In the context of decarbonizing the transportation industry, there has been a steady increase in the amount of electrical power onboard, necessitating improvements in the performance of electrical machines to achieve higher specific powers. Additionally, due to the high demands placed on components, the magneto-thermo-mechanical properties of materials play a crucial role in the design of highperformance machines. This paper proposes an analytical procedure to address the three key physics involved in the design of a highspeed induction machine with a rotor composed of a copper layer. The electromagnetic models developed utilize a multi-harmonic approach based on the magnetic vector potential. For thermal analysis, the two-dimensional cylindrical steady-state heat equation is solved using the separation of variable technique. This method is chosen for its compatibility with further coupling with other physics, compared to the classic lumped-parameter approach. The mechanical model enables the study the stresses induced by centrifugal forces, employing a plane deformation approach. Finite element models are employed to validate the various developed models, ensuring their accuracy and reliability in predicting the performance of the high-speed induction machine.