### Modelling of Crack Growth Using a New Fracture Criteria Based Peridynamics

Document Type: Research Paper

Author

Shanghai Baoye Group Corp., Ltd, Shanghai, 200941, China

Abstract

Peridynamics (PD) is a nonlocal continuum theory based on integro-differential equations without spatial derivatives. The elongation fracture criterion is implicitly incorporated in the PD theory, and fracture is a natural outcome of the simulation. On the other hand, a new fracture criterion based on the crack opening displacement combined with peridynamic (PD-COD) is proposed. When the relative deformation of the PD bond between two particles reaches the critical crack tip opening displacement of the fracture mechanics, we assume that the bond force vanishes. The new damage rule of fracture criteria similar to the local damage rule in conventional PD is introduced to simulate the fracture. In this paper, first, a comparative study between XFEM and PD is presented. Then, four examples, i.e., a bilateral crack problem, double parallel crack, monoclinic crack, and the double inclined crack are given to demonstrate the effectiveness of the new criterion.

Keywords

Main Subjects

[1] 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, 312 (2016) 254-275.
[2] Amiri F., Milan D., Shen Y., Rabczuk T., Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 69 (2014) 102-109.
[3] Areias P., Rabczuk T., Cesar de Sa J. A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement. Computational Mechanics, 58(6) (2016) 1003-1018.
[4] Quoc T.T., Rabczuk T., Meschke G., Bazilevs Y. A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 304 (2016) 584-604.
[5] Budarapu P., Gracie R., Bordas S., Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 53(6) (2014) 1129-1148.
[6] Budarapu P., Gracie R. Shih-Wei Y., Zhuang X., Rabczuk T. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 69 (2014) 126-143.
[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, 15 (2016) 18-23.
[8] Talebi H, Silani M., Rabczuk T. Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 80 (2015) 82-92.
[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, 74 (2014) 30-38.
[10] Talebi H., Silani M., Bordas S., Kerfriden P., Rabczuk T. A Computational Library for Multiscale Modelling of Material Failure. Computational Mechanics, 53(5) (2014) 1047-1071.
[11] 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, 312(C) (2016) 322-350.
[12] Areias P., Msekh M.A., Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 158 (2016) 116-143.
[13] Areias P.M.A., Rabczuk T., P.P. Camanho. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 72 (2014) 50-63.
[14] Areias P., Rabczuk T., Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 110 (2013) 113-137.
[15] Areias P., Rabczuk T., Camanho P.P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 52(4) (2013) 931-947.
[16] Areias P., Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotation. International Journal for Numerical Methods in Engineering, 94(12) (2013) 1099-1122.
[17] Dolbow J. E. An extended finite element method with discontinuous enrichment for applied mechanics, Northwestern University, 2000.
[18] 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, 84(3) (2010) 253-304.
[19] Ghorashi S., Valizadeh N., Mohammadi S., Rabczuk T. T-spline based XIGA for Fracture Analysis of Orthotropic Media. Computers Structures, 147 (2015) 138-146.
[20] 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, 284 (2015) 265-291.
[21] Shi G. H. Numerical manifold method and discontinuous deformation analysis. Chinese Journal of Rock Mechanics and Engineering, 16(3) (1988) 279-292.
[22] 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, 10(05) (2013) 1350028.
[23] Belytschko T, Lu Y Y, Gu L. Element‐free Galerkin methods. International Journal for Numerical Methods in Engineering, 37(2) (1994) 229-256.
[24] Liu W K, Jun S, Zhang Y F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 20(8‐9) (1995) 1081-1106.
[25] Li S, Liu W K. Meshfree and particle methods and their applications. Applied Mechanics Reviews, 55(1) (2002) 1-34.
[26] Nguyen, V. P., Rabczuk, T., Bordas, S., Duflot, M. Meshless methods: a review and computer implementation aspects. Mathematics and Computers in Simulation, 79(3) (2008) 763-813.
[27] 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, 92(11) (2012) 969-998.
[28] Amiri F., Anitescu C., Arroyo M, Bordas S., Rabczuk T. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 53(1) (2014) 45-57.
[29] Rabczuk T., Areias P.M.A. A meshfree thin shell for arbitrary evolving cracks based on an external enrichment. Cmes Computer Modeling in Engineering Ences, 16(2) (2006) 115-130.
[30] 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, 40(3) (2007) 473-495.
[31] Rabczuk T., Areias P.M.A., Belytschko T. A meshfree thin shell method for nonlinear dynamic fracture. International Journal for Numerical Methods in Engineering, 72(5) (2007) 524-548.
[32] Rabczuk T., Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 39(6) (2007) 743-760.
[33] Rabczuk T., Bordas S., Zi G. on three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 88(23-24) (2010) 1391-1411.
[34] 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, 75(16) (2008) 4740-4758.
[35] Rabczuk T., Gracie R., Song J.H., Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 81(1) (2010) 48-71.
[36] 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, 75(5) (2008) 943-960.
[37] Rabczuk T., Belytschko T. A three dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 196(29-30) (2007) 2777-2799.
[38] 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, 199(37-40) (2010) 2437-2455.
[39] Rabczuk T., Belytschko T. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 61(13) (2004) 2316-2343.
[40] Erdogan F, Sih G C. on the crack extension in plates under plane loading and transverse shear. Journal of Basic Engineering, 85(4) (1963) 519-527.
[41] Sih G C. Strain-energy-density factor applied to mixed mode crack problems. International Journal of Fracture, 10(3) (1974) 305-321.
[42] Hussain M A, Pu S L, Underwood J. Strain energy release rate for a crack under combined mode I and mode II. Fracture Analysis: Proceedings of the 1973 National Symposium on Fracture Mechanics, Part II. ASTM International, 1974.
[43] Ayatollahi M R, Abbasi H. Prediction of fracture using a strain based mechanism of crack growth. Building Research Journal, 49 (2011) 167-180.
[44] Sato S. Combined Mode Fracture Toigliess measurement by the Disk Test. Journal of Engineering Materials and Technology, 100(2) (1978) 175-182.
[45] 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, 26(6) (1987) 825-840.
[46] 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, 66(1) (2002) 79-97.
[47] 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, 5 (2006) 181-188.
[48] Richard H A, Benitz K. A loading device for the creation of mixed mode in fracture mechanics. International Journal of Fracture, 22(2) (1983) 55-58.
[49] Arcan M, Hashin Z, Voloshin A. A method to produce uniform plane-stress states with applications to fiber-reinforced materials. Experimental Mechanics, 18(4) (1978) 141-146.
[50] Zipf Jr R K, Bieniawski Z T. Mixed mode testing for fracture toughness of coal based on critical-energy-density. The 27th US Symposium on Rock Mechanics (USRMS). American Rock Mechanics Association, 1986.
[51] Belytschko, T., Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 45(5) (1999) 601-620.
[52] Sukumar, N., Belytschko, T. Arbitrary branched and intersecting cracks with the extended finite element method. International Journal for Numerical Methods in Engineering, 48 (2000) 1741-1760.
[53] Nagashima T, Omoto Y, Tani S. Stress intensity factor analysis of interface cracks using X‐FEM. International Journal for Numerical Methods in Engineering, 56(8) (2003) 1151-1173.
[54] Fang X J, Jin F, Wang J T. Cohesive crack model based on extended finite element method. Journal of Tsinghua University, 47(3) (2007) 344-347.
[55] Fang X, Jin F, Wang J. Seismic fracture simulation of the Koyna gravity dam using an extended finite element method. Journal of Tsinghua University, 48(12) (2008) 2065-2069.
[56] J-H Song, P.M.A. Areias, and T. Belytschko. A method for dynamic crack and shear band propagation with phantom nodes. International Journal for Numerical Methods in Engineering, 67(6) (2006) 868-893.
[57] A. Hansbo and P. Hansbo. A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 193(33-35) (2004) 3523-3540.
[58] Areias P M A, Belytschko T. A comment on the article A finite element method for simulation of strong and weak discontinuities in solid mechanics by A. Hansbo and P. Hansbo [Comput. Methods Appl. Mech. Engrg. 193 (2004) 3523–3540], Computer Methods in Applied Mechanics & Engineering, 195 (2006) 1275-1276.
[59] Rabczuk T., Zi G., Gerstenberger A., Wall W.A. A new crack tip element for the phantom node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 75(5) (2008) 577-599.
[60] Chau-Dinh T, Zi G, Lee P S, Rabczuk T, Song J H. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 92 (2012) 242-256.
[61] Zhuang X, Zhu H, Augarde C. An improved meshless Shepard and least squares method possessing the delta property and requiring no singular weight function. Computational Mechanics, 53(2) (2014) 343-357.
[62] Zhuang X, Cai Y, Augarde C. A meshless sub-region radial point interpolation method for accurate calculation of crack tip fields. Theoretical and Applied Fracture Mechanics, 69 (2014) 118-125.
[63] Ganzenmüller G.C., Hiermaier S., May M. on the similarity of meshless discretizations of peridynamics and Smoothed Particle Hydrodynamics. Computers & Structures, 150 (2015) 71-78.
[64] Bessa M A, Foster J T, Belytschko T, Liu W K. A meshfree unification: reproducing kernel peridynamics. Computational Mechanics, 53(6) (2014) 1251-1264.
[65] Silling S A. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 48(1) (2000) 175-209.
[66] Silling S A, Zimmermann M, Abeyaratne R. Deformation of a peridynamic bar. Journal of Elasticity, 73(1-3) (2003) 173-190.
[67] Silling, S.A., Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 48(1) (2000) 175-209.
[68] Silling, S.A., M. Zimmermann, and R. Abeyaratne, Deformation of a peridynamic bar. Journal of Elasticity, 73(1-3) (2003) 173-190.
[69] Silling, S.A. Askari, E. A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures, 83(17) (2005) 1526-1535.
[70] Ren H., Zhuang X., Cai Y., Rabczuk T. Dual-Horizon Peridynamics. International Journal for Numerical Methods in Engineering, 108 (2016) 1451-1476.
[71] Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 318 (2017) 762-782.
[72] Rabczuk T, Ren H. A peridynamics formulation for quasi-static fracture and contact in rock. Engineering Geology, 225C (2017) 42-48.
[73] Silling S A, Askari E. A meshfree method based on the peridynamic model of solid mechanics. Computers & structures, 83(17) (2005) 1526-1535.
[74] Silling, S.A. and F. Bobaru. Peridynamic modeling of membranes and fibers. International Journal of Non-Linear Mechanics, 40(2) (2005) 395-409.
[75] Weckner, O., Brunk, G., Epton, M. A., Silling, S. A., Askari, E. Green’s functions in non-local three-dimensional linear elasticity. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 465(2112) (2009) 3463-3487.
[76] 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, 19(4) (2011) 045003.
[77] Kilic B. Peridynamic theory for progressive failure prediction in homogeneous and heterogeneous materials. The University of Arizona, 2008.
[78] Foster J T, Silling S A, Chen W. An energy based failure criterion for use with peridynamic states. International Journal for Multiscale Computational Engineering, 9(6) (2011) 675–688.
[79] Silling S A, Bobaru F. Peridynamic modeling of membranes and fibers. International Journal of Non-Linear Mechanics, 40(2) (2005) 395-409.
[80] Ayatollahi M R, Aliha M R M. Analysis of a new specimen for mixed mode fracture tests on brittle materials. Engineering Fracture Mechanics, 76(11) (2009) 1563-1573.
[81] Shen F, Zhang Q, Huang D, et al. Damage and failure process of concrete structure under uni-axial tension based on peridynamics modeling. Chinese Journal of Mechanical Engineering, 30 (2013) 79-83.
[82] 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, 17(3) (2017) 04016086.
[83] Ren H, Zhuang X, Rabczuk T. A new peridynamic formulation with shear deformation for elastic solid. Journal of Micromechanics and Molecular Physics, 1(2) (2016) 1650009.
[84] Silani M., Talebi H., Ziaei-Rad S., Hamouda A.M.S., Zi G., Rabczuk T. A three dimensional Extended Arlequin Method for Dynamic Fracture. Computational Materials Science, 96 (2015) 425-431.
[85] 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, 11(6) (2013) 527-541.
[86] Yang S.W., Budarapu P.R., Mahapatra D.R., Bordas S.P.A., Zi G., Rabczuk T. A Meshless Adaptive Multiscale Method for Fracture. Computational Materials Science, 96 (2015) 382-395.
[87] Costa, T., Bond, S., Littlewood, D., Moore, S. Peridynamic Multiscale Finite Element Methods. No. SAND2015-10472. Sandia National Laboratories, Al-buquerque, United States, 2015.
[88] Jung, J. and J. Seok. Fatigue crack growth analysis in layered heterogeneous material systems using peridynamic approach. Composite Structures, 152 (2016) 403-407.
[89] Gu X, Zhou X P. The numerical simulation of tensile plate with circular hole using peridynamic theory. Chinese Journal of Mechanical Engineering, 36(5) (2015) 376–383.
[90] Cheng Z, Liu J. Fracture analysis of functionally graded materials under impact loading based on peridynamics. Chinese Journal of Applied Mechanics, 4 (2016) 634-639.
[91] Zhou, X. and Y. Shou, Numerical Simulation of Failure of Rock-Like Material Subjected to Compressive Loads Using Improved Peridynamic Method. International Journal of Geomechanics, 17(3) (2017) 04016086.
[92] Ren H, Zhuang X, Rabczuk T. Implementation of GTN Model in Dual-horizon Peridynamics. Procedia Engineering, (197) (2017) 224-232.
[93] Zeng S P, Bilateral crack growth behavior of Q345 steel under uniaxial tensile load. Dissertation for Master Degree Guangxi University, 2014.