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Anisotropic Damage in Elasto-plastic Materials with Structural Defects

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Multiscale Modelling in Sheet Metal Forming

Part of the book series: ESAFORM Bookseries on Material Forming ((EBMF))

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Abstract

The aims of the chapter is to develop elasto-plastic models with anistropic damage, which are mathematically coherent, physically motivated and consistent with certain dissipation principle. A concise and critical presentation of the contributions and results which led to basic ideas for the development of elasto-plastic anisotropic damaged materials is exposed in Sects. 6.2 and 6.3. Two types of constitutive models have been proposed in Sects. 6.4 and 6.5. The first model is based on the existence of the undamaged (fictitious) configuration and on the deformation-like damage tensor, which is involved in the deformation gradient multiplicative decomposition into its parts. Here F d characterizes the passage from the undamaged stress free configuration to the damaged stress free configuration. The second model is proposed within the second order elasto-plasticity, involving a symmetric defect density tensor, which is a measure of non-metricity of the so-called plastic connection. We extended to finite deformation the relationships between the continuum theory of lattice defect and the non-Euclidian geometry that has been provided within the small strain formalism. The constitutive and evolution equations are derived to be compatible with the free energy imbalance principle, which has been reformulated by Cleja-Ţigoiu (Int J Fract 147:67–812 (2007); Int J Fract 166:61–75 (2010)), when the stress and stress momentum are considered. We assumed that the plastic flow and the development of the microvoids and microcracks are distinct irreversible mechanism during the deformation process.

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Notes

  1. 1.

    D is denoted by \( \varPhi \) in Voyiadjis and Kattan (2005).

  2. 2.

    The correct written form of the above decomposition and which corresponds to the mentioned figures in the papers (Brünig 2003; Brünig and Ricci 2005) is \( {\mathbf{Q}} = {\mathbf{Q}}^{el} {\mathbf{Q}}^{pd} \).

  3. 3.

    The correct definition for \( {\dot{\mathbf{H}}}^{el} \) ought to be \( {\dot{\mathbf{H}}}^{el} = \frac{1}{2}(({\mathbf{Q}}^{el} )^{ - 1} \,{\dot{\mathbf{Q}}}^{el} + {\dot{\mathbf{Q}}}^{el} \,({\mathbf{Q}}^{el} )^{ - 1} ). \)

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Authors and Affiliations

  1. University of Bucharest, 14 Academiei, Sector 1, 010014, Bucharest, Romania

    Sanda Cleja-Ţigoiu

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  1. Sanda Cleja-Ţigoiu
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Correspondence to Sanda Cleja-Ţigoiu .

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  1. CERTETA Research Center, Technical University of Cluj Napoca, Cluj Napoca, Romania

    Dorel Banabic

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Cleja-Ţigoiu, S. (2016). Anisotropic Damage in Elasto-plastic Materials with Structural Defects. In: Banabic, D. (eds) Multiscale Modelling in Sheet Metal Forming. ESAFORM Bookseries on Material Forming. Springer, Cham. https://doi.org/10.1007/978-3-319-44070-5_6

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