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Original Articles

A phase field approach to fracture for hyperelastic and visco-hyperelastic materials applied to pre-stressed cylindrical structures

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Pages 749-768 | Received 10 Jul 2022, Accepted 01 Sep 2022, Published online: 18 Oct 2022

References

  • Y. Abou Msallem, F. Jacquemin, N. Boyard, A. Poitou, D. Delaunay, and S. Chatel, Material characterization and residual stresses simulation during the manufacturing process of epoxy matrix composites, Compos. A: Appl. Sci. Manuf., vol. 41, no. 1, pp. 108–115, 2010. DOI: 10.1016/j.compositesa.2009.09.025.
  • G.A. Holzapfel and R.W. Ogden, Modelling the layer-specific three-dimensional residual stresses in arteries, with an application to the human aorta, J. R. Soc. Interface, vol. 7, no. 46, pp. 787–799, 2010. DOI: 10.1098/rsif.2009.0357.
  • N.-K. Jha, J. Reinoso, H. Dehghani, and J. Merodio, A computational model for fiber-reinforced composites: Hyperelastic constitutive formulation including residual stresses and damage, Comput Mech., vol. 63, no. 5, pp. 931–948, 2019. DOI: 10.1007/s00466-018-1630-5.
  • J.D. Humphrey, Vascular adaptation and mechanical homeostasis at tissue, cellular, and sub-cellular levels, Cell Biochem. Biophys., vol. 50, no. 2, pp. 53–78, 2008. 10.1007/s12013-007-9002-3.
  • A. Alhayani, J. Giraldo, J. Rodríguez, and J. Merodio, Computational modelling of bulging of inflated cylindrical shells applicable to aneurysm formation and propagation in arterial wall tissue, Finite Elem. Anal. Des., vol. 73, pp. 20–29, 2013. DOI: 10.1016/j.finel.2013.05.001.
  • A. Alhayani, J. Rodríguez, and J. Merodio, Competition between radial expansion and axial propagation in bulging of inflated cylinders with application to aneurysms propagation in arterial wall tissue, Int. J. Eng. Sci., vol. 85, pp. 74–89, 2014. DOI: 10.1016/j.ijengsci.2014.08.008.
  • H. Demirkoparan and J. Merodio, Bulging bifurcation of inflated circular cylinders of doubly fiber-reinforced hyperelastic material under axial loading and swelling, Math. Mech. Solids, vol. 22, no. 4, pp. 666–682, 2017. DOI: 10.1177/1081286515600045.
  • J. Merodio and D. Haughton, Bifurcation of thick-walled cylindrical shells and the mechanical response of arterial tissue affected by marfan’s syndrome, Mech. Res. Commun., vol. 37, no. 1, pp. 1–6, 2010. DOI: 10.1016/j.mechrescom.2009.10.006.
  • N. Jha, J. Merodio, and J. Reinoso, A general non-local constitutive relation for residually stressed solids, Mech. Res. Commun., vol. 101, p. 103421, 2019. DOI: 10.1016/j.mechrescom.2019.103421.
  • C. Linder, M. Tkachuk, and C. Miehe, A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity, J. Mech. Phys. Solids, vol. 59, no. 10, pp. 2134–2156, 2011. DOI: 10.1016/j.jmps.2011.05.005.
  • C. Miehe and J. Keck, Superimposed finite elastic–viscoelastic–plastoelastic stress response with damage in filled rubbery polymers. experiments, modelling and algorithmic implementation, J. Mech. Phys. Solids, vol. 48, no. 2, pp. 323–365, 2000. DOI: 10.1016/S0022-5096(99)00017-4.
  • J. Simo, On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects, Comput. Methods Appl. Mech. Eng., vol. 60, no. 2, pp. 153–173, 1987. DOI: 10.1016/0045-7825(87)90107-1.
  • S. Govindjee and J.C. Simo, Mullins’ effect and the strain amplitude dependence of the storage modulus, Int. J. Solids Struct., vol. 29, no. 14–15, pp. 1737–1751, 1992. DOI: 10.1016/0020-7683(92)90167-R.
  • G.A. Holzapfel, On large strain viscoelasticity: Continuum formulation and finite element applications to elastomeric structures, Int. J. Numer. Methods Eng., vol. 39, no. 22, pp. 3903–3926, 1996. DOI: 10.1002/(SICI)1097-0207(19961130)39:22¡3903::AID-NME34¿3.0.CO;2-C.
  • M. Kaliske and H. Rothert, Formulation and implementation of three-dimensional viscoelasticity at small and finite strains, Comput. Mech., vol. 19, no. 3, pp. 228–239, 1997. DOI: 10.1007/s004660050171.
  • N. Jha, J. Reinoso, H. Dehghani, and J. Merodio, Constitutive modeling framework for residually stressed viscoelastic solids at finite strains, Mech. Res. Commun., vol. 95, pp. 79–84, 2019. DOI: 10.1016/j.mechrescom.2019.01.003.
  • J. Bergström and M. Boyce, Constitutive modeling of the large strain time-dependent behavior of elastomers, J. Mech. Phys. Solids, vol. 46, no. 5, pp. 931–954, 1998. DOI: 10.1016/S0022-5096(97)00075-6.
  • S. Reese and S. Govindjee, A theory of finite viscoelasticity and numerical aspects, Int. J. Solids Struct., vol. 35, no. 26–27, pp. 3455–3482, 1998. DOI: 10.1016/S0020-7683(97)00217-5.
  • J. Merodio and K.R. Rajagopal, On constitutive equations for anisotropic nonlinearly viscoelastic solids, Math. Mech. Solids, vol. 12, no. 2, pp. 131–147, 2007. DOI: 10.1177/1081286505055472.
  • O. Lopez-Pamies, M.I. Idiart, and T. Nakamura, Cavitation in elastomeric solids: I’a defect-growth theory, J. Mech. Phys. Solids, vol. 59, no. 8, pp. 1464–1487, 2011. DOI: 10.1016/j.jmps.2011.04.015.
  • W.G. Knauss, A review of fracture in viscoelastic materials, Int. J. Fract., vol. 196, no. 1–2, pp. 99–146, 2015. DOI: 10.1007/s10704-015-0058-6.
  • M.L. Cooke and D.D. Pollard, Fracture propagation paths under mixed mode loading within rectangular blocks of polymethyl methacrylate, J. Geophys. Res., vol. 101, no. B2, pp. 3387–3400, 1996. DOI: 10.1029/95JB02507.
  • A.J. Pons and A. Karma, Helical crack-front instability in mixed-mode fracture, Nature, vol. 464, no. 7285, pp. 85–89, 2010. 10.1038/nature08862.
  • N.A. Hocine, M.N. Abdelaziz, and A. Imad, Fracture problems of rubbers: J-integral estimation based upon η factors and an investigation on the strain energy density distribution as a local criterion, Int. J. Fract., vol. 117, no. 1, pp. 1–23, 2002. DOI: 10.1023/A:1020967429222.
  • R.A. Schapery, Correspondence principles and a generalizedj integral for large deformation and fracture analysis of viscoelastic media, Int. J. Fract., vol. 25, no. 3, pp. 195–223, 1984. DOI: 10.1007/BF01140837.
  • M. Kroon, Steady-state crack growth in rubber-like solids, Int. J. Fract., vol. 169, no. 1, pp. 49–60, 2011. DOI: 10.1007/s10704-010-9583-5.
  • G. Geißler, M. Kaliske, M. Nase, and W. Grellmann, Peel process simulation of sealed polymeric film computational modelling of experimental results, Eng. Comput., vol. 24, no. 6, pp. 586–607, 2007. DOI: 10.1108/02644400710774798.
  • I. Zreid, R. Fleischhauer, and M. Kaliske, A thermomechanically coupled viscoelastic cohesive zone model at large deformation, Int. J. Solids Struct., vol. 50, no. 25–26, pp. 4279–4291, 2013. DOI: 10.1016/j.ijsolstr.2013.08.031.
  • T.L. Warren, S.A. Silling, A. Askari, O. Weckner, M.A. Epton, and J. Xu, A non-ordinary state-based peridynamic method to model solid material deformation and fracture, Int. J. Solids Struct., vol. 46, no. 5, pp. 1186–1195, 2009. DOI: 10.1016/j.ijsolstr.2008.10.029.
  • Y. Huang, S. Oterkus, H. Hou, E. Oterkus, Z. Wei, and S. Zhang, Peridynamic model for visco-hyperelastic material deformation in different strain rates, Contin. Mech. Thermodyn., vol. 34, no. 4, pp. 977–1011, 2022. DOI: 10.1007/s00161-019-00849-0.
  • A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. A, vol. 221, pp. 163–198, 1920. DOI: 10.1098/rsta.1921.0006.
  • G. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, vol. 46, no. 8, pp. 1319–1342, 1998. DOI: 10.1016/S0022-5096(98)00034-9.
  • B. Bourdin, G.A. Francfort, and J.J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids, vol. 48, no. 4, pp. 797–826, 2000. DOI: 10.1016/S0022-5096(99)00028-9.
  • C. Miehe, F. Welshinger, and M. Hofacker, Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations, Int. J. Numer. Meth. Eng., vol. 83, no. 10, pp. 1273–1311, 2010. DOI: 10.1002/nme.2861.
  • C. Kuhn and R. Müller, A continuum phase field model for fracture, Eng. Fract. Mech., vol. 77, no. 18, pp. 3625–3634, 2010. DOI: 10.1016/j.engfracmech.2010.08.009.
  • J.-Y. Wu, A unified phase-field theory for the mechanics of damage and quasi-brittle failure, J. Mech. Phys. Solids, vol. 103, pp. 72–99, 2017. DOI: https://doi.org/10.1016/j.jmps.2017.03.015.
  • J.-Y. Wu and V.P. Nguyen, A length scale insensitive phase-field damage model for brittle fracture, J. Mech. Phys. Solids, vol. 119, pp. 20–42, 2018. DOI: https://doi.org/10.1016/j.jmps.2018.06.006.
  • F.P. Duda, A. Ciarbonetti, P.J. Sánchez, and A.E. Huespe, A phase-field/gradient damage model for brittle fracture in elastic-plastic solids, Int. J. Plast., vol. 65, pp. 269–296, 2015. DOI: 10.1016/j.ijplas.2014.09.005.
  • C. Kuhn, T. Noll, and R. Müller, On phase field modeling of ductile fracture, GAMM Mitteilungen., vol. 39, no. 1, pp. 35–54, 2016. DOI: 10.1002/gamm.201610003.
  • M.J. Borden, T.J.R. Hughes, C.M. Landis, A. Anvari, and I.J. Lee, A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects, Comput. Methods Appl. Mech. Eng., vol. 312, pp. 130–166, 2016. DOI: 10.1016/j.cma.2016.09.005.
  • R. Alessi, M. Ambati, T. Gerasimov, S. Vidoli, and L. De Lorenzis, Comparison of phase-field models of fracture coupled with plasticity. In: M.C.E. Oñate, D. Peric, E. de Souza-Neto (eds.), Advances in Computational Plasticity, Springer International Publishing, Cham, pp. 1–21, 2018. DOI: 10.1007/978-3-319-60885-3_1.
  • A. Dean, J. Reinoso, N. Jha, E. Mahdi, and R. Rolfes, A phase field approach for ductile fracture of short fibre reinforced composites, Theor. Appl. Fract. Mech., vol. 106, p. 102495, 2020. DOI: 10.1016/j.tafmec.2020.102495.
  • B. Yin and M. Kaliske, A ductile phase-field model based on degrading the fracture toughness: Theory and implementation at small strain, Comput. Methods Appl. Mech. Eng., vol. 366, p. 113068, 2020. DOI: 10.1016/j.cma.2020.113068.
  • S.S. Shishvan, S. Assadpour-Asl, and E. Martínez-Pañeda, A mechanism-based gradient damage model for metallic fracture, Eng. Fract. Mech., vol. 255, p. 107927, 2021. DOI: 10.1016/j.engfracmech.2021.107927.
  • A. Raina and C. Miehe, A phase-field model for fracture in biological tissues, Biomech. Model Mechanobiol., vol. 15, no. 3, pp. 479–496, 2016. 10.1007/s10237-015-0702-0.
  • B. Li, C. Peco, D. Millán, I. Arias, and M. Arroyo, Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy, Int. J. Numer. Methods Eng., vol. 102, no. 3–4, pp. 711–727, 2015. DOI: 10.1002/nme.4726.
  • O. Gültekin, H. Dal, and G.A. Holzapfel, A phase-field approach to model fracture of arterial walls: Theory and finite element analysis, Comput. Methods Appl. Mech. Eng., vol. 312, pp. 542–566, 2016. 10.1016/j.cma.2016.04.007.
  • M. Paggi, M. Corrado, and J. Reinoso, Fracture of solar-grade anisotropic polycrystalline silicon: A combined phase field–cohesive zone model approach, Comput. Methods Appl. Mech. Eng., vol. 330, pp. 123–148, 2018. DOI: 10.1016/j.cma.2017.10.021.
  • O. Gültekin, H. Dal, and G.A. Holzapfel, Numerical aspects of anisotropic failure in soft biological tissues favor energy-based criteria: A rate-dependent anisotropic crack phase-field model, Comput. Methods Appl. Mech. Eng., vol. 331, pp. 23–52, 2018. 10.1016/j.cma.2017.11.008.
  • J. Storm, D. Supriatna, and M. Kaliske, The concept of representative crack elements for phase-field fracture: Anisotropic elasticity and thermo-elasticity, Int. J. Numer. Methods Eng., vol. 121, no. 5, pp. 779–805, 2020. DOI: 10.1002/nme.6244.
  • B. Yin and M. Kaliske, An anisotropic phase-field model based on the equivalent crack surface energy density at finite strain, Comput. Methods Appl. Mech. Eng., vol. 369, p. 113202, 2020. DOI: 10.1016/j.cma.2020.113202.
  • H. Ren, X. Zhuang, C. Anitescu, and T. Rabczuk, An explicit phase field method for brittle dynamic fracture, Comput. Struct., vol. 217, pp. 45–56, 2019. DOI: 10.1016/j.compstruc.2019.03.005.
  • C. Steinke, K. Özenç, G. Chinaryan, and M. Kaliske, A comparative study of the r-adaptive material force approach and the phase-field method in dynamic fracture, Int. J. Fract., vol. 201, no. 1, pp. 97–118, 2016. DOI: 10.1007/s10704-016-0125-7.
  • P.K. Kristensen and E. Martínez-Pañeda, Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme, Theor. Appl. Fract. Mech., vol. 107, p. 102446, 2020. DOI: 10.1016/j.tafmec.2019.102446.
  • M. Paggi and J. Reinoso, Revisiting the problem of a crack impinging on an interface: A modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model, Comput. Methods Appl. Mech. Eng., vol. 321, pp. 145–172, 2017. DOI: 10.1016/j.cma.2017.04.004.
  • J. Reinoso, M. Paggi, and C. Linder, Phase field modeling of brittle fracture for enhanced assumed strain shells at large deformations: Formulation and finite element implementation, Comput. Mech., vol. 59, no. 6, pp. 981–1001, 2017. DOI: 10.1007/s00466-017-1386-3.
  • V. Carollo, J. Reinoso, and M. Paggi, A 3D finite strain model for intralayer and interlayer crack simulation coupling the phase field approach and cohesive zone model, Compos. Struct., vol. 182, pp. 636–651, 2017. DOI: 10.1016/j.compstruct.2017.08.095.
  • V. Carollo, T. Guillén-Hernández, J. Reinoso, and M. Paggi, Recent advancements on the phase field approach to brittle fracture for heterogeneous materials and structures, Adv. Model Simul. Eng. Sci., vol. 5, no. 1, p. 8, 2018. 10.1186/s40323-018-0102-y.
  • T. Guillén-Hernández, I. García, J. Reinoso, and M. Paggi, A micromechanical analysis of inter-fiber failure in long reinforced composites based on the phase field approach of fracture combined with the cohesive zone model, Int. J. Fract., vol. 220, no. 2, pp. 181–203, 2019. DOI: 10.1007/s10704-019-00384-8.
  • A. Quintanas-Corominas, J. Reinoso, E. Casoni, A. Turon, and J.A. Mayugo, A phase field approach to simulate intralaminar and translaminar fracture in long fiber composite materials, Compos. Struct., vol. 220, pp. 899–911, 2019. DOI: 10.1016/j.compstruct.2019.02.007.
  • S. Goswami, C. Anitescu, S. Chakraborty, and T. Rabczuk, Transfer learning enhanced physics informed neural network for phase-field modeling of fracture, Theor. Appl. Fract. Mech., vol. 106, p. 102447, 2020. DOI: 10.1016/j.tafmec.2019.102447.
  • R. Shen, H. Waisman, and L. Guo, Fracture of viscoelastic solids modeled with a modified phase field method, Comput. Methods Appl. Mech. Eng., vol. 346, pp. 862–890, 2019. DOI: 10.1016/j.cma.2018.09.018.
  • P.J. Loew, B. Peters, and L.A. Beex, Rate-dependent phase-field damage modeling of rubber and its experimental parameter identification, J. Mech. Phys. Solids, vol. 127, pp. 266–294, 2019. DOI: 10.1016/j.jmps.2019.03.022.
  • B. Yin and M. Kaliske, Fracture simulation of viscoelastic polymers by the phase-field method, Comput. Mech., vol. 65, no. 2, pp. 293–309, 2020. DOI: 10.1007/s00466-019-01769-1.
  • J. Song, et al., Coupling of phase field and viscoplasticity for modelling cyclic softening and crack growth under fatigue, Eur. J. Mech. A Solids, vol. 92, p. 104472, 2022. DOI: 10.1016/j.euromechsol.2021.104472.
  • F. Dammaß, M. Ambati, and M. Kästner, A unified phase-field model of fracture in viscoelastic materials, Contin. Mech. Thermodyn., vol. 33, no. 4, pp. 1907–1929, 2021. DOI: 10.1007/s00161-021-01013-3.
  • P. Thamburaja, K. Sarah, A. Srinivasa, and J. Reddy, Fracture of viscoelastic materials: Fem implementation of a non-local & rate form-based finite-deformation constitutive theory, Comput. Methods Appl. Mech. Eng., vol. 354, pp. 871–903, 2019. DOI: 10.1016/j.cma.2019.05.032.
  • K. Sarah, P. Thamburaja, A. Srinivasa, and J.N. Reddy, Numerical simulations of damage and fracture in viscoelastic solids using a nonlocal fracture criterion, Mech. Adv. Mater. Struct., vol. 27, no. 13, pp. 1085–1097, 2020. DOI: 10.1080/15376494.2020.1716414.
  • J. Rodríguez and J. Merodio, Helical buckling and postbuckling of pre-stressed cylindrical tubes under finite torsion, Finite Elem. Anal. Des., vol. 112, pp. 1–10, 2016. DOI: 10.1016/j.finel.2015.12.003.
  • J. Merodio and R.W. Ogden, Extension, inflation and torsion of a residually-stressed circular cylindrical tube, Contin. Mech. Thermodyn., vol. 28, no. 1–2, pp. 157–174, 2016. DOI: 10.1007/s00161-015-0411-z.
  • N. Nam, J. Merodio, R. Ogden, and P. Vinh, The effect of initial stress on the propagation of surface waves in a layered half-space, Int. J. Solids Struct., vol. 88–89, pp. 88–100, 2016. DOI: 10.1016/j.ijsolstr.2016.03.019.
  • J. Merodio and R. Ogden, Constitutive Modelling of Solid Continua, Solid Mechanics and its Applications, Springer, Berlin, vol. 262, 2020. DOI: 10.1007/978-3-030-31547-4.
  • H. Dehghani, D. Desena-Galarza, N. Jha, J. Reinoso, and J. Merodio, Bifurcation and post-bifurcation of an inflated and extended residually-stressed circular cylindrical tube with application to aneurysms initiation and propagation in arterial wall tissue, Finite Elem. Anal. Des., vol. 161, pp. 51–60, 2019. DOI: 10.1016/j.finel.2019.04.004.
  • A. Font, N. Jha, H. Dehghani, J. Reinoso, and J. Merodio, Modelling of residually stressed, extended and inflated cylinders with application to aneurysms, Mech. Res. Commun., vol. 111, p. 103643, 2021. DOI: 10.1016/j.mechrescom.2020.103643.
  • T. Li, Z. Huang, Z. Suo, S.P. Lacour, and S. Wagner, Stretchability of thin metal films on elastomer substrates, Appl. Phys. Lett., vol. 85, no. 16, pp. 3435–3437, 2004. DOI: 10.1063/1.1806275.
  • T. Li, Z. Zhang, and B. Michaux, Competing failure mechanisms of thin metal films on polymer substrates under tension, Theor. Appl. Mech. Lett., vol. 1, no. 4, p. 041002, 2011. DOI: 10.1063/2.1104102.
  • B. Bourdin, G.A. Francfort, and J.J. Marigo, The Variational Approach to Fracture, Springer, Netherlands, 2008. DOI: 10.1007/s10659-007-9107-3.
  • J. Merodio, R.W. Ogden, and J. Rodrguez, The influence of residual stress on finite deformation elastic response, Int. J. Non. Linear Mech., vol. 56, pp. 43–49, 2013. DOI: 10.1016/j.ijnonlinmec.2013.02.010.
  • D. Balzani, J. Schröder, and D. Gross, Numerical simulation of residual stresses in arterial walls, Comput. Mater. Sci., vol. 39, no. 1, pp. 117–123, 2007. DOI: 10.1016/j.commatsci.2005.11.014.
  • M.J. Borden, C.V. Verhoosel, M.A. Scott, T.J.R. Hughes, and C.M. Landis, A phase-field description of dynamic brittle fracture, Comput. Methods Appl. Mech. Eng., vol. 217–220, pp. 77–95, 2012. DOI: 10.1016/j.cma.2012.01.008.
  • H. Amor, J.J. Marigo, and C. Maurini, Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments, J. Mech. Phys. Solids, vol. 57, no. 8, pp. 1209–1229, 2009. DOI: 10.1016/j.jmps.2009.04.011.
  • R. Ostwald, E. Kuhl, and A. Menzel, On the implementation of finite deformation gradient-enhanced damage models, Comput. Mech., vol. 64, no. 3, pp. 847–877, 2019. DOI: 10.1007/s00466-019-01684-5.
  • Y. Navidtehrani, C. Betegón, and E. Martínez-Pañeda, A unified abaqus implementation of the phase field fracture method using only a user material subroutine, Materials, vol. 14, no. 8, p. 1913, 2021. DOI: 10.3390/ma14081913.
  • Y. Navidtehrani, C. Betegón, and E. Martínez-Pañeda, A simple and robust abaqus implementation of the phase field fracture method, Appl. Eng. Sci., vol. 6, p. 100050, 2021. DOI: 10.1016/j.apples.2021.100050.
  • C. Hortig, Local and non-local thermomechanical modeling and finite-element simulation of high-speed cutting, Ph.D. thesis, 2010. DOI: 10.17877/DE290R-15992.
  • C. Miehe and L.-M. Schänzel, Phase field modeling of fracture in rubbery polymers. Part I: Finite elasticity coupled with brittle failure, J. Mech. Phys. Solids, vol. 65, pp. 93–113, 2014. DOI: 10.1016/j.jmps.2013.06.007.
  • A. Hansen-Dörr, Phase field modelling and simulation of interface failure, Ph.D. thesis, 2017. DOI: 10.13140/RG.2.2.33904.35847.
  • D. Desena-Galarza, H. Dehghani, N. Jha, J. Reinoso, and J. Merodio, Computational bifurcation analysis for hyperelastic residually stressed tubes under combined inflation and extension and aneurysms in arterial tissue, Finite Elem. Anal. Des., vol. 197, p. 103636, 2021. DOI: 10.1016/j.finel.2021.103636.

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