Asymptotic behavior of a hard thin linear elastic interphase: An energy approach

Abstract

a b s t r a c tThe mechanical problem of two elastic bodies separated by a thin elastic film is studied here. The stiffness of the three bodies is assumed to be similar. The asymptotic behavior of the film as its thickness tends to zero is studied using a method based on asymptotic expansions and energy minimization. Several cases of interphase material symmetry are studied (from isotropy to triclinic symmetry). In each case, non-local relations are obtained relating the jumps in the displacements and stress vector fields at order one to these fields at order zero.

Mathematical Modeling of Weak and Strong Piezoelectric Interfaces

“…Le us choose test functions r = (v, ψ) ∈ X(Ω) × X(Ω) in problem (19). By letting ε tend to zero and using relations (20), we obtain, after some mathematical technicalities, that the weak limit verifies the limit problem…”

“…Within the theory of elasticity, the asymptotic analysis of a thin elastic interphase between two elastic materials has been deeply investigated through the years, by varying the rigidity ratios between the thin inclusion and the surrounding materials and by considering different geometry features. For instance, it is worth mentioning the pioneering work by E. Acerbi et al [1] on the variational behavior of the elastic energy of a thin inclusion using Γ -convergence, the contributions by G. Geymonat et al [16], F. Krasucki et al [17], and C. Licht and G. Michaille [20] for mathematical models for linear and non linear weak bonded joints, the works by F. Lebon and R. Rizzoni [18,19] for the case of thin interfaces with similar and hard rigidities, and, also, the works by A.-L. Bessoud et al [6][7][8] in which the authors studied the case of plate-like and shell-like inclusions with high rigidity in a rigorous functional framework. It is also noteworthy the work by W. Geis et al [13], in which the authors derive a reduced model for the problem of thin conductor plates embedded into a piezoelectric matrix with similar (undamaged electrodes) and weak (damaged electrodes) electromechanical rigidities, using functional convergence.…”

Normal and tangential stiffnesses of rough surfaces in contact via an imperfect interface model

“…On the first, the homogenization of the interphase region has been performed by adopting the wellestablished strategy by Kachanov (1994) , Tsukrov and Kachanov (20 0 0) , Kachanov and Sevostianov (2005) and Sevostianov and Kachanov (2013) , under the assumption that cracks do not interact each other (non-interacting approximation). On the second, in the limit of a vanishing interphase thickness, a matched asymptotic expansion method ( Lebon and Rizzoni, 2011;Lebon and Rizzoni, 2010;Lebon and Zaittouni, 2010;Rizzoni et al, 2014;Rizzoni and Lebon, 2013 ) has been considered. As a result, normal and tangential interface contact stiffnesses are consistently derived, introducing the dependency on the closure pressure p via a simple but accurate description for the evolution of the no-contact radius with p , and by enforcing as consistency requirements some physical constraints suggested by the classical Hertz theory.…”

“…Incremental normal and tangential equivalent stiffnesses at the nominal contact interface are derived, by assuming contact microgeometry be described by isolated internal cracks ( Sevostianov and Kachanov,20 08a;20 08b ) occurring in a thin interphase region. In detail, effective mechanical properties at the contact zone are consistently derived following the imperfect interface approach recently adopted by Lebon and coworkers ( Fouchal et al, 2014;Rekik and Lebon, 2010;2012 ), by coupling a homogenization approach for microcracked media based on the non-interacting approximation ( Kachanov, 1994;Kachanov and Sevostianov, 2005;Sevostianov and Kachanov, 2013;Tsukrov and Kachanov, 20 0 0 ) and the matched asymptotic expansion method, introduced by SanchezPalencia (1987) and Sanchez-Palencia and Sanchez-Hubert (1992) and recently employed by Lebon and Rizzoni (2011) , Rizzoni and Lebon (2013) , Rizzoni et al (2014) and Lebon and Zaittouni (2010) .…”

On modelling brick/mortar interface via a St. Venant-Kirchhoff orthotropic soft interface. Part II: in silico analysis

“…This model, hereinafter referred as the St. Venant-Kirchhoff (SVK) interface model, was obtained by coupling arguments of asymptotic analysis (Abdelmoula et al, 1998;Benveniste, 2006;Lebon and Rizzoni, 2008, 2011Rizzoni and Lebon, 2013;Rizzoni et al, 2014), extended to the finite strain theory Rizzoni et al, 2017), with a homogenisation method for microcracked media under the non-interacting approximation (Kachanov, 1994;Mauge and Kachanov, 1994;Tsukrov and Kachanov, 2000;Sevostianov and Kachanov, 2013). This second part of the study proposes a numerical validation of the SVK interface model.…”