8.2 Literature Review
301
Fig. 8.1 Geometry of loading and the introduced system of coordinates of the beam resonator (a)
and its cross section (b)
beam thicknesses which are compared with the results obtained based on the classical
theory of elasticity in both linear and nonlinear cases.
The influence of temperature field on the statics and dynamics of nonlinear EulerBernoulli nanobeams [96] fabricated based on the optimization method [97] has been
also investigated.
8.3 Formulation of the Problem
We introduce a rectangular system of coordinates to study a microbeam as shown in
Fig. 8.1. The origin of coordinates is fixed on the left microbeam end. The microbeam
has length L, thickness h, width b, and its transversal cross section parallel to the
Y O Z plane is denoted by A (see Fig. 8.1b).
8.3.1 Stress and Deformation Fields
Heat deformations occur in an elastic body due to a linear heat extension coefficient.
Therefore, the total deformation composed of mechanical and heat components is as
follows [76]:
ε i j = ε
(M)
i j + ε
(T )
i j
( i, j = 1, 2, 3) .
(8.1)
By θ = T − T 0 , we denote the temperature change with respect to the initial temperature T 0 , and we assume that the shear deformations generated by the temperature
301
Fig. 8.1 Geometry of loading and the introduced system of coordinates of the beam resonator (a)
and its cross section (b)
beam thicknesses which are compared with the results obtained based on the classical
theory of elasticity in both linear and nonlinear cases.
The influence of temperature field on the statics and dynamics of nonlinear EulerBernoulli nanobeams [96] fabricated based on the optimization method [97] has been
also investigated.
8.3 Formulation of the Problem
We introduce a rectangular system of coordinates to study a microbeam as shown in
Fig. 8.1. The origin of coordinates is fixed on the left microbeam end. The microbeam
has length L, thickness h, width b, and its transversal cross section parallel to the
Y O Z plane is denoted by A (see Fig. 8.1b).
8.3.1 Stress and Deformation Fields
Heat deformations occur in an elastic body due to a linear heat extension coefficient.
Therefore, the total deformation composed of mechanical and heat components is as
follows [76]:
ε i j = ε
(M)
i j + ε
(T )
i j
( i, j = 1, 2, 3) .
(8.1)
By θ = T − T 0 , we denote the temperature change with respect to the initial temperature T 0 , and we assume that the shear deformations generated by the temperature