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Coupled phase field and nonlocal integral elasticity analysis of stress-induced martensitic transformations at the nanoscale: boundary effects, limitations and contradictions

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Abstract

In this paper, the coupled phase field and local/nonlocal integral elasticity theories are used for stress-induced martensitic phase transformations (MPTs) at the nanoscale to investigate the limitations and contradictions of the nonlocal integral elasticity, which are due to the fact that the support of the nonlocal kernel exceeds the integration domain, i.e., the boundary effect. Different functions for the nonlocal kernel are compared. In order to compensate the boundary effect, a new nonlocal kernel, i.e., the compensated two-phase kernel, is introduced, in which a local part is added to the nonlocal part of the two-phase kernel to account for the boundary effect. In contrast to the previously introduced modified kernel, the compensated two-phase kernel does not lead to a purely nonlocal behavior in the core region, and hence no singular behavior, and consequently, no computational convergence issue is observed. The nonlinear finite element approach and the COMSOL code are used to solve the coupled system of Ginzburg–Landau and local/nonlocal integral elasticity equations. The numerical implementation of the phase field-local elasticity equations and the 2D nonlocal integral elasticity are verified. Boundary effect is investigated for MPT with both homogeneous and nonhomogeneous stress distributions. For the former, in contrast to the local elasticity, a nonhomogeneous phase transformation (PT) occurs in the nonlocal case with the two-phase kernel. Using the compensated two-phase kernel results in a homogeneous PT similar to the local elasticity. For the latter, the sample transforms to martensite except the adjacent region to the boundary for the local elasticity, while for the two-phase kernel, the entire sample transforms to martensite. The solution of the compensated two-phase kernel, however, is very similar to that of the local elasticity. The applicability of boundary symmetry in phase field problems is also investigated, which shows that it leads to incorrect results within the nonlocal integral elasticity. This is because when the symmetric portions of a sample are removed, the corresponding nonlocal effects on the remaining portion are neglected and the symmetric boundaries violate the normalization condition. An example is presented in which the results of a complete model with the two-phase kernel are different from those of its one-fourth model. In contrast, the compensated two-phase kernel can generate similar solutions for both the complete and one-fourth models. However, in general, none of the nonlocal kernels can overcome this issue. Therefore, the symmetrical models are not recommended for nonlocal integral elasticity based phase field simulations of MPTs. The current study helps for a better study of nonlocal elasticity based phase field problems for various phenomena such as various PTs.

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References

  1. Bhattacharya, K.: Microstructure of Martensite, Why It Forms and How It Gives Rise to The Shape-memory Effect. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  2. Wayman, C.M.: Introduction to the Crystallography of Martensitic Transformation. Macmillan, New York (1964)

    Google Scholar 

  3. Mamivand, M., Zaeem, M.A., El Kadiri, H.: A review on phase field modeling of martensitic phase transformation. Comput. Mater. Sci. 77, 304–11 (2013). https://doi.org/10.1016/j.commatsci.2013.04.059

    Article  Google Scholar 

  4. Levitas, V.I., Javanbakht, M.: Phase field approach to interaction of phase transformation and dislocation evolution. Appl. Phys. Lett. 102, 3–7 (2013). https://doi.org/10.1063/1.4812488

    Article  Google Scholar 

  5. Javanbakht, M., Ghaedi, M.S.: Nanovoid induced multivariant martensitic growth under negative pressure: Effect of misfit strain and temperature on PT threshold stress and phase evolution. Mech. Mater. (2020). https://doi.org/10.1016/j.mechmat.2020.103627

    Article  Google Scholar 

  6. Jacobs, A.E., Curnoe, S.H., Desai, R.C.: Simulations of cubic-tetragonal ferroelastics. Phys. Rev. B 68, 224104 (2003). https://doi.org/10.1103/PhysRevB.68.224104

    Article  ADS  Google Scholar 

  7. Artemev, A., Jin, Y., Khachaturyan, A.G.: Three-dimensional phase field model of proper martensitic transformation. Acta Mater. 49, 1165–77 (2001). https://doi.org/10.1016/S1359-6454(01)00021-0

    Article  ADS  Google Scholar 

  8. Levitas, V.I., Lee, D.W., Preston, D.L.: Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase transformations. Int. J. Plast 26, 395–422 (2010). https://doi.org/10.1016/j.ijplas.2009.08.003

    Article  MATH  Google Scholar 

  9. Yu, F., Wei, Y., Ji, Y., Chen, L.Q.: Phase field modeling of solidification microstructure evolution during welding. J. Mater. Process. Technol. 255, 285–93 (2018). https://doi.org/10.1016/j.jmatprotec.2017.12.007

    Article  Google Scholar 

  10. Park, J., Kang, J.-H., Oh, C.-S.: Phase-field simulations and microstructural analysis of epitaxial growth during rapid solidification of additively manufactured AlSi10Mg alloy. Mater. Des. 195, 108985 (2020). https://doi.org/10.1016/j.matdes.2020.108985

    Article  Google Scholar 

  11. Krill, C.E., Chen, L.Q.: Computer simulation of 3-D grain growth using a phase-field model. Acta Mater. 50, 3057–73 (2002). https://doi.org/10.1016/s1359-6454(02)00084-8

    Article  ADS  Google Scholar 

  12. Mikula, J., Joshi, S.P., Tay, T.E., Ahluwalia, R., Quek, S.S.: A phase field model of grain boundary migration and grain rotation under elasto-plastic anisotropies. Int. J. Solids Struct. 178–179, 1–18 (2019). https://doi.org/10.1016/j.ijsolstr.2019.06.014

    Article  Google Scholar 

  13. Rodney, D., Le Bouar, Y., Finel, A.: Phase field methods and dislocations. Acta Mater. (2003). https://doi.org/10.1016/S1359-6454(01)00379-2

    Article  Google Scholar 

  14. Levitas, V.I., Javanbakht, M.: Advanced phase-field approach to dislocation evolution. Phys. Rev. B - Condens. Matter. Mater. Phys. 86, 1–5 (2012). https://doi.org/10.1103/PhysRevB.86.140101

    Article  Google Scholar 

  15. Farrahi, G.H., Javanbakht, M., Jafarzadeh, H.: On the phase field modeling of crack growth and analytical treatment on the parameters. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0685-z

    Article  MATH  Google Scholar 

  16. Jafarzadeh, H., Farrahi, G.H., Javanbakht, M.: Phase field modeling of crack growth with double-well potential including surface effects. Contin. Mech. Thermodyn. (2020). https://doi.org/10.1007/s00161-019-00775-1

    Article  MathSciNet  MATH  Google Scholar 

  17. Javanbakht, M., Ghaedi, M.S.: Phase field approach for void dynamics with interface stresses at the nanoscale. Int. J. Eng. Sci. 154, 103279 (2020). https://doi.org/10.1016/j.ijengsci.2020.103279

    Article  MathSciNet  MATH  Google Scholar 

  18. Javanbakht, M., Sadegh, G.M.: Thermal induced nanovoid evolution in the vicinity of an immobile austenite-martensite interface. Comput. Mater. Sci. (2020). https://doi.org/10.1016/j.commatsci.2019.109339

    Article  Google Scholar 

  19. Chen, L.Q., Wang, Y., Khachaturyan, A.G.: Kinetics of tweed and twin formation during an ordering transition in a substitutional solid solution. Philos. Mag. Lett. 65, 15–23 (1992). https://doi.org/10.1080/09500839208215143

    Article  ADS  Google Scholar 

  20. Jin, Y.M., Artemev, A., Khachaturyan, A.G.: Three-dimensional phase field model of low-symmetry martensitic transformation in polycrystal: Simulation of \(\zeta ^{\prime }2\) martensite in AuCd alloys. Acta Mater. 49, 2309–20 (2001). https://doi.org/10.1016/S1359-6454(01)00108-2

    Article  ADS  Google Scholar 

  21. Artemev, A., Khachaturyan, A.G.: Phase field model and computer simulation of martensitic transformation under applied stresses. Mater. Sci. Forum 327, 347–50 (2000). https://doi.org/10.4028/www.scientific.net/msf.327-328.347

  22. Ahluwalia, R., Lookman, T., Saxena, A., Albers, R.C.: Landau theory for shape memory polycrystals. Acta Mater. 52, 209–18 (2004). https://doi.org/10.1016/j.actamat.2003.09.015

    Article  ADS  Google Scholar 

  23. Mirzakhani, S., Javanbakht, M.: Phase field-elasticity analysis of austenite-martensite phase transformation at the nanoscale: Finite element modeling. Comput. Mater. Sci. 154, 41–52 (2018). https://doi.org/10.1016/j.commatsci.2018.07.034

    Article  Google Scholar 

  24. Javanbakht, M., Rahbar, H., Ashourian, M.: Finite element implementation based on explicit, Galerkin and Crank-Nicolson methods to phase field theory for thermal- and surface- induced martensitic phase transformations. Contin Mech Thermodyn (2019). https://doi.org/10.1007/s00161-019-00838-3

    Article  Google Scholar 

  25. Artemev, A., Wang, Y., Khachaturyan, A.G.: Three-dimensional phase field model and simulation of martensitic transformation in multilayer systems under applied stresses. Acta Mater. 48, 2503–18 (2000). https://doi.org/10.1016/S1359-6454(00)00071-9

    Article  ADS  Google Scholar 

  26. Seol, D.J., Hu, S.Y., Li, Y.L., Chen, L.Q., Oh, K.H.: Computer simulation of martensitic transformation in constrained films. Mater. Sci. Forum 408–412, 1645–50 (2002). https://doi.org/10.4028/www.scientific.net/msf.408-412.1645

  27. Seol, D.J., Hu, S.Y., Li, Y.L., Chen, L.Q., Oh, K.H.: Cubic to tetragonal martensitic transformation in a thin film elastically constrained by a substrate. Met. Mater. Int. 9, 221–226 (2003). https://doi.org/10.1007/BF03027039

    Article  Google Scholar 

  28. Levitas, V.I., Preston, D.L.: Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. I. Austenite (formula presented) martensite. Phys. Rev. B Condens. Matter. Mater. Phys. 66, 1–9 (2002). https://doi.org/10.1103/PhysRevB.66.134206

    Article  Google Scholar 

  29. Levitas, V.I., Preston, D.L.: Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. II. Multivariant phase transformations and stress space analysis. Phys. Rev. B Condens. Matter. Mater. Phys. 66, 1–15 (2002). https://doi.org/10.1103/PhysRevB.66.134207

    Article  Google Scholar 

  30. Levitas, V.I., Preston, D.L., Lee, D.W.: Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. III. Alternative potentials, critical nuclei, kink solutions, and dislocation theory. Phys. Rev. B Condens. Matter. Mater. Phys. (2003). https://doi.org/10.1103/PhysRevB.68.134201

    Article  Google Scholar 

  31. Levitas, V.I., Javanbakht, M.: Surface tension and energy in multivariant martensitic transformations: phase-field theory, simulations, and model of coherent interface. Phys. Rev. Lett. 105, 1–4 (2010). https://doi.org/10.1103/PhysRevLett.105.165701

    Article  Google Scholar 

  32. Levitas, V.I., Javanbakht, M.: Surface-induced phase transformations: multiple scale and mechanics effects and morphological transitions. Phys. Rev. Lett. 107, 1–5 (2011). https://doi.org/10.1103/PhysRevLett.107.175701

    Article  Google Scholar 

  33. Javanbakht, M., Adaei, M.: Investigating the effect of elastic anisotropy on martensitic phase transformations at the nanoscale. Comput. Mater. Sci. 167, 168–82 (2019). https://doi.org/10.1016/j.commatsci.2019.05.047

    Article  Google Scholar 

  34. Javanbakht, M., Adaei, M.: Formation of stress- and thermal-induced martensitic nanostructures in a single crystal with phase-dependent elastic properties. J. Mater. Sci. 55, 2544–63 (2020). https://doi.org/10.1007/s10853-019-04067-6

    Article  ADS  Google Scholar 

  35. Levitas, V.I., Javanbakht, M.: Interaction between phase transformations and dislocations at the nanoscale. Part 1. General phase field approach. J. Mech. Phys. Solids 82, 287–319 (2015). https://doi.org/10.1016/j.jmps.2015.05.005

    Article  MathSciNet  ADS  Google Scholar 

  36. Javanbakht, M., Levitas, V.I.: Interaction between phase transformations and dislocations at the nanoscale. Part 2: Phase field simulation examples. J. Mech. Phys. Solids 82, 164–85 (2015). https://doi.org/10.1016/j.jmps.2015.05.006

    Article  MathSciNet  ADS  Google Scholar 

  37. Levin, V.A., Levitas, V.I., Zingerman, K.M., Freiman, E.I.: Phase-field simulation of stress-induced martensitic phase transformations at large strains. Int. J. Solids Struct. 50, 2914–28 (2013). https://doi.org/10.1016/j.ijsolstr.2013.05.003

    Article  Google Scholar 

  38. Javanbakht, M., Rahbar, H., Ashourian, M.: Explicit nonlinear finite element approach to the Lagrangian-based coupled phase field and elasticity equations for nanoscale thermal- and stress-induced martensitic transformations. Contin. Mech. Thermodyn. (2020). https://doi.org/10.1007/s00161-020-00912-1

    Article  Google Scholar 

  39. Javanbakht, M., Ghaedi, M.S., Barchiesi, E., Ciallella, A.: The effect of a pre-existing nanovoid on martensite formation and interface propagation: a phase field study. Math. Mech. Solids (2020). https://doi.org/10.1177/1081286520948118

    Article  MATH  Google Scholar 

  40. Javanbakht, M., Ghaedi, M.S.: Nanovoid induced martensitic growth under uniaxial stress: effect of misfit strain, temperature and nanovoid size on PT threshold stress and nanostructure in NiAl. Comput. Mater. Sci. 184, 109928 (2020). https://doi.org/10.1016/j.commatsci.2020.109928

    Article  Google Scholar 

  41. Bažant, Z.P., Jirásek, M.: Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128, 1119–1149 (2002). https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1119)

    Article  Google Scholar 

  42. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–24 (1968). https://doi.org/10.1016/0020-7683(68)90036-X

    Article  MATH  Google Scholar 

  43. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–508 (2003). https://doi.org/10.1016/S0022-5096(03)00053-X

    Article  ADS  MATH  Google Scholar 

  44. Alibert, J.J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8, 51–73 (2003). https://doi.org/10.1177/1081286503008001658

    Article  MathSciNet  MATH  Google Scholar 

  45. dell’Isola, F., Steigmann, D.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118, 113–125 (2015). https://doi.org/10.1007/s10659-014-9478-1

    Article  MathSciNet  MATH  Google Scholar 

  46. Placidi, L., Misra, A., Barchiesi, E.: Two-dimensional strain gradient damage modeling: a variational approach. Z. Angew. Math. Phys. 69, 56 (2018). https://doi.org/10.1007/s00033-018-0947-4

    Article  MathSciNet  MATH  Google Scholar 

  47. Placidi, L., Barchiesi, E., Misra, A.: A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Math. Mech. Complex Syst. 6, 77–100 (2018). https://doi.org/10.2140/memocs.2018.6.77

    Article  MathSciNet  MATH  Google Scholar 

  48. Barchiesi, E., Yang, H., Tran, C.A., Placidi, L., Müller, W.H.: Computation of brittle fracture propagation in strain gradient materials by the FEniCS library. Math. Mech. Solids 26, 325–340 (2020). https://doi.org/10.1177/1081286520954513

    Article  MathSciNet  MATH  Google Scholar 

  49. dell’Isola, F., Corte, A., Della, G.I.: Higher-gradient continua: the legacy of Piola Mindlin Sedov and Toupin and some future research perspectives. Math. Mech. Solids 22, 852–872 (2016). https://doi.org/10.1177/1081286515616034

    Article  MathSciNet  MATH  Google Scholar 

  50. Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A Math. Phys. Eng. Sci. 473, 20170636 (2017). https://doi.org/10.1098/rspa.2017.0636

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. Giorgio, I., Harrison, P., dell’Isola, F., Alsayednoor, J., Turco, E.: Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches. Proc. R. Soc. A Math. Phys. Eng. Sci. 474, 20180063 (2018). https://doi.org/10.1098/rspa.2018.0063

    Article  ADS  Google Scholar 

  52. Eugster, S.R., dell’Isola, F., Steigmann, D.J.: Continuum theory for mechanical metamaterials with a cubic lattice substructure. Math. Mech. Complex Syst. 7, 75–98 (2019). https://doi.org/10.2140/memocs.2019.7.75

    Article  MathSciNet  MATH  Google Scholar 

  53. Schulte, J., Dittmann, M., Eugster, S.R., Hesch, S., Reinicke, T., dell’Isola, F., et al.: Isogeometric analysis of fiber reinforced composites using Kirchhoff-Love shell elements. Comput. Methods Appl. Mech. Eng. 362, 112845 (2020). https://doi.org/10.1016/j.cma.2020.112845

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964). https://doi.org/10.1007/BF00253050

    Article  MathSciNet  MATH  Google Scholar 

  55. Hadjesfandiari, A.R., Dargush, G.F.: Couple stress theory for solids. Int. J. Solids Struct. 48, 2496–510 (2011). https://doi.org/10.1016/j.ijsolstr.2011.05.002

    Article  Google Scholar 

  56. Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct. 49, 1993–2005 (2012). https://doi.org/10.1016/j.ijsolstr.2012.04.007

    Article  Google Scholar 

  57. Grekova, E.F., Porubov, A.V., dell’Isola, F.: Reduced linear constrained elastic and viscoelastic homogeneous cosserat media as acoustic metamaterials. Symmetry 12, 521 (2020). https://doi.org/10.3390/SYM12040521

    Article  ADS  Google Scholar 

  58. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972). https://doi.org/10.1016/0020-7225(72)90039-0

    Article  MathSciNet  MATH  Google Scholar 

  59. Eringen, A.C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10, 425–435 (1972). https://doi.org/10.1016/0020-7225(72)90050-X

    Article  MATH  Google Scholar 

  60. dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, nonlocal and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids 20, 887–928 (2015). https://doi.org/10.1177/1081286513509811

    Article  MathSciNet  MATH  Google Scholar 

  61. dell’Isola, F., Della, C. A., Esposito, R., Russo, L.: Some cases of unrecognized transmission of scientific knowledge: From antiquity to gabrio piola’s peridynamics and generalized continuum theories. In: Altenbach, H., Forest, S., (eds.) Generalized Continua as Models for Classical and Advanced Materials, vol. 42, pp. 77-128. Springer , Cham (2016). https://doi.org/10.1007/978-3-319-31721-2_5

  62. Kröner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, 731–42 (1967). https://doi.org/10.1016/0020-7683(67)90049-2

    Article  MATH  Google Scholar 

  63. Kunin, I.A.: On foundations of the theory of elastic media with microstructure. Int. J. Eng. Sci. 22, 969–78 (1984). https://doi.org/10.1016/0020-7225(84)90098-3

    Article  MATH  Google Scholar 

  64. Krumhansl, J.A.: Some considerations of the relation between solid state physics and generalized continuum mechanics. In: Kröner, E., (ed.) Mechanics of generalized continua. p. 298–311. Springer, Berlin (1968). https://doi.org/10.1007/978-3-662-30257-6_37

  65. Edelen, D.G.B., Laws, N.: On the thermodynamics of systems with nonlocality. Arch. Ration. Mech. Anal. 43, 24–35 (1971). https://doi.org/10.1007/BF00251543

    Article  MathSciNet  MATH  Google Scholar 

  66. Eringen, A.C., Kim, B.S.: Stress concentration at the tip of crack. Mech. Res. Commun. 1, 233–7 (1974). https://doi.org/10.1016/0093-6413(74)90070-6

    Article  Google Scholar 

  67. Eringen, A.C., Speziale, C.G., Kim, B.S.: Crack-tip problem in non-local elasticity. J. Mech. Phys. Solids 25, 339–55 (1977). https://doi.org/10.1016/0022-5096(77)90002-3

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. Eringen, A.C.: Theory of nonlocal elasticity and some applications. Princeton Univ NJ Dept of Civil Engineering (1984)

  69. Altan, S.B.: Uniqueness of initial-boundary value problems in nonlocal elasticity. Int. J. Solids Struct. 25, 1271–1278 (1989). https://doi.org/10.1016/0020-7683(89)90091-7

    Article  MathSciNet  MATH  Google Scholar 

  70. Altan, S.B.: Existence in nonlocal elasticity. Arch. Mech. 41, 25–36 (1989)

    MathSciNet  MATH  Google Scholar 

  71. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703 (1983). https://doi.org/10.1063/1.332803

    Article  ADS  Google Scholar 

  72. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003). https://doi.org/10.1016/S0020-7225(02)00210-0

    Article  Google Scholar 

  73. Farajpour, A., Ghayesh, M.H., Farokhi, H.: A review on the mechanics of nanostructures. Int. J. Eng. Sci. 133, 231–263 (2018). https://doi.org/10.1016/j.ijengsci.2018.09.006

    Article  MathSciNet  MATH  Google Scholar 

  74. Polizzotto, C., Fuschi, P., Pisano, A.A.: A strain-difference-based nonlocal elasticity model. Int. J. Solids Struct. 41, 2383–2401 (2004). https://doi.org/10.1016/j.ijsolstr.2003.12.013

    Article  MATH  Google Scholar 

  75. Polizzotto, C., Fuschi, P., Pisano, A.A.: A nonhomogeneous nonlocal elasticity model. Eu. J. Mech. A/Solids 25, 308–333 (2006). https://doi.org/10.1016/j.euromechsol.2005.09.007

    Article  MathSciNet  ADS  MATH  Google Scholar 

  76. Pisano, A.A., Fuschi, P.: Closed form solution for a nonlocal elastic bar in tension. Int. J. Solids Struct. 40, 13–23 (2003). https://doi.org/10.1016/S0020-7683(02)00547-4

    Article  MathSciNet  MATH  Google Scholar 

  77. Tuna, M., Kirca, M.: Exact solution of Eringen’s nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams. Int. J. Eng. Sci. 105, 80–92 (2016). https://doi.org/10.1016/j.ijengsci.2016.05.001

    Article  MathSciNet  MATH  Google Scholar 

  78. Polizzotto, C.: Nonlocal elasticity and related variational principles. Int. J. Solids Struct. 38, 7359–80 (2001). https://doi.org/10.1016/S0020-7683(01)00039-7

    Article  MathSciNet  MATH  Google Scholar 

  79. Pisano, A.A., Sofi, A., Fuschi, P.: Nonlocal integral elasticity: 2D finite element based solutions. Int. J. Solids Struct. 46, 3836–49 (2009). https://doi.org/10.1016/j.ijsolstr.2009.07.009

    Article  MATH  Google Scholar 

  80. Borino, G., Failla, B., Parrinello, F.: A symmetric nonlocal damage theory. Int. J. Solids Struct. 40, 3621–3645 (2003). https://doi.org/10.1016/S0020-7683(03)00144-6

    Article  MATH  Google Scholar 

  81. Pijaudier-Cabot, G., Bažant, Z.P.: Nonlocal damage theory. J. Eng. Mech. ASCE 113, 1512–1533 (1987). https://doi.org/10.1061/(ASCE)0733-9399(1987)113:10(1512)

    Article  MATH  Google Scholar 

  82. Polizzotto, C.: Remarks on some aspects of nonlocal theories in solid mechanics. In: Proc. of the 6th Congress of Italian Society for Applied and Industrial Mathematics (SIMAI), Cagliari, Italy (2002)

  83. Danesh, H., Javanbakht, M., Aghdam, M.M.: A comparative study of 1D nonlocal integral Timoshenko beam and 2D nonlocal integral elasticity theories for bending of nanoscale beams. Contin. Mech. Thermodyn. (2021). https://doi.org/10.1007/s00161-021-00976-7

    Article  Google Scholar 

  84. Shaat, M., Ghavanloo, E., Fazelzadeh, S.A.: Review on nonlocal continuum mechanics: physics, material applicability, and mathematics. Mech. Mater. 150, 103587 (2020). https://doi.org/10.1016/j.mechmat.2020.103587

    Article  Google Scholar 

  85. Eringen, A.C.: Edge dislocation in nonlocal elasticity. Int. J. Eng. Sci. 15, 177–83 (1977). https://doi.org/10.1016/0020-7225(77)90003-9

    Article  MATH  Google Scholar 

  86. Pan, K.-L.: Interaction of a dislocation and an inclusion in nonlocal elasticity. Int. J. Eng. Sci. 34, 1675–1688 (1996). https://doi.org/10.1016/S0020-7225(96)00029-8

    Article  MATH  Google Scholar 

  87. Lazar, M., Agiasofitou, E.: Screw dislocation in nonlocal anisotropic elasticity. Int. J. Eng. Sci. 49, 1404–1414 (2011). https://doi.org/10.1016/j.ijengsci.2011.02.011

    Article  MathSciNet  MATH  Google Scholar 

  88. Doğgan, A.: Effect of nonlocal elasticity on internal friction peaks observed during martensite transformation. Pramana 44, 397–404 (1995). https://doi.org/10.1007/BF02848491

    Article  ADS  Google Scholar 

  89. Martowicz, A., Bryła, J., Staszewski, W.J., Ruzzene, M., Uhl, T.: Nonlocal elasticity in shape memory alloys modeled using peridynamics for solving dynamic problems. Nonlinear Dyn. 97, 1911–1935 (2019). https://doi.org/10.1007/s11071-019-04943-5

    Article  Google Scholar 

  90. Yang, W.D., Wang, X., Lu, G.: The evolution of void defects in metallic films based on a nonlocal phase field model. Eng. Fract. Mech. 127, 12–20 (2014). https://doi.org/10.1016/j.engfracmech.2014.04.018

    Article  Google Scholar 

  91. Danesh, H., Javanbakht, M., Mirzakhani, S.: Nonlocal integral elasticity based phase field modelling and simulations of nanoscale thermal- and stress-induced martensitic transformations using a boundary effect compensation kernel. Comput. Mater. Sci. 194, 110429 (2021). https://doi.org/10.1016/j.commatsci.2021.110429

    Article  Google Scholar 

  92. Abdollahi, R., Boroomand, B.: Benchmarks in nonlocal elasticity defined by Eringen’s integral model. Int. J. Solids Struct. 50, 2758–2771 (2013). https://doi.org/10.1016/j.ijsolstr.2013.04.027

    Article  Google Scholar 

  93. Romano, G., Barretta, R., Diaco, M., de Marotti, S.F.: Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int J Mech Sci 121, 151–156 (2017). https://doi.org/10.1016/j.ijmecsci.2016.10.036

    Article  Google Scholar 

  94. Pisano, A.A., Fuschi, P.: Structural symmetry and boundary conditions for nonlocal symmetrical problems. Meccanica 53, 629–38 (2018). https://doi.org/10.1007/s11012-017-0684-3

    Article  MathSciNet  MATH  Google Scholar 

  95. Golmakani, M.E., Malikan, M., Pour, S.G., Eremeyev, V.A.: Bending analysis of functionally graded nanoplates based on a higher-order shear deformation theory using dynamic relaxation method. Contin. Mech. Thermodyn. (2021). https://doi.org/10.1007/s00161-021-00995-4

    Article  Google Scholar 

  96. Mikhasev, G.: Free high-frequency vibrations of nonlocally elastic beam with varying cross-section area. Contin. Mech. Thermodyn. (2021). https://doi.org/10.1007/s00161-021-00977-6

    Article  MathSciNet  Google Scholar 

  97. Giorgio, I., Grygoruk, R., dell’Isola, F., Steigmann, D.J.: Pattern formation in the three-dimensional deformations of fibered sheets. Mech. Res. Commun. 69, 164–71 (2015). https://doi.org/10.1016/j.mechrescom.2015.08.005

    Article  Google Scholar 

  98. Scerrato, D., Giorgio, I., Rizzi, N.L.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Z. Angew. Math. Phys. 67, 53 (2016). https://doi.org/10.1007/s00033-016-0650-2

    Article  MathSciNet  MATH  Google Scholar 

  99. Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5, 127–162 (2017). https://doi.org/10.2140/memocs.2017.5.127

    Article  MathSciNet  MATH  Google Scholar 

  100. Andreaus, U., Spagnuolo, M., Lekszycki, T., Eugster, S.R.: A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams. Contin. Mech. Thermodyn. 30, 1103–1123 (2018). https://doi.org/10.1007/s00161-018-0665-3

    Article  MathSciNet  ADS  MATH  Google Scholar 

  101. dell’Isola, F., Seppecher, P., Spagnuolo, M., Barchiesi, E., Hild, F., Lekszycki, T., et al.: Advances in pantographic structures: design, manufacturing, models, experiments and image analyses. Contin. Mech. Thermodyn. 31, 1231–1282 (2019). https://doi.org/10.1007/s00161-019-00806-x

    Article  ADS  Google Scholar 

  102. dell’Isola, F., Seppecher, P., Alibert, J.J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., et al.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Thermodyn. 31, 851–884 (2019). https://doi.org/10.1007/s00161-018-0689-8

    Article  MathSciNet  ADS  Google Scholar 

  103. Turco, E., Misra, A., Sarikaya, R., Lekszycki, T.: Quantitative analysis of deformation mechanisms in pantographic substructures: experiments and modeling. Contin. Mech. Thermodyn. 31, 209–223 (2019). https://doi.org/10.1007/s00161-018-0678-y

    Article  MathSciNet  ADS  Google Scholar 

  104. Spagnuolo, M., Franciosi, P., dell’Isola, F.: A Green operator-based elastic modeling for two-phase pantographic-inspired bi-continuous materials. Int. J. Solids Struct. 188–189, 282–308 (2020). https://doi.org/10.1016/j.ijsolstr.2019.10.018

    Article  Google Scholar 

  105. Spagnuolo, M., Yildizdag, M.E., Andreaus, U., Cazzani, A.M.: Are higher-gradient models also capable of predicting mechanical behavior in the case of wide-knit pantographic structures? Math. Mech. Solids 26, 18–29 (2021). https://doi.org/10.1177/1081286520937339

    Article  MathSciNet  MATH  Google Scholar 

  106. Turco, E., Barchiesi, E., Giorgio, I., dell’Isola, F.: A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory. Int. J. Non Linear Mech. 123, 103481 (2020). https://doi.org/10.1016/j.ijnonlinmec.2020.103481

    Article  ADS  Google Scholar 

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Acknowledgements

The support of Isfahan University of Technology and Iran National Science Foundation is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Hooman Danesh & Mahdi Javanbakht

  2. Instituto de Investigación Científica, Universidad de Lima, Av. Javier Prado Este 4600, Santiago de Surco, 15023, Peru

    Emilio Barchiesi

  3. École Nationale d’Ingénieurs de Brest, ENIB, UMR CNRS 6027, IRDL, 29200, Brest, France

    Emilio Barchiesi & Nahiene Hamila

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  1. Hooman Danesh
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  2. Mahdi Javanbakht
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  3. Emilio Barchiesi
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  4. Nahiene Hamila
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Correspondence to Mahdi Javanbakht.

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Communicated by Andreas Öchsner.

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Danesh, H., Javanbakht, M., Barchiesi, E. et al. Coupled phase field and nonlocal integral elasticity analysis of stress-induced martensitic transformations at the nanoscale: boundary effects, limitations and contradictions. Continuum Mech. Thermodyn. 35, 1041–1062 (2023). https://doi.org/10.1007/s00161-021-01042-y

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