In this study, continuous contact problem in the functionally graded (FG) layer loaded with two rigid flat blocks resting on the elastic semi-infinite plane was analyzed by the finite element method. The two-dimensional numerical model of the FG layer was made with the software added to the ANSYS program. This software can be adapted to all contact problem types by making minor changes. The accuracy check of the program was performed by comparing with the analytical solution of the problem by homogeneous layer and its solution by the finite element method. So, fast and practical solutions can be obtained by the developed finite element method on many applications such as; automotive, aviation and space industry applications. The comparisons made showed that the proposed solution gave good results at acceptable levels. In the problem, it was thought that all surfaces were frictionless. The external loads P and Q were transmitted to the FG layer via two flat rigid blocks. Normal stresses between the FG layer and the elastic plane, initial separation loads, initial separation distances and contact stresses under the blocks were investigated for various dimensionless quantities.

We consider a viscoelastic body occupying a smooth bounded domain $\Omega\subset \mathbb{R}^3$ under the effect of a volumic traction force g . The macroscopic displacement vector from the equilibrium configuration is denoted by u. Inertial effects are considered; hence, the equation for u contains the second-order term u tt . On a part ΓD of the boundary of Ω, the body is fixed and no displacement may occur; on a second part Γ N ⊂ ∂Ω, the body can move freely; on a third portion Γ C ⊂ ∂Ω, the body is in adhesive contact with a solid support. The boundary forces acting on ΓC due to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a non-linear ordinary differential equation settled on ΓC and describing the evolution of the delamination order parameter z. Following the lines of a new approach outlined in Bonetti et al. (2015, arXiv:1503.01911) and based on duality methods in Sobolev–Bochner spaces, we define a suitable concept of weak solution to the resulting system of partial differential equations. Correspondingly, we prove an existence result on finite-time intervals of arbitrary length.

The paper deals with an iterative method for numerical solving frictionlesscontact problems for two elastic bodies. Each iterative step consists of aDirichlet problem for the one body, a contact problem for the other one and twoNeumann problems to coordinate contact stresses. Convergence is proved by theBanach fixed point theorem in both continuous and discrete case. Numericalexperiments indicate scalability of the algorithm for some choices of therelaxation parameter.