This first part of our study is concerned with the theoretical and variational formulations of the problem of elastic cyclic loading at very high frequency (or acoustic fatigue). The problem is treated by using the theory of longitudinal thermoelastic wave motion in a finite medium with and without running crack. Two methods are used to formulate the evolution problem: the first one deals with the use of classical time integration schema, and the second uses the Fourier transformation to solve the evolution problem in the frequency space. Comparison of our results with some closed form solutions of some classical problem is presented. In a second paper this method is used to calculate thermo-mechanical fields in specimens used in ultrasonic fatigue tests (endurance tests and crack growth tests).

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