A model is developed to calculate the impurity influx caused by induced radio-frequency (RF) sheaths near ion-cyclotron range of frequencies (ICRF) antennas, which accelerate ions to energies sufficient to sputter the Faraday screen and other adjacent material surfaces. The existence of several types of RF sheaths is discussed, and detailed models of Faraday-screen-gap and front-face sheaths are described which are relevant to the antenna geometry in the Joint European Torus (JET) experiment. Calculations of the local influx of metal impurities from the Faraday screen show good qualitative agreement with the observed dependence of the impurity influx in JET on phasing, magnetic field angle, and screen coatings, and the model provides a useful framework for understanding other aspects of the data. It is also shown that a great increase in the metal influx due to self-sputtering of the screen ('avalanche') can occur under certain conditions. However, the model calculations also demonstrate that proper antenna design can virtually eliminate the impurity influx from the screens.