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Standard gradient models and application to continuum damage in shell structures

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Abstract

The present paper deals with a gradient damage modeling framework for the case of general shell structures. The particularity of this formulation is that, as for the balance of linear momentum, the damage evolution is also governed by a boundary value problem. In a first step, the generalized standard formalism of continuum thermodynamics is developed for the particular case of continuum damage mechanics. A model is presented that fulfils the requirements of irreversible processes. Then, in a second step, the model is adapted to shell kinematics in terms of the two-dimensional mid-surface. On the numerical side, we discuss the problem of numerically integrating this coupled problem and its implementation within the context of the finite element method. For the spatial discretization, a node-based approach is performed for the damage field that is related to a staggered global solution procedure. Here the popular four-nodes shell finite element is used for illustrative purposes. Finally, a set of simulations highlights the efficiency of the proposed framework.

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References

  1. Abali, B.E., Klunker, A., Barchiesi, E., Placidi, L.: A novel phase-field approach to brittle damage mechanics of gradient metamaterials combining action formalism and history variables. ZAMM-Z. Angew. Math. Mech. 101(9), e202000289 (2021)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma \)-convergence. Commun. Pure Appl. Math. 43, 999–1036 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Awada, Z., Nedjar, B.: Finite viscoelastic modeling of yeast cells with an axisymmetrical shell approach. Mech. Res. Commun. 126, 104021 (2022)

    Article  Google Scholar 

  5. Awada, Z., Nedjar, B.: On a finite strain modeling of growth in budding yeast. Int. J. Numer. Methods Biomed. Eng. (2023). https://doi.org/10.1002/cnm.3710

  6. Borden, M.J., Hughes, T.J.R., 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. 321, 130–166 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chapelle, D., Bathe, K.J.: The Finite Element Analysis of Shells—Fundamentals. Springer, Berlin, Heidelberg (2011)

    Book  MATH  Google Scholar 

  8. Coleman, B., Gurtin, M.: Thermodynamics with internal variables. J. Chem. Phys. 47, 597–613 (1967)

    Article  Google Scholar 

  9. Dvorkin, E.N., Bathe, K.J.: A continuum mechanics based four-node shell element for general nonlinear analysis. Eng. Comput. 1, 77–88 (1984)

    Article  Google Scholar 

  10. Frémond, M., Nedjar, B.: Endommagement et principe des puissances virtuelles. Compt. Rend. l’Acad. Sci. Paris, Ser. II(317), 857–864 (1993)

    MATH  Google Scholar 

  11. Frémond, M., Nedjar, B.: Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33, 1083–1103 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Germain, P., Nguyen, Q.S., Suquet, P.: Continuum thermodynamics. ASME J. Appl. Mech. 50, 1010–1021 (1983)

    Article  MATH  Google Scholar 

  13. Holzapfel, G.: Nonlinear Solid Mechanics. A Continuum Approach for Engineering. Wiley, Chichester (2000)

    MATH  Google Scholar 

  14. Lemaitre, J., Chaboche, J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  15. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. 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, 1273–1311 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nedjar, B.: Elastoplastic-damage modelling including the gradient of damage. Formulation and computational aspects. Int. J. Solids Struct. 38, 5421–5451 (2001)

    Article  MATH  Google Scholar 

  19. Nedjar, B.: A theoretical and computational setting for geometrically nonlinear damage modelling framework. Comput. Mech. 30, 65–80 (2002)

    Article  MATH  Google Scholar 

  20. Nedjar, B.: On a concept of directional damage gradient in transversely isotropic materials. Int. J. Solids Struct. 88–89, 56–67 (2016)

    Article  Google Scholar 

  21. Nguyen, Q.S.: Some remarks on standard gradient models and gradient plasticity. Math. Mech. Solids 20(6), 760–769 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Oñate, E.: Structural analysis with the finite element method. In: Linear Statics. Volume 2. Beams, Plates and Shells. International Center for Numerical Methods in Engineering (CIMNE), Barcelona (2013)

  23. Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M., de Vree, J.H.P.: Gradient-enhanced damage for quasi-brittle materials. Int. J. Numer. Methods Eng. 39, 3391–3403 (1986)

    Article  MATH  Google Scholar 

  24. Placidi, L., Barchiesi, E., dell’Isola, F., Maksimov, V., Misra, A., Rezaei, N., Scrofani, A., Timofeev, D.: On a hemi-variational formulation for 2D elasto-plastic-damage strain gradient solid with granular microstructure. Math. Eng. 5(1), 1–24 (2022)

    Article  MathSciNet  Google Scholar 

  25. Simo, J.C., Fox, D.D., Rifai, M.S.: On a stress resultant geometrically exact shell model. Part II: the linear theory. Computational aspects. Comput. Methods Appl. Mech. Eng. 73, 53–92 (1989)

    Article  MATH  Google Scholar 

  26. Simo, J.C., Ju, J.: Strain- and stress-based continuum damage models. Part I: formulation. Int. J. Solids Struct. 23(7), 821–840 (1987)

    Article  MATH  Google Scholar 

  27. Simo, J.C., Ju, J.: Strain- and stress-based continuum damage models. Part II: computational aspects. Int. J. Solids Struct. 23(7), 841–869 (1987)

    Article  MATH  Google Scholar 

  28. Spencer, A.: Constitutive theory for strongly anisotropic solids. In: A. Spencer (ed.) Continuum Theory of the Mechanics of Fibre-Reinforced Composites, CISM Courses and Lectures, vol. 282. Springer, Wien (1984)

  29. Steinmann, P.: Formulation and computation of geometrically non-linear gradient damage. Int. J. Numer. Methods Eng. 46, 757–779 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Teichtmeister, S., Kienle, D., Aldakheel, F., Keip, M.A.: Phase-field modeling of fracture in anisotropic solids. Int. J. Non-Linear Mech. 97, 1–21 (2017)

    Article  Google Scholar 

  31. Wick, T.: Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation. Comput. Methods Appl. Mech. Eng. 325, 577–611 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wriggers, P.: Nonlinear Finite Element Methods. Springer, Berlin, Heidelberg (2008)

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

    Article  MathSciNet  Google Scholar 

  34. Zienkiewicz, O., Taylor, R.: The Finite Element Method, vol. 1, 5th edn. Butterworth-Heinemann, Oxford (2000)

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Authors and Affiliations

  1. Université Gustave Eiffel, MAST-EMGCU 5 Boulevard Descartes, 77454, Marne-la-Vallée Cedex 2, France

    Boumediene Nedjar

  2. Institut Jean Le Rond d’Alembert, Sorbonne Université, CNRS UMR 7190, 4 Place Jussieu, 75005, Paris, France

    Zeinab Awada

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  1. Boumediene Nedjar
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  2. Zeinab Awada
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B. Nedjar wrote the main manuscript and Z. Awada prepared and performed the numerical simulations

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Correspondence to Boumediene Nedjar.

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Nedjar, B., Awada, Z. Standard gradient models and application to continuum damage in shell structures. Z. Angew. Math. Phys. 74, 163 (2023). https://doi.org/10.1007/s00033-023-02055-0

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