Approaches to Generalized Continua
Decades ago, it has been recognized that for some materials the kinematics on meso- and micro-structural scale needs to be considered, if the material’s resistance to deformation exhibits a finite radius of interaction on atomic or molecule level, e.g. (1
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1
*
and Sebastian Skatulla
†
Department of Civil Engineering, INSA Rennes, France; on leave from The University of Nottingham, United Kingdom † CERECAM, Department of Civil Engineering, University of Cape Town, South Africa
Introduction
Decades ago, it has been recognized that for some materials the kinematics on meso- and micro-structural scale needs to be considered, if the material’s resistance to deformation exhibits a finite radius of interaction on atomic or molecule level, e.g. Kr¨oner (1963); Mindlin (1964) outlined that this is the case if the deformation wave length approaches micro-structural length scale. Differently said, if the external loading corresponds material entities smaller than the representative volume element (RVE), then the statistical average of the macro-scopical material behaviour does not hold anymore. In this sense the fluctuation of deformation on micro-structural level as well as relative motion of micro-structural constituens, such as granule, crystalline or other heterogeneous aggregates, influence the material response on macro-structural level. Consequently, field equations based on the assumption of micro-scopically homogeneous material have to be supplemented and enriched to also include non-local and higher-order contributions. In particular, generalized continua aim to describe material behaviour based on a deeper understanding of the kinematics at smaller scales rather than by pure phenomenological approximation of experimental data obtained at macro-scopical level. The meso- or micro-structural kinematics and its nonlocal nature is then treated either by incorporating higher-order gradients or by introducing extra degrees of freedom. For the latter, the small-scale kinematics at each material point can be thought to be equipped with a set of directors which specify the orientation and deformation of a surrounding a micro-space. The resulting theories can be categorized into three main groups: (1) rigid body motion where the directors only rotate but do not deform or change their angles between each other referring to C. Sansour et al. (eds.), Generalized Continua and Dislocation Theory © CISM, Udine 2012
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the so-called Cosserat (Cosserat and Cosserat, 1909) or micropolar continuum (Eringen, 1967); (2) directors undergoing additional to rotation also stretch referring to the so-called micro-stretch continuum (Eringen, 1969); (3) rotation, stretch and change of angles between directors referring to the so-called micromorphic continuum (Ericksen and Truesdell, 1957; Eringen, 1972). Accordingly, the number of extra degrees of freedom depends on the dimension of macro- and small-scale spaces but also what kind of meso- or micro-deformation modes are considered and the order of the coresponding ansatz formulations. Thus, the number of additional degrees of freedom can vary between 1 and 18 or more (Eringen, 1999; Sansour and Skatulla, 2010) which in turn leads to large diversity of derived strain and stress measures describing deformation phenom
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