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Three-dimensional Cosserat homogenization of masonry structures: elasticity

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Abstract

Masonry is a two-phase composite material formed by regularly distributed bricks and mortar. The homogenization procedure followed here extends the 2D approach of Sulem and Mühlhaus (Mech Cohesive Frict Mater 2:31–46, 1997) and leads to an anisotropic 3D Cosserat continuum. The enriched kinematics of the Cosserat continuum allows us to model microelement systems undergoing in-plane and out-of-plane rotations. The domain of validity of the derived Cosserat continuum is discussed by comparing the dispersion function of the discrete system of blocks with the continuous one and is found to be in good agreement.

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Notes

  1. The reader is invited to download the Mathematica Working files from: http://www.geolab.mechan.ntua.gr/people/stefano

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Acknowledgments

This research is sponsored by the General Secretariat for Research and Technology in Greece and the French Ministry of Foreign Affairs in the frame of the bilateral S & T cooperation between the French and Hellenic Republic (2005–2007): “Nouvelles méthodes d’analyse numérique du comportement mécanique des monuments anciens - Application à l’Acropole”. The authors would like to thank K. Sab for fruitful discussions and the two reviewers for their constructive remarks and suggestions.

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Authors and Affiliations

  1. Department of Applied Mechanics and Physics, National Technical University of Athens, Athens, Greece

    I. Stefanou & I. Vardoulakis

  2. CERMES, Ecole Nationale des Ponts et Chaussèes/LCPC, Institut Navier, Paris, France

    J. Sulem

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  1. I. Stefanou
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  2. J. Sulem
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  3. I. Vardoulakis
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Correspondence to I. Stefanou.

Appendix

Appendix

1.1 Lattice equations of motion

The equilibrium of forces and moments acting on block (I,J) yields to the following six equations:

$$ \begin{aligned} m{\left({{\ddot{u}}_{1}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} &= A_{V} c_{N} {\left({{\left({u_{1}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} - 2{\left({u_{1}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({u_{1}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)}\\ & \quad + A_{H} c_{Q} {\left({{\left({u_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({u_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - 4{\left({u_{1}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({u_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({u_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad + \frac{1}{2}A_{H} c_{Q} b{\left({- {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ m{\left({{\ddot{u}}_{2}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} &= A_{H} c_{N} {\left({{\left({u_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({u_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - 4{\left({u_{2}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({u_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({u_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad + A_{V} c_{Q} {\left({{\left({u_{2}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} - 2{\left({u_{2}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({u_{2}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)} + \frac{1}{2}A_{V} c_{Q} a{\left({{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} - {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)} \\ & \quad + \frac{1}{4}A_{H} c_{N} a{\left({{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} - {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\m{\left({{\ddot{u}}_{3}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} &= A_{H} c_{Q} {\left({{\left({u_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({u_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - 4{\left({u_{3}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({u_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({u_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad + A_{V} c_{Q} {\left({{\left({u_{3}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} - 2{\left({u_{3}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({u_{3}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)}\\ & \quad + \frac{1}{2}A_{H} c_{Q} b{\left({{\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} - {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} - {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad + \frac{1}{4}A_{H} c_{Q} a{\left({- {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} - {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad + \frac{1}{2}A_{V} c_{Q} a{\left({- {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)} \\J_{1} {\left({{\ddot{\omega}}_{1}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} &= A_{V} c_{{{{\rm MV1}}}} {\left({{\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} - 2{\left({\omega_{1}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)} \\ & \quad + A_{H} c_{{{{\rm MH1}}}} {\left({{\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - 4{\left({\omega_{1}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{4}A_{H} c_{Q} b^{2} {\left({+ {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + 4{\left({\omega_{1}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{8}A_{H} c_{Q} ba{\left({- {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} - {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{2}A_{H} c_{Q} b{\left({{\left({u_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} - {\left({u_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + {\left({u_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} - {\left({u_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ J_{2} {\left({{\ddot{\omega}}_{2}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} &= A_{V} c_{{{{\rm MV2}}}} {\left({{\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} - 2{\left({\omega_{2}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)} \\ & \quad + A_{H} c_{{{{\rm MH2}}}} {\left({{\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - 4{\left({\omega_{2}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{4}A_{V} c_{Q} a^{2} {\left({{\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} + 2{\left({\omega_{2}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)} \\ & \quad - \frac{1}{{16}}A_{H} c_{Q} a^{2} {\left({+ {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + 4{\left({\omega_{2}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({\omega_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{8}A_{H} c_{Q} ab{\left({- {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} - {\left({\omega_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{2}A_{V} c_{Q} a{\left({- {\left({u_{3}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} + {\left({u_{3}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)} \\ & \quad - \frac{1}{4}A_{H} ac_{Q} {\left({- {\left({u_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} - {\left({u_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + {\left({u_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({u_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\J_{3} {\left({{\ddot{\omega}}_{3}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} &= A_{V} c_{{{{\rm MV3}}}} {\left({{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} - 2{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)} \\ & \quad + c_{{{{\rm MH3}}}} A_{H} {\left({{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - 4{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{{16}}A_{H} c_{N} a^{2} {\left({{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + 4{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{4}A_{H} c_{Q} b^{2} {\left({{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} + 4{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + b{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)}\\ & \quad - \frac{1}{4}A_{V} c_{Q} a^{2} {\left({{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} + 2{\left({\omega_{3}} \right)}^{b}_{{{{\rm I}},{{\rm J}}}} + {\left({\omega_{3}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)} \\ & \quad - \frac{1}{4}A_{H} c_{N} a{\left({{\left({u_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({u_{2}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - {\left({u_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} - {\left({u_{2}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{2}A_{H} c_{Q} b{\left({- {\left({u_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} - 1}} + {\left({u_{1}} \right)}^{b}_{{{{\rm I}} - 1,{{\rm J}} + 1}} - {\left({u_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} - 1}} + {\left({u_{1}} \right)}^{b}_{{{{\rm I}} + 1,{{\rm J}} + 1}}} \right)} \\ & \quad - \frac{1}{2}A_{V} c_{Q} a{\left({{\left({u_{2}} \right)}^{b}_{{{{\rm I}} - 2,{{\rm J}}}} - {\left({u_{2}} \right)}^{b}_{{{{\rm I}} + 2,{{\rm J}}}}} \right)}\end{aligned} $$
(33)

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Stefanou, I., Sulem, J. & Vardoulakis, I. Three-dimensional Cosserat homogenization of masonry structures: elasticity. Acta Geotech. 3, 71–83 (2008). https://doi.org/10.1007/s11440-007-0051-y

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