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8 Thermoelastic Vibrations of Timoshenko Microbeams
8.3.2 Beam Equation of Motion Based on the Modified
Couple Stress Theory
In the modified couple stress theory [62], the accumulated energy of deformation U
in a linear elastic body occupying space is defined as follows:
U =
1
2
V
σ i j ε i j + m i j χ i j
dV ,
(8.10)
where V is the volume of elastic matter, m i j are the components of a deviator part
of the symmetric tensor momentum of higher order and χ i j are the components of
symmetric curvature tensor [53], which are defined in the following way:
m i j = 2Gl
2
χ i j , χ i j =
1
2
ϕ i, j + ϕ j, i
, ϕ i =
1
2
(rot (u)) i ,
(8.11)
where l is the material length parameter characterizing the influence of a higher order
moment. Substituting (8.7) into (8.11) yields the following non-zero components of
ϕ i , χ i j , m i j :
ϕ 2 = −
1
2
w , x − ψ
, χ 12 = χ 21 =
1
4
ψ , x − w , xx
,
m 12 = m 21 =
1
2
Gl
2
ψ , x − w , xx
.
(8.12)
In the case of plane stress state after substitution of (8.8) and (8.12) into (8.10),
the full deformation energy of the bended beam takes the following form:
U =
1
2
L
0
h/2
−h/2
E
u , x +
1
2
w , x
2 + zψ , x
2
− E
u , x +
1
2
w , x
2 + zψ , x
α θ+
+k s G
ψ + w , x
2 +
1
4
Gl
2
ψ , x − w , x x
2
b dz dx .
(8.13)
After computation of integrals with regard to z and a few simple transformations
one gets
U =
1
2
L
0
k 1
u , x +
1
2
w , x
2
2
+ k 2
ψ , x
2 − 2N
T
u , x +
1
2
w , x
2
−
− 2M
T
ψ , x + k 3
ψ + w , x
2 + k 4
ψ , x − w , xx
2
dx ,
(8.14)