SOME NEW MATHEMATICAL MODELINGS OF JUNCTIONS

Abstract

Most of the structures in Civil Engineering consists in assemblies of

deformable bodies, thus it is of interest to dispose of efficient models of

junctions between deformable solids. The classical schemes of Continuum

Mechanics lead to boundary value problems involving several parameters,

one being essential: the (low) thickness of the layer filled by the adhesive.

For usual behaviors of the adherents and the adhesive, it is not difficult

to prove existence of solutions, but their numerical approximations may

be difficult due to the rather low thickness of the adhesive implying a too

fine mesh. We propose a simplified but accurate mathematical modeling

by a rigorous study of the asymptotic behavior of the three-dimensional

adhesive when its thickness goes to zero. Depending on the stiffness of

the adhesive, the limit model will replace the thin adhesive layer by either

a mechanical constraint along the surface the layer shrinks toward or a

material surface; the structure of the constitutive equations of the con

straint or of the material surface keeping the memory of the mechanical

behavior of the adhesive.

The mathematical techniques used in these studies, carried out for

more than 25 years, involve variational convergences and the Trotter

theory of convergence of semi-groups of operators. We will present clas

sical results concerning standard elastic or dissipative behaviors of the

adhesive and some new ones devoted to microscopic aspects, imperfectly

bonded adhesive joints, loaded joints, etc. . .