Keywords: Mori-Tanaka method, self-consistent method, differential scheme, alcali activated fly ash, natural wood, metallic foam, image analysis, nanoindentation, multi-scale homogenization.

The field of composite materials now offers a tremendous variability and complexity of microstructures depending on their particular application. Nevertheless, it still shows a number of common features which brings various material systems to the same footing, at least from their analysis point of view. The latter issue has been repeatedly exploited particularly in connection with systems described by limited data often reduced to volume fractions and material properties of individual phases and assumptions as to their statistically uniform arrangement.

Introduction of images of real microstructures into the analysis opened the way for more rigorous quantification of microstructures as well as more advanced modelling strategies based on statistically equivalent representation of microstructural details using computational models [1] often formulated in the hierarchical manner to account for multiple scales [2]. The use of advanced computational strategies was further supported by novel techniques such as nanoindentation for the determination of material properties of composite constituents on the level of microns. Note however that combining image analysis, nanoindentation and hierarchical modelling is by no means limited to complex and time consuming computations. As a contrast, the use of analytical models such as self-consistent and Mori-Tanaka methods is sufficient in many practical applications particularly if benefiting from the above three features. Combining these features (image analysis, nanoindentation and analytical micromechanical models) leads to reliable estimates of the bulk response of heterogeneous materials which is the principal objective of this paper.

Since learning from examples is the most simple way towards understanding, this paper may also serve as a tutorial for the application of various averaging schemes to the derivation of effective properties of heterogeneous materials which show multiple scales. To that end, several vastly dissimilar material systems are examined. In particular, the calculation of Young's modulus as a function of the degree of hydration of alkali-activated fly ash [3] is discussed first followed by the application of micromechanical modelling of natural wood [4] and closes with a reference to metallic foams [5].

By discussing the quality of the theoretical predictions in comparison with available experimental data the paper introduces the reader to the concept of a virtual testing tool as an integrated set of models, algorithms and procedures for the prediction of mechanical properties on an arbitrary scale.

1

J. Zeman, M. Šejnoha, "From random microstructures to representative volume elements", Modelling and Simulation in Materials Science and Engineering, 15(4), S325-S335, 2007. doi:10.1088/0965-0393/15/4/S01

2

J. Sýkora, T. Krejcí, J. Kruis, M. Šejnoha, "Computational homogenization of non-stationary transport processes in masonry structures", Journal of Computational and Applied Mathematics, 2012. (Accepted) doi:10.1016/j.cam.2012.02.031

3

V. Šmilauer, P. Hlavácek, Škvára F., L. Kopecký, J. Nemecek, "Micromechanical multiscale model for alkali activation of fly ash and metakaolin", Journal of Material Sciece, 46(20), 6545-6555, 2011.

4

K. Hofstetter, C. Hellmich, J. Eberhardsteiner, "Continuum micromechanics estimation of wood strength", Proc. Appl. Math. Mech, 6, 75-78, 2006. doi:10.1002/pamm.200610020

5

J. Nemecek, V. Králík, J. Vondrejc, J. Nemeceková, "Identification of micromechanical properties on metal foams using nanoindentation", in B. Topping, Y. Tsompanakis, (Editors), "Proceedings of the Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 125, 2011. doi:10.4203/ccp.96.125

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