Higher order model for soft and hard elastic interfaces
Résumé
The present paper deals with the derivation of a higher order theory of interface models. In particular, it is
studied the problem of two bodies joined by an adhesive interphase for which ‘‘soft’’ and ‘‘hard’’ linear
elastic constitutive laws are considered. For the adhesive, interface models are determined by using
two different methods. The first method is based on the matched asymptotic expansion technique, which
adopts the strong formulation of classical continuum mechanics equations (compatibility, constitutive
and equilibrium equations). The second method adopts a suitable variational (weak) formulation, based
on the minimization of the potential energy. First and higher order interface models are derived for soft
and hard adhesives. In particular, it is shown that the two approaches, strong and weak formulations, lead
to the same asymptotic equations governing the limit behavior of the adhesive as its thickness vanishes.
The governing equations derived at zero order are then put in comparison with the ones accounting for
the first order of the asymptotic expansion, thus remarking the influence of the higher order terms and of
the higher order derivatives on the interface response. Moreover, it is shown how the elastic properties of
the adhesive enter the higher order terms. The effects taken into account by the latter ones could play an
important role in the nonlinear response of the interface, herein not investigated. Finally, two simple
applications are developed in order to illustrate the differences among the interface theories at the
different orders.