Long memory effects arising from homogenization and internal variables
Résumé
This talk is devoted to the homogenization of viscoelastic composite materials or polycrystals. It is well known that short memory effects in the individual constituents give rise, after homogenization, to long memory effects in the composite (Sanchez-Hubert and Sanchez- Palencia, 1978; Suquet, 1986). In other words the constitutive equations, which are ordinary differential equations at the level of individual constituents, become integro- differential equations after homogenization. It is also well-known that the main ingredient for establishing this result is the Laplace transform which does not apply any more when the constituents becomes nonlinear or simply aging (time dependent material properties). The present talk will report on less well-known recent progress made on specific properties of the kernel characterizing the long memory effects and its relation with constitutive theories based on internal variables. First, four relations satisfied by this kernel at large and short times have been derived (Suquet, 2012). Second, in practice, the implementation of functional constitutive relations requires the storage of all the stress (or strain) history in the composite from the initial time to the present time t and is therefore of limited use. However, in certain circumstances this long memory effect can be expressed by means of a finite number of internal variables (Ricaud and Masson, 2009). The constitutive relations remain ordinary differential equations for a larger, but still finite, number of variables than for the original constituents. When this is not the case, an approximate (but often accurate) model involving two internal variables can be proposed, based on the above- mentioned four relations. With the question of nonlinear viscoelastic composites in mind, the end of the talk will discuss another approach of the problem, valid both for linear and nonlinear composites. Incremental variational principles can be used to introduce the approximation of effective internal variables (Lahellec and Suquet, 2007). The accuracy of two versions of this approximation will be discussed.