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The adaptive coupling of dual-horizon peridynamic element and finite element for the progressive failure of materials

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Abstract

The peridynamic correspondence model (PDCM) provides the stress–strain relation that can introduce many classical constitutive models, however, the high computational consumption and zero-energy mode of PDCM certainly limit its further application to practical engineering crack problems. To solve these limitations and exploit the advantage of PDCM, we propose a simple and effective method that adaptively couples dual-horizon peridynamic element (DH-PDE) with finite element (FE) to simulate the quasi-static fracture problems. To this end, a stabilized dual-horizon peridynamic element for DH-PDCM is firstly developed that the peridynamic strain matrices for the bond and material point are constructed respectively. The nonlocal ordinary and correctional peridynamic element stiffness matrices are derived in detail and calculated by the proposed dual-assembly algorithm. Subsequently, a unified variational weak form of this adaptive coupling of DH-PDE and FE is proposed based on the convergence of peridynamics to the classical model in the limit of vanishing horizon. Therefore, the integrals of the peridynamic element and finite element in this coupling method are completely decoupled in the viewpoint of numerical implementation, which makes it easier to realize the proposed adaptive coupling by switching integral element. Moreover, the proposed adaptive coupling is implemented in Abaqus/UEL to optimize the calculational efficiency and real-time visualization of calculated results, which has potential for dealing with the engineering crack problems. Two-dimensional numerical examples involving mode-I and mixed-mode crack problems are used to demonstrate the effectiveness of this adaptive coupling in addressing the quasi-static fracture of cohesive materials.

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References

  • Amani J, Oterkus E, Areias P et al (2016) A non-ordinary state-based peridynamics formulation for thermoplastic fracture. Int J Impact Eng 87:83–94

    Article  Google Scholar 

  • Azdoud Y, Han F, Lubineau G (2014) The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture. Comput Mech 54:711–722

    Article  MathSciNet  Google Scholar 

  • Belytschko T, Chen H, Xu J, Zi G (2003) Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int J Numer Methods Eng 58:1873–1905

    Article  Google Scholar 

  • Bie YH, Cui XY, Li ZC (2018) A coupling approach of state-based peridynamics with node-based smoothed finite element method. Comput Methods Appl Mech Eng 331:675–700

    Article  ADS  MathSciNet  Google Scholar 

  • Bie YH, Li S, Hu X, Cui XY (2019) An implicit dual-based approach to couple peridynamics with classical continuum mechanics. Int J Numer Methods Eng 120:1349–1379

    Article  MathSciNet  Google Scholar 

  • Bie YH, Liu ZM, Yang H, Cui XY (2020) Abaqus implementation of dual peridynamics for brittle fracture. Comput Methods Appl Mech Eng 372:113398

    Article  ADS  MathSciNet  Google Scholar 

  • Bobaru F, Duangpanya M (2010) The peridynamic formulation for transient heat conduction. Int J Heat Mass Transf 53:4047–4059

    Article  Google Scholar 

  • Bobaru F, Zhang G (2015) Why do cracks branch? A peridynamic investigation of dynamic brittle fracture. Int J Fract 196:59–98. https://doi.org/10.1007/s10704-015-0056-8

    Article  Google Scholar 

  • Bobaru F, Foster JT, Geubelle PH (2016) Handbook of Peridynamic Modeling. Crc Press.

  • Chen ZG, Bobaru F (2015) Peridynamic modeling of pitting corrosion damage. J Mech Phys Solids 78:352–381

    Article  ADS  MathSciNet  Google Scholar 

  • Chen ZG, Jafarzadeh S, Zhao J, Bobaru F (2021) A coupled mechano-chemical peridynamic model for pit-to-crack transition in stress-corrosion cracking. J Mech Phys Solids 146:104203

    Article  MathSciNet  CAS  Google Scholar 

  • Elices M, Guinea GV, Gómez J, Planas J (2001) The cohesive zone model: advantages, limitations and challenges. Eng Fract Mech 69:137–163

    Article  Google Scholar 

  • Galvanetto U, Mudric T, Shojaei A, Zaccariotto M (2016) An effective way to couple FEM meshes and peridynamics grids for the solution of static equilibrium problems. Mech Res Commun 76:41–47

    Article  Google Scholar 

  • Gu X, Zhang Q, Xia X (2017) Voronoi-based peridynamics and cracking analysis with adaptive refinement. Int J Numer Methods Eng 112(13):2087–2109

    Article  MathSciNet  Google Scholar 

  • Ha YD, Bobaru F (2011) Characteristics of dynamic brittle fracture captured with peridynamics. Eng Fract Mech 78:1156–1168

    Article  Google Scholar 

  • Han F, Lubineau G (2012) Coupling of nonlocal and local continuum models by the arlequin approach. Int J Numer Methods Eng 89(6):671–685

    Article  MathSciNet  Google Scholar 

  • Han F, Lubineau G, Azdoud Y (2016a) Adaptive coupling between damage mechanics and peridynamics: a route for objective simulation of material degradation up to complete failure. J Mech Phys Solids 94:453–472

    Article  ADS  MathSciNet  Google Scholar 

  • Han F, Lubineau G, Azdoud Y, Askari A (2016b) A morphing approach to couple state-based peridynamics with classical continuum mechanics. Comput Methods Appl Mech Eng 301:336–358

    Article  ADS  MathSciNet  Google Scholar 

  • Jafarzadeh S, Wang L, Larios A, Bobaru F (2021) A fast convolution-based method for peridynamic transient diffusion in arbitrary domains. Comput Methods Appl Mech Eng 375:113633

    Article  ADS  MathSciNet  Google Scholar 

  • Jafarzadeh S, Mousavi F, Larios A, Bobaru F (2022) A general and fast convolution-based method for peridynamics: applications to elasticity and brittle fracture. Comput Methods Appl Mech Eng 392:114666

    Article  ADS  MathSciNet  Google Scholar 

  • Lai X, Liu L, Li S et al (2018) A non-ordinary state-based peridynamics modeling of fractures in quasi-brittle materials. Int J Impact Eng 111:130–146

    Article  Google Scholar 

  • Li P, Hao ZM, Zhen WQ (2018) A stabilized non-ordinary state-based peridynamic model. Comput Methods Appl Mech Eng 339:262–280

    Article  ADS  MathSciNet  Google Scholar 

  • Li P, Hao Z, Yu S, Zhen W (2020) Implicit implementation of the stabilized non-ordinary state-based peridynamic model. Int J Numer Methods Eng 121:571–587

    Article  MathSciNet  Google Scholar 

  • Littlewood D, Silling SA, Mitchell JA (2015) Strong local–nonlocal coupling for integrated fracture modeling, Sandia Report SAND2015–7998, Sandia National Laboratories.

  • Liu W, Hong JW (2012) A coupling approach of discretized peridynamics with finite element method. Comput Methods Appl Mech Eng 245–246:163–175

    Article  ADS  MathSciNet  Google Scholar 

  • Liu ZM, Bie YH, Cui ZQ, Cui XY (2020) Ordinary state-based peridynamics for nonlinear hardening plastic materials’ deformation and its fracture process. Eng Fract Mech 223:106782

    Article  Google Scholar 

  • Liu S, Fang G, Fu M et al (2022) A coupling model of element-based peridynamics and finite element method for elastic-plastic deformation and fracture analysis. Int J Mech Sci 220:107170

    Article  Google Scholar 

  • Lubineau G, Azdoud Y, Han F et al (2012) A morphing strategy to couple non-local to local continuum mechanics. J Mech Phys Solids 60:1088–1102

    Article  ADS  MathSciNet  Google Scholar 

  • Luo J, Sundararaghavan V (2018) Stress-point method for stabilizing zero-energy modes in non-ordinary state-based peridynamics. Int J Solids Struct 150:197–207

    Article  Google Scholar 

  • Madenci E, Oterkus S (2016) Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening. J Mech Phys Solids 86:192–219

    Article  ADS  MathSciNet  Google Scholar 

  • Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45–48):2765–2778

    Article  ADS  MathSciNet  Google Scholar 

  • Molnár G, Gravouil A (2017) 2D and 3D Abaqus implementation of a robust staggered phase-field solution for modeling brittle fracture. Finite Elem Anal Des 130:27–38

    Article  Google Scholar 

  • Nguyen VP, Nguyen GD, Nguyen CT et al (2017) Modelling complex cracks with finite elements: a kinematically enriched constitutive model. Int J Fract 203:21–39

    Article  CAS  Google Scholar 

  • Ni T, Zhu Q, Zhao LY, Li PF (2018) Peridynamic simulation of fracture in quasi brittle solids using irregular finite element mesh. Eng Fract Mech 188:320–343

    Article  Google Scholar 

  • Ni T, Pesavento F, Zaccariotto M et al (2020) Hybrid FEM and peridynamic simulation of hydraulic fracture propagation in saturated porous media. Comput Methods Appl Mech Eng 366:113101

    Article  ADS  MathSciNet  Google Scholar 

  • Ni T, Pesavento F, Zaccariotto M et al (2021) Numerical simulation of forerunning fracture in saturated porous solids with hybrid FEM/peridynamic model. Comput Geotech 133:104024

    Article  Google Scholar 

  • Niazi S, Chen Z, Bobaru F (2021) Crack nucleation in brittle and quasi-brittle materials: a peridynamic analysis. Theor Appl Fract Mech 112:102855. https://doi.org/10.1016/j.tafmec.2020.102855

    Article  Google Scholar 

  • Nooru-Mohamed MB, Schlangen E, van Mier JGM (1993) Experimental and numerical study on the behavior of concrete subjected to biaxial tension and shear. Adv Cem Based Mater 1:22–37

    Article  CAS  Google Scholar 

  • Oterkus S, Madenci E, Agwai A (2014) Fully coupled peridynamic thermomechanics. J Mech Phys Solids 64:1–23

    Article  ADS  MathSciNet  Google Scholar 

  • Ou X, Yao X, Han F (2023) An adaptive coupling modeling between peridynamics and classical continuum mechanics for dynamic crack propagation and crack branching. Eng Fract Mech 281:109096

    Article  Google Scholar 

  • Ren H, Zhuang X, Cai Y, Rabczuk T (2016) Dual-horizon peridynamics. Int J Numer Methods Eng 108:1451–1476

    Article  MathSciNet  Google Scholar 

  • Ren H, Zhuang X, Rabczuk T (2017a) Dual-horizon peridynamics: a stable solution to varying horizons. Comput Methods Appl Mech Eng 318:762–782

    Article  ADS  MathSciNet  Google Scholar 

  • Ren H, Zhuang X, Rabczuk T (2017b) Implementation of GTN model in dual-horizon peridynamics. Procedia Eng 197:224–232

    Article  Google Scholar 

  • Ren H, Zhuang X, Rabczuk T (2020a) A nonlocal operator method for solving partial differential equations. Comput Methods Appl Mech Eng 358:112621

    Article  ADS  MathSciNet  Google Scholar 

  • Ren H, Zhuang X, Rabczuk T (2020b) A higher order nonlocal operator method for solving partial differential equations. Comput Methods Appl Mech Eng 367:113132

    Article  ADS  MathSciNet  Google Scholar 

  • Seleson P, Beneddine S, Prudhomme S (2013) A force-based coupling scheme for peridynamics and classical elasticity. Comput Mater Sci 66:34–49

    Article  Google Scholar 

  • Shao JF, Rudnicki JW (2000) Microcrack-based continuous damage model for brittle geomaterials. Mech Mater 32:607–619

    Article  Google Scholar 

  • Shojaei A, Mudric T, Zaccariotto M, Galvanetto U (2016) A coupled meshless finite point/peridynamic method for 2D dynamic fracture analysis. Int J Mech Sci 119:419–431

    Article  Google Scholar 

  • Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175–209

    Article  ADS  MathSciNet  Google Scholar 

  • Silling SA (2010) Linearized theory of peridynamic states. J Elast 99:85–111

    Article  MathSciNet  Google Scholar 

  • Silling SA (2017) Stability of peridynamic correspondence material models and their particle discretizations. Comput Methods Appl Mech Eng 322:42–57

    Article  ADS  MathSciNet  Google Scholar 

  • Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83:1526–1535

    Article  Google Scholar 

  • Silling SA, Lehoucq RB (2008) Convergence of peridynamics to classical elasticity theory. J Elast 93:13–37

    Article  MathSciNet  Google Scholar 

  • Silling SA, Epton M, Weckner O et al (2007) Peridynamic states and constitutive modeling. J Elast 88:151–184

    Article  MathSciNet  Google Scholar 

  • Silling S, Littlewood D, Seleson P (2015) Variable horizon in a peridynamic medium. J Mech Mater 10:591–612

    MathSciNet  Google Scholar 

  • Su X, Yang Z, Liu G (2010) Finite element modelling of complex 3D static and dynamic crack propagation by embedding cohesive elements in abaqus. Acta Mech Solida Sin 23:271–282

    Article  Google Scholar 

  • Sun W, Fish J (2021a) Coupling of non-ordinary state-based peridynamics and finite element method for fracture propagation in saturated porous media. Int J Numer Anal Methods Geomech 45:1260–1281

    Article  Google Scholar 

  • Sun W, Fish J (2021b) Superposition-based concurrent multiscale approaches for poromechanics. Int J Numer Methods Eng 122:1–26

    Article  MathSciNet  Google Scholar 

  • Tong Y, Shen WQ, Shao JF (2020a) An adaptive coupling method of state-based peridynamics theory and finite element method for modeling progressive failure process in cohesive materials. Comput Methods Appl Mech Eng 370:113248

    Article  ADS  MathSciNet  Google Scholar 

  • Tong Y, Shen W, Shao J, Chen J (2020b) A new bond model in peridynamics theory for progressive failure in cohesive brittle materials. Eng Fract Mech 223:106767

    Article  Google Scholar 

  • Wang Y, Han F, Lubineau G (2021) Strength-induced peridynamic modeling and simulation of fractures in brittle materials. Comput Methods Appl Mech Eng 374:113558

    Article  ADS  MathSciNet  Google Scholar 

  • Wang L, Mehrmashhadi J, Bobaru F (2023) Interfaces in dynamic brittle fracture of PMMA: a peridynamic analysis. Int J Fract. https://doi.org/10.1007/s10704-023-00731-w

    Article  Google Scholar 

  • Warren TL, Silling SA, Askari A et al (2009) A non-ordinary state-based peridynamic method to model solid material deformation and fracture. Int J Solids Struct 46:1186–1195

    Article  Google Scholar 

  • Wu JY, Qiu JF, Nguyen VP, Mandal TK, Zhuang LJ (2019) Computational modeling of localized failure in solids: XFEM vs PF-CZM. Comput Methods Appl Mech Eng 345:618–643

    Article  ADS  MathSciNet  Google Scholar 

  • Wu P, Zhao J, Chen Z, Bobaru F (2020) Validation of a stochastically homogenized peridynamic model for quasi-static fracture in concrete. Eng Fract Mech 237:107293

    Article  Google Scholar 

  • Wu P, Yang F, Chen Z, Bobaru F (2021) Stochastically homogenized peridynamic model for dynamic fracture analysis of concrete. Eng Fract Mech 253:107863

    Article  Google Scholar 

  • Yang L, Yang Y, Zheng H, Wu Z (2021) An explicit representation of cracks in the variational phase field method for brittle fractures. Comput Methods Appl Mech Eng 387:114127

    Article  ADS  MathSciNet  Google Scholar 

  • Yu H, Sun Y (2021) Bridging the gap between local and nonlocal numerical methods—a unified variational framework for non-ordinary state-based peridynamics. Comput Methods Appl Mech Eng 384:113962

    Article  ADS  MathSciNet  Google Scholar 

  • Yu H, Chen X, Sun Y (2020) A generalized bond-based peridynamic model for quasi-brittle materials enriched with bond tension–rotation–shear coupling effects. Comput Methods Appl Mech Eng 372:113405

    Article  ADS  MathSciNet  Google Scholar 

  • Zaccariotto M, Luongo F, Sarego G, Galvanetto U (2015) Examples of applications of the peridynamic theory to the solution of static equilibrium problems. Aeronaut J 119:677–700

    Article  Google Scholar 

  • Zaccariotto M, Tomasi D, Galvanetto U (2017) An enhanced coupling of PD grids to FE meshes. Mech Res Commun 84:125–135

    Article  Google Scholar 

  • Zaccariotto M, Mudric T, Tomasi D et al (2018) Coupling of FEM meshes with peridynamic grids. Comput Methods Appl Mech Eng 330:471–497

    Article  ADS  MathSciNet  Google Scholar 

  • Zhang Y, Lackner R, Zeiml M, Mang HA (2015) Strong discontinuity embedded approach with standard SOS formulation: element formulation, energy-based crack-tracking strategy, and validations. Comput Methods Appl Mech Eng 287:335–366

    Article  ADS  MathSciNet  Google Scholar 

  • Zhao LY, Shao JF, Zhu QZ (2018) Analysis of localized cracking in quasi-brittle materials with a micro-mechanics based friction-damage approach. J Mech Phys Solids 119:163–187

    Article  ADS  MathSciNet  Google Scholar 

  • Zhu QZ, Ni T (2017) Peridynamic formulations enriched with bond rotation effects. Int J Eng Sci 121:118–129

    Article  MathSciNet  Google Scholar 

  • Zi G, Belytschko T (2003) New crack-tip elements for XFEM and applications to cohesive cracks. Int J Numer Methods Eng 57:2221–2240

    Article  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11890681, 12032001 and 11521202). The first author would like to thank Pro. Ziguang Chen from Huazhong University of Science and Technology for kind support and encouragement.

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Authors and Affiliations

  1. Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871, People’s Republic of China

    Yehui Bie, Kuanjie Ding, Zhifu Zhao & Yueguang Wei

  2. LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Zhifu Zhao

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  1. Yehui Bie
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  2. Kuanjie Ding
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  3. Zhifu Zhao
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  4. Yueguang Wei
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Contributions

Yehui Bie: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. Kuanjie Ding: Investigation, Writing – original draft, Writing – review & editing. Zhifu Zhao: Writing – original draft, Writing – review & editing, Investigation. Yueguang Wei: Funding acquisition, Supervision, Writing – original draft, Writing – review & editing.

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Correspondence to Yueguang Wei.

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Appendix B. Warm Reminder: Please open the following videos with Windows Media Player. Supplementary file1 (mp4 913kb)

Appendix A

Appendix A

The derivation and calculation of the nonlocal correctional peridynamic stiffness matrices for plane problems

The correctional peridynamic force vector state of the stabilized DH-PDCM is expressed as

$$ \begin{gathered} \underline{{\mathbf{T}}}^{s} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\mathbf{x}}_{j} - {\mathbf{x}}_{i} } \right\rangle = \frac{1}{2}\underline {\omega } (\left| {{\mathbf{x}}_{j} - {\mathbf{x}}_{i} } \right|) \cdot {\mathbf{C}}\left\langle {{\mathbf{x}}_{j} - {\mathbf{x}}_{i} } \right\rangle \cdot {\mathbf{z}}\left\langle {{\mathbf{x}}_{j} - {\mathbf{x}}_{i} } \right\rangle - \frac{1}{2}\underline {\omega } (\left| {{\mathbf{x}}_{j} - {\mathbf{x}}_{i} } \right|) \hfill \\ \, \cdot \left( {\left. {\int_{{H_{i} }} {\underline {\omega } (\left| {{\mathbf{x}}_{k} - {\mathbf{x}}_{i} } \right|)} \cdot {\mathbf{C}}\left\langle {{\mathbf{x}}_{k} - {\mathbf{x}}_{i} } \right\rangle \cdot {\mathbf{z}}\left\langle {{\mathbf{x}}_{k} - {\mathbf{x}}_{i} } \right\rangle \otimes {(}{\mathbf{x}}_{k} - {\mathbf{x}}_{i} {\text{)d}}V_{k} } \right) \cdot {\mathbf{K}}_{i}^{ - 1} \cdot ({\mathbf{x}}_{j} - {\mathbf{x}}_{i} )} \right. \hfill \\ \end{gathered} $$
(A.1)

Substituting Eqs. (34 and 35) into Eq. (A.1), we have

$$ \begin{gathered} \underline{{\mathbf{T}}}^{s} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle = \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot \left[ {{\mathbf{C}}\left\langle {{\varvec{\upxi}}} \right\rangle \cdot {\mathbf{z}}\left\langle {{\varvec{\upxi}}} \right\rangle - \left( {\left. {\int_{{H_{i} }} {\underline {\omega } (\left| {{\varvec{\upzeta}}} \right|)} \cdot {\mathbf{C}}\left\langle {{\varvec{\upzeta}}} \right\rangle \cdot {\mathbf{z}}\left\langle {{\varvec{\upzeta}}} \right\rangle \otimes {(}{{\varvec{\upzeta}}}{\text{)d}}V_{k} } \right) \cdot {\mathbf{K}}_{i}^{ - 1} \cdot {{\varvec{\upxi}}}} \right.} \right] \hfill \\ \, = \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot c_{0} \cdot \frac{{{{\varvec{\upxi}}} \otimes {{\varvec{\upxi}}}}}{{\left| {{\varvec{\upxi}}} \right|^{3} }} \cdot \left( {{{\varvec{\upeta}}}^{ij} - \overline{\nabla } \otimes {\mathbf{u}}_{i} \cdot {{\varvec{\upxi}}}} \right) \hfill \\ \, - \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot \left( {\left. {\int_{{H_{i} }} {\underline {\omega } (\left| {{\varvec{\upzeta}}} \right|)} \cdot c_{0} \cdot \frac{{{{\varvec{\upzeta}}} \otimes {{\varvec{\upzeta}}}}}{{\left| {{\varvec{\upzeta}}} \right|^{3} }} \cdot \left( {{{\varvec{\upeta}}}^{ik} - \overline{\nabla } \otimes {\mathbf{u}}_{i} \cdot {{\varvec{\upzeta}}}} \right) \otimes {{\varvec{\upzeta}}}{\text{d}}V_{k} } \right) \cdot {\mathbf{K}}_{i}^{ - 1} \cdot {{\varvec{\upxi}}}} \right. \hfill \\ \end{gathered} $$
(A.2)

where the vectors \({{\varvec{\upxi}}}\), \({{\varvec{\upzeta}}}\), \({{\varvec{\upeta}}}^{ij}\), the matrix \({\mathbf{K}}_{i}^{ - 1}\) and their components in Eq. (A.2) are defined as

$$ \begin{gathered} {{\varvec{\upxi}}} = {\mathbf{x}}_{j} - {\mathbf{x}}_{i} , \, {{\varvec{\upxi}}} = [\xi_{1} \, \xi_{2} ]^{T} . \, {{\varvec{\upzeta}}} = {\mathbf{x}}_{k} - {\mathbf{x}}_{i} , \, {{\varvec{\upzeta}}} = [\zeta_{1} \, \zeta_{2} ]^{T} . \, \hfill \\ {{\varvec{\upeta}}}^{ij} = {\mathbf{u}}_{j} - {\mathbf{u}}_{i} , \, {{\varvec{\upeta}}}^{ij} = [\eta_{1}^{ij} \, \eta_{2}^{ij} ]^{T} . \, {{\varvec{\upeta}}}^{ik} = {\mathbf{u}}_{k} - {\mathbf{u}}_{i} , \, {{\varvec{\upeta}}}^{ik} = [\eta_{1}^{ik} \, \eta_{2}^{ik} ]^{T} . \hfill \\ \end{gathered} $$
(A.3)
$$ {\mathbf{K}}_{i}^{ - 1} = \left[ \begin{gathered} k_{11} \, k_{12} \hfill \\ k_{21} \, k_{22} \hfill \\ \end{gathered} \right] $$
(A.4)

To make it clear, the correctional force vector state is divided into four parts that are derived separately, that is

$$ \underline{{\mathbf{T}}}^{s} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle = \underline{{\mathbf{T}}}^{s1} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle - \underline{{\mathbf{T}}}^{s2} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle - \underline{{\mathbf{T}}}^{s3} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle + \underline{{\mathbf{T}}}^{s4} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle $$
(A.5)
$$ \underline{{\mathbf{T}}}^{s1} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle = \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot c_{0} \cdot \frac{{{{\varvec{\upxi}}} \otimes {{\varvec{\upxi}}}}}{{\left| {{\varvec{\upxi}}} \right|^{3} }} \cdot {{\varvec{\upeta}}}^{ij} = \underline {K}^{s1} \left[ {{\mathbf{x}}_{i} } \right]_{{2 \times \left( {2 \cdot N + 2} \right)}} \cdot {\mathbf{U}}\left[ {{\mathbf{x}}_{i} } \right]_{{\left( {2 \cdot N + 2} \right) \times 1}} $$
(A.6)
$$ \underline{{\mathbf{T}}}^{s2} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle = \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot c_{0} \cdot \frac{{{{\varvec{\upxi}}} \otimes {{\varvec{\upxi}}}}}{{\left| {{\varvec{\upxi}}} \right|^{3} }} \cdot \overline{\nabla } \otimes {\mathbf{u}}_{i} \cdot {{\varvec{\upxi}}} = \underline {K}^{s2} \left[ {{\mathbf{x}}_{i} } \right]_{{2 \times \left( {2 \cdot N + 2} \right)}} \cdot {\mathbf{U}}\left[ {{\mathbf{x}}_{i} } \right]_{{\left( {2 \cdot N + 2} \right) \times 1}} $$
(A.7)
$$ \begin{gathered} \underline{{\mathbf{T}}}^{s3} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle = \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot \left( {\left. {\int_{{H_{i} }} {\underline {\omega } (\left| {{\varvec{\upzeta}}} \right|)} \cdot c_{0} \cdot \frac{{{{\varvec{\upzeta}}} \otimes {{\varvec{\upzeta}}}}}{{\left| {{\varvec{\upzeta}}} \right|^{3} }} \cdot {{\varvec{\upeta}}}^{ik} \otimes {{\varvec{\upzeta}}}{\text{d}}V_{k} } \right) \cdot {\mathbf{K}}_{i}^{ - 1} \cdot {{\varvec{\upxi}}}} \right. \hfill \\ \, = \underline {K}^{s3} \left[ {{\mathbf{x}}_{i} } \right]_{{2 \times \left( {2 \cdot N + 2} \right)}} \cdot {\mathbf{U}}\left[ {{\mathbf{x}}_{i} } \right]_{{\left( {2 \cdot N + 2} \right) \times 1}} \hfill \\ \end{gathered} $$
(A.8)
$$ \begin{gathered} \underline{{\mathbf{T}}}^{s4} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle = \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot \left( {\left. {\int_{{H_{i} }} {\underline {\omega } (\left| {{\varvec{\upzeta}}} \right|)} \cdot c_{0} \cdot \frac{{{{\varvec{\upzeta}}} \otimes {{\varvec{\upzeta}}}}}{{\left| {{\varvec{\upzeta}}} \right|^{3} }} \cdot \overline{\nabla } \otimes {\mathbf{u}}_{i} \cdot {{\varvec{\upzeta}}} \otimes {{\varvec{\upzeta}}}{\text{d}}V_{k} } \right) \cdot {\mathbf{K}}_{i}^{ - 1} \cdot {{\varvec{\upxi}}}} \right. \hfill \\ \, = \underline {K}^{s4} \left[ {{\mathbf{x}}_{i} } \right]_{{2 \times \left( {2 \cdot N + 2} \right)}} \cdot {\mathbf{U}}\left[ {{\mathbf{x}}_{i} } \right]_{{\left( {2 \cdot N + 2} \right) \times 1}} \hfill \\ \end{gathered} $$
(A.9)

Therefore, the nonlocal correctional peridynamic stiffness matrix can be expressed as

$$ \underline {K}^{s} \left[ {{\mathbf{x}}_{i} } \right] = \underline {K}^{s1} \left[ {{\mathbf{x}}_{i} } \right] - \underline {K}^{s2} \left[ {{\mathbf{x}}_{i} } \right] - \underline {K}^{s3} \left[ {{\mathbf{x}}_{i} } \right] + \underline {K}^{s4} \left[ {{\mathbf{x}}_{i} } \right] $$
(A.10)

To derive \(\underline {K}^{s1} \left[ {{\mathbf{x}}_{i} } \right]\), \(\underline {K}^{s2} \left[ {{\mathbf{x}}_{i} } \right]\), \(\underline {K}^{s3} \left[ {{\mathbf{x}}_{i} } \right]\) and \(\underline {K}^{s4} \left[ {{\mathbf{x}}_{i} } \right]\), the nonlocal stabilized base vector \({\mathbf{B}}^{ij}\) and base matrix \({\mathbf{C}}^{ij}\) for the bond \( \left\langle {{\mathbf{x}}_{j} - {\mathbf{x}}_{i} } \right\rangle \) is defined in the form of

$$ {\mathbf{B}}^{ij} = \left[ \begin{gathered} C_{11}^{ij} \cdot \eta_{1}^{ij} + C_{12}^{ij} \cdot \eta_{2}^{ij} \hfill \\ C_{21}^{ij} \cdot \eta_{1}^{ij} + C_{22}^{ij} \cdot \eta_{2}^{ij} \hfill \\ \end{gathered} \right]_{2 \times 1}^{T} \, {\mathbf{C}}^{ij} = \left[ \begin{gathered} C_{11}^{ij} \, C_{12}^{ij} \hfill \\ C_{21}^{ij} \, C_{22}^{ij} \hfill \\ \end{gathered} \right]_{2 \times 2} \, {\mathbf{C}}^{ij} = \frac{{\partial {\mathbf{B}}^{ij} }}{{\partial {{\varvec{\upeta}}}^{ij} }} $$
(A.11)

It will be seen that the base matrix is symmetric in the following part. For the first part of the correctional force vector state \(\underline{{\mathbf{T}}}^{s1} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle\)

$$ \begin{gathered} \underline{{\mathbf{T}}} ^{{s1}} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {\mathbf{\xi }} \right\rangle = {\text{ }}\mathchar'26\mkern-10mu\lambda _{0} \cdot {\mathbf{B}}^{{ij}} ,{\text{ }}{\mathbf{B}}^{{ij}} = {\mathbf{\xi }} \otimes {\mathbf{\xi }} \cdot {\mathbf{\eta }}^{{ij}} = \left[ \begin{gathered} \xi _{1} \cdot \xi _{1} \cdot \eta _{1}^{{ij}} + \xi _{1} \cdot \xi _{2} \cdot \eta _{2}^{{ij}} \hfill \\ \xi _{1} \cdot \xi _{2} \cdot \eta _{1}^{{ij}} + \xi _{2} \cdot \xi _{2} \cdot \eta _{2}^{{ij}} \hfill \\ \end{gathered} \right] \hfill \\ {\text{ }}\mathchar'26\mkern-10mu\lambda _{0} {\text{ = }}\frac{1}{2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\xi }} \right|) \cdot \frac{{c_{0} }}{{\left| {\mathbf{\xi }} \right|^{3} }},{\text{ }}{\mathbf{C}}^{{ij}} = \frac{{\partial {\mathbf{B}}^{{ij}} }}{{\partial {\mathbf{\eta }}^{{ij}} }} = \left[ \begin{gathered} C_{{11}}^{{ij}} {\text{ }}C_{{12}}^{{ij}} \hfill \\ C_{{21}}^{{ij}} {\text{ }}C_{{22}}^{{ij}} \hfill \\ \end{gathered} \right]_{{2 \times 2}} = \left[ \begin{gathered} \xi _{1} \cdot \xi _{1} {\text{ }}\xi _{1} \cdot \xi _{2} \hfill \\ \xi _{1} \cdot \xi _{2} {\text{ }}\xi _{2} \cdot \xi _{2} \hfill \\ \end{gathered} \right]_{{2 \times 2}} \hfill \\ \end{gathered} $$
(A.12)

where \( \mathchar'26\mkern-10mu\lambda _{0} \) is called the scalar bond state in this paper.

It can be seen that the nonlocal stabilized base vector \({\mathbf{B}}^{ij}\) and base matrix \({\mathbf{C}}^{ij}\) can be very easy to be derived without any tool. Once the base matrix is calculated, the nonlocal correctional stiffness matrix \(\underline {K}^{s1} \left[ {{\mathbf{x}}_{i} } \right]\) of the first part can be obtained in Algorithm A.1.

Algorithm A.1
figure b

The calculation of the nonlocal correctional stiffness matrix of the first part

Full size image

For the second part of the correctional force vector state \(\underline{{\mathbf{T}}}^{s2} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle\), the nonlocal stabilized base vector \({\mathbf{B}}^{ik}\) and base matrix \({\mathbf{C}}^{ik}\) can be expressed as

$$ \begin{gathered} \underline{{\mathbf{T}}}^{s2} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle = \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot c_{0} \cdot \frac{{{{\varvec{\upxi}}} \otimes {{\varvec{\upxi}}}}}{{\left| {{\varvec{\upxi}}} \right|^{3} }} \cdot \overline{\nabla } \otimes {\mathbf{u}}_{i} \cdot {{\varvec{\upxi}}} \hfill \\ = \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot c_{0} \cdot \frac{{{{\varvec{\upxi}}} \otimes {{\varvec{\upxi}}}}}{{\left| {{\varvec{\upxi}}} \right|^{3} }} \cdot \left( {\int_{{H_{i} }} {\underline{\omega } } (\left| {{\varvec{\upzeta}}} \right|) \cdot {{\varvec{\upeta}}}^{ik} \otimes {{\varvec{\upzeta}}}{\text{d}}V_{k} \cdot {\mathbf{K}}_{i}^{ - 1} } \right) \cdot {{\varvec{\upxi}}} \hfill \\ = \frac{1}{2}\underline {\omega } (\left| {{\varvec{\upxi}}} \right|) \cdot c_{0} \cdot \frac{{{{\varvec{\upxi}}} \otimes {{\varvec{\upxi}}}}}{{\left| {{\varvec{\upxi}}} \right|^{3} }} \cdot \left( {\sum\limits_{{H_{i} }} {\underline {\omega } (\left| {{\varvec{\upzeta}}} \right|) \cdot {{\varvec{\upeta}}}^{ik} \otimes {{\varvec{\upzeta}}} \cdot V_{k} \cdot {\mathbf{K}}_{i}^{ - 1} } } \right) \cdot {{\varvec{\upxi}}} \hfill \\ \end{gathered} $$
(A.13)
$$ \begin{gathered} \underline{{\mathbf{T}}} ^{{s2}} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {\mathbf{\xi }} \right\rangle = \sum\limits_{{H_{i} }} {{\text{ }}\mathchar'26\mkern-10mu\lambda _{0} \cdot {\mathbf{B}}^{{ik}} } ,{\text{ }}{\mathbf{B}}^{{ik}} = {\mathbf{\xi }} \otimes {\mathbf{\xi }} \cdot \left( {{\mathbf{\eta }}^{{ik}} \otimes {\mathbf{\zeta }} \cdot {\mathbf{K}}_{i} ^{{ - 1}} } \right) \cdot {\mathbf{\xi }} \hfill \\ {\text{ }}\mathchar'26\mkern-10mu\lambda _{0} {\text{ = }}\frac{1}{2} \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\xi }} \right|) \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\zeta }} \right|) \cdot \frac{{c_{0} }}{{\left| {\mathbf{\xi }} \right|^{3} }} \cdot V_{k} ,{\text{ }}{\mathbf{C}}^{{ik}} = \frac{{\partial {\mathbf{B}}^{{ik}} }}{{\partial {\mathbf{\eta }}^{{ik}} }} = \left[ \begin{gathered} C_{{11}}^{{ik}} {\text{ }}C_{{12}}^{{ik}} \hfill \\ C_{{21}}^{{ik}} {\text{ }}C_{{22}}^{{ik}} \hfill \\ \end{gathered} \right]_{{2 \times 2}} \hfill \\ \end{gathered} $$
(A.14)

One can see from Eq. (A.14), it is more complicated than Eq. (A.12) to obtain the base matrix \({\mathbf{C}}^{ik}\). Using the software Matlab, it is convenient to derive the mathematical expression of the base matrix.

$$ \begin{gathered} {\mathbf{C}}^{{ik}} = \frac{{\partial {\mathbf{B}}^{{ik}} }}{{\partial {\mathbf{\eta }}^{{ik}} }} = \left[ \begin{gathered} C_{{11}}^{{ik}} {\text{ }}C_{{12}}^{{ik}} \hfill \\ C_{{21}}^{{ik}} {\text{ }}C_{{22}}^{{ik}} \hfill \\ \end{gathered} \right]_{{2 \times 2}} = {\text{ }}\mathchar'26\mkern-10mu\lambda _{1} \cdot \left[ \begin{gathered} \xi _{1} \cdot \xi _{1} {\text{ }}\xi _{1} \cdot \xi _{2} \hfill \\ \xi _{1} \cdot \xi _{2} {\text{ }}\xi _{2} \cdot \xi _{2} \hfill \\ \end{gathered} \right]_{{2 \times 2}} \hfill \\ \mathchar'26\mkern-10mu\lambda _{1} = \left( {k_{{11}} \cdot \xi _{1} \cdot \zeta _{1} + k_{{12}} \cdot \zeta _{1} \cdot \xi _{2} + k_{{21}} \cdot \xi _{1} \cdot \zeta _{2} + k_{{22}} \cdot \xi _{2} \cdot \zeta _{2} } \right) \hfill \\ \end{gathered} $$
(A.15)

Similarly, the nonlocal correctional stiffness matrix \(\underline {K}^{s2} \left[ {{\mathbf{x}}_{i} } \right]\) of the second part is obtained in Algorithm A.2.

Algorithm A.2
figure c

The calculation of the nonlocal correctional stiffness matrix of the second part

Full size image

For the third part of the correctional force vector state \(\underline{{\mathbf{T}}}^{s3} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle\), the nonlocal stabilized base vector \({\mathbf{B}}^{ik}\) and the component of base matrix \({\mathbf{C}}^{ik}\) can be calculated as

$$ \begin{gathered} \underline{{\mathbf{T}}} ^{{s3}} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {\mathbf{\xi }} \right\rangle = \frac{1}{2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\xi }} \right|) \cdot \left( {\left. {\int_{{H_{i} }} {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\zeta }} \right|)} \cdot c_{0} \cdot \frac{{{\mathbf{\zeta }} \otimes {\mathbf{\zeta }}}}{{\left| {\mathbf{\zeta }} \right|^{3} }} \cdot {\mathbf{\eta }}^{{ik}} \otimes {\mathbf{\zeta }}{\text{d}}V_{k} } \right) \cdot {\mathbf{K}}_{i} ^{{ - 1}} \cdot {\mathbf{\xi }}} \right. = \sum\limits_{{H_{i} }} {{\text{ }}\mathchar'26\mkern-10mu\lambda _{0} \cdot {\mathbf{B}}^{{ik}} } \hfill \\ {\text{ }}\mathchar'26\mkern-10mu\lambda _{0} {\text{ = }}\frac{1}{2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\xi }} \right|) \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\zeta }} \right|) \cdot \frac{{c_{0} }}{{\left| {\mathbf{\xi }} \right|^{3} }} \cdot V_{k} ,\,{\mathbf{B}}^{{ik}} = {\mathbf{\zeta }} \otimes {\mathbf{\zeta }} \cdot \left( {{\mathbf{\eta }}^{{ik}} \otimes {\mathbf{\zeta }}} \right) \cdot {\mathbf{K}}_{i} ^{{ - 1}} \cdot {\mathbf{\xi }} \hfill \\ \end{gathered} $$
(A.16)
$$ \begin{gathered} C_{11}^{ik} = \xi_{1} \cdot \left( {k_{11} \zeta_{1}^{3} + k_{21} \zeta_{2} \zeta_{1}^{2} } \right) + \xi_{2} \cdot \left( {k_{12} \zeta_{1}^{3} + k_{22} \zeta_{2} \zeta_{1}^{2} } \right) \hfill \\ C_{12}^{ik} = \xi_{1} \cdot \left( {k_{11} \zeta_{2} \zeta_{1}^{2} + k_{21} \zeta_{1} \zeta_{2}^{2} } \right) + \xi_{2} \cdot \left( {k_{12} \zeta_{2} \zeta_{1}^{2} + k_{22} \zeta_{1} \zeta_{2}^{2} } \right) \hfill \\ C_{21}^{ik} = \xi_{1} \cdot \left( {k_{11} \zeta_{2} \zeta_{1}^{2} + k_{21} \zeta_{1} \zeta_{2}^{2} } \right) + \xi_{2} \cdot \left( {k_{12} \zeta_{2} \zeta_{1}^{2} + k_{22} \zeta_{1} \zeta_{2}^{2} } \right) \hfill \\ C_{22}^{ik} = \xi_{1} \cdot \left( {k_{11} \zeta_{2}^{3} + k_{21} \zeta_{1} \zeta_{2}^{2} } \right) + \xi_{2} \cdot \left( {k_{12} \zeta_{2}^{3} + k_{22} \zeta_{1} \zeta_{2}^{2} } \right) \hfill \\ \end{gathered} $$
(A.17)

The construction algorithm of the nonlocal correctional stiffness matrix \(\underline {K}^{s3} \left[ {{\mathbf{x}}_{i} } \right]\) is the same as that of \(\underline {K}^{s2} \left[ {{\mathbf{x}}_{i} } \right]\) in Algorithm A.2.

For the last part of the correctional force vector state \(\underline{{\mathbf{T}}}^{s4} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {{\varvec{\upxi}}} \right\rangle\), the nonlocal stabilized base vector \({\mathbf{B}}^{il}\) and the component of base matrix \({\mathbf{C}}^{il}\) can be calculated as

$$ \begin{gathered} \underline{{\mathbf{T}}} ^{{s4}} \left[ {{\mathbf{x}}_{i} } \right]\left\langle {\mathbf{\xi }} \right\rangle = \frac{1}{2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\xi }} \right|) \cdot \left( {\left. {\int_{{H_{i} }} {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\zeta }} \right|)} \cdot c_{0} \cdot \frac{{{\mathbf{\zeta }} \otimes {\mathbf{\zeta }}}}{{\left| {\mathbf{\zeta }} \right|^{3} }} \cdot \bar{\nabla } \otimes {\mathbf{u}}_{i} \cdot {\mathbf{\zeta }} \otimes {\mathbf{\zeta }}{\text{d}}V_{k} } \right) \cdot {\mathbf{K}}_{i} ^{{ - 1}} \cdot {\mathbf{\xi }}} \right. \hfill \\ {\text{ }} = \frac{1}{2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\xi }} \right|) \cdot \left( {\left. {\int_{{H_{i} }} {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\zeta }} \right|)} \cdot c_{0} \cdot \frac{{{\mathbf{\zeta }} \otimes {\mathbf{\zeta }}}}{{\left| {\mathbf{\zeta }} \right|^{3} }} \cdot \left( {\int_{{H_{i} }} {\underline{\omega } } (\left| {\mathbf{\chi }} \right|) \cdot {\mathbf{\eta }}^{{il}} \otimes {\mathbf{\chi }}{\text{d}}V_{l} \cdot {\mathbf{K}}_{i} ^{{ - 1}} } \right) \cdot {\mathbf{\zeta }} \otimes {\mathbf{\zeta }}{\text{d}}V_{k} } \right) \cdot {\mathbf{K}}_{i} ^{{ - 1}} \cdot {\mathbf{\xi }}} \right. \hfill \\ {\text{ }} = \sum\limits_{{H_{i} }} {\left( {\sum\limits_{{H_{i} }} {{\text{ }}\mathchar'26\mkern-10mu\lambda _{0} \cdot {\mathbf{B}}^{{il}} } } \right)} \hfill \\ {\text{ }}\mathchar'26\mkern-10mu\lambda _{0} {\text{ = }}\frac{1}{2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\xi }} \right|) \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\zeta }} \right|) \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } (\left| {\mathbf{\chi }} \right|) \cdot \frac{{c_{0} }}{{\left| {\mathbf{\xi }} \right|^{3} }}V_{l} \cdot V_{k} ,{\mathbf{B}}^{{il}} = {\mathbf{\zeta }} \otimes {\mathbf{\zeta }} \cdot \left( {{\mathbf{\eta }}^{{il}} \otimes {\mathbf{\chi }} \cdot {\mathbf{K}}_{i} ^{{ - 1}} } \right) \cdot {\mathbf{\zeta }} \otimes {\mathbf{\zeta }} \cdot {\mathbf{K}}_{i} ^{{ - 1}} \cdot {\mathbf{\xi }} \hfill \\ \end{gathered} $$
(A.18)
$$ \begin{gathered} {\mathbf{C}}^{{il}} = \frac{{\partial {\mathbf{B}}^{{il}} }}{{\partial {\mathbf{\eta }}^{{il}} }} = \left[ \begin{gathered} C_{{11}}^{{ik}} {\text{ }}C_{{12}}^{{ik}} \hfill \\ C_{{21}}^{{ik}} {\text{ }}C_{{22}}^{{ik}} \hfill \\ \end{gathered} \right]_{{2 \times 2}} = {\text{ }}\mathchar'26\mkern-10mu\lambda _{1} \cdot {\text{ }}\mathchar'26\mkern-10mu\lambda _{2} \cdot \left[ \begin{gathered} \zeta _{1} \cdot \zeta _{1} {\text{ }}\zeta _{1} \cdot \zeta _{2} \hfill \\ \zeta _{1} \cdot \zeta _{2} {\text{ }}\zeta _{2} \cdot \zeta _{2} \hfill \\ \end{gathered} \right]_{{2 \times 2}} \hfill \\ \mathchar'26\mkern-10mu\lambda _{1} \left( {k_{{11}} \cdot \xi _{1} \cdot \zeta _{1} + k_{{12}} \cdot \xi _{2} \cdot \zeta _{1} + k_{{21}} \cdot \xi _{1} \cdot \zeta _{2} + k_{{22}} \cdot \xi _{2} \cdot \zeta _{2} } \right) \hfill \\ \mathchar'26\mkern-10mu\lambda _{2} = \left( {k_{{11}} \cdot \zeta _{1} \cdot \chi _{1} + k_{{12}} \cdot \zeta _{2} \cdot \chi _{1} + k_{{21}} \cdot \zeta _{1} \cdot \chi _{2} + k_{{22}} \cdot \zeta _{2} \cdot \chi _{2} } \right) \hfill \\ \end{gathered} $$
(A.19)

where \({{\varvec{\upchi}}} = {\mathbf{x}}_{l} - {\mathbf{x}}_{i} , \, {{\varvec{\upchi}}} = [\chi_{1} \, \chi_{2} ]^{T}\).

Similarly, the nonlocal correctional stiffness matrix \(\underline {K}^{s4} \left[ {{\mathbf{x}}_{i} } \right]\) of the last part is obtained in Algorithm A.3.

Algorithm A.3
figure d

The calculation of the nonlocal correctional stiffness matrix of the last part

Full size image

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Bie, Y., Ding, K., Zhao, Z. et al. The adaptive coupling of dual-horizon peridynamic element and finite element for the progressive failure of materials. Int J Fract 245, 89–114 (2024). https://doi.org/10.1007/s10704-023-00758-z

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