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Surface corrections for peridynamic models in elasticity and fracture. (English) Zbl 1446.74084
Summary: Peridynamic models are derived by assuming that a material point is located in the bulk. Near a surface or boundary, material points do not have a full non-local neighborhood. This leads to effective material properties near the surface of a peridynamic model to be slightly different from those in the bulk. A number of methods/algorithms have been proposed recently for correcting this peridynamic surface effect. In this study, we investigate the efficacy and computational cost of peridynamic surface correction methods for elasticity and fracture. We provide practical suggestions for reducing the peridynamic surface effect.

##### MSC:
 74A70 Peridynamics 74A45 Theories of fracture and damage
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##### References:
 [1] Silling, SA, Reformulation of elasticity theory for discontinuities and long-range forces, J Mech Phys Solids, 48, 175-209, (2000) · Zbl 0970.74030 [2] Silling, SA; Askari, E, A meshfree method based on the peridynamic model of solid mechanics, Comput. Struct., 83, 1526-1535, (2005) [3] Silling, SA; Epton, M; Weckner, O; Xu, J; Askari, E, Peridynamic states and constitutive modeling, J Elast, 88, 151-184, (2007) · Zbl 1120.74003 [4] Silling, SA; Lehoucq, RB, Convergence of peridynamics to classical elasticity theory, J Elast, 93, 13-37, (2008) · Zbl 1159.74316 [5] Emmrich, E; Weckner, O, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity, Commun Math Sci, 5, 851-864, (2007) · Zbl 1133.35098 [6] Han, W; Liu, WK, Flexible piecewise approximations based on partition of unity, Adv Comput Math, 23, 191-199, (2005) · Zbl 1067.65125 [7] Kim, DW; Liu, WK, Maximum principle and convergence analysis for the meshfree point collocation method, SIAM J Numer Anal, 44, 515-539, (2006) · Zbl 1155.65090 [8] Liu, WK; Han, W; Lu, H; Li, S; Cao, J, Reproducing kernel element method. part I: theoretical formulation, Comput Methods Appl Mech Eng, 193, 933-951, (2004) · Zbl 1060.74670 [9] Li, S; Lu, H; Han, W; Liu, WK; Simkins, DC, Reproducing kernel element method part II: globally conforming $$I^m/C^n$$ hierarchies, Comput Methods Appl Mech Eng, 193, 953-987, (2004) · Zbl 1093.74062 [10] Lu, H; Li, S; Simkins, DC; Liu, WK; Cao, J, Reproducing kernel element method part III: generalized enrichment and applications, Comput Methods Appl Mech Eng, 193, 989-1011, (2004) · Zbl 1060.74671 [11] Simkins, DC; Li, S; Lu, H; Liu, WK, Reproducing kernel element method. part IV: globally compatible $$\text{C}^{{\rm n}} ({\rm n}⩾ 1)$$ triangular hierarchy, Comput Methods Appl Mech Eng, 193, 1013-1034, (2004) · Zbl 1093.74064 [12] Gerstle W H, Sau N, Silling S A (2005) Peridynamic modeling of plain and reinforced concrete structures. In: Presented at the 18th international conference on structural mechanics in reactor technology, Beijing, China [13] Bobaru, F; Ha, YD, Adaptive refinement and multiscale modeling in 2D peridynamics, J Multisc Comput Eng, 9, 635-660, (2011) [14] Chen, Z; Bakenhus, D; Bobaru, F, A constructive peridynamic kernel for elasticity, Comput Methods Appl Mech Eng, 311, 356-373, (2016) [15] Chen, Z; Bobaru, F, Selecting the kernel in a peridynamic formulation: a study for transient heat diffusion, Comput Phys Commun, 197, 51-60, (2015) [16] Du, Q; Tian, L; Zhao, X, A convergent adaptive finite element algorithm for nonlocal diffusion and peridynamic models, SIAM J Num Anal, 51, 1211-1234, (2013) · Zbl 1271.82011 [17] Tian, XC; Du, Q, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J Num Anal, 51, 3458-3482, (2013) · Zbl 1295.82021 [18] Macek, RW; Silling, SA, Peridynamics via finite element analysis, Fin Elem Anal Design, 43, 1169-1178, (2007) [19] Gunzburger, M; Lehoucq, RB, A nonlocal vector calculus with application to nonlocal boundary value problems, Multisc Model Simul, 8, 1581-1598, (2010) · Zbl 1210.35057 [20] Silling, SA; Lehoucq, RB, Peridynamic theory of solid mechanics, Adv Appl Mech, 44, 73-168, (2010) [21] Bobaru F, Foster JT, Geubelle PH, Silling SA (2017) Handbook of peridynamic modeling. CRC Press, Taylor & Francis Group, Boca Raton · Zbl 1351.74001 [22] Le, QV; Chan, WK; Schwartz, J, A two-dimensional ordinary, state-based peridynamic model for linearly elastic solids, Int J Numer Methods Eng, 98, 547-561, (2014) · Zbl 1352.74040 [23] Sarego, G; Le, QV; Bobaru, F; Zaccariotto, M; Galvanetto, U, Linearized state-based peridynamics for 2D problems, Int J Numer Methods Eng, 108, 1174-1197, (2016) [24] Madenci E, Oterkus E (2014) Coupling of the peridynamic theory and finite element method. In: Peridynamic theory and its applications, Springer, ed New York, pp 191-202 · Zbl 1295.74001 [25] Oterkus E (2010) Peridynamic theory for modeling three-dimensional damage growth in metallic and composite structures, Ph.D. thesis, The University of Arizona [26] Mitchell, JA; Silling, SA; Littlewood, DJ, A position-aware linear solid constitutive model for peridynamics, J Mech Mater Struct, 10, 539-557, (2015) [27] Emmrich, E; Weckner, O, The peridynamic equation and its spatial discretisation, Math Model Anal, 12, 17-27, (2007) · Zbl 1121.65073 [28] Seleson, P; Parks, ML; Gunzburger, M; Lehoucq, RB, Peridynamics as an upscaling of molecular dynamics, Multisc Model Simul, 8, 204-227, (2009) · Zbl 1375.82073 [29] Chen, Z; Bobaru, F, Peridynamic modeling of pitting corrosion damage, J Mech Phys Solids, 78, 352-381, (2015) [30] Seleson, P, Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations, Comput Methods Appl Mech Eng, 282, 184-217, (2014) · Zbl 1423.74143 [31] Henke, SF; Shanbhag, S, Mesh sensitivity in peridynamic simulations, Comput Phys Commun, 185, 181-193, (2014) · Zbl 1344.65115 [32] Hu W, Ha Y D, Bobaru F (2010) Numerical integration in peridynamics. In: Technical report, University of Nebraska-Lincoln [33] Du, Q; Gunzburger, M; Lehoucq, RB; Zhou, K, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math Models Methods Appl Sci, 23, 493-540, (2013) · Zbl 1266.26020 [34] Tao, Y; Tian, X; Du, Q, Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations, Appl Math Comput, 305, 282-298, (2017) [35] Shewchuk JR (1994) An introduction to the conjugate gradient method without the agonizing pain. In: Technical report school of computer science. Carnegie Mellon University [36] Silling, SA, Linearized theory of peridynamic states, J Elast, 99, 85-111, (2010) · Zbl 1188.74008 [37] Kilic, B; Madenci, E, Peridynamic theory for thermomechanical analysis, IEEE Trans Adv Packag, 33, 97-105, (2010) [38] Kilic, B; Agwai, A; Madenci, E, Peridynamic theory for progressive damage prediction in center-cracked composite laminates, Compos Struct, 90, 141-151, (2009) [39] Bobaru, F; Yang, M; Alves, LF; Silling, SA; Askari, E; Xu, J, Convergence, adaptive refinement, and scaling in 1D peridynamics, Int J Numer Methods Eng, 77, 852-877, (2009) · Zbl 1156.74399 [40] 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) [41] Hu, W; Ha, YD; Bobaru, F, Modeling dynamic fracture and damage in a fiber-reinforced composite lamina with peridynamics, Intl J Multisc Comput Eng, 9, 707-726, (2011) [42] Gerstle WH (2016) Introduction to Practical Peridynamics. World Scientific Publishing Co., Singapore [43] Ha, Y; Bobaru, F, Studies of dynamic crack propagation and crack branching with peridynamics, Int J Fract, 162, 229-244, (2010) · Zbl 1425.74416 [44] Madenci E, Oterkus E (2014) Peridynamic theory and its applications. Springer, Berlin · Zbl 1295.74001 [45] Timoshenko S, Goodier JN (1969) Theory of elasticity. McGraw-Hill, New York City · Zbl 0045.26402 [46] Oterkus, S; Madenci, E; Agwai, A, Peridynamic thermal diffusion, J Comput Phys, 265, 71-96, (2014) · Zbl 1349.80020 [47] Oterkus S (2015) Peridynamics for the solution of multiphysics problems. Ph.D. thesis, The University of Arizona [48] Rice, JR, A path independent integral and the approximate analysis of strain concentration by notches and cracks, J Appl Mech, 35, 379-386, (1968) [49] Hu, W; Ha, Y; Bobaru, F; Silling, S, The formulation and computation of the nonlocal J-integral in bond-based peridynamics, Int J Fract, 176, 195-206, (2012) [50] Silling, SA, Origin and effect of nonlocality in a composite, J Mech Mater Struct, 9, 245-258, (2014) [51] Dipasquale, D; Zaccariotto, M; Galvanetto, U, Crack propagation with adaptive grid refinement in 2D peridynamics, Int J Fract, 190, 1-22, (2014)
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