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An a priori irreversible phase-field formulation for ductile fracture at finite strains based on the Allen–Cahn theory: a variational approach and FE-implementation

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In this paper, a new crack surface energy for the simulation of ductile fracture is proposed, which is based on the Allen–Cahn theory of diffuse interfaces. In contrast to existing fracture approaches, here, the crack surface energy density is a double-well potential based on a new interpretation of the crack surface. That is, the energy associated with the whole diffuse region between the fully cracked and intact regions is interpreted as crack surface energy. This kind of formulation, on the one hand, results in the balance of micromechanical forces and on the other hand, is a priori thermodynamically consistent. Furthermore, the proposed formulation is based on a gamma-convergent interface energy and it is in agreement with the classical solution of Irwin (Appl Mech Trans ASME E24:351–369, 1957). It is shown that in contrast to existing models, crack irreversibility is automatically fulfilled and no further constraints related to neither local nor global irreversibility are needed. To also account for potential plastic shear band localization, the approach is extended by a micromorphic plasticity model. By analyzing two different classical numerical benchmark problems, the proposed formulation is shown to enable mesh-independent results which are in agreement with the results of competing approaches.

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  1. ABAQUS/Standard. Dassault Systèmes Simulia Corp, Providence, RI (2022)

  2. Aldakheel, F., Kienle, D., Keip, M.-A., Miehe, C.: Phase field modeling of ductile fracture in soil mechanics. PAMM 17(1), 383–384 (2017).

    Article  Google Scholar 

  3. Aldakheel, F., Wriggers, P., Miehe, C.: A modified Gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling. Comput. Mech. 62(4), 815–833 (2018).

    Article  MathSciNet  Google Scholar 

  4. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27(6), 1085–1095 (1979).

    Article  CAS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Ambati, M., Kruse, R., De Lorenzis, L.: A phase-field model for ductile fracture at finite strains and its experimental verification. Comput. Mech. 57(1), 149–167 (2015).

    Article  MathSciNet  Google Scholar 

  7. Balzani, D., Ortiz, M.: Relaxed incremental variational formulation for damage at large strains with application to fiber-reinforced materials and materials with truss-like microstructures. Comput. Methods Appl. Mech. Eng. 92, 551–570 (2012)

    MathSciNet  Google Scholar 

  8. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45(5), 601–620 (1999)

    Article  Google Scholar 

  9. Borden, M.J., Hughes, T.J., Landis, C.M., Anvari, A., Lee, I.J.: 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 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  10. Bourdin, B.: Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound. 9, 411–430 (2007).

    Article  MathSciNet  Google Scholar 

  11. Braides, A.: Gamma-Convergence for Beginners. Oxford University Press, London (2002).

    Book  Google Scholar 

  12. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958).

    Article  ADS  CAS  Google Scholar 

  13. Cornetti, P., Pugno, N., Carpinteri, A., Taylor, D.: Finite fracture mechanics: a coupled stress and energy failure criterion. Eng. Fract. Mech. 73(14), 2021–2033 (2006).

    Article  Google Scholar 

  14. de Souza Neto, E.: The exact derivative of the exponential of an unsymmetric tensor. Comput. Methods Appl. Mech. Eng. 190(18–19), 2377–2383 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  15. Dimitrijevic, B., Hackl, K.: A method for gradient enhancement of continuum damage models. Tech. Mech. 28(1), 43–52 (2008)

    Google Scholar 

  16. Dimitrijevic, B.J., Hackl, K.: A regularization framework for damage-plasticity models via gradient enhancement of the free energy. Int. J. Numer. Methods Biomed. Eng. 27(8), 1199–1210 (2011).

    Article  MathSciNet  Google Scholar 

  17. 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).

    Article  Google Scholar 

  18. Eringen, A.C., Suhubi, E.: Nonlinear theory of simple micro-elastic solids–i. Int. J. Eng. Sci. 2(2), 189–203 (1964).

    Article  MathSciNet  Google Scholar 

  19. Forest, S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135(3), 117–131 (2009).

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  21. Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Ser. A 221, 163–198 (1921)

    Article  ADS  Google Scholar 

  22. Gültekin, O., Dal, H., Holzapfel, G.A.: A phase-field approach to model fracture of arterial walls: theory and finite element analysis. Comput. Methods Appl. Mech. Eng. 312, 542–566 (2016).

    Article  ADS  MathSciNet  PubMed  PubMed Central  Google Scholar 

  23. Gültekin, O., Dal, H., Holzapfel, G.A.: Numerical aspects of anisotropic failure in soft biological tissues favor energy-based criteria: a rate-dependent anisotropic crack phase-field model. Comput. Methods Appl. Mech. Eng. 331, 23–52 (2018).

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  24. Gurtin, M.E.: Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Phys. D 92(3–4), 178–192 (1996).

    Article  MathSciNet  CAS  Google Scholar 

  25. Hakim, V., Karma, A.: Laws of crack motion and phase-field models of fracture. J. Mech. Phys. Solids 57(2), 342–368 (2009).

    Article  ADS  CAS  Google Scholar 

  26. Hashin, Z.: Finite thermoelastic fracture criterion with application to laminate cracking analysis. J. Mech. Phys. Solids 44(7), 1129–1145 (1996).

    Article  ADS  CAS  Google Scholar 

  27. Hurtado, D., Stainier, L., Ortiz, M.: The special-linear update: an application of differential manifold theory to the update of isochoric plasticity flow rules. Int. J. Numer. Methods Eng. 97(4), 298–312 (2019).

    Article  MathSciNet  Google Scholar 

  28. Inglis, C.E.: Stresses in a plate due to the presence of cracks and sharp corners. Trans. Inst. Naval Archit. 55, 219–241 (1913)

    Google Scholar 

  29. Irwin, G.R.: Analysis of stresses and strains near the end of a crack traversing a plate. Appl. Mech. Trans. ASME E24, 351–369 (1957)

    Google Scholar 

  30. Junker, P., Schwarz, S., Jantos, D., Hackl, K.: A fast and robust numerical treatment of a gradient-enhanced model for brittle damage. Int. J. Multiscale Comput. Eng 17(2) (2019)

  31. Junker, P., Riesselmann, J., Balzani, D.: Efficient and robust numerical treatment of a gradient-enhanced damage model at large deformations. Int. J. Numer. Methods Eng. 123, 774–793 (2022).

    Article  MathSciNet  Google Scholar 

  32. Karma, A., Kessler, D.A., Levine, H.: Phase-field model of mode III dynamic fracture. Phys. Rev. Lett. 87(4), 045501 (2001).

    Article  ADS  CAS  PubMed  Google Scholar 

  33. Kirsch. Die theorie der elastizität und die bedürfnisse der festigkeitslehre. Zeitschrift des Vereins deutscher Ingenieure, pp. 797–807 (1898)

  34. Köhler, M., Balzani, D.: Evolving microstructures in relaxed continuum damage mechanics for the modeling of strain softening. J. Mech. Phys. Solids 173, 105199 (2023)

    Article  MathSciNet  Google Scholar 

  35. Kuhn, C., Müller, R.: A continuum phase field model for fracture. Eng. Fract. Mech. 77(18), 3625–3634 (2010).

    Article  Google Scholar 

  36. Kuhn, C., Schlüter, A., Müller, R.: On degradation functions in phase field fracture models. Comput. Mater. Sci. 108, 374–384 (2015).

    Article  Google Scholar 

  37. Kuhn, C., Noll, T., Müller, R.: On phase field modeling of ductile fracture. GAMM-Mitt. 39(1), 35–54 (2016).

    Article  MathSciNet  Google Scholar 

  38. Langenfeld, K., Mosler, J.: A micromorphic approach for gradient-enhanced anisotropic ductile damage. Comput. Methods Appl. Mech. Eng. 360, 112717 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  39. Langenfeld, K., Kurzeja, P., Mosler, J.: How regularization concepts interfere with (quasi-)brittle damage: a comparison based on a unified variational framework. Continuum Mech. Thermodyn. 34(6), 1517–1544 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  40. Lee, E.: Elasto-plastic deformation at finite strains. J. Appl. Mech. 36, 1–6 (1969)

    Article  ADS  Google Scholar 

  41. Leguillon, D.: Strength or toughness? A criterion for crack onset at a notch. Eur. J. Mech. A. Solids 21(1), 61–72 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  42. Linse, T., Hennig, P., Kästner, M., de Borst, R.: A convergence study of phase-field models for brittle fracture. Eng. Fract. Mech. 184, 307–318 (2017).

    Article  Google Scholar 

  43. Lubliner, J.: On the thermodynamic foundations of non-linear solid mechanics. Int. J. Non-Linear Mech. 7(3), 237–254 (1972).

    Article  ADS  Google Scholar 

  44. J. Mandel. Plasticité classique et viscoplasticité. In: CISM Courses and Lectures No. 97. Springer (1972)

  45. May, S., Vignollet, J., De Borst, R.: A numerical assessment of phase-field models for brittle and cohesive fracture: \(\gamma \)-convergence and stress oscillations. Eur. J. Mech. A. Solids 52, 72–84 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  46. Miehe, C., Schänzel, L.-M.: Phase field modeling of fracture in rubbery polymers. Part I: finite elasticity coupled with brittle failure. J. Mech. Phys. Solids 65, 93–113 (2014).

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  48. Miehe, C., Schänzel, L.-M., Ulmer, H.: 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, 449–485 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  49. Miehe, C., Aldakheel, F., Raina, A.: Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory. Int. J. Plast. 84, 1–32 (2016).

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

  51. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964).

    Article  MathSciNet  Google Scholar 

  52. Modica, L., Mortola, S.: Un esempio di \(\gamma \)-convergenza. Bollettino dell’Unione Matematica Italiana 14–B, 285–299 (1977)

    MathSciNet  Google Scholar 

  53. Mosler, M., Shchyglo, O., Montazer Hojjat, H.: A novel homogenization method for phase field approaches based on partial rank-one relaxation. J. Mech. Phys. Solids 68, 251–266 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  54. Msekh, M.A., Sargado, J.M., Jamshidian, M., Areias, P.M., Rabczuk, T.: Abaqus implementation of phase-field model for brittle fracture. Comput. Mater. Sci. 96, 472–484 (2014).

    Article  Google Scholar 

  55. Mughrabi, H.: Assessment of fatigue damage on the basis of nonlinear compliance effects. In: Handbook of Materials Behavior Models, pp. 622–632. Academic Press (2001)

  56. Orowan, E.: Fracture and strength of solids. Rep. Prog. Phys. 12(1), 185–232 (1949).

    Article  ADS  Google Scholar 

  57. Pandolfi, A., Ortiz, M.: An Eigenerosion approach to brittle fracture. Int. J. Numer. Methods Eng. 92(8), 694–714 (2012).

    Article  MathSciNet  Google Scholar 

  58. Polindara, C., Waffenschmidt, T., Menzel, A.: Simulation of balloon angioplasty in residually stressed blood vessels–application of a gradient-enhanced fibre damage model. J. Biomech. 49(12), 2341–2348 (2016).

    Article  PubMed  Google Scholar 

  59. Proserpio, D., Ambati, M., De Lorenzis, L., Kiendl, J.: A framework for efficient isogeometric computations of phase-field brittle fracture in multipatch shell structures. Comput. Methods Appl. Mech. Eng. 372, 113363 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  60. Raina, A., Miehe, C.: A phase-field model for fracture in biological tissues. Biomech. Model. Mechanobiol. 15(3), 479–496 (2015).

    Article  PubMed  Google Scholar 

  61. Riesselmann, J., Balzani, D.: A simple and efficient Lagrange multiplier based mixed finite element for gradient damage. Comput. Struct. 281, 107030 (2023).

    Article  Google Scholar 

  62. Schmidt, T., Balzani, D.: Relaxed incremental variational approach for the modeling of damage-induced stress hysteresis in arterial walls. J. Mech. Behav. Biomed. Mater. 58, 149–162 (2016).

    Article  PubMed  Google Scholar 

  63. Seupel, A., Hütter, G., Kuna, M.: An efficient FE-implementation of implicit gradient-enhanced damage models to simulate ductile failure. Eng. Fract. Mech. 199, 41–60 (2018).

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  65. Spetz, A., Denzer, R., Tudisco, E., Dahlblom, O.: Phase-field fracture modelling of crack nucleation and propagation in porous rock. Int. J. Fract. 224, 31–46 (2020).

    Article  CAS  Google Scholar 

  66. Steinbach, I.: Phase-field model for microstructure evolution at the mescoscopic scale. Annu. Rev. Mater. Res. 43(1), 89–107 (2013).

    Article  ADS  CAS  Google Scholar 

  67. Torabi, A.R., Berto, F., Sapora, A.: Finite fracture mechanics assessment in moderate and large scale yielding regimes. Metals 9(5), 602 (2019).

    Article  CAS  Google Scholar 

  68. Voce, E.: A practical strain hardening function. Metallurgia 51, 219–226 (1955)

    Google Scholar 

  69. Waffenschmidt, T., Polindara, C., Menzel, A., Blanco, S.: A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials. Comput. Methods Appl. Mech. Eng. 268, 801–842 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  70. Weißgraeber, P., Leguillon, D., Becker, B.: A review of finite fracture mechanics: crack initiation at singular and non-singular stress raisers. Arch. Appl. Mech. 86(1–2), 375–401 (2015).

    Article  Google Scholar 

  71. Wingender, D., Balzani, D.: Simulation of crack propagation through voxel-based, heterogeneous structures based on Eigenerosion and finite cells. Comput. Mech. 70, 385–406 (2022).

  72. Wingender, D., Balzani, D.: Simulation of crack propagation based on Eigenerosion in brittle and ductile materials subject to finite strains. Arch. Appl. Mech. (2022).

    Article  Google Scholar 

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Hereby, the authors H. Montazer Hojjat and D. Balzani would like to appreciate financial funding by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Collaborative Research Center 837 (SFB 837), Project C6 ”Interaction Modeling in Mechanized Tunneling”. The first author would like to thank to the chief executive officer of the SFB 837, Jörg Sahlmen, for his kind support during the stay at Ruhr University Bochum. The author S. Kozinov would like to thank the DFG for funding under Grant KO 6356/1-1.

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A Continuum mechanical framework for ductile materials

Here, the continuum mechanical basis for the phase field model proposed in this paper as well as the considered elasto-plastic material model are described. Generally, a fictitiously undamaged strain energy density functional \(\psi _0\), defined per unit reference volume, can be written as a function of a deformation gradient \(\varvec{F}=\text {Grad}_{\varvec{X}}(\varvec{\varphi })=\nabla _{\varvec{X}}\varvec{\varphi }\). The deformation mapping \(\varvec{\varphi }\) maps the point \(\varvec{X}\in \Omega \) in the reference (undeformed) configuration to the point \(\varvec{x}\in \varvec{\varphi }(\Omega )\) in the current (deformed) configuration. In addition, the plastic deformation is an inelastic stress-free process. Therefore, an intermediate stress-free configuration is considered, which results in a multiplicative decomposition of the deformation gradient into a plastic and elastic one, i.e.,

$$\begin{aligned} \varvec{F} =\varvec{F}^{\textrm{e}}\cdot \varvec{F}^{\textrm{p}}, \end{aligned}$$

cf. [40]. Based on this decomposition, it is clear that the plastic deformation gradient has no contribution to the stored elastic energy. Furthermore, in some materials such as metals, the plastic deformation is volume preserving, i.e., det\((\varvec{F^{\textrm{p}}})= 1\). Therefore, the plastic incompressibility condition has to be satisfied. The strain energy density can thus be decomposed additively into an elastic and plastic term

$$\begin{aligned} \psi _0(\varvec{F},\varvec{F}^{\textrm{p}},\alpha ) =\psi _0^{\textrm{e}}(\varvec{F},\varvec{F}^{\textrm{p}})+\psi _0^{\textrm{p}}(\alpha ). \end{aligned}$$

Herein, \(\alpha \) is a strain-like internal variable. To fulfill objectivity, the elastic part is rewritten as a function of the elastic right Cauchy-Green tensor \(\varvec{C}^{\textrm{e}}=\varvec{F}^{\mathrm {e^T}}\cdot \varvec{F}^{\textrm{e}} =\varvec{F}^{\mathrm {p^{-T}}}\cdot \varvec{C}\cdot \varvec{F}^{\mathrm {p^{-1}}}\) with the right Cauchy-Green tensor \(\varvec{C}:=\varvec{F}^{\textrm{T}}\cdot \varvec{F}\), such that \(\psi _0^{\textrm{e}}=\psi _0^{\textrm{e}}(\varvec{F}^{\mathrm {e^{T}}}\cdot \varvec{F}^{\textrm{e}}) = \psi _0(\varvec{C}^{\textrm{e}})\). By means of the postulate of minimum total potential energy and assuming conservative external forces, the deformation mapping is computed by

$$\begin{aligned} \varvec{\varphi }=\text {arg}\,\Biggl \{\text {inf}\left( \int _\Omega \psi _0\left( \varvec{C}^{\textrm{e}},\alpha \right) \,{\text {d}}V-\int _{\Omega }\rho _0\,\varvec{b}\cdot \varvec{\varphi } {\text {d}}V-\int _{\partial _{\varvec{N}}\Omega }\varvec{T}\cdot \varvec{\varphi }\,{\text {d}}A\right) \Biggr \}, \end{aligned}$$

where \(\rho _0\), \(\rho _0\varvec{b}\), \(\varvec{T}\) are the material density in reference configuration, body forces and the traction vector, respectively. Considering the second law of thermodynamics for isothermal conditions

$$\begin{aligned} \mathscr {D}=\varvec{P}_0:\dot{\varvec{F}}-\dot{\psi }_0\ge 0, \end{aligned}$$

with the first Piola–Kirchhoff stress tensor \(\varvec{P}_0\). Inserting the time derivative of \(\psi _0\) yields

$$\begin{aligned} \left( \varvec{P}_0 - \dfrac{\partial \psi _0}{\partial \varvec{F}}\right) :\dot{\varvec{F}} - \dfrac{\partial \psi _0}{\partial \varvec{F}^{\textrm{p}}}:\dot{\varvec{F}}^{\textrm{p}} - \dfrac{\partial \psi _0}{\partial \alpha }\dot{\alpha } \ge 0. \end{aligned}$$

Applying the Coleman–Noll procedure leads to the constitutive equation for the first Piola–Kirchhoff stress tensor

$$\begin{aligned} \varvec{P}_0=\dfrac{\partial \psi _0^{\textrm{e}}}{\partial {\varvec{F}}} = \varvec{P}_0^{\textrm{e}}\cdot \varvec{F}^{\textrm{p}^{\mathrm {-T}}} \end{aligned}$$

wherein \(\varvec{P}^{\textrm{e}}_0:=\partial _{\varvec{F}^{\textrm{e}}}\psi _0^{\textrm{e}}\) and \(\partial _{\varvec{F}}\varvec{F}^{\textrm{e}} = \varvec{F}^{\textrm{p}^{\mathrm {-T}}}\). Defining the Mandel stress tensor [44] in the intermediate configuration \(\varvec{\Sigma }^{\textrm{e}}=\varvec{F}^{\mathrm {e^T}}\cdot \varvec{P}^{\textrm{e}}\) and the plastic spatial velocity gradient \(\varvec{L}^{\textrm{p}}=\dot{\varvec{F}^{\textrm{p}}}\cdot \varvec{F}^{{\textrm{p}}^{-1}}\) yields the reduced dissipation inequality

$$\begin{aligned} \mathscr {D}=\varvec{\Sigma }^{\textrm{e}}:\varvec{L}^{\textrm{p}} -Q\dot{\alpha }\ge 0, \end{aligned}$$

where Q is the thermodynamic force of the internal variable \(\alpha \), i.e.,

$$\begin{aligned} Q:= \dfrac{\partial \psi _0}{\partial \alpha }=\dfrac{\partial \psi _0^{\textrm{p}}}{\partial \alpha }. \end{aligned}$$

For the description of elasto-plasticity, an admissible domain is defined in terms of the yield function \(\phi \le 0\). If the yield function is less than zero, i.e., \(\phi <0\), no plastic deformation takes place [43]. Furthermore, \(\phi =0\) represents the yield surface, where plastic deformations evolve. In case of metal plasticity with isotropic hardening, this state function is considered to have the form

$$\begin{aligned} \phi (\varvec{\Sigma ^{\textrm{e}}},Q):=||dev\varvec{\Sigma }^{\textrm{e}}||-Q. \end{aligned}$$

Application of the principle of maximum dissipation and considering an associative flow rule, the evolution equations are obtained as

$$\begin{aligned} \varvec{L}^{\textrm{p}}=\dot{\lambda }\varvec{N}, \,\,\,\,\, \dot{\lambda } = \dot{\alpha }, \end{aligned}$$

where \(\varvec{N}:=\partial _{\varvec{\Sigma }^{\textrm{e}}}\phi \) is a second-order tensor imposing a constraint on the direction of plastic deformation and \(\lambda \) is the plastic multiplier. Applying the principle of maximum dissipation yields the well-known Kuhn-Tucker conditions

$$\begin{aligned} \dot{\lambda }\ge 0, \,\,\,\,\, \dot{\lambda }\dot{\phi }\ge 0, \end{aligned}$$

which need to be fulfilled. In order to fulfill the plastic incompressibility condition observed in metals, exponential integration is often used, cf. de Souza Neto [14]. An alternative can be obtained by noting that the plastic incompressibility condition is automatically fulfilled if \(\varvec{N}\) is traceless [27]. In our model this will be the case. As a specification of the model, we consider a Neo-Hookean type energy density function

$$\begin{aligned} \psi _0^{\textrm{e}} = \dfrac{\mu }{2}\left( {\text {tr}}\left( \varvec{C}^{\textrm{e}}\right) -3\right) +\lambda \dfrac{J^{{\textrm{e}}^{2}} -1}{4} -\left( \dfrac{\lambda }{2}+\mu \right) \ln (J^{\textrm{e}}), \,\,\,\,\,\,\, J^{\textrm{e}}={\text {det}}(\varvec{F}^{\textrm{e}})>0. \end{aligned}$$

The plastic energy density is assumed to have the form [68]

$$\begin{aligned} \psi _0^{\textrm{p}}=\dfrac{1}{2} h \alpha ^2 + (\sigma _{\infty }-\sigma _y)\left[ \alpha +\dfrac{\exp (-\zeta \alpha )-1}{\zeta }\right] + \sigma _y\alpha , \end{aligned}$$

considering a saturation parameter \(\zeta \), a linear hardening parameter h, the initial yield stress \(\sigma _y\), and a further parameter \(\sigma _{\infty }\), which is associated with the modeling of the transition between an initially negative exponential hardening to a linear hardening. Note that the method proposed in this paper is not restricted to this choice of \(\psi _0^{\textrm{e}}\) and \(\psi _0^{\textrm{p}}\).

B Details of phase-field modeling for brittle fracture

For the case of a hyperelastic material, the stored energy can be obtained by multiplying the degradation function with the Helmholtz free energy density of the intact material \(\psi _0\). This leads to the stored energy of the body given by

$$\begin{aligned} E (\varvec{F},d) = \int _{\Omega } g(d)\psi _0(\varvec{F}) \,{\text {d}}V, \end{aligned}$$

Moreover, using the chain rule and the time derivative of (68), the rate of degraded stored energy reads

$$\begin{aligned} \mathscr {E}(\varvec{F},d, \dot{\varvec{F}}, \dot{d}) =\int _{\Omega } \left( \varvec{P}:\dot{\varvec{F}} -f^f \,\dot{d}\right) \,{\text {d}}V, \end{aligned}$$

where the first Piola–Kirchhoff stress tensor is given by \(\varvec{P} = g(d)\varvec{P}_0\). The fracture driving force \(f^f\) being the work conjugate to the phase-field parameter [22, 47] reads

$$\begin{aligned} f^f:=-\dfrac{\partial \psi }{\partial d} =-g^{\prime }(d)\psi _0. \end{aligned}$$

In addition, considering the long and short-range forces acting on the body, the external power is obtained as

$$\begin{aligned} \mathscr {P} = \int _{\Omega } \rho _0\, \varvec{b} \cdot \dot{\varvec{\varphi }} \,{\text {d}}V + \int _{\partial _{\varvec{N}}\Omega }\varvec{T}\cdot \dot{\varvec{\varphi }} \,{\text {d}}A. \end{aligned}$$

Furthermore, the regularized crack surface energy is defined using the critical fracture energy constant \(g_c\). This energy is required to convert a fully intact matter into a fully cracked one. Considering constant \(g_c\) and the crack surface density functional, the crack energy is obtained

$$\begin{aligned} D (d) = \int _{\Omega } g_c\gamma (d, \nabla d)\, {\text {d}}V. \end{aligned}$$

Therefore, using the time derivative and the chain rule, the dissipation functional \(\mathscr {D}\) for elastic materials reads

$$\begin{aligned} \mathscr {D}(\dot{d}) = \int _{\Omega } g_c\dot{\gamma }\, {\text {d}}V = \int _{\Omega } g_c \frac{\partial \gamma }{\partial d}\dot{d} \,{\text {d}}V, \end{aligned}$$

where according to the second law of thermodynamics, only the non-negative values of the dissipation functional are admissible, i.e., \(\mathscr {D}(\dot{d})\ge 0\). Furthermore, Miehe et al. [47] enforced an irreversibility condition locally by defining a ramp-type energy function, that explodes for a negative evolution of the phase-field parameter. That means, the phase-field parameter is not allowed to reduce at any material point. Furthermore, the derivative of the crack surface density functional with respect to the phase-field parameter reads

$$\begin{aligned} \frac{\partial \gamma }{\partial d} = \frac{1}{l_s} \left( d - l_f^2 \Delta d\right) , \end{aligned}$$

where \(\Delta d\) is the material Laplacian of the phase-field parameter. At this stage, using the rate of stored energy functional (69), the external power (71) and the dissipation functional (73), the balance of mechanical power can be described as

$$\begin{aligned} \Pi (\dot{\varvec{\varphi }}, \dot{d}):= \mathscr {E}(\dot{\varvec{\varphi }}, \dot{d}) + \mathscr {D}(\dot{d}) - \mathscr {P}(\dot{\varvec{\varphi }}). \end{aligned}$$

Hence, the rates of deformation and damage parameter can be obtained from the variational principle

$$\begin{aligned} (\dot{\varvec{\varphi }}, \dot{d}) = \arg \{\inf _{\dot{\varvec{\varphi }}\in \mathscr {W}_{\varvec{\varphi }}}\,\inf _{\dot{d}\in \mathscr {W}_{d}}\,\Pi (\dot{\varvec{\varphi }}, \dot{d}) \}, \end{aligned}$$

where the following Dirichlet-type boundary condition is satisfied for the state variables

$$\begin{aligned} \varvec{\dot{\varphi }} \in \mathscr {W}_{\varvec{\varphi }}:=\Biggl \{\varvec{\dot{\varphi }}| \varvec{\dot{\varphi }}=\varvec{0}\, \,\textrm{on}\, \partial _{\varvec{N}} \Omega _{\varvec{\varphi }}\Bigg \} \,\,\,\,\,\textrm{and} \,\,\,\,\, \dot{d}\in \mathscr {W}_{d}:=\Biggl \{\dot{d}| \dot{d}=0\, \,\textrm{on}\, \partial _{\varvec{N}} \Omega _{d}\Bigg \}. \end{aligned}$$

The variation of the balance of mechanical power leads to two equations: the balance of linear momentum

$$\begin{aligned} \text {Div}\, \varvec{P} +\rho _0\, \varvec{b} = \varvec{0}. \end{aligned}$$

and the Kuhn–Tucker complementary conditions \(\dot{d} \ge 0\), \(f^f - g_c \delta _d \gamma \le 0\), and \((f^f - g_c \delta _d \gamma )\dot{d}=0\). That means the crack does not propagate as soon as \(f^f - g_c \delta _d \gamma < 0\). On the other hand, as soon as the driving force reaches the critical value \(f^f = g_c \delta _d \gamma \), the crack propagates. Remembering \(f^f = -g^{\prime }(d)\psi _0\), i.e., (70), the case distinction can be formulated as

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{d} = 0 \,\,\,\,\,\,\, \text {if} \,\,\,\,\,\,\, -g^{\prime }(d)\psi _0 - g_c \delta _d \gamma < 0, \\ \dot{d} > 0 \,\,\,\,\,\,\, \text {if} \,\,\,\,\,\,\, -g^{\prime }(d)\psi _0 = g_c \delta _d \gamma . \end{array}\right. } \end{aligned}$$

Due to the local irreversibility, this type of formulation may lead to an unrealistic evolution of the phase-field parameter far away from the localization area in the case of ductile fracture. Moreover, May et al. [45] have shown numerically that discretized forms of such type of formulation are not necessarily \(\Gamma \)-convergent. This is also rooted in the local irreversibility condition, see [42]. This flaw was to some extent removed in Miehe et al. [50] by considering a fracture energy threshold. This threshold is imposed by a material constant \(w_c\). That means, the phase-field parameter would evolve just in the material points, which possess energies beyond this threshold. Furthermore, the local irreversibility constraint is only enforced within this region which will vary throughout the crack formation in the general case. Taking this threshold into account, the thermodynamic force in (70) is modified to

$$\begin{aligned} f^f:=-g^{\prime }(d)(\psi _0 - w_c). \end{aligned}$$

This formulation is sensitive to the definition of the material parameter \(w_c\). In other words, the phase-field parameter would only evolve in the region, primarily determined by this material parameter. Not only this, but also the damage parameter is not allowed to get more localized within this region. Additionally, the discrete form of this formulation is not proven to be \(\Gamma \)-convergent.

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Montazer Hojjat, H., Kozinov, S. & Balzani, D. An a priori irreversible phase-field formulation for ductile fracture at finite strains based on the Allen–Cahn theory: a variational approach and FE-implementation. Arch Appl Mech 94, 365–390 (2024).

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