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A computational framework for the constitutive modeling of nonlinear micropolar media

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Abstract

Despite the large number of applications with micropolar models, the aspects of their implementation have been rarely addressed in the literature. In the present paper, a strategy for the computational modeling of micropolar media with elasto-plasticity and elastic degradation is investigated. The proposed strategy is based on the Object-Oriented Paradigm (OOP) and on the use of tensor objects. The presence of tensor objects inside the code allows to obtain a constitutive models framework that, with respect to existent implementations, is independent on both the adopted analysis model and numerical method. The OOP, with its properties of abstraction, inheritance, and polymorphism, leads to a framework highly modular and easy to expand. The theoretical basis is a compact tensorial representation for the micropolar equations that makes them formally identical to the ones of the classic continuum theory. This compatibility has been here extended to their computational expressions, making possible to use the same code structure for both the continuum models, taking advantage of existing implementations of classic constitutive models.

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Notes

  1. The development code is available at the Git repository http://git.insane.dees.ufmg.br/insane/insane.git.

  2. The approach followed here is the one which Forest and Sievert [53] refer to as single-criterion approach. The same authors point out that in order to obtain two un-coupled rate expressions, it is necessary to adopt at least two loading functions, one dependent on the sole strain tensor and the other on the sole micro-curvature tensor. This situation falls in the case of multidissipative models; despite it could be easily introduced in the framework proposed in this paper, this is not the objective of the present work, which concentrates on the implementation of micropolar monodissipative models.

  3. In the present approach, the same approximation functions are used to discretize both the displacement field and the micro-rotation field of the micropolar medium. From following Eq. 51 it can be seen that, differently from a classic medium, in this case also the approximation functions must be integrated together with their spatial derivatives, hence requiring a larger number of integration points with respect to a classic continuum model. In order to overcome this issue, a possible solution suggested by De Borst [2] is to use a different order of interpolation for the two state variables. For example, in a six-nodes triangular element, a number of integration points equal to three are sufficient if the micro-rotation are linearly interpolated, using their nodal values at only three nodes.

  4. An important issue when dealing with micropolar models is the proper calibration of the additional material parameters. Since the main aim of the paper is about the implementation aspects of the micropolar theory, this issue has not been addressed; in the examples of the present section, the values of the Cosserat’s shear modulus and internal bending length have been simply chosen considering values usually adopted in the literature (see, e.g., [2]). The reader interested on the topic may refer to the existing literature cited by Hassanpour and Heppler [60], which can be complemented with the works by Chang et al. devoted to granular materials [9, 61, 62], and other focused on regular and random composites [10, 17,18,19,20, 63, 64].

  5. The input files for the software INSANE of the simulations presented in this section can be found in [65].

  6. More detailed information on this characteristic of the software can be found in [34].

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Acknowledgements

The authors gratefully acknowledge the support of the Brazilian research agencies CAPES (in Portuguese: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), FAPEMIG (in Portuguese: Fundação de Amparo à Pesquisa do Estado de Minas Gerais) and CNPq (in Portuguese: Conselho Nacional de Desenvolvimento Científico e Tecnológico - Grant 309515/2017-3).

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  1. Department of Structural Engineering, Engineering School, Federal University of Minas Gerais (UFMG), Avenida Antônio Carlos, 6627, Pampulha, Belo Horizonte, MG, 31270-901, Brazil

    Lapo Gori, Samuel Silva Penna & Roque Luiz da Silva Pitangueira

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  1. Lapo Gori
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Gori, L., Silva Penna, S. & da Silva Pitangueira, R.L. A computational framework for the constitutive modeling of nonlinear micropolar media. J Braz. Soc. Mech. Sci. Eng. 41, 275 (2019). https://doi.org/10.1007/s40430-019-1779-7

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