2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
29
of microbeam to find e 0 by monitoring free vibrations of nonlocal beams. They
obtained analytical formula for e 0 based on the geometric properties and modes of
vibration. Zhang et al. [76–78] proposed the microstructural beam mesh model to
find e 0 under consideration of free vibrations of nonlocal beams [76], stability loss
and free vibrations of nonlocal plates [77]. It has been found that the value e 0 depends
on the initial stress, rotation inertia, vibration modes and a ratio of the rectangular
plate sides. In the general case, owing to the conservative estimations, the value of
the nonlocal parameter of the single-walled carbon nanotubes is e 0 <2.0 nm [79].
We consider a particular kernel function k [33] transformed the nonlocal state
equation into the differential counterpart form
(1 − μ∇
2
)σ i j = σ
L
i j ,
(2.2)
where μ = κ
2 and ∇
2 stands for the Laplacian. Explicit form of equation (2.2) can
be derived for three case studies dealing with an isotropic material.
In the case of 1D problems, we have
σ xx − μ
d
2
σ xx
dx 2 = Eε xx ,
(2.3)
σ xz − μ
d
2
σ xz
dx 2 = Eε xz .
(2.4)
In the case of (2D) problems, one gets
σ xx − μ
d
2
σ xx
dx 2 +
d
2
σ xx
d y 2
=
E
1 − ν 2 (ε xx + νε yy ),
(2.5)
σ xx − μ
d
2
σ yy
dx 2 +
d
2
σ yy
d y 2
=
E
1 − ν 2 (νε xx + νε yy ),
(2.6)
σ xy − μ
d
2
σ xy
dx 2 +
d
2
σ xy
d y 2
=
E
1 + ν
ε xy ,
(2.7)
σ xz − μ
d
2
σ xz
dx 2 +
d
2
σ xz
d y 2
=
E
1 + ν
ε xz ,
(2.8)
σ yz − μ
d
2
σ yz
dx 2 +
d
2
σ yz
d y 2
=
E
1 + ν
ε yz .
(2.9)
Finally, in the case of 3D problems, the following formulas hold:
σ xx − μ
d
2
σ xx
dx 2 +
d
2
σ xx
d y 2 +
d
2
σ xx
dz 2
=
E
1 − ν 2 (ε xx + νε yy + νε zz ),
(2.10)
29
of microbeam to find e 0 by monitoring free vibrations of nonlocal beams. They
obtained analytical formula for e 0 based on the geometric properties and modes of
vibration. Zhang et al. [76–78] proposed the microstructural beam mesh model to
find e 0 under consideration of free vibrations of nonlocal beams [76], stability loss
and free vibrations of nonlocal plates [77]. It has been found that the value e 0 depends
on the initial stress, rotation inertia, vibration modes and a ratio of the rectangular
plate sides. In the general case, owing to the conservative estimations, the value of
the nonlocal parameter of the single-walled carbon nanotubes is e 0 <2.0 nm [79].
We consider a particular kernel function k [33] transformed the nonlocal state
equation into the differential counterpart form
(1 − μ∇
2
)σ i j = σ
L
i j ,
(2.2)
where μ = κ
2 and ∇
2 stands for the Laplacian. Explicit form of equation (2.2) can
be derived for three case studies dealing with an isotropic material.
In the case of 1D problems, we have
σ xx − μ
d
2
σ xx
dx 2 = Eε xx ,
(2.3)
σ xz − μ
d
2
σ xz
dx 2 = Eε xz .
(2.4)
In the case of (2D) problems, one gets
σ xx − μ
d
2
σ xx
dx 2 +
d
2
σ xx
d y 2
=
E
1 − ν 2 (ε xx + νε yy ),
(2.5)
σ xx − μ
d
2
σ yy
dx 2 +
d
2
σ yy
d y 2
=
E
1 − ν 2 (νε xx + νε yy ),
(2.6)
σ xy − μ
d
2
σ xy
dx 2 +
d
2
σ xy
d y 2
=
E
1 + ν
ε xy ,
(2.7)
σ xz − μ
d
2
σ xz
dx 2 +
d
2
σ xz
d y 2
=
E
1 + ν
ε xz ,
(2.8)
σ yz − μ
d
2
σ yz
dx 2 +
d
2
σ yz
d y 2
=
E
1 + ν
ε yz .
(2.9)
Finally, in the case of 3D problems, the following formulas hold:
σ xx − μ
d
2
σ xx
dx 2 +
d
2
σ xx
d y 2 +
d
2
σ xx
dz 2
=
E
1 − ν 2 (ε xx + νε yy + νε zz ),
(2.10)