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A two-field semi-Lagrangian reproducing kernel model for impact and penetration simulation into geo-materials

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Abstract

In this paper, a displacement–pressure (u–p) semi-Lagrangian reproducing kernel (RK) is introduced to study the penetration depth of various projectile types into dry and fully saturated geo-materials. To describe the poromechanics of saturated materials, Biot theory is incorporated into the semi-Lagrangian formulation and a damage model is embedded into Drucker–Prager constitutive model to simulate the soil behavior and separation during the impact and penetration process. The stabilized nodal domain integration is developed in the two-field semi-Lagrangian RK formulation to ensure numerical stability, and the kernel contact algorithm is implemented to model the interaction between soil and projectile bodies. Several examples are studied to validate and assess the proposed method’s performance in predicting the final penetration depth, and the results are compared to those reported in the literature.

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References

  1. Backman ME, Goldsmith W (1978) The mechanics of penetration of projectiles into targets. Int J Eng Sci 16(1):1–99

    Google Scholar 

  2. Byers RK, Yarrington P, Chabai AJ (1978) Dynamic penetration of soil media by slender projectiles. Int J Eng Sci 16:835–844

    Google Scholar 

  3. Borg JP et al (2013) In situ velocity and stress characterization of a projectile penetrating a sand target: experimental measurements and continuum simulations. Int J Impact Eng 51:23–35

    Google Scholar 

  4. Omidvar M, Iskander M, Bless S (2014) Response of granular media to rapid penetration. Int J Impact Eng 66:60–82

    Google Scholar 

  5. Forrestal MJ, Luk VK (1992) Penetration into soil targets. Int J Impact Eng 12(3):427–444

    Google Scholar 

  6. Savvateev AF et al (2001) High-speed penetration into sand. Int J Impact Eng 26(1):675–681

    Google Scholar 

  7. Bless SJ et al (2009) Sand penetration by high speed projectiles. AIP Conference Proc 1195(1):1361–1364

    Google Scholar 

  8. Seguin A et al (2009) Sphere penetration by impact in a granular medium: a collisional process. EPL (Europhys Lett) 88(4):44002

    Google Scholar 

  9. Orphal DL (2006) Explosions and impacts. Int J Impact Eng 33(1):496–545

    Google Scholar 

  10. Xu J et al (2014) A study on the ricochet of concrete debris on sand. Int J Impact Eng 65:56–68

    Google Scholar 

  11. Guzman IL, Iskander M, Bless S (2015) Observations of projectile penetration into a transparent soil. Mech Res Commun 70:4–11

    Google Scholar 

  12. Finno R (1994) Analytical interpretation of dilatometer penetration through saturated cohesive soils. Geotechnique 43(2):241–254

    Google Scholar 

  13. Teh CI, Houlsby GT (1991) An analytical study of the cone penetration test in clay. Geotechnique 41(1):17–34

    Google Scholar 

  14. Boguslavskii Y, Drabkin S, Salman A (1996) Analysis of vertical projectile penetration in granular soils. J Phys D Appl Phys 29(3):905–916

    Google Scholar 

  15. Yankelevsky DZ, Gluck J (1980) Nose shape effect on high velocity soil penetration. Int J Mech Sci 22(5):297–311

    Google Scholar 

  16. Rubin MB (2012) Analytical formulas for penetration of a long rigid projectile including the effect of cavitation. Int J Impact Eng 40–41:1–9

    Google Scholar 

  17. van den Peter B, Borst R, Huetink H (1996) An Eulerian finite element model for penetration in layered soil. Int J Numer Anal Meth Geomech 20:865–886

    Google Scholar 

  18. Walker J, Yu HS (2006) Adaptive finite element analysis of cone penetration in clay. Acta Geotechnica 1:43–57

    Google Scholar 

  19. Wang D et al (2015) Large deformation finite element analyses in geotechnical engineering. Comput Geotech 65:104–114

    Google Scholar 

  20. Huang W et al (2004) Finite element analysis of cone penetration in cohesionless soil. Comput Geotech 31(7):517–528

    Google Scholar 

  21. Børvik T et al (2002) Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses: Part I: experimental study. Int J Impact Eng 27(1):19–35

    Google Scholar 

  22. Arias A, Rodríguez-Martínez JA, Rusinek A (2008) Numerical simulations of impact behaviour of thin steel plates subjected to cylindrical, conical and hemispherical non-deformable projectiles. Eng Fract Mech 75(6):1635–1656

    Google Scholar 

  23. Scheffler DR (2005) Modeling non-eroding perforation of an oblique aluminum target using the Eulerian CTH hydrocode. Int J Impact Eng 32(1):461–472

    Google Scholar 

  24. Jiang MJ, Yu H-S, Harris D (2006) Discrete element modelling of deep penetration in granular soils. Int J Numer Anal Meth Geomech 30(4):335–361

    MATH  Google Scholar 

  25. Balevičius R, Džiugys A, Kačianauskas R (2004) Discrete element method and its application to the analysis of penetration into granular media. J Civ Eng Manag 10(1):3–14

    Google Scholar 

  26. Børvik T, Dey S, Olovsson L (2015) Penetration of granular materials by small-arms bullets. Int J Impact Eng 75:123–139

    Google Scholar 

  27. Tran QA, Chevalier B, Breul P (2016) Discrete modeling of penetration tests in constant velocity and impact conditions. Comput Geotech 71:12–18

    Google Scholar 

  28. Pica Ciamarra M et al (2004) Dynamics of drag and force distributions for projectile impact in a granular medium. Phys Rev Lett 92(19):194301

    Google Scholar 

  29. Li S, Liu WK (2004) Meshfree particle methods. Springer, Berlin

    MATH  Google Scholar 

  30. Johnson GR, Cook WH (1993) Lagrangian EPIC code computations for oblique, yawed-rod impacts onto thin-plate and spaced-plate targets at various velocities. Int J Impact Eng 14(1):373–383

    Google Scholar 

  31. Moxnes JF et al (2016) On the study of ricochet and penetration in sand, water and gelatin by spheres, 7.62 mm APM2, and 25 mm projectiles. Def Technol 12(2):159–170

    Google Scholar 

  32. Holmen JK, Olovsson L, Børvik T (2017) Discrete modeling of low-velocity penetration in sand. Comput Geotech 86:21–32

    Google Scholar 

  33. Johnson GR, Stryk RA, Beissel SR (1996) SPH for high velocity impact computations. Comput Methods Appl Mech Eng 139(1):347–373

    MATH  Google Scholar 

  34. Johnson GR (1994) Linking of Lagrangian particle methods to standard finite element methods for high velocity impact computations. Nucl Eng Des 150(2):265–274

    MathSciNet  Google Scholar 

  35. Bui HH et al (2008) Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic–plastic soil constitutive model. Int J Numer Anal Meth Geomech 32(12):1537–1570

    MATH  Google Scholar 

  36. Norouz Oliaei M, Soga K, Pak A (2009) Some numerical issues using element-free Galerkin mesh-less method for coupled hydro-mechanical problems. Int J Numer Anal Meth Geomech 33:915–938

    MATH  Google Scholar 

  37. Wang D et al (2019) A three-dimensional two-level gradient smoothing meshfree method for rainfall induced landslide simulations. Front Struct Civ Eng 13(2):337–352

    Google Scholar 

  38. Shibata T, Murakami A (2011) A stabilization procedure for soil-water coupled problems using the element-free Galerkin method. Comput Geotech 38(5):585–597

    Google Scholar 

  39. Wei H, Chen J-S, Hillman M (2016) A stabilized nodally integrated meshfree formulation for fully coupled hydro-mechanical analysis of fluid-saturated porous media. Comput Fluids 141:105–115

    MathSciNet  MATH  Google Scholar 

  40. Xie Y, Wang G (2014) A stabilized iterative scheme for coupled hydro-mechanical systems using reproducing kernel particle method. Int J Numer Meth Eng 99(11):819–843

    MathSciNet  MATH  Google Scholar 

  41. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Meth Fluids 20(8–9):1081–1106

    MathSciNet  MATH  Google Scholar 

  42. Chen J-S et al (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139(1):195–227

    MathSciNet  MATH  Google Scholar 

  43. Chi S-W et al (2015) A level set enhanced natural kernel contact algorithm for impact and penetration modeling. Int J Numer Meth Eng 102(3–4):839–866

    MathSciNet  MATH  Google Scholar 

  44. Siriaksorn T et al (2018) u–p semi-Lagrangian reproducing kernel formulation for landslide modeling. Int J Numer Meth Eng Geomech 42(2):231–255

    Google Scholar 

  45. Chen J-S, Hillman M, Chi S-W (2017) Meshfree methods: progress made after 20 years. J Eng Mech 143(4):04017001

    Google Scholar 

  46. Guan PC et al (2011) Semi-Lagrangian reproducing kernel particle method for fragment-impact problems. Int J Impact Eng 38(12):1033–1047

    Google Scholar 

  47. Sherburn JA et al (2015) Meshfree modeling of concrete slab perforation using a reproducing kernel particle impact and penetration formulation. Int J Impact Eng 86:96–110

    Google Scholar 

  48. Bessa MA et al (2014) A meshfree unification: reproducing kernel peridynamics. Comput Mech 53(6):1251–1264

    MathSciNet  MATH  Google Scholar 

  49. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209

    MathSciNet  MATH  Google Scholar 

  50. Silling SA, Lehoucq RB (2008) Convergence of peridynamics to classical elasticity theory. J Elast 93(1):13–37

    MathSciNet  MATH  Google Scholar 

  51. Wu CT et al (2017) Three-dimensional concrete impact and penetration simulations using the smoothed particle Galerkin method. Int J Impact Eng 106:1–17

    Google Scholar 

  52. Wu Y, Wu CT (2018) Simulation of impact penetration and perforation of metal targets using the smoothed particle Galerkin method. J Eng Mech 144(8):04018057

    Google Scholar 

  53. Dolbow J, Belytschko T (1999) Numerical integration of the Galerkin weak form in meshfree methods. Comput Mech 23(3):219–230

    MathSciNet  MATH  Google Scholar 

  54. Chen J-S et al (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Meth Eng 50(2):435–466

    MATH  Google Scholar 

  55. Chen J-S, Yoon S, Wu C-T (2002) Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Meth Eng 53(12):587–2615

    Google Scholar 

  56. Chen J-S, Hillman M, Rüter M (2013) An arbitrary order variationally consistent integration for Galerkin meshfree methods. Int J Numer Meth Eng 95(5):387–418

    MathSciNet  MATH  Google Scholar 

  57. Hillman M, Chen J-S, Chi S-W (2014) Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems. Comput Part Mech 1(3):245–256

    Google Scholar 

  58. Chen JS et al (2007) Strain smoothing for stabilization and regularization of Galerkin meshfree methods. In: Griebel M, Schweitzer MA (eds) Meshfree methods for partial differential equations III. Springer, Berlin, pp 57–75

    Google Scholar 

  59. Wu C-T et al (2016) Strain gradient stabilization with dual stress points for the meshfree nodal integration method in inelastic analyses. Int J Numer Meth Eng 107(1):3–30

    MathSciNet  MATH  Google Scholar 

  60. Hillman M, Chen J-S (2016) An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. Int J Numer Meth Eng 107(7):603–630

    MathSciNet  MATH  Google Scholar 

  61. Wu CT et al (2018) Numerical and experimental validation of a particle Galerkin method for metal grinding simulation. Comput Mech 61(3):365–383

    MathSciNet  MATH  Google Scholar 

  62. Chen J-S, Wu Y (2007) Stability in Lagrangian and semi-Lagrangian reproducing kernel discretizations using nodal integration in nonlinear solid mechanics. Springer, Dordrecht

    MATH  Google Scholar 

  63. Biot MA (1963) Theory of stability and consolidation of a porous medium under initial stress. J Math Mech 12(4):521–541

    MathSciNet  MATH  Google Scholar 

  64. Chi S-W, Siriaksorn T, Lin S-P (2016) Von Neumann stability analysis of the u–p reproducing kernel formulation for saturated porous media. Comput Mech 59(2):335–357

    MathSciNet  MATH  Google Scholar 

  65. Puso MA et al (2008) Meshfree and finite element nodal integration methods. Int J Numer Meth Eng 74(3):416–446

    MathSciNet  MATH  Google Scholar 

  66. Simo JC, Ju JW (1987) Strain- and stress-based continuum damage models—I. Formulation. Int J Solids Struct 23(7):821–840

    MATH  Google Scholar 

  67. Castellanos A et al (2009) Cohesion and internal friction of fine glass beads as affected by small intensity vertical vibration. AIP Conference Proc 1145(1):707–710

    Google Scholar 

  68. Khan M (2015) Mechanics of projectile penetration into non-cohesive soil targets. Int J Civil Eng 13:28–39

    Google Scholar 

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Acknowledgements

The support from the U.S. Strategic Environmental Research and Development Program (SERDP) under contract number W912HQ18C0099 is gratefully acknowledged.

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  1. Department of Civil and Material Engineering, University of Illinois at Chicago, Chicago, 60607, IL, USA

    Ashkan Mahdavi, Sheng-Wei Chi & Mohammed M. Atif

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  1. Ashkan Mahdavi
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  2. Sheng-Wei Chi
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Correspondence to Sheng-Wei Chi.

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Mahdavi, A., Chi, SW. & Atif, M.M. A two-field semi-Lagrangian reproducing kernel model for impact and penetration simulation into geo-materials. Comp. Part. Mech. 7, 351–364 (2020). https://doi.org/10.1007/s40571-019-00253-0

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