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Variational Modelling of Strain Localization in Solids: A Computational Mechanics Point of View

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Abstract

Strain localization is one of the most challenging phenomena in solid mechanics. It occurs when strains concentrate in very narrow bands within a solid, typically referred to as localization bands. This behaviour is related to different types of failure mechanisms: fracture, shear bands and slip lines. Being a dissipative process, the modelling of strain localization could seem to preclude the use of variational methods. However, a sound framework to deal with this issue has been developed in the last few decades. In this contribution, we review the modelling of strain localization by means of variational methods systematically, presenting the main underlying theoretical concepts. The issue of irreversibility is approached by means of the theory of generalized standard materials, which constitutes the basis for the variational approach. Then, typical problems occurring in the modelling of strain localization are analyzed: the tendency to localize in a band of zero thickness with no dissipation, the determination of the geometry of the localization band, and the orientation bias of such band with respect to mesh alignment in finite element discretizations. We discuss solutions for these problems, focusing on the approach that tackles the description of the localization band using phase fields.

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References

  1. Alessi R (2013) Variational approach to fracture mechanics with plasticity. PhD thesis, Université Pierre et Marie Curie-Paris

  2. Alessi R, Marigo JJ, Vidoli S (2014) Gradient damage models coupled with plasticity and nucleation of cohesive cracks. Arch Ration Mech Anal 214(2):575–615

    MathSciNet  MATH  Google Scholar 

  3. Alessi R, Marigo JJ, Maurini C, Vidoli S (2017) Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: one-dimensional examples. Int J Mech Sci 149:559–576

    Google Scholar 

  4. Alessi R, Ambati M, Gerasimov T, Vidoli S, De Lorenzis L (2018) Comparison of phase-field models of fracture coupled with plasticity. In: Oñate E, Peric D, de Souza Neto E, Chiumenti M (eds) Advances in computational plasticity. Springer, Cham, pp 1–21

    Google Scholar 

  5. Ambati M, Gerasimov T, De Lorenzis L (2015) Phase-field modeling of ductile fracture. Comput Mech 55(5):1017–1040

    MathSciNet  MATH  Google Scholar 

  6. Ambati M, Kruse R, De Lorenzis L (2016) A phase-field model for ductile fracture at finite strains and its experimental verification. Comput Mech 57(1):149–167

    MathSciNet  MATH  Google Scholar 

  7. Ambrosio L, Tortorelli VM (1990) Approximation of functional depending on jumps by elliptic functional via \(\gamma \)-convergence. Commun Pure Appl Math 43(8):999–1036

    MathSciNet  MATH  Google Scholar 

  8. Ambrosio L, Lemenant A, Royer-Carfagni G (2013) A variational model for plastic slip and its regularization via\(\gamma \)-convergence. J Elast 110(2):201–235

    MathSciNet  MATH  Google Scholar 

  9. 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):1209–1229

    MATH  Google Scholar 

  10. Armero F (2018) Elastoplastic and viscoplastic deformations in solids and structures. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, 2nd edn. Wiley, Hoboken, pp 1–41

  11. Borden MJ, Verhoosel CV, Scott MA, Hughes TJ, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217:77–95

    MathSciNet  MATH  Google Scholar 

  12. Borden MJ, Hughes TJ, Landis CM, Anvari A, Lee IJ (2016) A phase-field formulation for fracture in ductile materials: finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Comput Methods Appl Mech Eng 312:130–166

    MathSciNet  MATH  Google Scholar 

  13. Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826

    MathSciNet  MATH  Google Scholar 

  14. Bourdin B, Francfort GA, Marigo JJ (2008) The variational approach to fracture. J Elast 91(1–3):5–148

    MathSciNet  MATH  Google Scholar 

  15. Braides A, Dal Maso G, Garroni A (1999) Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch Ration Mech Anal 146(1):23–58

    MathSciNet  MATH  Google Scholar 

  16. Conti S, Focardi M, Iurlano F (2016) Phase field approximation of cohesive fracture models. Ann Inst Henri Poincare Non Linear Anal 33:1033–1067

    MathSciNet  MATH  Google Scholar 

  17. Dal Maso G, Toader R (2002) A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch Ration Mech Anal 162(2):101–135

    MathSciNet  MATH  Google Scholar 

  18. Dal Maso G, Orlando G, Toader R (2016) Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case. Calc Var Partial Differ Equ 55(3):45

    MathSciNet  MATH  Google Scholar 

  19. De Borst R, Mühlhaus HB (1992) Gradient-dependent plasticity: formulation and algorithmic aspects. Int J Numer Methods Eng 35:521–539

    MATH  Google Scholar 

  20. de Borst R, Verhoosel CV (2016) Gradient damage vs phase-field approaches for fracture: similarities and differences. Comput Methods Appl Mech Eng 312:78–94

    MathSciNet  MATH  Google Scholar 

  21. de Borst R, Pamin J, Geers MG (1999) On coupled gradient-dependent plasticity and damage theories with a view to localization analysis. Eur J Mech A Solids 18(6):939–962

    MATH  Google Scholar 

  22. Del Piero G (2013) A variational approach to fracture and other inelastic phenomena. J Elast 112(1):3–77

    MathSciNet  MATH  Google Scholar 

  23. Faria R, Oliver J, Cervera M (2004) Modeling material failure in concrete structures under cyclic actions. J Struct Eng 130(12):1997–2005

    Google Scholar 

  24. Forest S, Lorentz E (2004) Localization phenomena and regularization methods. In: Besson (ed) Local approach to fracture, vol 1. Presses des Mines, Paris, pp 311–371

    Google Scholar 

  25. Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342

    MathSciNet  MATH  Google Scholar 

  26. Fraternali F, Negri M, Ortiz M (2010) On the convergence of 3D free discontinuity models in variational fracture. Int J Fract 166(1–2):3–11

    MATH  Google Scholar 

  27. Gerasimov T, De Lorenzis L (2016) A line search assisted monolithic approach for phase-field computing of brittle fracture. Comput Methods Appl Mech Eng 312:276–303

    MathSciNet  MATH  Google Scholar 

  28. Germain P, Nguyen QS, Suquet P (1983) Continuum thermodynamics. J Appl Mech 50(4b):1010–1020

    MATH  Google Scholar 

  29. Halphen B, Nguyen Q (1975) Generalized standard materials. J Mec 14(1):39–63

    Google Scholar 

  30. Jirásek M (2000) Comparative study on finite elements with embedded discontinuities. Comput Methods Appl Mech Eng 188(1–3):307–330

    MATH  Google Scholar 

  31. Jirásek M (2002) Objective modeling of strain localization. Revue française de génie civil 6(6):1119–1132

    Google Scholar 

  32. Jirásek M (2007) Mathematical analysis of strain localization. Revue européenne de génie civil 11(7–8):977–991

    Google Scholar 

  33. Loret B, Prevost JH (1991) Dynamic strain localization in fluid-saturated porous media. J Eng Mech 117(4):907–922

    Google Scholar 

  34. Lussardi L, Negri M (2007) Convergence of nonlocal finite element energies for fracture mechanics. Numer Funct Anal Optim 28(1–2):83–109

    MathSciNet  MATH  Google Scholar 

  35. Mandal TK, Nguyen VP, Wu JY (2019) Length scale and mesh bias sensitivity of phase-field models for brittle and cohesive fracture. Eng Fract Mech 217:106532

    Google Scholar 

  36. Marigo JJ (2000) From Clausius-Duhem and Drucker-Ilyushin inequalities to standard materials. In: Maugin GA, Drouot R, Sidoroff F (eds) Continuum thermomechanics. Springer, Dordrecht, pp 289–300

    Google Scholar 

  37. Marigo JJ, Geromel Fischer A (2018) Gradient damage models coupled with plasticity and their application to dynamic fragmentation. In: Lambert DE, Pasiliao CL, Erzar B, Revil-Baudard B, Cazacu O (eds) Dynamic damage and fragmentation. Wiley, Hoboken, pp 95–141

    Google Scholar 

  38. Marigo JJ, Maurini C, Pham K (2016) An overview of the modelling of fracture by gradient damage models. Meccanica 51(12):3107–3128

    MathSciNet  MATH  Google Scholar 

  39. Mesgarnejad A, Bourdin B, Khonsari M (2015) Validation simulations for the variational approach to fracture. Comput Methods Appl Mech Eng 290:420–437

    MathSciNet  MATH  Google Scholar 

  40. Miehe C, Lambrecht M (2003a) Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: large-strain theory for standard dissipative solids. Int J Numer Methods Eng 58(1):1–41

    MathSciNet  MATH  Google Scholar 

  41. Miehe C, Lambrecht M (2003b) A two-scale finite element relaxation analysis of shear bands in non-convex inelastic solids: small-strain theory for standard dissipative materials. Comput Methods Appl Mech Eng 192(5–6):473–508

    MathSciNet  MATH  Google Scholar 

  42. Miehe C, Hofacker M, Welschinger F (2010a) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45):2765–2778

    MathSciNet  MATH  Google Scholar 

  43. Miehe C, Welschinger F, Hofacker M (2010b) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83(10):1273–1311

    MathSciNet  MATH  Google Scholar 

  44. Miehe C, Teichtmeister S, Aldakheel F (2016) Phase-field modelling of ductile fracture: a variational gradient-extended plasticity-damage theory and its micromorphic regularization. Philos Trans R Soc A 374(2066):20150170

    MathSciNet  MATH  Google Scholar 

  45. Miehe C, Aldakheel F, Teichtmeister S (2017) Phase-field modeling of ductile fracture at finite strains: a robust variational-based numerical implementation of a gradient-extended theory by micromorphic regularization. Int J Numer Methods Eng 111:816–863

    MathSciNet  Google Scholar 

  46. Mielke A (2006) A mathematical framework for generalized standard materials in the rate-independent case. In: Helmig R, Mielke A, Wohlmuth BI (eds) Multifield problems in solid and fluid mechanics, vol 28. Springer, Berlin, p 399

    MATH  Google Scholar 

  47. Moës N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fract Mech 69(7):813–833

    Google Scholar 

  48. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150

    MathSciNet  MATH  Google Scholar 

  49. Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42(5):577–685

    MathSciNet  MATH  Google Scholar 

  50. Negri M (2007) Convergence analysis for a smeared crack approach in brittle fracture. Interfaces Free Bound 9(3):307–330

    MathSciNet  MATH  Google Scholar 

  51. Negri M (2014) Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics. ESAIM Control Optim Calc Var 20(4):983–1008

    MathSciNet  MATH  Google Scholar 

  52. Nguyen QS (2000) Standard dissipative systems and stability analysis. In: Maugin GA, Drouot R, Sidoroff F (eds) Continuum thermomechanics. Springer, Dordrecht, pp 343–354

    Google Scholar 

  53. Oliver J (1989) A consistent characteristic length for smeared cracking models. Int J Numer Methods Eng 28(2):461–474

    MATH  Google Scholar 

  54. Oliver J, Huespe A, Samaniego E, Chaves E (2002) On strategies for tracking strong discontinuities in computational failure mechanics. In: Fifth world congress on computational mechanics, pp 7–12

  55. Oliver J, Huespe A, Samaniego E (2003) A study on finite elements for capturing strong discontinuities. Int J Numer Methods Eng 56(14):2135–2161

    MathSciNet  MATH  Google Scholar 

  56. Oliver J, Huespe A, Samaniego E, Chaves E (2004) Continuum approach to the numerical simulation of material failure in concrete. Int J Numer Anal Methods Geomech 28(7–8):609–632

    MATH  Google Scholar 

  57. Oliver X, Agelet de Saracibar C (2017) Continuum mechanics for engineers. In: Theory and problems, 2nd edn. https://doi.org/10.13140/RG.2.2.25821.20961

    Chapter  Google Scholar 

  58. Peerlings R, Geers M, De Borst R, Brekelmans W (2001) A critical comparison of nonlocal and gradient-enhanced softening continua. Int J Solids Struct 38(44–45):7723–7746

    MATH  Google Scholar 

  59. Pham K, Marigo JJ (2010) Approche variationnelle de l’endommagement: I. Les concepts fondamentaux. C R Mec 338(4):191–198

    MATH  Google Scholar 

  60. Pham K, Marigo JJ (2010) Approche variationnelle de l’endommagement: II. Les modèles à gradient. C R Mec 338(4):199–206

    MATH  Google Scholar 

  61. Pham K, Amor H, Marigo JJ, Maurini C (2011) Gradient damage models and their use to approximate brittle fracture. Int J Damage Mech 20(4):618–652

    Google Scholar 

  62. Rabczuk T, Bordas S, Zi G (2010) On three-dimensional modelling of crack growth using partition of unity methods. Comput Struct 88(23–24):1391–1411

    Google Scholar 

  63. Rodriguez P, Ulloa J, Samaniego C, Samaniego E (2018) A variational approach to the phase field modeling of brittle and ductile fracture. Int J Mech Sci 144:502–517

    Google Scholar 

  64. Samaniego E, Belytschko T (2005) Continuum–discontinuum modelling of shear bands. Int J Numer Methods Eng 62(13):1857–1872

    MathSciNet  MATH  Google Scholar 

  65. Simo JC, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 12(5):277–296

    MathSciNet  MATH  Google Scholar 

  66. Tanne E (2017) Variational phase-field models from brittle to ductile fracture: nucleation and propagation. PhD thesis, Paris Saclay

  67. Ulloa J, Rodríguez P, Samaniego E (2016) On the modeling of dissipative mechanisms in a ductile softening bar. J Mech Mater Struct 11(4):463–490

    MathSciNet  Google Scholar 

  68. Ulloa J, Rodríguez P, Samaniego C, Samaniego E (2019) Phase-field modeling of fracture for quasi-brittle materials. Undergr Space 4(1):10–21

    Google Scholar 

  69. Vigueras G, Sket F, Samaniego C, Wu L, Noels L, Tjahjanto D, Casoni E, Houzeaux G, Makradi A, Molina-Aldareguia JM et al (2015) An XFEM/CZM implementation for massively parallel simulations of composites fracture. Compos Struct 125:542–557

    Google Scholar 

  70. Wu JY (2017) A unified phase-field theory for the mechanics of damage and quasi-brittle failure. J Mech Phys Solids 103:72–99

    MathSciNet  Google Scholar 

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Acknowledgements

The first author would like to acknowledge that this work was completed while being in a sabbatical leave.

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Authors and Affiliations

  1. Facultad de Ingeniería y Departamento de Recursos Hídricos y Ciencias Ambientales, Universidad de Cuenca, Cuenca, Ecuador

    Esteban Samaniego

  2. Department of Civil Engineering, KU Leuven, 3001, Leuven, Belgium

    Jacinto Ulloa

  3. Institut für Kontinuumsmechanik, Leibniz Universität Hannover, 30167, Hannover, Germany

    Patricio Rodríguez

  4. Barcelona Supercomputing Center, Barcelona, Spain

    Cristóbal Samaniego

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  1. Esteban Samaniego
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  2. Jacinto Ulloa
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Correspondence to Esteban Samaniego.

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Samaniego, E., Ulloa, J., Rodríguez, P. et al. Variational Modelling of Strain Localization in Solids: A Computational Mechanics Point of View. Arch Computat Methods Eng 28, 1183–1203 (2021). https://doi.org/10.1007/s11831-020-09410-8

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