282
7 Mathematical Models of Functionally Graded Beams in Temperature Field
ω
2
∗ = λ
4
1
λ
2
1 k ˜
ϑ + 1
1 +
d h
h
2
kλ
2
1
1 +
a h
h
2
+ 1
.
(7.186)
In the classical case, we have [76]
ω
2
clas = λ
4
1
λ
2
1 kϑ + 1
kλ
2
1 + 1
.
(7.187)
Let us investigate the ratio r (λ
2
1 ) = ω
2
∗ /ω
2
clas expressed explicitly in the form
r (λ
2
1 ) =
λ
2
1 k ˜
ϑ + 1
1 +
d h
h
2
kλ
2
1
1 +
a h
h
2
+ 1
kλ
2
1 + 1
λ
2
1 kϑ + 1
,
(7.188)
where λ
2
1 plays a role of the control parameter.
7.7.6 Numerical Results and Their Validation
Owing to the literature reports [117, 133, 174], the values of the size-dependent
parameter l are quite different and still awaiting the estimation for many materials.
Since the three-layer beams can be metallic, polymer and made from the biological
tissues, therefore the scalar length l parameter can change within a rather large
interval. For instance, in the case of Al and Si materials, the scalar parameter regarding
length l equals to 10
−10 m. The results of the molecular modelling reported in the
Ref. [14] show that the gradient effects occur the thickness of amount l ≈ 10
−9 m.
In Ref. [44], in the case of investigation of the copper wire, the value of the length
parameter achieves l = 3 · 10
−3 m. On the other hand, in the case of the rubber
epoxides materials, in Ref. [172] has been shown that the scalar parameter has been
found experimentally as l = 12 · 10
−3 m for Young’s moduli E = 1.44· GPa and
Poisson’s coefficient ν = 0.38.
As an example, we consider a three-layer beam with the microstructural effect
by taking into account the following length parameters l 1 = l 2 , l 3 for the given
values t 1 = t 2 = 0.125, t 3 = 0.75, the thickness h = 32 · 10
−6 m and length L =
240 · 10
−6 m, being under action of the constant and uniformly distributed load q 0 =
1 N/m.
The numerical examples have been carried out for the three-layer microbeam
composed of two copper made external layers and the middle layer made from the
rubber epoxide material:
E 1 = E 2 = 120 GPa, ν 1 = ν 2 = 0.38, l 1 = l 2 = 3 · 10
−3 m,
E 3 = 1.44 GPa, ν 3 = 0.38, l 3 = 12 · 10
−3 m.
7 Mathematical Models of Functionally Graded Beams in Temperature Field
ω
2
∗ = λ
4
1
λ
2
1 k ˜
ϑ + 1
1 +
d h
h
2
kλ
2
1
1 +
a h
h
2
+ 1
.
(7.186)
In the classical case, we have [76]
ω
2
clas = λ
4
1
λ
2
1 kϑ + 1
kλ
2
1 + 1
.
(7.187)
Let us investigate the ratio r (λ
2
1 ) = ω
2
∗ /ω
2
clas expressed explicitly in the form
r (λ
2
1 ) =
λ
2
1 k ˜
ϑ + 1
1 +
d h
h
2
kλ
2
1
1 +
a h
h
2
+ 1
kλ
2
1 + 1
λ
2
1 kϑ + 1
,
(7.188)
where λ
2
1 plays a role of the control parameter.
7.7.6 Numerical Results and Their Validation
Owing to the literature reports [117, 133, 174], the values of the size-dependent
parameter l are quite different and still awaiting the estimation for many materials.
Since the three-layer beams can be metallic, polymer and made from the biological
tissues, therefore the scalar length l parameter can change within a rather large
interval. For instance, in the case of Al and Si materials, the scalar parameter regarding
length l equals to 10
−10 m. The results of the molecular modelling reported in the
Ref. [14] show that the gradient effects occur the thickness of amount l ≈ 10
−9 m.
In Ref. [44], in the case of investigation of the copper wire, the value of the length
parameter achieves l = 3 · 10
−3 m. On the other hand, in the case of the rubber
epoxides materials, in Ref. [172] has been shown that the scalar parameter has been
found experimentally as l = 12 · 10
−3 m for Young’s moduli E = 1.44· GPa and
Poisson’s coefficient ν = 0.38.
As an example, we consider a three-layer beam with the microstructural effect
by taking into account the following length parameters l 1 = l 2 , l 3 for the given
values t 1 = t 2 = 0.125, t 3 = 0.75, the thickness h = 32 · 10
−6 m and length L =
240 · 10
−6 m, being under action of the constant and uniformly distributed load q 0 =
1 N/m.
The numerical examples have been carried out for the three-layer microbeam
composed of two copper made external layers and the middle layer made from the
rubber epoxide material:
E 1 = E 2 = 120 GPa, ν 1 = ν 2 = 0.38, l 1 = l 2 = 3 · 10
−3 m,
E 3 = 1.44 GPa, ν 3 = 0.38, l 3 = 12 · 10
−3 m.