By H. F. Tiersten (auth.)
Read or Download Linear Piezoelectric Plate Vibrations: Elements of the Linear Theory of Piezoelectricity and the Vibrations Piezoelectric Plates PDF
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Additional info for Linear Piezoelectric Plate Vibrations: Elements of the Linear Theory of Piezoelectricity and the Vibrations Piezoelectric Plates
TJ f b a F(x) dx = fbaF(x) dx - a fb F(x) dx = fba F(x) dx - fba F(x) dx = - fba [F(x) F(x)] dx = 41 fba tJF(x) dx. 4) 42 Hamilton's Principle [Ch. 6 This shows that the variation of a definite integral is equal to the definite integral of the variation. In practice we are not interested in the simple cases we have just discussed, in which F is a prescribed function of the independent variable x. Instead we are interested in the situation in which F is a prescribed function of a dependent variable, say, y, and its first derivative, y', which in turn depends on the independent variable x.
14) B··=O ',I. 16) 2. 19) Sec. 3] Poynting's Theorem and Electromagnetic Energy Flux 27 Note that Au is arbitrary. Different selections of Au determine different gauges. 20) where P'is any scalar, could also have been used as the vector potential, since VxVP' O. 19), which we rewrite here as eijkHk,j 1. 25) Di,i = 4nee· Although A and hence ((! are not unique, E, H, B, D, and J are unique. 3. 27) (ljc)H/J i , and subtracting, we obtain 1 = C - (ejikEiHk),j = C eijkEiHk,j - eijkHiEk,j 1 . EiDi .
20) arise as a consequence of the constraints on the variations imposed in the principle. Thus it is clear that if we can remove the constraints, we can remove the limitation. Now, it is well known in variational calculus [(15), Chapter IV] (and in stationary problems in general) that a constraint on a variation may be removed by adding to the Lagrangian each constraint as a zero times a Lagrange multiplier and then treating all variations of the field variables as unconstrained [(15), Fig. 5. Diagram of a bounded region containing an internal surface of discontinuity.