s00466-013-0891-2
30
Rabczuk T, Areias P M A. A meshfree thin shell for arbitrary evolving cracks based on an external enrichment. CMES-Computer Modeling in Engineering and Sciences, 2006, 16(2): 115–130
31
Rabczuk T, Bordas S, Zi G. A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495 https://doi.org/10.1007/s00466-006-0122-1
32
Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for nonlinear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548 https://doi.org/10.1002/nme.2013
33
Rabczuk T, Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39(6): 743–760 https://doi.org/10.1007/s00466-006-0067-4
34
Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23-24): 1391–1411 https://doi.org/10.1016/j.compstruc.2008.08.010
35
Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically non-linear three dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758 https://doi.org/10.1016/j.engfracmech.2008.06.019
36
Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71
37
Bordas S, Rabczuk T, Zi G. Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by extrinsic discontinuous enrichment of meshfree methods without asymptotic enrichment. Engineering Fracture Mechanics, 2008, 75(5): 943–960 https://doi.org/10.1016/j.engfracmech.2007.05.010
38
Rabczuk T, Song J H, Belytschko T. Simulations of instability in dynamic fracture by the cracking particles method. Engineering Fracture Mechanics, 2009, 76(6): 730–741 https://doi.org/10.1016/j.engfracmech.2008.06.002
39
Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37-40): 2437–2455 https://doi.org/10.1016/j.cma.2010.03.031
40
Rabczuk T, Belytschko T. Cracking particles: a simpli_ed meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343 https://doi.org/10.1002/nme.1151
41
Rabczuk T, Belytschko T. A three dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29-30): 2777–2799 https://doi.org/10.1016/j.cma.2006.06.020
42
Erdogan F, Sih G C. On the crack extension in plates under plane loading and transverse shear. Journal of Basic Engineering, 1963, 85(4): 519–527 https://doi.org/10.1115/1.3656897
43
Sih G C. Strain-energy-density factor applied to mixed mode crack problems. International Journal of Fracture, 1974, 10(3): 305–321 https://doi.org/10.1007/BF00035493
44
Hussain M A, Pu S L, Underwood J. Strain energy release rate for a crack under combined mode I and mode II. In: Proceedings of the 1973 National Symposium on Fracture Mechanics, Part II. ASTM International, 1974.
45
Ayatollahi M R, Abbasi H. Prediction of fracture using a strain based mechanism of crack growth. Building Research Journal, 2001, 49: 167–180
46
Awaji H, Sato S. Combined Mode Fracture Toigliess measurement by the Disk Test. Journal of Engineering Materials and Technology, 1978, 100(2): 175 https://doi.org/10.1115/1.3443468
47
Shetty D K, Rosenfield A R, Duckworth W H. Mixed-mode fracture in biaxial stress state: application of the diametral-compression (Brazilian disk) test. Engineering Fracture Mechanics, 1987, 26(6): 825–840 https://doi.org/10.1016/0013-7944(87)90032-4
48
Chang S H, Lee C I, Jeon S. Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Engineering Geology, 2002, 66(1-2): 79–97 https://doi.org/10.1016/S0013-7952(02)00033-9
49
Aliha M R M, Ashtari R, Ayatollahi M R. Mode I and mode II fracture toughness testing for a coarse grain marble. Applied Mechanics and Materials, 2006, 5: 181–188.
50
Richard H A, Benitz K.A loading device for the creation of mixed mode in fracture mechanics. international Journal of Fracture, 1983, 22(2): R55–R58.
51
Arcan M, Hashin Z, Voloshin A. A method to produce uniform plane-stress states with applications to fiber-reinforced materials. Experimental Mechanics, 1978, 18(4): 141–146 https://doi.org/10.1007/BF02324146
52
Zipf R K Jr, Bieniawski Z T. Mixed mode testing for fracture toughness of coal based on critical-energy-density. In: The 27th US Symposium on Rock Mechanics (USRMS). American Rock Mechanics Association, 1986.
53
Silling S A. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 2000, 48(1): 175–209 https://doi.org/10.1016/S0022-5096(99)00029-0
Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-Horizon Peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476 https://doi.org/10.1002/nme.5257
56
Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782 https://doi.org/10.1016/j.cma.2016.12.031
57
Silling S A. Dynamic fracture modeling with a meshfree peridynamic code. Computational Fluid & Solid Mechamcs,. 2003, 641–644
58
Weckner O, Brunk G, Epton M A, Silling S A. Green’s functions in non-local three-dimensional linear elasticity In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society, 2009
59
Yu K, Xin X J, Lease K B. A new adaptive integration method for the peridynamic theory. Modelling and Simulation in Materials Science and Engineering, 2011, 19(4): 045003 https://doi.org/10.1088/0965-0393/19/4/045003
60
Kilic B. Peridynamic theory for progressive failure prediction in homogeneous and heterogeneous materials. ProQuest, 2008.
61
Silling S A. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 2000, 48(1): 175–209 https://doi.org/10.1016/S0022-5096(99)00029-0
62
Silling S A, Askari E. A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures, 2005, 83(17-18): 1526–1535 https://doi.org/10.1016/j.compstruc.2004.11.026
Silling S A, Askari E. A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures, 2005, 83(17-18): 1526–1535 https://doi.org/10.1016/j.compstruc.2004.11.026
65
Foster J T, Silling S A, Chen W. An energy based failure criterion for use with peridynamic states. International Journal for Multiscale Computational Engineering, 2011, 9(6): 675–688 https://doi.org/10.1615/IntJMultCompEng.2011002407
Ayatollahi M R, Aliha M R M. Analysis of a new specimen for mixed mode fracture tests on brittle materials. Engineering Fracture Mechanics, 2009, 76(11): 1563–1573 https://doi.org/10.1016/j.engfracmech.2009.02.016
68
Shen F, Zhang Q, Huang D. Damage and failure process of concrete structure under uni-axial tension based on peridynamics modeling. Chinese Journal of Computational Mechanics, 2013, 30: 79–83
69
Zhou X P, Shou Y D. Numerical Simulation of Failure of Rock-Like Material Subjected to Compressive Loads Using Improved Peridynamic Method. International Journal of Geomechanics, 2016: 04016086
70
Ren H, Zhuang X, Rabczuk T. A new peridynamic formulation with shear deformation for elastic solid. Journal of Micromechanics and Molecular Physics, 2016, 1(02): 1650009 https://doi.org/10.1142/S2424913016500090
71
Oterkus E. Peridynamic theory for modeling three-dimensional damage growth in metallic and composite structures. University of Arizona, 2010.
72
Bilby B A, Cottrell A H, Swinden K H. The spread of plastic yield from a notch. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society, 1963, 272(1350): 304–314
73
Bilby B A, Cottrell A H, Smith E, Swinden K H. Plastic yielding from sharp notches. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society, 1964, 279(1376): 1–9