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A non-ordinary state-based peridynamics framework for anisotropic materials. (English) Zbl 1440.74026
Summary: Peridynamics (PD) represents a new approach for modelling fracture mechanics, where a continuum domain is modelled through particles connected via physical interactions. This formulation allows us to model crack initiation, propagation, branching and coalescence without special assumptions. Up to date, anisotropic materials were modelled in the PD framework as different isotropic materials (for instance, fibre and matrix of a composite laminate), where the stiffness of the bond depends on its orientation. In this work we propose a non-ordinary state-based formulation to model general anisotropic materials. The material properties for each particle are defined using the material constitutive matrix, rather than being defined through the bond stiffness between adjacent particles. We propose a damage criterion for composite materials to model the crack propagation behaviour for anisotropic materials. We validate the model using benchmark problems obtained with established numerical methods or experimental results. The proposed approach enables the use of general classes of material models including rocks, concrete and biomaterials.

MSC:
74A45 Theories of fracture and damage
74E10 Anisotropy in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
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[1] Sih, G. C.; Paris, P. C.; Irwin, G. R., On cracks in rectilinearly anisotropic bodies, Int. J. Fract., 1, 189-203 (1965)
[2] Nobile, L.; Carloni, C., Fracture analysis for orthotropic cracked plates, Compos. Struct., 68, 285-293 (2005)
[3] Motamedi, D.; Milani, A. S.; Komeili, M.; Bureau, M. N.; Thibault, F.; Trudel-Boucher, D., A stochastic XFEM model to study delamination in PPS/Glass UD composites: Effect of uncertain fracture properties, Appl. Compos. Mater., 21, 341-358 (2014)
[4] García-Sánchez, F.; Sáez, A.; Domínguez, J., Anisotropic and piezoelectric materials fracture analysis by BEM, Comput. Struct., 83, 804-820 (2005)
[5] Wünsche, M.; Zhang, C.; García-Sánchez, F.; Sáez, A.; Sladek, J.; Sladek, V., Dynamic crack analysis in piezoelectric solids with non-linear electrical and mechanical boundary conditions by a time-domain BEM, Comput. Methods Appl. Mech. Engrg., 200, 2848-2858 (2011) · Zbl 1230.74211
[6] Milazzo, A., An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics, Compos. Struct., 94, 2078-2086 (2012)
[7] Wu, T. L.; Huang, J. H., Closed-form solutions for the magnetoelectric coupling coefficients in fibrous composites with piezoelectric and piezomagnetic phases, Int. J. Solids Struct., 37, 2981-3009 (2000) · Zbl 0997.74019
[8] Giordano, C.; Zappalà, S.; Kleiven, S., Anisotropic finite element models for brain injury prediction: the sensitivity of axonal strain to white matter tract inter-subject variability, Biomech. Model. Mechanobiol., 16, 1269-1293 (2017)
[9] Santiuste, C.; Rodríguez-Millán, M.; Giner, E.; Miguélez, H., The influence of anisotropy in numerical modeling of orthogonal cutting of cortical bone, Compos. Struct., 116, 423-431 (2014)
[10] Giraud, A.; Hoxha, D.; Huynh, Q. V.; Do, D. P.; Magnenet, V., Effective porothermoelastic properties of transversely isotropic rock-like composites, Int. J. Eng. Sci., 46, 527-550 (2008) · Zbl 1213.74098
[11] Hattori, G.; Trevelyan, J.; Augarde, C. E.; Cooms, W. M.; Aplin, A. C., Numerical simulation of fracking in shale rocks: Current state and future approaches, Arch. Comput. Methods Eng., 24, 281-317 (2017) · Zbl 1364.76221
[12] Griffith, A. A., The phenomena of rupture and flow in solids, Phil. Trans. R. Soc. London A, 221, 163-198 (1921)
[13] Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity (1953), Noordhoff: Noordhoff Leiden · Zbl 0052.41402
[14] Sih, G. C., Mechanics of Fracture Initiation and Propagation (1991), Springer
[15] Bouhala, L.; Makradi, A.; Belouettar, S.; Kiefer-Kamal, H.; Fréres, P., Modelling of failure in long fibres reinforced composites by X-FEM and cohesive zone model, Composites B, 55, 352-361 (2013)
[16] Motamedi, D.; Mohammadi, S., Fracture analysis of composites by time independent moving-crack orthotropic XFEM, Int. J. Mech. Sci., 54, 20-37 (2012)
[17] Hattori, G.; Alatawi, I. A.; Trevelyan, J., An extended boundary element method formulation for the direct calculation of the stress intensity factors in fully anisotropic materials, Internat. J. Numer. Methods Engrg., 109, 965-981 (2017)
[18] Biner, S.; Hu, S. Y., Simulation of damage evolution in composites: A phase-field model, Acta Mater., 57, 2088-2097 (2009) · Zbl 1400.74099
[19] Henry, H., Study of the branching instability using a phase field model of inplane crack propagation, Europhys. Lett., 83, Article 16004 pp. (2008)
[20] Schlüter, A.; Willenbücher, A.; Kuhn, C.; Müller, R., Phase field approximation of dynamic brittle fracture, Comput. Mech., 54, 1141-1161 (2014) · Zbl 1311.74106
[21] Silling, S. A., Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48, 175-209 (2000) · Zbl 0970.74030
[22] Eringen, A. C.; Edelen, D., On nonlocal elasticity, Int. J. Eng. Sci., 10, 233-248 (1972) · Zbl 0247.73005
[23] Lorentz, E.; Andrieux, S., Analysis of non-local models through energetic formulations, Int. J. Solids Struct., 40, 2905-2936 (2003) · Zbl 1038.74506
[24] Silling, S. A.; Epton, M.; Weckner, O.; Xu, J.; Askari, E., Peridynamic states and constitutive modeling, J. Elasticity, 88, 151-184 (2007) · Zbl 1120.74003
[25] Silling, S. A.; Lehoucq, R. B., Peridynamic theory of solid mechanics, Adv. Appl. Mech., 44, 73-166 (2010)
[26] Madenci, E.; Oterkus, E., Peridynamic Theory and its Applications (2014), Springer · Zbl 1295.74001
[27] Warren, T. L.; Silling, S. A.; Askari, A.; Weckner, O.; Epton, M. A.; Xu, J., A non-ordinary state-based peridynamic method to model solid material deformation and fracture, Int. J. Solids Struct., 46, 1186-1195 (2009) · Zbl 1236.74012
[28] Breitenfeld, M.; Geubelle, P.; Weckner, O.; Silling, S., Non-ordinary state-based peridynamic analysis of stationary crack problems, Comput. Methods Appl. Mech. Engrg., 272, 233-250 (2014) · Zbl 1296.74099
[29] Yaghoobi, A.; Chorzepa, M. G., Meshless modeling framework for fiber reinforced concrete structures, Comput. Struct., 161, 43-54 (2015)
[30] Wu, C.; Ren, B., A stabilized non-ordinary state-based peridynamics for the nonlocal ductile material failure analysis in metal machining process, Comput. Methods Appl. Mech. Engrg., 291, 197-215 (2015) · Zbl 1423.74067
[31] Wang, Y.; Zhou, X.; Xu, X., Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics, Eng. Fract. Mech., 163, 248-273 (2016)
[32] Hu, W.; Ha, Y. D.; Bobaru, F., Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites, Comput. Methods Appl. Mech. Engrg., 217, 247-261 (2012) · Zbl 1253.74008
[33] Oterkus, E.; Madenci, E., Peridynamic analysis of fiber-reinforced composite materials, J. Mech. Mater. Struct., 7, 45-84 (2012)
[34] Oterkus, E.; Madenci, E.; Weckner, O.; Silling, S.; Bogert, P.; Tessler, A., Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot, Compos. Struct., 94, 839-850 (2012)
[35] Hu, Y.l.; Yu, Y.; Wang, H., Peridynamic analytical method for progressive damage in notched composite laminates, Compos. Struct., 108, 801-810 (2014)
[36] Ghajari, M.; Iannucci, L.; Curtis, P., A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media, Comput. Methods Appl. Mech. Engrg., 276, 431-452 (2014) · Zbl 1423.74882
[37] Holzapfel, G. A., Nonlinear Solid Mechanics (2000), John Wiley & Sons Ltd · Zbl 0980.74001
[38] Macek, R. W.; Silling, S. A., Peridynamics via finite element analysis, Finite Elem. Anal. Des., 43, 1169-1178 (2007)
[39] Gerstle, W.; Sau, N.; Silling, S., Peridynamic modeling of concrete structures, Nucl. Eng. Des., 237, 1250-1258 (2007)
[40] Silling, S. A.; Littlewood, D.; Seleson, P., Variable horizon in a peridynamic medium, J. Mech. Mater. Struct., 10, 591-612 (2015)
[41] Le, Q. V.; Bobaru, F., Surface corrections for peridynamic models in elasticity and fracture, Comput. Mech. (2017) · Zbl 1446.74084
[42] Queiruga, A. F.; Moridis, G., Numerical experiments on the convergence properties of state-based peridynamic laws and influence functions in two-dimensional problems, Comput. Methods Appl. Mech. Engrg., 322, 97-122 (2017)
[43] Silling, S. A.; Lehoucq, R. B., Convergence of peridynamics to classical elasticity theory, J. Elasticity, 93, 13-37 (2008) · Zbl 1159.74316
[44] Pensée, V.; Kondo, D.; Dormieux, L., Micromechanical analysis of anisotropic damage in brittle materials, J. Eng. Mech., 128, 889-897 (2002)
[45] Zhou, X. P.; Wang, Y. T., Numerical simulation of crack propagation and coalescence in pre-cracked rock-like brazilian disks using the non-ordinary state-based peridynamics, Int. J. Rock Mech. Min. Sci., 89, 235-249 (2016)
[46] Liu, W.; Hong, J. W., A coupling approach of discretized peridynamics with finite element method, Comput. Methods Appl. Mech. Engrg., 245, 163-175 (2012) · Zbl 1354.74284
[47] Zaccariotto, M.; Mudric, T.; Tomasi, D.; Shojaei, A.; Galvanetto, U., Coupling of FEM meshes with peridynamic grids, Comput. Methods Appl. Mech. Engrg., 330, 471-497 (2018)
[48] Mossaiby, F.; Shojaei, A.; Zaccariotto, M.; Galvanetto, U., OpenCL implementation of a high performance 3D peridynamic model on graphics accelerators, Comput. Math. Appl., 74, 1856-1870 (2017) · Zbl 06928457
[49] Liu, W.; Hong, J. W., Discretized peridynamics for linear elastic solids, Comput. Mech., 50, 579-590 (2012) · Zbl 1312.74030
[50] Silling, S. A.; Askari, E., A meshfree method based on the peridynamic model of solid mechanics, Comput. Struct., 83, 1526-1535 (2005)
[51] Courant, R.; Friedrichs, K.; Lewy, H., Über die partiellen differenzengleichungen der mathematischen Physik, Math. Ann., 100, 32-74 (1928) · JFM 54.0486.01
[52] Seleson, P., Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations, Comput. Methods Appl. Mech. Engrg., 282, 184-217 (2014) · Zbl 1423.74143
[53] Ting, T. C.T., Anisotropic Elasticity (1996), Oxford University Press: Oxford University Press New York · Zbl 0883.73001
[54] Du, Q., Local limits and asymptotically compatible discretizations, (Bobaru, F.; Foster, J. T.; Geubelle, P. H.; Silling, S. A., Handbook of Peridynamic Modeling (2017), CRC Press: CRC Press Boca Raton), 87-108, (Chapter 4)
[55] Bobaru, F.; Yang, M.; Alves, L. F.; Silling, S. A.; Askari, E.; Xu, J., Convergence, adaptive refinement, and scaling in 1D peridynamics, Internat. J. Numer. Methods Engrg., 77, 852-877 (2009) · Zbl 1156.74399
[56] Cahill, L. M.A.; Natarajan, S.; Bordas, S. P.A.; O’Higgins, R. M.; McCarthy, C. T., An experimental/numerical investigation into the main driving force for crack propagation in uni-directional fibre-reinforced composite laminae, Compos. Struct., 107, 119-130 (2014)
[57] Dipasquale, D.; Sarego, G.; Zaccariotto, M.; Galvanetto, U., Dependence of crack paths on the orientation of regular 2D peridynamic grids, Eng. Fract. Mech., 160, 248-263 (2016)
[58] Engwirda, D., Locally-Optimal Delaunay-Refinement and Optimisation-Based Mesh Generation (2014), (Ph.D. thesis)
[59] Bobaru, F.; Hu, W., The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials, Int. J. Fract., 176, 215-222 (2012)
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