Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies

Abstract Previous results of the author on the influence of size and boundary-conditions on the apparent properties of elastic heterogeneous materials are recalled and extended to the viscoelastic case. Materials with random microstructures, possibly with anisotropy, defects or damage, are considered. In place of the classical concept of effective properties, which holds for the representative volume only, the concept of apparent properties is recalled. It makes use of special kinds of boundary-conditions and appropriate stochastic averages for which the required Hill condition is still valid while the ergodic assumption – legitimating the use of the concept of equivalent homogeneous medium – is relaxed. Statistical apparent properties are defined on various partitions of a given initial specimen D 0 into a set of coarse specimens on the one hand, and of smaller specimens on the other hand. The boundary-condition and size-effects hierarchies derived in Huet (1990. Journal of Mechanical Physics and Solids 38, 813; 1997. Engineering Fracture Mechanics, (special issue) 58 (5–6) 459) are recalled. Examples of numerical verification by micromechanical simulations are shown using 3D finite element models of elastic granular composites. Some salient features of the viscoelastic behavior of non-metallic construction materials used in civil engineering are recalled. 3D simulations on viscoelastic micromechanical models are also shown, with results exhibiting trends similar to the elastic case and for which theoretical justification and generalisation is seeked in the paper. Use is made of a Statistical Continuum Thermodynamics approach through which the concept of equivalent homogeneous medium and effective properties is first discussed for materials with constituents having dissipative constitutive equations of any kind and from which natural definitions of dissipative potential and complementary energies are derived. Dissipative extensions of the so-called Hill condition are obtained. Then, new minimum theorems for viscoelasticity are derived from a slight modification of the pseudo-convolutive procedure used in Huet (1992. European Journal of Mechanics, A/Solids 5 (11) 653; In: Bazant, Z.P., Carol, I. (Eds.), Creep and Shrinkage of Concrete. Spon, London, pp. 189–200). From this, theoretical bounds and size-effect hierarchies on the statistical viscoelastic stiffness and compliance function tensors and their rates are obtained in the time domain. It is found that viscoelastic heterogeneous bodies smaller than the representative volume exhibit, for the two classes of uniform boundary-conditions, the same kind of size-effects as the elastic ones. Possible extensions to other kinds of time-dependent physical properties are quoted.