Some New Mathematical Modelings of Junctions

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 constraint 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 classical 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. . .