2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
31
χ yy =
∂θ y
∂ x
,
(2.18)
χ zz =
∂θ z
∂ x
,
(2.19)
χ xy =
1
2
∂θ x
∂ x
+
∂θ y
∂ y
,
(2.20)
χ xz =
1
2
∂θ x
∂ x
+
∂θ z
∂z
,
(2.21)
χ yz =
1
2
∂θ y
∂ y
+
∂θ z
∂z
.
(2.22)
Here, the vector of rotation θ is defined via displacements field (u x , u y , u z ) in the
following way:
θ x =
1
2
∂u z
∂z
−
∂u y
∂ y
,
(2.23)
θ y =
1
2
∂u x
∂z
−
∂u z
∂ x
,
(2.24)
θ x =
1
2
∂u y
∂ x
−
∂u x
∂ y
.
(2.25)
In the case of linear elastic material, we have
m i j =
E
1 + ν
χ i j l
2
,
(2.26)
where l stands for the internal length material parameter. Estimation of the parameter
l can be found in Refs. [85, 86].
2.3.3 Modified Theory of a Gradient of Deformations
In this theory [18], the deformation energy includes two additional parts of the
gradient, i.e. the gradient of dilatation γ and the deviatory gradient of extension η
as a supplement to the symmetric curvature χ i j . Therefore, energy of deformation is
[18]
U =
1
2
V
(σ i j ε i j + p i γ i + τ i jk η i jk + m i j χ i j )dV ,
(2.27)