32
2 Size-Dependent Theories of Beams, Plates and Shells
where the symmetric tensor of curvature χ i j is defined by Eq. (2.16). The vector of
the dilatation gradient γ i and the deviatory tensor of the extension gradient η i jk are
defined via Eqs. (2.28)–(2.30) and (2.31)–(2.40) using the following relations:
γ x =
∂ε xx
∂ x
+
∂ε yy
∂ x
+
∂ε zz
∂ x
,
(2.28)
γ y =
∂ε xx
∂ y
+
∂ε yy
∂ y
+
∂ε zz
∂ y
,
(2.29)
γ z =
∂ε xx
∂z
+
∂ε yy
∂z
+
∂ε zz
∂z
,
(2.30)
η xxx =
∂ε xx
∂ x
−
1
5
γ x + 3
∂ε xx
∂ x
+
∂ε xy
∂ y
+
∂ε xz
∂z
,
(2.31)
η yyy =
∂ε yy
∂ y
−
1
5
γ y + 3
∂ε xy
∂ x
+
∂ε yy
∂ y
+
∂ε yz
∂z
,
(2.32)
η zzz =
∂ε zz
∂z
−
1
5
γ z + 3
∂ε xz
∂ x
+
∂ε yz
∂ y
+
∂ε zz
∂z
,
(2.33)
η yyx = η yx y = η xyy =
1
3
η xxx + 2
∂ε xy
∂ y
+
∂
ε yy − ε xx
∂ x
,
(2.34)
η zzx = η zxz = η xzz =
1
3
η xxx + 2
∂ε xz
∂z
+
∂ (ε zz − ε xx )
∂ x
,
(2.35)
η xxy = η xyx = η yx x =
1
3
η yyy + 2
∂ε xy
∂ x
+
∂
ε xx − ε yy
∂ x
,
(2.36)
η zzy = η zyz = η yzz =
1
3
η yyy + 2
∂ε yz
∂z
+
∂
ε zz − ε yy
∂ y
,
(2.37)
η xxz = η xzx = η zx x =
1
3
η zzz + 2
∂ε xz
∂ x
+
∂ (ε xx − ε zz )
∂z
,
(2.38)
η yyz = η yzy = η zyy =
1
3
η zzz + 2
∂ε yz
∂ y
+
∂
ε yy − ε zz
∂z
,
(2.39)