A thermoelastic model with two relaxations for the vibration of a microbeam resting on elastic foundations

doi:10.1007/s10483-025-3241-8

Abstract

Abstract:

The Euler-Bernoulli (E-B) beam theory is combined with Green-Lindsay's (G-L) generalized thermoelasticity theory to analyze the vibration of microbeams. The frequency control equation, based on the two-parameter Winkler-Pasternak elastic foundation for simply-supported microbeams, is presented. This study investigates the effects of the side-to-thickness ratio and relaxation time parameters on the vibrational natural frequency of thermoelastic microbeam resonators. =0.28 em plus 0.1em minus 0.1em The frequencies derived from the present model are compared with those from Lord and Shulman's (L-S) theory. The fourth-order solutions for natural vibration frequencies are graphically displayed for comparison. Therefore, attention should be paid to the use of effective foundations to prevent microbeam damage caused by contraction and expansion problems caused by high temperatures.

Key words: Green-Lindsay's (G-L) model, thermoelastic vibration, Euler-Bernoulli (E-B) theory, elastic foundation

2010 MSC Number:

O343

Z. S. HAFED, A. M. ZENKOUR. A thermoelastic model with two relaxations for the vibration of a microbeam resting on elastic foundations. Applied Mathematics and Mechanics (English Edition), 2025, 46(4): 711-722.

TrendMD

Figures/Tables 14

Fig. 1

Fig. 2

Fig. 3

Table 1

The natural frequency ϖ1 versus the first relaxation time τ0 and the length-to-thickness ratio Lh (τ1=0.6, and k1=k2=0)"

$L h$	$τ 0$	$ϖ 1$
$L h$	$τ 0$	$n = 1$	$n = 2$	$n = 3$	$n = 4$	$n = 5$
5	0.1	1.772 147 (0.569 822) $∗$	1.870 380 (2.279 288)	2.023 535 (5.128 397)	2.220 276 (9.117 150)	2.450 124 (14.245 547)
	0.2	1.253 097	1.322 559	1.430 856	1.569 985	1.732 708
	0.3	1.023 150	1.079 867	1.168 325	1.281 884	1.415 455
	0.5	0.792 528	0.836 464	0.904 946	0.993 040	1.093 283
10	0.1	1.235 121 (0.284 911)	1.253 095 (1.139 644)	1.282 497 (2.564 198)	1.322 576 (4.558 575)	1.372 336 (7.122 773)
	0.2	0.873 363	0.886 073	0.906 858	0.935 136	0.970 080
	0.3	0.713 097	0.723 479	0.740 474	0.763 680	0.792 779
	0.5	0.552 363	0.560 395	0.573 588	0.590 874	0.614 520
20	0.1	0.870 156 (0.142 455)	0.873 363 (0.569 822)	0.878 686 (1.282 099)	0.886 067 (2.279 288)	0.895 509 (3.561 387)
	0.2	0.615 294	0.617 587	0.621 295	0.626 366	0.632 518
	0.3	0.502 385	0.504 231	0.507 349	0.511 265	0.514 787
	0.5	0.389 148	0.390 559	0.392 926	0.396 995	0.409 948
$(⋅) ∗$ represents the value of the natural frequency $ϖ 3$

Table 1

Fig. 4

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Fig. 13

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