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Biological materials and systems are hierarchically organized.The main motivation for hierarchical biomechanics is that the wide variability of mechanical properties encountered at the macroscopic scale may be traced back to just a few universal. i.e. tissue-invariant, mechanical properties of elementary components at a sufficiently small scale (such as collagen, elastin, and water in case of soft tissues; complemented by hydroxyapatite in case of hard tissues), and to the nano and microstructures which the latter build up. This challenging task requires a physically rigorous and mathematically sound basis, as provided by Finite Element and Fast Fourier Transform methods, as well as by continuum micromechanics resting on (semi-)analytical solutions for Eshelby-type matrix-inclusion problems. Corresponding numerical and analytical mathematical models have undergone diligent experimental validation, by means of data stemming from a variety of biophysical, biochemical, and biomechanical testing methods, such as light and electron microscopy, ultrasonic testing and scanning acoustic microscopy, as well as physico-chemical tests associated with dehydration, demineralization, decollagenization, ashing, and weighing in air and fluid. While elastic scale transition and homogenization methods have attained a high maturity level, the hierarchical nature of dissipative (i.e. viscous or strength) properties is still a vibrant field of research. This applies even more to hierarchical approaches elucidating the interface between biological cells and extracellular matrices, and to the highly undiscovered mechanics unfolding within biological cells.

The mechanical interactions of C-(N-)A-S-H (Calcium-sodium-aluminum-silicate-hydrate) gel with slag and fly ash inclusions in alkali-activated materials (AAM) are quantified through image-supported grid nanoindentation. Nonuniform distributions of indent-specific indentation properties reveal that the elasticity-related domain is up to 130 times the contact indentation depth, while the hardness-related domain, in turn, is by a factor of two to three smaller. These rather large domains are consistent with the slag/fly-ash inclusions being much stiffer and harder than the surrounding C-(N-)A-S-H gel. Corresponding Hashin-Shtirkman bounds for the overall AAM stiffness consistently frame ultrasonic data characterizing this homogenized material scale. This confirms our new testing protocol.

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