Computational Mechanics

, Volume 59, Issue 6, pp 981–1001 | Cite as

Phase field modeling of brittle fracture for enhanced assumed strain shells at large deformations: formulation and finite element implementation

  • J. Reinoso
  • M. Paggi
  • C. Linder
Original Paper


Fracture of technological thin-walled components can notably limit the performance of their corresponding engineering systems. With the aim of achieving reliable fracture predictions of thin structures, this work presents a new phase field model of brittle fracture for large deformation analysis of shells relying on a mixed enhanced assumed strain (EAS) formulation. The kinematic description of the shell body is constructed according to the solid shell concept. This enables the use of fully three-dimensional constitutive models for the material. The proposed phase field formulation integrates the use of the (EAS) method to alleviate locking pathologies, especially Poisson thickness and volumetric locking. This technique is further combined with the assumed natural strain method to efficiently derive a locking-free solid shell element. On the computational side, a fully coupled monolithic framework is consistently formulated. Specific details regarding the corresponding finite element formulation and the main aspects associated with its implementation in the general purpose packages FEAP and ABAQUS are addressed. Finally, the applicability of the current strategy is demonstrated through several numerical examples involving different loading conditions, and including linear and nonlinear hyperelastic constitutive models.


Shells Fracture mechanics Phase field fracture Finite elements Mixed formulation 



MP and JR gratefully acknowledge financial support of the European Research Council (ERC), Grant No. 306622 through the ERC Starting Grant “Multi-field and multi-scale Computational Approach to Design and Durability of PhotoVoltaic Modules” - CA2PVM. JR is also grateful to the support of the Spanish Ministry of Economy and Competitiveness (Projects MAT2015-71036-P and MAT2015-71309-P) and Andalusian Government (Project of Excellence No. TEP-7093).


  1. 1.
    Reinoso J, Paggi M, Areias P (2016) A finite element framework for the interplay between delamination and buckling of rubber-like bi-material systems and stretchable electronics. J Eur Ceram Soc 36:2371–2382CrossRefGoogle Scholar
  2. 2.
    Paggi M, Corrado M, Rodriguez MA (2013) A multi-physics and multi-scale numerical approach to microcracking and power-loss in photovoltaic modules. Compos Struct 95:630–638CrossRefGoogle Scholar
  3. 3.
    van der Sluis O, Abdallah AA, Bouten PCP, Timmermans PHM, den Toonder JMJ, de With G (2011) Effect of a hard coat layer on buckle delamination of thin ITO layers on a compliant elasto-plastic substrate: an experimental-numerical approach. Eng Fract Mech 78:877–889CrossRefGoogle Scholar
  4. 4.
    Rogers JA, Someya T, Huang Y (2010) Materials and mechanics for stretchable electronics. Science 327:1603–1607CrossRefGoogle Scholar
  5. 5.
    Jirásek M (1998) Nonlocal models for damage and fracture: comparison of approaches. Int J Solids Struct 35:4133–4145MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lemaitre J, Chaboche JL (1990) Mechanics of solid materials, vol 40. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  7. 7.
    Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP (1996) Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39:3391–3403CrossRefzbMATHGoogle Scholar
  8. 8.
    Pijaudier-Cabot G, Bazant Z (1987) Nonlocal damage theory. J Eng Mech 113:1512–1533CrossRefzbMATHGoogle Scholar
  9. 9.
    Linder C, Armero F (2007) Finite elements with embedded strong discontinuities for the modeling of failure in solids. Int J Numer Methods Eng 72:1391–1433MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150CrossRefzbMATHGoogle Scholar
  11. 11.
    Moës N, Stolz C, Bernard P-E, Chevaugeon N (2011) A level set based model for crack growth: thick level set method. Int J Numer Methods Eng 86(3):358–380CrossRefzbMATHGoogle Scholar
  12. 12.
    Reinoso J, Paggi M (2014) A consistent interface element formulation for geometrical and material nonlinearities. Comput Mech 54(6):1569–1581MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Forest S (2009) Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J Eng Mech 135:117–131CrossRefGoogle Scholar
  14. 14.
    Linder C, Zhang X (2013) A marching cubes based failure surface propagation concept for 3D finite elements with non-planar embedded strong discontinuities of higher order kinematics. Int J Numer Methods Eng 96:339–372CrossRefzbMATHGoogle Scholar
  15. 15.
    Waffenschmidt T, Polindara C, Menzel A, Blanco S (2014) A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials. Comput Methods Appl Mech Eng 268:801–842MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dolbow J, Moës N, Belytschko T (2000) Modeling fracture in MindlinReissner with the extended finite element method. Int J Solids Struct 33:7161–83CrossRefzbMATHGoogle Scholar
  17. 17.
    Areias PMA, Belytschko T (2005) Non-linear analysis of shells with arbitrary evolving cracks using XFEM. Int J Numer Methods Eng 62:384–415CrossRefzbMATHGoogle Scholar
  18. 18.
    Areias PMA, Song JH, Belytschko T (2006) Analysis of fracture in thin shells by overlapping paired elements. Comput Methods Appl Mech Eng 195:5343–5360CrossRefzbMATHGoogle Scholar
  19. 19.
    Rabczuk T, Areias PMA (2006) A meshfree thin shell for arbitrary evolving cracks based on an extrinsic basis. Comput Model Eng Sci 16:115–130Google Scholar
  20. 20.
    Rabczuk T, Zi G (2010) A meshfree method based on the local partition of unity for cohesive cracks. Comput Mech 39(6):743–760CrossRefzbMATHGoogle Scholar
  21. 21.
    Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193:3523–3540MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chau-Dinh T, Zi G, Lee PS, Rabczuk T, Song JH (2012) Phantom-node method for shell models with arbitrary cracks. Comput Struct 92:242–256CrossRefGoogle Scholar
  23. 23.
    Remmers JJC, Wells GN, de Borst R (2003) A solid-like shell element allowing for arbitrary delaminations. Int J Numer Methods Eng 58:2013–2040CrossRefzbMATHGoogle Scholar
  24. 24.
    Hosseini S, Remmers JJ, Borst R (2014) The incorporation of gradient damage models in shell elements. Int J Numer Methods Eng 98(6):391–398MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ambrosio L, Tortorelli VM (1992) On the approximation of free discontinuity problems. Boll Un Mat Italy B(7)6(1):105–123Google Scholar
  26. 26.
    Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Amor H, Marigo JJ, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229CrossRefzbMATHGoogle Scholar
  28. 28.
    Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond A 221:163–198CrossRefGoogle Scholar
  30. 30.
    Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rateindependent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45–48):2765–2778CrossRefzbMATHGoogle Scholar
  31. 31.
    Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase field models of fracture: variational principles and multi-field Fe-implementations. Int J Numer Methods Eng 83(10):1273–1311MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Bourdin B, Francfort GA, Marigo JJ (2008) The variational approach to fracture. J Elast 91(1–3):5–148MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Miehe C, Schänzel L (2014) Phase field modeling of fracture in rubbery polymers. Part I: finite elasticity coupled with brittle failure. J Mech Phys Solids 65:93–113MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Miehe C, Schänzel L, Ulmer H (2015) Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput Methods Appl Mech Eng 294:449–485MathSciNetCrossRefGoogle Scholar
  35. 35.
    Miehe C, Kienle D, Aldakheel F, Teichtmeister S (2016) Phase field modeling of fracture in porous plasticity: a variational gradient-extended Eulerian framework for the macroscopic analysis of ductile failure. Comput Methods Appl Mech Eng 312:3–50MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zhang X, Krischok A, Linder C (2016) A variational framework to model diffusion induced large plastic deformation and phase field fracture during initial two-phase lithiation of silicon electrodes. Comput Methods Appl Mech Eng 312:51–77MathSciNetCrossRefGoogle Scholar
  37. 37.
    Miehe C, Schänzel L, Ulmer H (2015) Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elasticplastic solids. Comput Methods Appl Mech Eng 294:486–522CrossRefGoogle Scholar
  38. 38.
    Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217220:7795MathSciNetzbMATHGoogle Scholar
  39. 39.
    Hesch C, Weinberg K (2014) Thermodynamically consistent algorithms for a finite-deformation phase field approach to fracture. Int J Numer Methods Eng 99(12):906–924MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Hofacker M, Miehe C (2013) A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns. Int J Numer Methods Eng 93(3):276–301MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Keip MA, Kiefer B, Schröder J, Linder C (2016) Special Issue on Phase Field Approaches to Fracture. In: Memory of Professor Christian Miehe (1956–2016). Comput Methods Appl Mech Eng 312:1–2Google Scholar
  42. 42.
    Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M (2014) Phase-field modeling of fracture in linear thin shells. Theor Appl Fract Mech 69:102–109CrossRefGoogle Scholar
  43. 43.
    Ulmer H, Hofacker M, Miehe C (2013) Phase field modeling of fracture in plates and shells. PAMM 12(1):171–172CrossRefGoogle Scholar
  44. 44.
    Ambati M, De Lorenzis L (2016) Phase-field modeling of brittle and ductile fracture in shells with isogeometric NURBS-based solid-shell elements. Comput Methods Appl Mech Eng 312:351–373Google Scholar
  45. 45.
    Areias P, Rabczuk T, Msekh MA (2016) Phase-field analysis of finite-strain plates and shells including element subdivision. Comput Methods Appl Mech Eng 312:322–350Google Scholar
  46. 46.
    Kiendl J, Ambati M, De Lorenzis L, Gomez H, Reali A (2016) Phase-field description of brittle fracture in plates and shells. Comput Methods Appl Mech Eng. doi: 10.1016/j.cma.2016.09.011 MathSciNetGoogle Scholar
  47. 47.
    Bischoff M, Ramm E (1997) Shear deformable shell elements for large strains and rotations. Int J Numer Methods Eng 40:4427–4449CrossRefzbMATHGoogle Scholar
  48. 48.
    Hauptmann R, Schweizerhof K (1998) A systematic development of solid-shell element formulations for linear and non-linear analyses employing only displacement degrees of freedom. Int J Numer Methods Eng 42:49–69CrossRefzbMATHGoogle Scholar
  49. 49.
    Klinkel S, Wagner W (1997) A geometrical non-linear brick element based on the EAS-method. Int J Numer Methods Eng 40:4529–4545CrossRefzbMATHGoogle Scholar
  50. 50.
    Miehe C (1998) A theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains. Comput Methods Appl Mech Eng 155:193–233CrossRefzbMATHGoogle Scholar
  51. 51.
    Simo JC, Rifai S (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29:1595–1638MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Simo JC, Armero F (1992) Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Methods Eng 33:1413–1449MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Dvorkin EN, Bathe KJ (1984) Continuum mechanics based fournode shell element for general non-linear analysis. Eng Comput 1:77–88CrossRefGoogle Scholar
  54. 54.
    Betsch P, Stein E (1995) An assumed strain approach avoiding arti- ficial thickness straining for a nonlinear 4-node shell element. Commun Numer Methods Eng 11:899–909CrossRefzbMATHGoogle Scholar
  55. 55.
    Vu-Quoc L, Tan XG (2003) Optimal solid shells for non-linear analyses of multilayer composites. I. Statics. Comput Methods Appl Mech Eng 192:975–1016CrossRefzbMATHGoogle Scholar
  56. 56.
    Msekh MA, Sargado M, Jamshidian M, Areias P, Rabczuk T (2015) Abaqus implementation of phase field model for brittle fracture. Comput Mater Sci 96(B):472–484CrossRefGoogle Scholar
  57. 57.
    Hauptmann R, Schweizerhof K, Doll S (2000) Extension of the solid-shell concept for application to large elastic and large elastoplastic deformations. Int J Numer Methods Eng 49:1121–1141CrossRefzbMATHGoogle Scholar
  58. 58.
    Schwarze M, Reese S (2011) A reduced integration solid-shell finite element based on the EAS and the ANS concept-large deformation problems. Int J Numer Methods Eng 85:289–329MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Holzapfel G (2000) Nonlinear solid mechanics. Wiley, New York ISBN: 978–0–471–82319–3zbMATHGoogle Scholar
  60. 60.
    Rah K, Van Paepegem W, Habraken AM, Degrieck J, de Sousa RA, Valente RAF (2013) Optimal low-order fully integrated solid-shell elements. Comput Mech 51(3):309–326MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Büchter N, Ramm E, Roehl D (1994) Three-dimensional extension of nonlinear shell formulation based on the enhanced assumed strain concept. Int J Numer Meth Eng 37:2551–2568CrossRefzbMATHGoogle Scholar
  62. 62.
    Zienkiewicz OC, Taylor RL (2000) The finite element method. Butterworth–Heinemann, Woburn, 5th ed, Vol. I. ISBN: 0750650494Google Scholar
  63. 63.
    Reinoso J, Blázquez A (2016) Application and finite element implementation of 7-parameter shell element for geometrically nonlinear analysis of layered CFRP composites. Compos Struct 139:263–276CrossRefGoogle Scholar
  64. 64.
    Simo JC, Armero F, Taylor RL (1993) Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comput Meth Appl Mech Eng 110(3):359–386CrossRefzbMATHGoogle Scholar
  65. 65.
    Hocine N, Abdelaziz M, Imad A (2002) Fracture problems of rubbers: J-integral estimation based upon factors and an investigation on the strain energy density distribution as a local criterion. Int J Fract 117:1–23CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Elasticity and Strength of Materials Group, School of EngineeringUniversity of SevilleSevilleSpain
  2. 2.IMT School for Advanced Studies LuccaLuccaItaly
  3. 3.Department of Civil and Environmental EngineeringStanford UniversityStanfordUSA

Personalised recommendations