Spatial energy estimates in dynamical problems for a semi-infinite piezoelectric beam
AbstractThe behaviour of the total energy for a semi-infinite piezoelectric beam, with uniform or variable cross-section, in dynamical conditions is investigated. Precisely, we obtain some estimates in terms of the data for the energy of the portion of the body at distance greater than z from the base and for its norm in L 1 (0, t ) (where t is an arbitrary positive time). We obtain some estimates which depend on the initial data if t ⩽ z/V (where V is a computable positive material constant); if t > z/V , by using also the Korn inequality, we show that the bounds depend on all the data. Under mild hypotheses on the initial data we derive the asymptotic behaviour of the energy as z → +∞. All possible combinations of boundary conditions are examined and the kind of the estimate is formally the same for all the problems whether the beam is a cylinder or not.