Multiscale Homogenization Analysis of the Effective Elastic Properties of Masonry Structures
Publication: Journal of Materials in Civil Engineering
Volume 28, Issue 8
Abstract
The asymptotic homogenization method of differential equations with rapidly oscillating coefficients is used to evaluate the effective elastic stiffness of masonry structures. The equilibrium equations as well as the constitutive relationships at different scales are determined from the asymptotic expansion of the displacement field. The formulation is developed for a general three-dimensional problem. It is then particularized for a simple one-dimensional (1D) model that is applicable to available experimental results of a masonry prism. Key concepts of the method such as periodicity of the fields are highlighted in this 1D problem. A multiscale finite-element model was developed for a two-dimensional unit cell to solve the canonical cell equation that arises in homogenization, which provides numerical solution of the effective elastic moduli. In particular, the numerical study focuses on the in-plane action of the masonry panels that are used as the experimental specimens in laboratory tests. Comparing experimental and analytical (from classical computational homogenization) results, the multiscale homogenization model for linear elastic masonry structures is validated. The main advantage of this model is its straightforward extension for the study of the dynamic homogenization of masonry structures. Furthermore, the developed model is suitable for a comprehensive study of the stress localization in masonry structures for the determination of its strength-based limiting properties.
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References
Allaire, G., and Brizzi, R. (2004). “A multiscale finite element method for numerical homogenization.” SIAM Multiscale Model. Simul., 4(3), 790–812.
Anthoine, A. (1995). “Derivation of the in-plane elastic characteristics of masonry through homogenization theory.” Int. J. Solids Struct., 32(2), 137–163.
Anthoine, A. (1997). “Homogenization of periodic masonry: Plane stress, generalized plane strain or 3D modeling?” Commun. Numer. Methods Eng., 13(5), 319–326.
Briccoli Bati, S., Ranocchiai, G., and Rovero, L. (1999). “A micromechanical model for linear homogenization of brick masonry.” Mater. Struct./Matériaux et Constr., 32(1), 22–30.
Cecchi, A., and Rizzi, N. L. (2001). “Heterogeneous elastic solids: A mixed homogenization-rigidification technique.” Int. J. Solids Struct., 38(1), 29–36.
Efendiev, Y., and Hou, T. Y. (2008). Multiscale finite element methods: Theory and applications, Springer, New York.
Eshelby, J. D. (1957). “The determination of the elastic field of an ellipsoidal inclusion, and related problems.” Proc. R. Soc. London, Ser. A, 241(1226), 376–396.
Fredholm, E. I. (1903). “Sur une classe d’equations fonctionnelles.” Acta Math., 27(1), 365–390.
Hashemi, A., and Mosalam, K. M. (2006). “Shake-table experiment on reinforced concrete structure containing masonry infill wall.” Earthquake Eng. Struct. Dyn., 35(14), 1827–1852.
Hashemi, A., and Mosalam, K. M. (2007). “Seismic evaluation of reinforced concrete buildings including effects of masonry infill walls.”, Pacific Earthquake Engineering Research Center, College of Engineering, Univ. of California, Berkeley, CA.
Hou, T. Y., and Wu, X. (1997). “A multiscale finite element method for elliptic problems in composite materials and porous media.” J. Comput. Phys., 134(1), 169–189.
Kawa, M., Pietruszcza, S., and Shieh-Beygi, B. (2008). “Limit state of brick masonry based on homogenization approach.” Int. J. Solids Struct., 45(3–4), 998–1016.
Lee, K., Moorthy, S., and Gosh, S. (1999). “Multiple scale computational model for damage in composite materials.” Comput. Methods Appl. Mech. Eng., 172(1–4), 175–201.
Li, S., and Wang, G. (2008). Introduction to micromechanics and nanomechanics, World Scientific, Singapore.
Lourenço, P. B., Milani, G., Tralli, A., and Zucchini, A. (2007). “Analysis of masonry structures: Review of and recent trends in homogenization techniques.” Canada J. Civ. Eng., 34(11), 1443–1457.
Michel, J. C., Moulinee, H., and Suquet, P. (1999). “Effective properties of composite materials with periodic microstructure: A computational approach.” Comput. Methods Applied Mech. Eng., 172(1–4), 109–143.
Mori, T., and Tanaka, K. (1973). “Average stress in matrix and average elastic energy of materials with misfitting inclusions.” Acta Metall., 21(5), 571–574.
Mosalam, K. M. (1996). “Experimental and computational strategies for the seismic behavior evaluation of frames with infill walls.” Ph.D. dissertation, Cornell Univ., NY.
Pande, G. N., Liang, J. X., and Middleton, J. (1989). “Equivalent elastic moduli for brick masonry.” Comput. Geotech., 8(3), 243–265.
Pietruszczak, S., and Niu, X. (1992). “A mathematical description of macroscopic behavior of brick masonry.” Int. J. Solids Struct., 29(5), 531–546.
Sanchez-Palencia, E. (1983). “Homogenization method for the study of composite media.” Asymptotic analysis II, F. Verhulst, ed., Vol. 945, Springer, Berlin.
Sanchez-Palencia, E. (1983). Non-homogeneous media and vibration theory, Springer, Berlin.
Singh, A. K. (1993). “The lattice strain in a specimen (cubic system) compressed nonhydrostatically in an opposed anvil device.” J. Appl. Phys., 73(9), 4278–4286.
Wang, G., Li, S., Nguyen, H., and Sitar, N. (2007). “Effective elastic stiffness for periodic masonry structures via eigenstrain homogenization.” J. Mater. Civ. Eng., 269–277.
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© 2016 American Society of Civil Engineers.
History
Received: Apr 1, 2015
Accepted: Dec 1, 2015
Published online: Mar 3, 2016
Published in print: Aug 1, 2016
Discussion open until: Aug 3, 2016
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