Scale transitions in solid mechanics based on computational ... - Cism

then. A survey of activities over the past forty years is given <strong>in</strong> [23].Homogenization techniques were first developed with<strong>in</strong> the framework of elasticity, as anexcellent tool to predict the effective or apparent l<strong>in</strong>ear elastic properties of heterogeneousmaterials. Several closed-form homogenization techniques have been proposed <strong>in</strong> this context,e.g. the Voigt-Reuss-Hill bounds, the Hash<strong>in</strong>-Shtrikman variational pr<strong>in</strong>ciple, theself-consistent method, etc., see [24] for an overview. Asymptotic or mathematical homogenizationschemes have been used frequently to assess effective properties of elasticheterogeneous materials [25, 26]. Extensions towards higher-order and nonlocal constitutiveequations have been considered as well, e.g. developments <strong>in</strong>clud<strong>in</strong>g Cosserat media[27], couple stress theory [28], nonlocal effective cont<strong>in</strong>ua [29] or higher-order gradienthomogenized elastic materials [30, 31, 32]. Interest<strong>in</strong>g approaches towards the analysisof random (physically nonl<strong>in</strong>ear) microstructures [33, 34, 35] are the Taylor-Bishop-Hillestimates, several generalizations of self-consistent schemes. The mathematical asymptotichomogenization theory [36, 37] applies an asymptotic expansion of displacementand stress fields on a “natural scale parameter”, which is the ratio of a characteristicsize of the heterogeneities and a measure of the macrostructure [38, 39, 40, 41, 42]. Theasymptotic homogenization approach provides effective overall properties as well as localstress and stra<strong>in</strong> values. However, usually the considerations are restricted to verysimple microscopic geometries and simple material models, mostly at small stra<strong>in</strong>s. Acomprehensive overview of different homogenization methods may be found <strong>in</strong> a book byNemat-Nasser and Hori. [24]. Homogenization of <strong>solid</strong>s <strong>in</strong> a geometrically and physicallynonl<strong>in</strong>ear regime is clearly more cumbersome.The <strong>in</strong>creas<strong>in</strong>g complexity of microstructural mechanical and physical behaviour, alongwith the development of computational methods, made the class of so-called unit cellmethods attractive. These approaches have been used <strong>in</strong> a great number of differentapplications [43, 44, 45, 46, 47, 48, 49]. A selection of examples <strong>in</strong> the field of metalmatrix composites has been collected, for example, <strong>in</strong> a book by Suresh et al.[50] Theunit cell methods serve a twofold purpose: they provide valuable <strong>in</strong>formation on the localmicrostructural fields as well as the effective material properties. These properties aregenerally determ<strong>in</strong>ed by fitt<strong>in</strong>g the averaged microscopical stress-stra<strong>in</strong> fields, result<strong>in</strong>gfrom the analysis of a microstructural representative cell subjected to a certa<strong>in</strong> load<strong>in</strong>gpath, on macroscopic closed-form phenomenological constitutive equations <strong>in</strong> a formatestablished a priori. Once the constitutive behaviour becomes non-l<strong>in</strong>ear (geometrically,physically or both), it becomes <strong>in</strong>tr<strong>in</strong>sically difficult to make a well-motivated assumptionon a suitable macroscopic constitutive format. For example, McHugh et al. [51] havedemonstrated that, when a composite is characterized by power-law slip system harden<strong>in</strong>g,the power-law harden<strong>in</strong>g behaviour is not preserved at the macroscale. Hence, most ofthe known homogenization techniques are not suitable for large deformations nor complexload<strong>in</strong>g paths, neither do they account for the geometrical and physical changes of themicrostructure (which is relevant, for example, when deal<strong>in</strong>g with phase <strong>transitions</strong>).Closed-form homogenization towards constitutive material frameworks or effective (orrather apparent) material properties of composites turns out to be really cumbersomeif one wishes to take <strong>in</strong>to account more complex physics, geometrical nonl<strong>in</strong>earities ordamage and localization.Another class of hierarchical techniques are generally known as variational multi-scalemethods [52, 53]. In here, the weak form of the govern<strong>in</strong>g equations is the po<strong>in</strong>t ofdeparture, which can be separated <strong>in</strong> a coarse and a f<strong>in</strong>e scale part on the basis of suitable4