# Mixed Variational Formulations for Micro-cracked Continua in the Multifield Framework

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## Abstract

**:**

**2001**, 190, 5657–5677) and propose a few novel variational formulations of mixed type along with relevant mixed FEM discretizations. To this goal, suitably extended Hellinger-Reissner principles of primal and dual type are derived. A few numerical studies are presented that include an investigation on the interaction between a single cohesive macrocrack and diffuse microcracks (Mariano, P.M. and Stazi, F.L., Strain localization due to crack–microcrack interactions: X–FEM for a multifield approach, Comp. Methods Appl. Mech. Engrg.

**2004**, 193, 5035–5062).

## 1. Introduction and modeling

#### 1.1. Introductory remarks

#### 1.2. The constitutive model

t | = | specimen thickness |

$RV{E}^{M}$ | = | material reference volume element = ${\ell}_{M}^{2}t$ |

$RV{E}^{m}$ | = | micro-fracture reference volume element = ${\ell}_{m}^{2}t$ |

E | = | Young modulus of the bonds between macro-spheres |

${E}^{*}$ | = | Young modulus of the bonds between macro-spheres and micro-holes |

$\mathcal{W}$ | = | $EA/{\ell}_{M}$ = macro-lattice stiffness |

$\mathcal{Q}$ | = | $EA/(\pi {\ell}_{c})$ = mean stiffness of the ellipsoidal micro-holes |

$\mathcal{H}$ | = | $2EA/\left[\sqrt{2}({\ell}_{m}-{\ell}_{M})\right]$ = stiffness of the bonds between macro and meso lattices |

$\widehat{A}$ | = | cross section of rods between adjacent micro-cracks. |

## 2. Mixed variational formulations

#### 2.1. Primal Hellinger–Reissner formulation

#### 2.2. Dual Hellinger–Reissner formulation

#### 2.3. Implementation

## 3. Numerical studies

#### 3.1. A clamped square lamina

${\ell}_{m}$ $\left[mm\right]$ | ${\ell}_{M}$ $\left[mm\right]$ | δ $\left[mm\right]$ | E $[N/m{m}^{2}]$ | A $\left[m{m}^{2}\right]$ | $\widehat{A}$ $\left[m{m}^{2}\right]$ | ${\ell}_{C}$ $mm$ | $\chi $ |

200 | 1 | 0.1 | ${10}^{5}$ | 1 | 0.314 | 1 | 50 |

#### 3.2. Interaction between a macrocrack and diffuse microcracks

${\ell}_{m}$ $\left[mm\right]$ | ${\ell}_{M}$ $\left[mm\right]$ | E $[N/m{m}^{2}]$ | A $\left[m{m}^{2}\right]$ | $\widehat{A}$ $\left[m{m}^{2}\right]$ | ${\ell}_{C}$ $mm$ | χ |

75 | 5 | ${10}^{3}$ | 1 | 0.0314 | 1 | 50 |

## 4. Conclusions and future work

## Acknowledgements

## References and Notes

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© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Bruggi, M.; Venini, P. Mixed Variational Formulations for Micro-cracked Continua in the Multifield Framework. *Algorithms* **2009**, *2*, 606-622.
https://doi.org/10.3390/a2010606

**AMA Style**

Bruggi M, Venini P. Mixed Variational Formulations for Micro-cracked Continua in the Multifield Framework. *Algorithms*. 2009; 2(1):606-622.
https://doi.org/10.3390/a2010606

**Chicago/Turabian Style**

Bruggi, Matteo, and Paolo Venini. 2009. "Mixed Variational Formulations for Micro-cracked Continua in the Multifield Framework" *Algorithms* 2, no. 1: 606-622.
https://doi.org/10.3390/a2010606