2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
33
η xyz = η yxz = η zx y = η xzy = η zyx = η yxz =
1
3
∂ε yz
∂ x
+
∂ε xz
∂ y
+
∂ε xz
∂z
. (2.40)
In the case of linear elastic material, the stresses of higher order ( p i , τ i jk , m i j ) are
as follows:
p i = 2ηl
2
0 ε mm, j , τ
(1)
i jk = 2ηl
2
1 η
(1)
i jk ,
m
s
i j = 2ηl
2
2 χ
(s)
i j
p i =
E
1 + ν
l
2
0 ε mm, j ,
(2.41)
τ
(1)
i jk =
E
1 + ν
l
2
1 η
(1)
i jk ,
(2.42)
m
s
i j =
E
1 + ν
l
2
2 χ
(s)
i j ,
(2.43)
where l 0 , l 1 , l 2 are internal parameters of the material length coupled with the dilatation gradient, the deviatory gradient of extension and the gradient of symmetric
curvature, respectively.
2.3.4 Surface Theory of Elasticity
It is known that a nature of the chemical coupling and equilibrium atomic localization
of the surface atoms is different from the nature of internal atom localization (see
[87, 88]). In result, material properties in vicinity of a surface principally differ from
the atoms located inside the solid body volume.
The surface energy being necessary for isothermic creation of a unit part of the
new surface can be used as measure of chemical coupling in the surface vicinity [89].
In the case of fluid, the surface energy is equal to surface tension, but it is different
for the case of solid body. In the case when the surface energy does not depend on
deformation, Shuttleworth [90] and Herring [91] proposed the following formula
(see also [92, 93]):
τ αβ = δ αβ +
∂∂
∂ε αβ
,
(2.44)
where —surface energy; τ ab , ε αβ , δ αβ —surface tension, surface deformation and
Kronecker’s symbol, respectively. In formula (2.44) and later, we employ standard
notation of indices together with Einstein summation concept where each Greek
index takes the value from 1 to 2, and each Latin symbol changes from 1 to 3.
In the theory of surface elasticity, a surface is considered as a thin membrane [61,
94] exhibiting bending stiffness [95, 96], which is ideally coupled with volume material. It allows to include the elastic constant parameters for the surface and to satisfy
33
η xyz = η yxz = η zx y = η xzy = η zyx = η yxz =
1
3
∂ε yz
∂ x
+
∂ε xz
∂ y
+
∂ε xz
∂z
. (2.40)
In the case of linear elastic material, the stresses of higher order ( p i , τ i jk , m i j ) are
as follows:
p i = 2ηl
2
0 ε mm, j , τ
(1)
i jk = 2ηl
2
1 η
(1)
i jk ,
m
s
i j = 2ηl
2
2 χ
(s)
i j
p i =
E
1 + ν
l
2
0 ε mm, j ,
(2.41)
τ
(1)
i jk =
E
1 + ν
l
2
1 η
(1)
i jk ,
(2.42)
m
s
i j =
E
1 + ν
l
2
2 χ
(s)
i j ,
(2.43)
where l 0 , l 1 , l 2 are internal parameters of the material length coupled with the dilatation gradient, the deviatory gradient of extension and the gradient of symmetric
curvature, respectively.
2.3.4 Surface Theory of Elasticity
It is known that a nature of the chemical coupling and equilibrium atomic localization
of the surface atoms is different from the nature of internal atom localization (see
[87, 88]). In result, material properties in vicinity of a surface principally differ from
the atoms located inside the solid body volume.
The surface energy being necessary for isothermic creation of a unit part of the
new surface can be used as measure of chemical coupling in the surface vicinity [89].
In the case of fluid, the surface energy is equal to surface tension, but it is different
for the case of solid body. In the case when the surface energy does not depend on
deformation, Shuttleworth [90] and Herring [91] proposed the following formula
(see also [92, 93]):
τ αβ = δ αβ +
∂∂
∂ε αβ
,
(2.44)
where —surface energy; τ ab , ε αβ , δ αβ —surface tension, surface deformation and
Kronecker’s symbol, respectively. In formula (2.44) and later, we employ standard
notation of indices together with Einstein summation concept where each Greek
index takes the value from 1 to 2, and each Latin symbol changes from 1 to 3.
In the theory of surface elasticity, a surface is considered as a thin membrane [61,
94] exhibiting bending stiffness [95, 96], which is ideally coupled with volume material. It allows to include the elastic constant parameters for the surface and to satisfy