Two-dimensional strain gradient damage modeling: a variational approach

  • Luca Placidi
  • Anil Misra
  • Emilio Barchiesi


In this paper, we formulate a linear elastic second gradient isotropic two-dimensional continuum model accounting for irreversible damage. The failure is defined as the condition in which the damage parameter reaches 1, at least in one point of the domain. The quasi-static approximation is done, i.e., the kinetic energy is assumed to be negligible. In order to deal with dissipation, a damage dissipation term is considered in the deformation energy functional. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush–Kuhn–Tucker conditions. Finally, numerical simulations for exemplary problems are discussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element-free Galerkin method and moving least square shape functions have been employed.


Strain gradient 2D continua Damage mechanics Variational procedure Karush–Kuhn–Tucker conditions 

Mathematics Subject Classification




This work was supported by a grant from the Government of the Russian Federation (Contract No. 14.Y26.31.0031)


  1. 1.
    Alibert, J.-J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Altenbach, H., Eremeyev, V.: On the linear theory of micropolar plates. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 89(4), 242–256 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    AminPour, H., Rizzi, N.: A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis. Math. Mech. Solids 21(2), 168–181 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Amor, H., Marigo, J.-J., Maurini, C.: Reguralized formulation of the variational brittle fracture with unilateral contact: numerical experiment. J. Mech. Phys. Solids 57, 1209–1229 (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Andreaus, U., Giorgio, I., Lekszycki, T.: A 2D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time. Zeitschrift für Angewandte Mathematik und Mechanik 13, 7 (2013)zbMATHGoogle Scholar
  7. 7.
    Andreaus, U., Giorgio, I., Madeo, A.: Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids. Zeitschrift für angewandte Mathematik und Physik 66(1), 209–237 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Aslan, O., Forest, S.: The micromorphic versus phase field approach to gradient plasticity and damage with application to cracking in metal single crystals. In: Multiscale Methods in Computational Mechanics, pp. 135–153. Springer (2011)Google Scholar
  9. 9.
    Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37(2), 229–256 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bilotta, A., Formica, G., Turco, E.: Performance of a high-continuity finite element in three-dimensional elasticity. Int. J. Numer. Methods Biomed. Eng. 26(9), 1155–1175 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Bourdin, B., Francfort, G.A., Marigo, J.-J.: The variational approach to fracture. J. Elast. 91, 5–148 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Carcaterra, A., Akay, A., Bernardini, C.: Trapping of vibration energy into a set of resonators: theory and application to aerospace structures. Mech. Syst. Signal Process. 26, 1–14 (2012)CrossRefGoogle Scholar
  14. 14.
    Carcaterra, A., dell’Isola, F., Esposito, R., Pulvirenti, M.: Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials. Arch. Ration. Mech. Anal. 218(3), 1239–1262 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Contin. Mech. Thermodyn. 28(1–2), 139–156 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 96(10), 1220–1244 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cecchi, A., Rizzi, N.: Heterogeneous elastic solids: a mixed homogenization-rigidification technique. Int. J. Solids Struct. 38(1), 29–36 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chen, J., Ouyang, L., Rulis, P., Misra, A., Ching, W.Y.: Complex nonlinear deformation of nanometer intergranular glassy films in \(\beta \)-\(Si_{3}{N}_{4}\). Phys. Rev. Lett. 95(256103), 25 (2005)Google Scholar
  19. 19.
    Contrafatto, L., Cuomo, M., Fazio, F.: An enriched finite element for crack opening and rebar slip in reinforced concrete members. Int. J. Fract. 178(1–2), 33–50 (2012)CrossRefGoogle Scholar
  20. 20.
    Contrafatto, L., Cuomo, M., Gazzo, S.: A concrete homogenisation technique at meso-scale level accounting for damaging behaviour of cement paste and aggregates. Comput. Struct. 173, 1–18 (2016)CrossRefGoogle Scholar
  21. 21.
    Contrafatto, L., Cuomo, M., Greco, L.: Meso-scale simulation of concrete multiaxial behaviour. Eur. J. Environ. Civ. Eng. 21(7–8), 896–911 (2017)CrossRefGoogle Scholar
  22. 22.
    Cuomo, M., Contrafatto, L., Greco, L.: A variational model based on isogeometric interpolation for the analysis of cracked bodies. Int. J. Eng. Sci. 80, 173–188 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    de Felice, G., Rizzi, N.: Macroscopic modelling of cosserat media. Trends Appl. Math. Mech. Monogr. Surv. Pure Appl. Math. 106, 58–65 (1999)MathSciNetzbMATHGoogle Scholar
  24. 24.
    dell’Isola, F., d’Agostino, M., Madeo, A., Boisse, P., Steigmann, D.: Minimization of shear energy in two dimensional continua with two orthogonal families of inextensible fibers: the case of standard bias extension test. J. Elast. 122(2), 131–155 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    dell’Isola, F., Della Corte, A., Giorgio, I.: Higher-gradient continua: the legacy of piola, mindlin, sedov and toupin and some future research perspectives. Math. Mech. Solids 22(4), 1–21 (2017)MathSciNetzbMATHGoogle Scholar
  26. 26.
    dell’Isola, F., Della Corte, A., Greco, L., Luongo, A.: Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with lagrange multipliers and a perturbation solution. Int. J. Solids Struct. 81, 1–12 (2016)CrossRefGoogle Scholar
  27. 27.
    dell’Isola, F., Giorgio, I., Andreaus, U.: Elastic pantographic 2D lattices: a numerical analysis on static response and wave propagation. Proc. Est. Acad. Sci. 64, 219–225 (2015)CrossRefGoogle Scholar
  28. 28.
    dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. In: Proc. R. Soc. A, vol. 472, p. 20150790. The Royal Society (2016)Google Scholar
  29. 29.
    dell’Isola, F., Placidi, L.: Variational principles are a powerful tool also for formulating field theories. In: Variational Models and Methods in Solid and Fluid Mechanics, pp. 1–15. Springer (2011)Google Scholar
  30. 30.
    dell’Isola, F., Seppecher, P., Della Corte, A.: The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. In: Proc. R. Soc. A, vol. 471, p. 20150415. The Royal Society (2015)Google Scholar
  31. 31.
    dell’Isola, F., Steigmann, D.J.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 18, 113–125 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Di Carlo, A., Rizzi, N., Tatone, A.: Continuum modelling of a beam-like latticed truss: identification of the constitutive functions for the contact and inertial actions. Meccanica 25(3), 168–174 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Duda, F.P., Ciarbonetti, A., Sánchez, P.J., Huespe, A.E.: A phase-field/gradient damage model for brittle fracture in elastic–plastic solids. Int. J. Plast. 65, 269–296 (2015)CrossRefGoogle Scholar
  34. 34.
    Ferretti, M., Madeo, A., dell’Isola, F., Boisse, P.: Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory. Zeitschrift für angewandte Mathematik und Physik 65(3), 587–612 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Forest, S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135(3), 117–131 (2009)CrossRefGoogle Scholar
  36. 36.
    Giorgio, I.: Numerical identification procedure between a micro-cauchy model and a macro-second gradient model for planar pantographic structures. Zeitschrift für angewandte Mathematik und Physik 67(4), 95 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Giorgio, I., Andreaus, U., Lekszycki, T., Della Corte, A.: The influence of different geometries of matrix/scaffold on the remodeling process of a bone and bioresorbable material mixture with voids. Math. Mech. Solids 22(5), 969–987 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Giorgio, I., Grygoruk, R., dell’Isola, F., Steigmann, D.J.: Pattern formation in the three-dimensional deformations of fibered sheets. Mech. Res. Commun. 69, 164–171 (2015)CrossRefGoogle Scholar
  39. 39.
    Goda, I., Assidi, M., Ganghoffer, J.F.: A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech. Model. Mechanobiol. 13, 53–83 (2014)CrossRefGoogle Scholar
  40. 40.
    Greco, L., Cuomo, M.: B-spline interpolation of Kirchhoff-Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Greco, L., Cuomo, M.: An implicit G1 multi patch B-spline interpolation for Kirchhoff-Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)CrossRefzbMATHGoogle Scholar
  42. 42.
    Grillo, A., Wittum, G., Tomic, A., Federico, S.: Remodelling in statistically oriented fibre-reinforced composites and biological tissues. Math. Mech. Solids 20, 1107–1129 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Harrison, P.: Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh. Compos. A Appl. Sci. Manuf. 81, 145–157 (2016)CrossRefGoogle Scholar
  44. 44.
    Harrison, P., Clifford, M.J., Long, A.C.: Shear characterisation of viscous woven textile composites: a comparison between picture frame and bias extension experiments. Compos. Sci. Technol. 64(10), 1453–1465 (2004)CrossRefGoogle Scholar
  45. 45.
    Liu, G.-R., Gu, Y.-T.: An Introduction to Meshfree Methods and Their Programming. Springer, Berlin (2005)Google Scholar
  46. 46.
    Lorentz, E., Andrieux, S.: Analysis of non-local models through energetic formulations. Int. J. Solids Struct. 40(12), 2905–2936 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Misra, A., Ouyang, L., Chen, J., Ching, W.Y.: Ab initio calculations of strain fields and failure patterns in silicon nitride intergranular glassy films. Philos. Mag. 87(25), 3839–3852 (2007)CrossRefGoogle Scholar
  49. 49.
    Misra, A., Poorsolhjouy, P.: Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics. Math. Mech. Solids (2015).
  50. 50.
    Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Contin. Mech. Thermodyn. 28(1–2), 215–234 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Misra, A., Poorsolhjouy, P.: Granular micromechanics model of anisotropic elasticity derived from Gibbs potential. Acta Mech. 227(5), 1393–1413 (2016)CrossRefzbMATHGoogle Scholar
  52. 52.
    Misra, A., Singh, V.: Micromechanical model for viscoelastic materials undergoing damage. Contin. Mech. Thermodyn. 25(2–4), 343–358 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Misra, A., Singh, V.: Thermomechanics-based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model. Contin. Mech. Thermodyn. 27(4–5), 787–817 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Peerlings, R.H.J., Geers, M.G.D., De Borst, R., Brekelmans, W.A.M.: A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38(44), 7723–7746 (2001)CrossRefzbMATHGoogle Scholar
  55. 55.
    Pham, K., Marigo, J.-J.: Approche variationnelle de l’endommagement: I. les concepts fondamentaux. C.R. Mécanique 338, 191–198 (2010)CrossRefzbMATHGoogle Scholar
  56. 56.
    Pham, K., Marigo, J.-J.: Approche variationnelle de l’endommagement: II. les modèles à gradient. C.R. Mécanique 338, 199–206 (2010)CrossRefzbMATHGoogle Scholar
  57. 57.
    Pham, K., Marigo, J.-J.: From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models. Contin. Mech. Thermodyn. 25(2–4), 147–171 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Pham, K., Marigo, J.-J., Maurini, C.: The issue of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models. J. Mech. Phys. Solids 59, 1163–1190 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Piccardo, G., Ranzi, G., Luongo, A.: A complete dynamic approach to the generalized beam theory cross-section analysis including extension and shear modes. Math. Mech. Solids 19, 900–924 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Pietraszkiewicz, W., Eremeyev, V.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46(3), 774–787 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Placidi, L.: A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Contin. Mech. Thermodyn. 27(4–5), 623–638 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Placidi, L.: A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model. Contin. Mech. Thermodyn. 28(1–2), 119–137 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Placidi, L., Andreaus, U., Della Corte, A., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Zeitschrift für angewandte Mathematik und Physik 66(6), 3699–3725 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J. Eng. Math. (2017). MathSciNetzbMATHGoogle Scholar
  65. 65.
    Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain gradient modelling. Proc. R. Soc. Math. Phys. Eng. Sci. 474, 20170878 (2018)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Placidi, L., El Dhaba, A.: Semi-inverse method à la saint-venant for two-dimensional linear isotropic homogeneous second-gradient elasticity. Math. Mech. Solids 22(5), 919–937 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.L.: A second gradient formulation for a 2D fabric sheet with inextensible fibres. Zeitschrift für angewandte Mathematik und Physik 67(5), 114 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Placidi, L., Greve, R., Seddik, H., Faria, S.: Continuum-mechanical, anisotropic flow model for polar ice masses, based on an anisotropic flow enhancement factor. Contin. Mech. Thermodyn. 22(3), 221–237 (2010)CrossRefzbMATHGoogle Scholar
  69. 69.
    Placidi, L., Hutter, K.: Thermodynamics of polycrystalline materials treated by the theory of mixtures with continuous diversity. Contin. Mech. Thermodyn. 17(6), 409–451 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Poorsolhjouy, P., Misra, A.: Effect of intermediate principal stress and loading-path on failure of cementitious materials using granular micromechanics. Int. J. Solids Struct. 108, 139–152 (2017)CrossRefGoogle Scholar
  71. 71.
    Rinaldi, A., Placidi, L.: A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 94(10), 862–877 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Scerrato, D., Giorgio, I., Rizzi, N.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für angewandte Mathematik und Physik 67(3), 1–19 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Scerrato, D., Zhurba Eremeeva, I.A., Lekszycki, T., Rizzi, N.L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 96(11), 1268–1279 (2016)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Seddik, H., Greve, R., Placidi, L., Hamann, I., Gagliardini, O.: Application of a continuum-mechanical model for the flow of anisotropic polar ice to the EDML core, Antarctica. J. Glaciol. 54(187), 631–642 (2008)CrossRefGoogle Scholar
  75. 75.
    Sicsic, P., Marigo, J.-J.: From gradient damage laws to Griffith’s theory of crack propagation. J. Elast. 113(1), 55–74 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Voyiadjis, G.Z., Mozaffari, N.: Nonlocal damage model using the phase field method: theory and applications. Int. J. Solids Struct. 50(20), 3136–3151 (2013)CrossRefGoogle Scholar
  77. 77.
    Yang, Y., Ching, W.Y., Misra, A.: Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film. J. Nanomech. Micromech. 1(2), 60–71 (2011)CrossRefGoogle Scholar
  78. 78.
    Yang, Y., Misra, A.: Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. Int. J. Solids Struct. 49(18), 2500–2514 (2012)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Faculty of EngineeringInternational Telematic University UninettunoRomeItaly
  2. 2.Civil, Environmental and Architectural Engineering DepartmentThe University of KansasLawrenceUSA
  3. 3.Dipartimento di Ingegneria Strutturale e GeotecnicaUniversità degli Studi di Roma “La Sapienza”RomeItaly

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