Generalized viscothermoelasticity theory of dual-phase-lagging model for damping analysis in circular micro-plate resonators

Vibration of circular micro-ceramic (Si3N4) plate resonators in the context of the generalized viscothermoelastic dual-phase-lagging theory

“…and l* and m* are Lame`parameters, and a T is the coefficient of thermal expansion. The stress components are as follows 18,27,28,[30][31][32]…”

“…The neutral surface is kept on the (r, u) plane, and the zaxis taken normal to the neutral surface, as shown in Figure 1. 27 Hence, the displacement components are as follows 18…”

“…The generalized heat conduction equation in the context of the viscothermoelastic DPL model is given by 11,27…”

“…We will consider the edge of the micro-beams r = a is clamped and thermally loaded by thermal shock as follows 27 w a, t…”

“…In the following numerical calculations, the circular micro-ceramic (Si 3 N 4 ) plate is assumed. The thermal and mechanical properties are given by the following 18,27 K = 8:0 W=(m K) , a T = 3:0(10) À6 K À1 , r = 3200 kg =m 3 , T 0 = 300 K , C y = 937:5 J=(kg K) , E 0 = 250 GPa , E 1 = 6:7 3 10 À12 s, and y = 0:44 : Figures 2 and 3 represent the temperature increment and the lateral deflection for the non-dimensional parameters a=h =5, t =20, z =h=2, t q =0:02, t T =0:9t q , E 1 =0:213 10 3 , and for different values of the mechanical relaxation times E 1 =0 for the thermoelastic materials while E 1 6 ¼ 0 for the viscothermoelastic materials. It is noted that the effect of the parameter E 1 is not significant on the temperature increment distribution, while its effect is significant on the lateral deflection distribution.…”