302
8 Thermoelastic Vibrations of Timoshenko Microbeams
change can be neglected. Then, in the case of a linear, isotropic and homogeneous
body, heat (T ) and mechanical (M) deformations are governed by the following
formula [98]:
ε
(T )
i j = α θ δ i j ,
ε
(M)
i j
=
1 + ν
E
σ i j −
ν
E
σ kk δ i j ,
(8.2)
where α—temperature coefficient of a linear extension; σ i j —stress tensor; δ i j —
Kronecker’s symbol; E—Young modulus, and ν is the Poisson coefficient. Owing
to (8.1) and (8.2), the total deformation field is as follows:
ε i j =
1 + ν
E
σ i j +
α θ −
ν
E
σ kk
δ i j .
(8.3)
8.3.1.1 Plane Stress State Condition
In the case when beam thickness (in direction of the O Z axis) and its width (in
direction of the OY axis) is sufficiently small in comparison to the beam length (in
direction of the O X axis), then assumption of the plane stress state implies that all
components of the stress tensor in directions of the axes OY and O Z are equal to
zero (σ 12 = σ 22 = σ 32 = 0, σ 23 = σ 33 = 0). However, in the case of the Timoshenko
beams we have σ 13 = 0 [98]. In the latter case, components of the deformations tensor
can be recast to the following simple form:
ε 11 =
σ 11
E
+ α θ, ε 22 = ε 33 = −ν ε 11 + (1 + ν) α θ, ε 12 = ε 23 = 0. (8.4)
A stress tensor under the plane stress state assumption has only two components
different from zero, which is expressed by the components of the deformation tensor
in the following way:
σ 11 = Eε 11 − Eα θ, σ 13 = 2k s Gε 13 ,
(8.5)
where k s —correction coefficient is introduced due to the assumption of
non-homogeneous shear deformation with respect to the transverse beam cross
section and it depends on the forms of the beam cross sections; G—shear modulus.
Owing to the known Cauchy-Green deformations
ε i j =
1
2
u i, j + u j, i + u m, i u m, j
,
(8.6)
the kinematic relations of the theory of Timoshenko beams can be presented in the
following form:
u 1 = u (x, t) + zψ(x, t), u 2 = 0, u 3 = w (x, t) ,
(8.7)
8 Thermoelastic Vibrations of Timoshenko Microbeams
change can be neglected. Then, in the case of a linear, isotropic and homogeneous
body, heat (T ) and mechanical (M) deformations are governed by the following
formula [98]:
ε
(T )
i j = α θ δ i j ,
ε
(M)
i j
=
1 + ν
E
σ i j −
ν
E
σ kk δ i j ,
(8.2)
where α—temperature coefficient of a linear extension; σ i j —stress tensor; δ i j —
Kronecker’s symbol; E—Young modulus, and ν is the Poisson coefficient. Owing
to (8.1) and (8.2), the total deformation field is as follows:
ε i j =
1 + ν
E
σ i j +
α θ −
ν
E
σ kk
δ i j .
(8.3)
8.3.1.1 Plane Stress State Condition
In the case when beam thickness (in direction of the O Z axis) and its width (in
direction of the OY axis) is sufficiently small in comparison to the beam length (in
direction of the O X axis), then assumption of the plane stress state implies that all
components of the stress tensor in directions of the axes OY and O Z are equal to
zero (σ 12 = σ 22 = σ 32 = 0, σ 23 = σ 33 = 0). However, in the case of the Timoshenko
beams we have σ 13 = 0 [98]. In the latter case, components of the deformations tensor
can be recast to the following simple form:
ε 11 =
σ 11
E
+ α θ, ε 22 = ε 33 = −ν ε 11 + (1 + ν) α θ, ε 12 = ε 23 = 0. (8.4)
A stress tensor under the plane stress state assumption has only two components
different from zero, which is expressed by the components of the deformation tensor
in the following way:
σ 11 = Eε 11 − Eα θ, σ 13 = 2k s Gε 13 ,
(8.5)
where k s —correction coefficient is introduced due to the assumption of
non-homogeneous shear deformation with respect to the transverse beam cross
section and it depends on the forms of the beam cross sections; G—shear modulus.
Owing to the known Cauchy-Green deformations
ε i j =
1
2
u i, j + u j, i + u m, i u m, j
,
(8.6)
the kinematic relations of the theory of Timoshenko beams can be presented in the
following form:
u 1 = u (x, t) + zψ(x, t), u 2 = 0, u 3 = w (x, t) ,
(8.7)