A semi-analytical method for the tridimensional thermal-elastic-plastic contact between two hemispherical asperities is proposed. This first part of the paper describes the algorithm used to deal with the normal contact, which can be either load-driven (ld) or displacement-driven (dd). Both formulations use the Conjugate Gradient (CG) method and the Discrete Convolution and Fast Fourier Transform (DC-FFT) technique. A validation of the code is made in the case of the displacement-driven formulation for an elastic-plastic body in contact with a rigid punch, simulating a nano-indentation test. For both formulations (ld and dd) a very good convergence rate is found. Another new feature is the treatment of the contact between two elastic-plastic bodies. The model is first validated through comparison with the Finite Element Method (FEM). The contact pressure distribution, the hydrostatic pressure and the equivalent plastic strain state below the contacting surfaces are also found to be strongly modified in comparison to the case of an elastic-plastic body in contact with a purely elastic body. An application to the tugging between two spherical asperities in simple sliding (dd formulation) is made in part II of the paper.

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