A new fracture criterion for peridynamic and dual-horizon peridynamics

Abstract

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.

Keywords

Castigliano’s theorem / breaking energy / critical extension / XFEM / COD / PD-COD

References

[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, 2016, 312(C): 254–275

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[24] Belytschko T, Lu Y Y, Gu L. Element‐free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256

[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

[26] Li S, Liu W K. Meshfree and particle methods and their applications. Applied Mechanics Reviews, 2002, 55(1): 1–34

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[43] Sih G C. Strain-energy-density factor applied to mixed mode crack problems. International Journal of Fracture, 1974, 10(3): 305–321

[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

[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

[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

[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

[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

[54] Silling S A, Zimmermann M, Abeyaratne R. Deformation of a peridynamic bar. Journal of Elasticity, 2003, 73(1-3): 173–190

[55] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-Horizon Peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476

[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

[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

[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

[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

[63] Silling S A, Bobaru F. Peridynamic modeling of membranes and fibers. International Journal of Non-linear Mechanics, 2005, 40(2-3): 395–409

[64] Silling S A, Askari E. A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures, 2005, 83(17-18): 1526–1535

[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

[66] Silling S A, Bobaru F. Peridynamic modeling of membranes and fibers. International Journal of Non-linear Mechanics, 2005, 40(2-3): 395–409

[67] 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

[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

[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

[74] Dugdale D S. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids, 1960, 8(2): 100–104

[75] Barenblatt G I. The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, 1962, 7: 55–129

[76] Zeng S. Bilateral crack growth behavior of Q345 steel under uniaxial tensile load. Guangxi University, 2014

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg