A computational framework for the constitutive modeling of nonlinear micropolar media

  • Lapo GoriEmail author
  • Samuel Silva Penna
  • Roque Luiz da Silva Pitangueira
Technical Paper


Despite the large number of applications with micropolar models, the aspects of their implementation have been rarely addressed in the literature. In the present paper, a strategy for the computational modeling of micropolar media with elasto-plasticity and elastic degradation is investigated. The proposed strategy is based on the Object-Oriented Paradigm (OOP) and on the use of tensor objects. The presence of tensor objects inside the code allows to obtain a constitutive models framework that, with respect to existent implementations, is independent on both the adopted analysis model and numerical method. The OOP, with its properties of abstraction, inheritance, and polymorphism, leads to a framework highly modular and easy to expand. The theoretical basis is a compact tensorial representation for the micropolar equations that makes them formally identical to the ones of the classic continuum theory. This compatibility has been here extended to their computational expressions, making possible to use the same code structure for both the continuum models, taking advantage of existing implementations of classic constitutive models.


Finite element method Object-oriented programming Micropolar media Elastic degradation Elasto-plasticity Continuum damage mechanics 



The authors gratefully acknowledge the support of the Brazilian research agencies CAPES (in Portuguese: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), FAPEMIG (in Portuguese: Fundação de Amparo à Pesquisa do Estado de Minas Gerais) and CNPq (in Portuguese: Conselho Nacional de Desenvolvimento Científico e Tecnológico - Grant 309515/2017-3).


  1. 1.
    de Borst R, Sluys LJ (1991) Localization in a Cosserat continuum under static and dynamic loading. Comput Methods Appl Mech Eng 90(1–3):805CrossRefGoogle Scholar
  2. 2.
    De Borst R (1991) Simulation of strain localization: a reappraisal of the Cosserat continuum. Eng Comput 8(4):317. CrossRefGoogle Scholar
  3. 3.
    de Borst R (1993) A generalization of j2-flow theory for polar continua. Comput Methods Appl Mech Eng 103:347CrossRefGoogle Scholar
  4. 4.
    Sluys LJ (1992) Wave propagation, localization and dispersion in softening solids. Ph.D. thesis, Technische Universiteit Delft, Delft, The NederlandsGoogle Scholar
  5. 5.
    Dietsche A, Steinmann P, Willam K (1993) Micropolar elastoplasticity and its role in localization. Int J Plast 9:813CrossRefGoogle Scholar
  6. 6.
    Iordache MM, Willam K (1998) Localized failure analysis in elastoplastic Cosserat continua. Comput Methods Appl Mech Eng 121(3–4):559CrossRefGoogle Scholar
  7. 7.
    Gori L, Penna SS, Pitangueira RLS (2017) An enhanced tensorial formulation for elastic degradation in micropolar continua. Appl Math Model 41:299. MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gori L, Penna SS, da Silva Pitangueira RL (2018) Discontinuous failure in micropolar elastic-degrading models. Int J Damage Mech 27(10):1482–1515. CrossRefGoogle Scholar
  9. 9.
    Chang CS, Ma L (1991) A micromechanical-based micropolar theory for deformation of granular solids. Int J Solids Struct 28(1):67. CrossRefzbMATHGoogle Scholar
  10. 10.
    Suiker ASJ, De Borst R, Chang CS (2001) Micro-mechanical modelling of granular material. Part 1: derivation of a second-gradient micro-polar constitutive theory. Acta Mech 149:161. CrossRefzbMATHGoogle Scholar
  11. 11.
    Walsh SDC, Tordesillas A (2006) Finite element methods for micropolar models of granular materials. Appl Math Model 30:1043. CrossRefzbMATHGoogle Scholar
  12. 12.
    Arslan H, Willam KJ (2007) Analytical and geometrical representation of localization in granular material. Acta Mech 194:159CrossRefGoogle Scholar
  13. 13.
    Arslan H, Sture S (2008) Finite element simulation of localization in granular materials by micropolar continuum approach. Comput Geotech. CrossRefzbMATHGoogle Scholar
  14. 14.
    Casolo S (2006) Macroscopic modelling of structured materials: relationship between orthotropic Cosserat continuum and rigid elements. Int J Solids Struct 43(3–4):475. CrossRefzbMATHGoogle Scholar
  15. 15.
    Dos Reis F, Ganghoffer JF (2012) Construction of micropolar continua from the asymptotic homogenization of beam lattices. Comput Struct 112–113:354. CrossRefGoogle Scholar
  16. 16.
    Addessi D, De Bellis ML, Sacco E (2013) Micromechanical analysis of heterogeneous materials subjected to overall Cosserat strains. Mech Res Commun 54:27. CrossRefGoogle Scholar
  17. 17.
    De Bellis ML, Addessi D (2014) A micromechanical approach for the micropolar modeling of heterogeneous periodic media. Fract Struct Integr 8(29):37. CrossRefGoogle Scholar
  18. 18.
    Trovalusci P, Ostoja-Starzewski M, De Bellis ML, Murrali A (2015) Scale-dependent homogenization of random composites as micropolar continua. Eur J Mech A Solids 49:396. CrossRefGoogle Scholar
  19. 19.
    Addessi D, De Bellis ML, Sacco E (2016) A micromechanical approach for the Cosserat modeling of composites. Meccanica 51:569. MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Trovalusci P, De Bellis ML, Masiani R (2017) A multiscale description of particle composites: from lattice microstructures to micropolar continua. Compos Part B Eng 128:164. CrossRefGoogle Scholar
  21. 21.
    Trovalusci P, Masiani R (2003) Non-linear micropolar and classical continua for anisotropic discontinuous materials. Int J Solids Struct 40(5):1281. CrossRefzbMATHGoogle Scholar
  22. 22.
    Steinmann P (1995) Theory and numerics of ductile micropolar elastoplastic damage. Int J Numer Methods Eng 38:583MathSciNetCrossRefGoogle Scholar
  23. 23.
    Addessi D (2014) A 2D Cosserat finite element based on a damage-plastic model for brittle materials. Compute Struct 135:20. CrossRefGoogle Scholar
  24. 24.
    Xotta G, Beizaee S, Willam KJ (2016) Bifurcation investigation of coupled damage-plasticity models for concrete materials. Comput Methods Appl Mech Eng 298:428MathSciNetCrossRefGoogle Scholar
  25. 25.
    Carol I, Rizzi E, Willam K (1994) A unified theory of elastic degradation and damage based on a loading surface. Int J Solids Struct 31(20):2835. CrossRefzbMATHGoogle Scholar
  26. 26.
    Rizzi E (1995) Sulla localizzazione delle deformazioni in materiali e strutture. Ph.D. thesis, Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Milano, Italy (in italian) Google Scholar
  27. 27.
    Rizzi E, Carol I, Willam K (1995) Localization analysis of elastic degradation with application to scalar damage. J Eng Mech 121(4):541. CrossRefGoogle Scholar
  28. 28.
    Carol I (1996). In: Mecánica Computacional, vol XVII(4). Solid Mechanics (A)Google Scholar
  29. 29.
    Carol I, Willam K (1996) Spurious energy dissipation/generation in stiffness recovery models for elastic degradation and damage. Int J Solids Struct 33(20–22):2939–2957. CrossRefzbMATHGoogle Scholar
  30. 30.
    Carol I (1999). In: MECOM 99 (Mendoza Argentina)Google Scholar
  31. 31.
    Hansen E, Willam K, Carol I (2001). In Proceedings of fracture mechanics of concrete materials—Framcos-4 Conferece. France, ParisGoogle Scholar
  32. 32.
    Penna SS (2011) Formulação multipotencial para modelos de degradação elástica: unificação teórica, proposta de novo modelo, implementação computational e modelagem de estruturas de concreto. Ph.D. thesis, UFMG - Federal University of Minas Gerais, Belo Horizonte, Brazil (in portuguese) Google Scholar
  33. 33.
    Rizzi E, Carol I (2001) A formulation of anisotropic elastic damage using compact tensor formalism. J Elast 64(2):85. MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gori L, Penna SS, da Silva Pitangueira RL (2017) A computational framework for constitutive modelling. Comput Struct 187:1. CrossRefGoogle Scholar
  35. 35.
    INSANE Project. Accessed June 2019
  36. 36.
    Jeremić B, Sture S (1998) Tensor objects in finite element programming. Int J Numer Methods Eng 41(1):113.;2-4 CrossRefzbMATHGoogle Scholar
  37. 37.
    Jeremić B, Runesson K, Sture S (1999) Object-oriented approach to hyperelasticity. Eng Comput 15(1):2. CrossRefzbMATHGoogle Scholar
  38. 38.
    Jeremić B, Yang Z (2002) Template elastic-plastic computations in geomechanics. Int J Numer Anal Methods Geomech 26(14):1407. CrossRefzbMATHGoogle Scholar
  39. 39.
    Peixoto R, Anacleto F, Ribeiro G, Pitangueira R, Penna S (2016) A solution strategy for non-linear implicit BEM formulation using a unified constitutive modelling framework. Eng Anal Bound Elem 64:295. MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Monteiro A, Wolenski A, Barros F, Pitangueira R, Penna S (2017) A computational framework for G/XFEM material nonlinear analysis. Adv Eng Softw 114:380. CrossRefGoogle Scholar
  41. 41.
    Pinheiro D, Barros F, Pitangueira R, Penna S (2017) High regularity partition of unity for structural physically non-linear analysis. Eng Anal Bound Elem 83:43. MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Gori L, Penna S, Pitangueira R (2019) Smoothed point interpolation methods for the regularization of material instabilities in scalar damage models. Int J Numer Methods Eng 117:729–755. MathSciNetCrossRefGoogle Scholar
  43. 43.
    Gori L, Pitangueira RLS, Penna SS, Fuina JS (2015) A generalized elasto-plastic micro-polar constitutive model. Appl Mech Mater 798:505CrossRefGoogle Scholar
  44. 44.
    Eremeyev VA (2005) Acceleration waves in micropolar elastic media. Dokl Phys 50(4):204CrossRefGoogle Scholar
  45. 45.
    Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New YorkzbMATHGoogle Scholar
  46. 46.
    Mazars J, Lemaitre J (1984). In: Shah SP (ed) Application of fracture mechanics to cementitious composites, NATO ASI series, vol 94. Springer, Dordrecht, pp 507–520Google Scholar
  47. 47.
    Simo JC, Ju JW (1987) Strain- and stress-based continuum damage models—I. Int J Solids Struct Formul 23(7):821. CrossRefzbMATHGoogle Scholar
  48. 48.
    Marigo J (1985) Modelling of brittle and fatigue damage for elastic material by growth of microvoids. Eng Fract Mech 21(4):861. CrossRefGoogle Scholar
  49. 49.
    Lemaitre J, Chaboche JL (1990) Mechanics of solid materials. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  50. 50.
    Mazars J (1984) Application de le mécanique de l’endommagement au comportement non lineaire et à la rupture du béton de structure. Ph.D. thesis, Université Pierre et Marie Curie - Laboratoire de Mécanique et Technologie, Paris, France (in french) Google Scholar
  51. 51.
    de Borst R, Gutiérrez MA (1999) A unified framework for concrete damage and fracture models including size effects. Int J Fract 95(1–4):261. CrossRefGoogle Scholar
  52. 52.
    Rahaman M, Deepu S, Roy D, Reddy J (2015) A micropolar cohesive damage model for delamination of composites. Compos Struct 131:425CrossRefGoogle Scholar
  53. 53.
    Forest S, Sievert R (2003) Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech 160(1–2):71CrossRefGoogle Scholar
  54. 54.
    Alves PD, Barros FB, Pitangueira RLS (2013) An object-oriented approach to the generalized finite element method. Adv Eng Softw 59:1. CrossRefGoogle Scholar
  55. 55.
    Malekan M, Barros FB, Pitangueira RLDS, Alves PD (2016) An object-oriented class organization for global-local generalized finite element method. Lat Am J Solids Struct 13:2529. CrossRefGoogle Scholar
  56. 56.
    Gamma E, Helm R, Johnson R, Vlissides J (1994) Design patterns: elements of reusable object-oriented software. Addison-Wesley Professional computing series. Pearson Education, LondonGoogle Scholar
  57. 57.
    Batoz JL, Dhatt G (1979) Incremental displacement algorithms for nonlinear problems. Int J Numer Methods Eng 14(8):1262. MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Yang YB, Shieh MS (1990) Solution method for nonlinear problems with multiple critical points. AIIA J 28(12):2110. CrossRefGoogle Scholar
  59. 59.
    Weller HG, Tabor G, Jasak H, Fureby C (1998) A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput Phys 12(6):620. CrossRefGoogle Scholar
  60. 60.
    Hassanpour S, Heppler G (2017) Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations. Math Mech Solids 22(2):224. MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Chang CS, Liao CL (1990) Constitutive relation for a particulate medium with the effect of particle rotation. Int J Solids Struct 26(4):437. CrossRefzbMATHGoogle Scholar
  62. 62.
    Chang CS, Ma L (1992) Elastic material constants for isotropic granular solids with particle rotation. Int J Solids Struct 29(8):1001. CrossRefzbMATHGoogle Scholar
  63. 63.
    Bigoni D, Drugan WJ (2007) Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J Appl Mech 74:741. MathSciNetCrossRefGoogle Scholar
  64. 64.
    Willoughby N, Parnell WJ, Hazel AL, Abrahams ID (2012) Homogenization methods to approximate the effective response of random fibre-reinforced composites. Int J Solids Struct 49:1421. CrossRefGoogle Scholar
  65. 65.
    Gori L Insane input files of the paper “A computational framework for the constitutive modelling of non-linear micropolar media”. Mendeley Data, v1.
  66. 66.
    Petersson PE (1981) Crack growth and development of fracture zones in plain concrete and similar materials. 28 TVBM-1006, Division of Building Materials, Lund Institute of Technology, Lund, SwedenGoogle Scholar
  67. 67.
    Winkler B, Hofstetter G, Lehar H (2004) Application of a constitutive model for concrete to the analysis of a precast segmental tunnel lining. Int J Numer Anal Methods Geomech 28(7–8):797CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Structural Engineering, Engineering SchoolFederal University of Minas Gerais (UFMG)Belo HorizonteBrazil

Personalised recommendations