**6. Conclusions**

The work reported herein advances a numerical model for the fretting contact between dissimilarly elastic materials. A numerical approach is required to simulate this type of contact process, as analytical models can incorporate neither the loading history, which must be reproduced when friction is accounted for, nor the coupling of normal and tangential effects.

The implemented algorithm is based on three levels of iterations, fully incorporating the interconnectivity between normal and tangential tractions. The innermost level solves, in a conjugate gradient type loop, either the normal, or the tangential uncoupled contact problem, while the intermediate loop iterates between these solutions, until pressure convergence is reached. The outer level reproduces the loading history, and is based on the assumption that irreversibility of friction requires simulation of all previous states in an incremental load process.

The strong points of this algorithm consist in reduced computational complexity, compared to finite element analysis, as well as in its ability to handle arbitrarily shaped contact geometries or imposed frictional coefficient maps. Comparison with existing closed-form solutions for the spherical contact undergoing a fretting loop or torsion, when a uniform frictional coefficient is assumed, gives confidence in the newly advanced model.

Evolution of stress state during a fretting loop in the spherical contact between similarly elastic materials is assessed. It is found that for specific combinations of loading levels and frictional coefficients, the most severely stressed region can be found on the bounding surfaces, at the trailing edge of the contact.

Numerical simulations suggest that the fretting contact between dissimilarly elastic materials exhibits a unique path in the first two loading cycles, which stabilizes with subsequent oscillating loading to a fixed trajectory, as in case of similarly elastic materials.

The newly advanced algorithm is expected to solve a large variety of contact problems involving interfacial friction, leading to a better understanding of complex multidisciplinary phenomena like fretting wear and fretting fatigue, in which transient contact tractions and induced subsurface stresses play an important role, as well as to a more accurate prediction of contact failure through yield inception or crack nucleation. Study of partial slip elasticplastic contact is anticipated for future contributions, by addition of the residual term, related to development of plastic strains.
