Keywords: interface laws, asymptotic study, elasto-plasticity, non-monotone behavior, finite elements.

Contact and interface mechanics intervenes more and more often in computational structures. To have reliable tools to size more and more complex systems, it is indispensable to take into account in a precise way the connections between the various solids of the structure. This consideration can be very complex because of the strongly non-linear character and possibly imperfect and very localized of these connections. For examples, in the modelling of the processes of metal forming, consideration of the friction is necessary because it pilots the whole process; the modelling of the glue in the case of masonry structures is necessary to estimate in a precise way the mechanical characteristics of the structures, their risks of collapse, … The purpose of this lecture is to show how it is possible to obtain families of interface laws from the mechanical behaviour of thin layers. The consideration of bonded joints in real structures can lead from a numerical point of view to problems of too large sizes, especially if these joints have a non-linear behaviour [1,2,3,4,5]. From a general way, one is going to consider joints of weak thickness and weak rigidity with regard to those of the substrata. One has then to deal with problems taking into account at least two small parameters (the thickness, the rigidity). An asymptotic study (a micro-macro passage), completed by numerical calculations, leads to so-called "asymptotic contact laws" who allow describing the (macroscopic) mechanical behaviour of the interfaces (see references in [1]). We consider in this lecture various kinds of non-linear behaviour for the thin layers:

• visco-elastic of Norton type (one explicits the results expressed in [2]),
• elasto-plastic of Mohr-Coulomb or Drucker-Prager type (figure 65.1),
• non-monotone relationship in the strain-stress diagram,
• contact conditions between the adhesive and the adherents [3].

One can show in the two first cases, that the interface laws obtained are of Tresca or Coulomb kind. The third case can model instabilities on the contact boundary. The last case permits to show how it is possible to add various behaviour laws.

1

F. Lebon, A. Ould Khaoua, C. Licht "Numerical study of soft adhesively bonded joints in finite elasticity", Computational Mechanics, 21, 134-140, 1998. doi:10.1007/s004660050289

2

C. Licht "Comportement asymptotique d'une bande dissipative mince de faible rigidité", Comptes Rendus Académie des Sciences Paris, Série I, 322, 295-300, 1992.

3

F. Lebon, C. Licht, F. Zaittouni "Asymptotic modelling of interfaces taking into account contact conditions, Part 1 Asymptotic expansions and numerical implementation", International Journal of Solids and Structures, to appear, 2002.

4

P. Suquet "Discontinuities and plasticity", in Nonsmooth mechanics and applications, J.J. Moreau and P.D. Panagiotopoulos Eds., CISM Courses and Lectures, 302, Springer-Verlag, 279-340, 1988.

5

F. Zaittouni, F. Lebon, C. Licht "Etude théorique et numérique du comportement d'un assemblage de plaques", Comptes Rendus Académie des Sciences Paris, Série Mécanique, 330, 359-364, 2002.

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