7.6 Stability of Curvilinear Euler-Bernoulli Beams in Temperature
Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.6.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.6.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.6.4 Influence of Imperfections . . . . . . . . . . . . . . . . . . . . . . . . 255
7.7 Mathematical Model of Three-Layer Micro- and Nano-Beams
Based on the Hypotheses of the Grigolyuk-Chulkov
and the Modified Couple Stress Theory . . . . . . . . . . . . . . . . . . . . 261
7.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.7.2 Theory of Bending Including Shear Effects
and the Modified Couple Stress Theory . . . . . . . . . . . . . . 267
7.7.3 Equations of Motion and Boundary Conditions
for the Three-Layer Beam . . . . . . . . . . . . . . . . . . . . . . . . 273
7.7.4 Static Transversal Bending of the Three-Layer Beam . . . . . 276
7.7.5 Vibrations of a Three-Layer Beam . . . . . . . . . . . . . . . . . . 279
7.7.6 Numerical Results and Their Validation . . . . . . . . . . . . . . 282
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8 Thermoelastic Vibrations of Timoshenko Microbeams
(Modified Couple Stress Theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
8.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
8.3 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
8.3.1 Stress and Deformation Fields . . . . . . . . . . . . . . . . . . . . . 301
8.3.2 Beam Equation of Motion Based on the Modified
Couple Stress Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
8.3.3 Coupled Problem of Thermoelasticity for Timoshenko
Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
8.4 Quality Factor of the Resonator with Thermoelastic Damping . . . . 308
8.5 Numerical and Analytical Results for a Rectangular CrossSectional Beam Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
8.5.1 Analytical Solution for the Linear Size-Dependent
Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
8.5.2 Analytical Solution for the Nonlinear Size-Dependent
Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
9 Vibrations of Size-Dependent Beams Under Topologic
Optimization and Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . 333
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
9.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
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