Kinematics
In the previous chapter our investigations were based on a comparison between the current configuration and the reference configuration, being concerned essentially with the geometrical point of view, without considering the intermediate states of the sys
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Kinematics
'train rate. Rat of v Ium dilatation. 'pin t nsor. Rat of rotation vector. orticity. lalNial cl rivati\"(~. ·onv tiv t rm. onscrvation of mass. ontinuit quation.
J. Salençon, Handbook of Continuum Mechanics © Springer-Verlag Berlin Heidelberg 2001
Chapter III. Kinematics
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In Brief
The evolution of a system may be viewed within either the Lagrangian or the Eulerian framework. In the Lagrangian formulation, the partial time derivative is the material derivative or convective derivative that follows the changes in a quantity associated with a given particle, a discrete set of particles or, more generally, some material element (Beets. 2 and 4). Geometrically speaking, the kinematics of the continuum can be inferred direetly from the convective transport, the transformation and the deformation occurring between some initial reference configuration and the current configuration. The notions of Lagrangian rate of extension, volume dilatation, and strain thus arise quite naturally as time derivatives of the corresponding quantities defined by comparing the current configuration with the referenee configuration. The disadvantage, however, is that they must refer to values of quantities at the initial time in order to eharacterise the eoming infinitesimal evolution at the present time (Beet. 2). The ineremental viewpoint is most consistent in, and indeed provides the cornerstone for the Eulerian formulation. At eaeh instant of time the eoming infinitesimal evolution is defined from the eurrent configuration. Geometrieally speaking, as the motion is given by the velocity field, it is the gradient of this field in the eurrent configuration which defines the infinitesimal transformation loeally. The (Eulerian) strain rate tensor, whieh is the symmetrie part of this gradient, eharacterises the way the strain evolves, always relative to the eurrent configuration. In this manner, at eaeh instant of time, it is the eurrent configuration that plays the role of referenee configuration. The antisymmetric part of the gradient of the velo city field is the spin tensor. This defines loeally the infinitesimal rotational motion of the matter, to which must be added the infinitesimal motion due to stretch, defined by the strain rate tensor (Beet. 3). The Eulerian deseription defines quantities in the eurrent configuration as a function of the spatial variables and time, and does not identify any material elements. The material derivative must therefore be evaluated as the time derivative following the particle or material element in question. This explains the structure of the corresponding formulas: they systematieally include a term corresponding to the partial time derivative holding the spatial variables constant (fixed geometrieal point or region), together with
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Chapter IH. Kinematics
a convective term. The latter is the contribution arising from the convective transport of the particle or material element with which the relevant quantity is associated (Sect. 4). Special attention is paid to the material der
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