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Variational estimates for the creep behaviour of polycrystals. (English) Zbl 0829.73077
This is a very interesting paper regarding the prediction of the effective constitutive behaviour of polycrystalline aggregates undergoing high-temperature creep, directly from known information about constitutive properties and geometric arrangement of their constituent single-crystal grains. The paper starts with a detailed presentation of the history of the problem and the main steps and methods of its approaching. The definition of the effective behaviour of “generalized polycrystals” and the characterization of the creep behaviour of single crystals and polycrystals are given in Sect. 2. A polycrystal is an aggregate of a large number of identical single-crystal grains with generally distinct orientations. It is assumed that polycrystals have no texture, i.e. the single-crystals have no prefered orientations within polycrystal, and that the geometric arrangement of the grains within composite is statistically uniform and isotropic.
In Sect. 3 the authors present the main contribution of the paper consisting in the development of an extension of the variational procedure of the second author [J. Mech. Phys. Solids 39, 45-71 (1991; Zbl 0734.73052) and ibid. 40, 1757-1788 (1992; Zbl 0764.73103)] for nonlinear heterogeneous materials with locally isotropic behaviour to classes of nonlinear heterogeneous materials with a special type of locally anisotropic behaviour enough to include the nonlinear polycrystalline aggregates. The variational statement is expressed in terms of the effective potentials of appropriate classes of linear comparison polycrystals. The result is expressed as a finite-dimensional optimization problem over $$N\times K$$ variables corresponding to the “slip compliances” of the linear comparison polycrystal with $$N$$ different crystals and $$K$$ distinct “slip systems” for each crystal.
Sect. 4 contains the applications of the variational procedure developed to obtain bounds for the effective potential classes of nonlinear polycrystals. It is shown that the new variational procedure can deliver at least the same amount of information as the classical variational principle, and that it provides other types of bounds that are not directly available by classical methods. By using the previous results, in Sect. 5 the bounds are obtained for the effective creep behaviour of the class of untextured, statistically isotropic face-centred-cubic polycrystals, with “isotropic hardening” of the power-law type. Interesting concluding remarks are contained in the final Sect. 6.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74P10 Optimization of other properties in solid mechanics 74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
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