30
2 Size-Dependent Theories of Beams, Plates and Shells
σ yy − μ
d
2
σ yy
dx 2 +
d
2
σ yy
d y 2 +
d
2
σ yy
dz 2
=
E
1 − ν 2 (ε xx + νε yy + νε zz ),
(2.11)
σ zz − μ
d
2
σ zz
dx 2 +
d
2
σ zz
d y 2 +
d
2
σ zz
dz 2
=
E
1 − ν 2 (ε xx + νε yy + νε zz ),
(2.12)
σ xy − μ
d
2
σ xy
dx 2 +
d
2
σ xy
d y 2 +
d
2
σ xy
dz 2
=
E
1 + ν
ε xy ,
(2.13)
σ xz − μ
d
2
σ xz
dx 2 +
d
2
σ xz
d y 2 +
d
2
σ xz
dz 2
=
E
1 + ν
ε xz ,
(2.14)
σ yz − μ
d
2
σ yz
dx 2 +
d
2
σ yz
d y 2 +
d
2
σ yz
dz 2
=
E
1 + ν
ε xz .
(2.15)
Here, ε i j are components of the deformation tensor and E is the Young modulus,
whereas ν is Poisson’s coefficient. The differential model is simple and is widely
employed for nanostructures investigation. However, the latter model can give sometimes paradoxal results, for example, in the problems of bending and vibration of
cantilever beams. The detailed information about paradoxes of the differential models
can be found in [80–83].
2.3.2 Modified Couple Stress Theory
The modified couple stress theory has been proposed by Yang et al. [15], in result of
modification of the classical couple stress theory developed by Toupin [12], Mindlin
and Tiersten [84] and Koiter [14]. Introducing the additional condition of balance
of stress couples in order to guarantee symmetry of the couple stress tensor implied
decrease of a number of additional internal material length parameters from two to
one. The latter result improved efficiency of the modified couple stress theory, since
estimation of the internal parameter of material length does not belong to easy task.
Energy of deformation U , owing to that theory, is a function of both deformation
and curvature [15] and takes the form
U =
1
2
V
(σ i j ε i j + m i j χ i j )dV ,
(2.16)
where m i, j are components of deviation part of the symmetric tensor of moments,
whereas χ i, j are components of the tensor of symmetric curvature defined by the
following formulas:
χ xx =
∂θ x
∂ x
,
(2.17)
2 Size-Dependent Theories of Beams, Plates and Shells
σ yy − μ
d
2
σ yy
dx 2 +
d
2
σ yy
d y 2 +
d
2
σ yy
dz 2
=
E
1 − ν 2 (ε xx + νε yy + νε zz ),
(2.11)
σ zz − μ
d
2
σ zz
dx 2 +
d
2
σ zz
d y 2 +
d
2
σ zz
dz 2
=
E
1 − ν 2 (ε xx + νε yy + νε zz ),
(2.12)
σ xy − μ
d
2
σ xy
dx 2 +
d
2
σ xy
d y 2 +
d
2
σ xy
dz 2
=
E
1 + ν
ε xy ,
(2.13)
σ xz − μ
d
2
σ xz
dx 2 +
d
2
σ xz
d y 2 +
d
2
σ xz
dz 2
=
E
1 + ν
ε xz ,
(2.14)
σ yz − μ
d
2
σ yz
dx 2 +
d
2
σ yz
d y 2 +
d
2
σ yz
dz 2
=
E
1 + ν
ε xz .
(2.15)
Here, ε i j are components of the deformation tensor and E is the Young modulus,
whereas ν is Poisson’s coefficient. The differential model is simple and is widely
employed for nanostructures investigation. However, the latter model can give sometimes paradoxal results, for example, in the problems of bending and vibration of
cantilever beams. The detailed information about paradoxes of the differential models
can be found in [80–83].
2.3.2 Modified Couple Stress Theory
The modified couple stress theory has been proposed by Yang et al. [15], in result of
modification of the classical couple stress theory developed by Toupin [12], Mindlin
and Tiersten [84] and Koiter [14]. Introducing the additional condition of balance
of stress couples in order to guarantee symmetry of the couple stress tensor implied
decrease of a number of additional internal material length parameters from two to
one. The latter result improved efficiency of the modified couple stress theory, since
estimation of the internal parameter of material length does not belong to easy task.
Energy of deformation U , owing to that theory, is a function of both deformation
and curvature [15] and takes the form
U =
1
2
V
(σ i j ε i j + m i j χ i j )dV ,
(2.16)
where m i, j are components of deviation part of the symmetric tensor of moments,
whereas χ i, j are components of the tensor of symmetric curvature defined by the
following formulas:
χ xx =
∂θ x
∂ x
,
(2.17)