428
32 On Boundary Value Problems of Inelastic Body Mechanics
ω A 2 A 1 =
∂
∂y
∂γ xy
∂z
+
∂γ yz
∂x
+
∂γ xz
∂y
−
∂ 2 ε y
∂x∂z
dxdy.
The value in the square brackets defines the torsion component (p. 422) of the
incompatibility tensor S ij , (i = j) and expresses, as already mentioned, the fiber
twist angle per unit area parallel to the j -th axis, for a frame located in a plane with
the normal i, i.e.
S zx =
∂
∂y
∂γ xy
∂z
+
∂γ yz
∂x
+
∂γ xz
∂y
−
∂ 2 ε y
∂x∂z
.
(32.46)
Similarly, we can obtain that the formula
S zz =
∂ 2 ε x
∂y 2 +
∂ 2 ε y
∂x 2 − 2
∂ 2
∂x∂y
γ xy
(32.47)
determines the bending component of the incompatibility tensor S zz , which is
proportional to the rotation angle relative to the Oz axis of the linear elements of
the frame located in the plane with the normal Oz.
By changing the axis indexes and bearing in mind notations like
γ xx = ε x , 2γ xy =
∂u
∂y
+
∂v
∂x
,
we can obtain six components of the incompatibility tensor, and
S xy = S yx , S xz = S zx , S yz = S zy .
(32.48)
The components (32.46)–(32.48) can be written differently as
S mn = e mki e nsj
∂ 2 γ ks
∂x i ∂x j
;
(32.49)
here, as mentioned above in this chapter, summation is carried out over repeated
(dummy) indexes; e mki and e nsj —Levi–Civita symbols; they are equal to zero if
there are duplicate indexes, equal to 1 in the following arrangement of indexes:
123, 321, 231, and equal to −1 for any other order of the indexes.