On a hemi-variational formulation for a 2D elasto-plastic-damage strain gradient solid with granular microstructure

Luca Placidi; Emilio Barchiesi; Francesco dell'Isola; Valerii Maksimov; Anil Misra; Nasrin Rezaei; Angelo Scrofani; Dmitry Timofeev; Luca Placidi; Emilio Barchiesi; Francesco dell'Isola; Valerii Maksimov; Anil Misra; Nasrin Rezaei; Angelo Scrofani; Dmitry Timofeev

doi:10.3934/mine.2023021

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-24. doi: 10.3934/mine.2023021

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On a hemi-variational formulation for a 2D elasto-plastic-damage strain gradient solid with granular microstructure

1.
Faculty of Engineering, International Telematic University UNINETTUNO, Rome, Italy
2.
International Research Center on Mathematics and Mechanics of Complex Systems (M & MoCS), Università degli Studi dell'Aquila, L'Aquila, Italy
3.
École Nationale d'Ingénieurs de Brest, 760 ENIB, UMR CNRS 6027, IRDL, F-29200 Brest, France
4.
Civil, Environmental and Architectural Engineering Department, The University of Kansas, Kansas City, Kansas, USA
5.
Université de Bretagne Sud, Rue de Saint Maudé - BP 92116 56321, Lorient, Cedex, France

Received: 25 September 2021 Revised: 21 January 2022 Accepted: 26 January 2022 Published: 29 March 2022

We report a continuum theory for 2D strain gradient materials accounting for a class of dissipation phenomena. The continuum description is constructed by means of a (reversible) placement function and by (irreversible) damage and plastic functions. Besides, expressions of elastic and dissipation energies have been assumed as well as the postulation of a hemi-variational principle. No flow rules have been assumed and plastic deformation is also compatible, that means it can be derived by a placement function. Strain gradient Partial Differential Equations (PDEs), boundary conditions (BCs) and Karush-Kuhn-Tucker (KKT) type conditions are derived by a hemi variational principle. PDEs and BCs govern the evolution of the placement descriptor and KKT conditions that of damage and plastic variables. Numerical experiments for the investigated homogeneous cases do not need the use of Finite Element simulations and have been performed to show the applicability of the model. In particular, the induced anisotropy of the response has been investigated and the coupling between damage and plasticity evolution has been shown.
- damage mechanics,
- granular microstructures,
- Karush-Kuhn-Tucker conditions,
- strain gradient,
- 2D continua
Citation: Luca Placidi, Emilio Barchiesi, Francesco dell'Isola, Valerii Maksimov, Anil Misra, Nasrin Rezaei, Angelo Scrofani, Dmitry Timofeev. On a hemi-variational formulation for a 2D elasto-plastic-damage strain gradient solid with granular microstructure[J]. Mathematics in Engineering, 2023, 5(1): 1-24. doi: 10.3934/mine.2023021

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Abstract

We report a continuum theory for 2D strain gradient materials accounting for a class of dissipation phenomena. The continuum description is constructed by means of a (reversible) placement function and by (irreversible) damage and plastic functions. Besides, expressions of elastic and dissipation energies have been assumed as well as the postulation of a hemi-variational principle. No flow rules have been assumed and plastic deformation is also compatible, that means it can be derived by a placement function. Strain gradient Partial Differential Equations (PDEs), boundary conditions (BCs) and Karush-Kuhn-Tucker (KKT) type conditions are derived by a hemi variational principle. PDEs and BCs govern the evolution of the placement descriptor and KKT conditions that of damage and plastic variables. Numerical experiments for the investigated homogeneous cases do not need the use of Finite Element simulations and have been performed to show the applicability of the model. In particular, the induced anisotropy of the response has been investigated and the coupling between damage and plasticity evolution has been shown.

References

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