Work on deriving clearly quantified expressions of Saint-Venant's principles for elastic and other materials has been ongoing for some decades. Rigorous proofs of of such a 'principles' for particular classes of linear viscoelastic solids have been provided in the past by many authors [see e. g. [26] and references cited therein]. In spite of the general shape of the analyzed bodies, main issues, such as proving a Saint-Venant principle for general dissipative relaxation functions, have not been solved yet. A very intuitive form of the Saint-Venant principle for a linear elastic cylinder maybe seen in several papers, such as. In these cases, such a cylinder is assumed to be free from constraints and loaded on one basis only by a self equilibrated traction field; the spatial decay properties of the stored energy are then investigated. In particular, the state of points on the cross sections of the cylinder are considered. The rate of spatial decay of the energy is determined along the direction of the axis. In the case of a semi-infinite solid this argument shows that the energy stored in the solid delimited by the loaded basis and a given cross section approaches its value at the natural state, as the distance of the given cross section increases from the loaded basis. It follows that the corresponding state of points on the same cross section approaches the natural state. In the context just described, the state, called elastic state, is given by the triple {u(x, t), E(x, t), T(x, t)} (see [2]). As far as linear viscoelastic materials are concerned, two replacements have to be done in order to establish a Saint Venant principle: (ⅰ) the stored energy has to be replaced by some free energy, and at the same time (ⅱ) a notion of linear viscoelastic state has to be provided. About (ⅰ), it is very well known that there are different (but related) possibilities of defining the free energy for a linear viscoelastic material [4]pir3, and the issue of its non-uniqueness arises, About (ⅱ), a notion of state for linear viscoleastic materials has been provided in [4] by particularizing the concept of state proposed by Noll.
展开▼
