424
32 On Boundary Value Problems of Inelastic Body Mechanics
32.7 More About Incompatible Deformations
32.7.1 Distributed Wedge Dislocations
Imagine a rectangular contour ABCD (Fig. 32.13) inside an elastic plane. Mentally
free this contour (frame) from communication with the body and cut it (frame)
(Fig. 32.14). If there were wedge-shaped dislocations inside the frame, then the
vertical elements in the section nn (Fig. 32.13) will rotate relative to each other
by some angle (ω). Assume that the frame in question is cut from a body subject to
spatial deformation. Let us define the rotation angles of linear elements in the frame
section in this general case. The frame element parallel to the Oy axis is rotated in
the xOy plane by the angle whose tangent will be
at the point B
−
∂u
∂y
x +
x
2
, y −
y
2
;
at the point C
−
∂u
∂y
x +
x
2
, y +
y
2
;
Fig. 32.13 For the study of
deformations of the contour
C
B
A
D
Δy
n ∼ n
Δx
(x,y )
+
Fig. 32.14 Deformations of
the cut frame
y
D
C
x
B
ω
A 2
A 1
32 On Boundary Value Problems of Inelastic Body Mechanics
32.7 More About Incompatible Deformations
32.7.1 Distributed Wedge Dislocations
Imagine a rectangular contour ABCD (Fig. 32.13) inside an elastic plane. Mentally
free this contour (frame) from communication with the body and cut it (frame)
(Fig. 32.14). If there were wedge-shaped dislocations inside the frame, then the
vertical elements in the section nn (Fig. 32.13) will rotate relative to each other
by some angle (ω). Assume that the frame in question is cut from a body subject to
spatial deformation. Let us define the rotation angles of linear elements in the frame
section in this general case. The frame element parallel to the Oy axis is rotated in
the xOy plane by the angle whose tangent will be
at the point B
−
∂u
∂y
x +
x
2
, y −
y
2
;
at the point C
−
∂u
∂y
x +
x
2
, y +
y
2
;
Fig. 32.13 For the study of
deformations of the contour
C
B
A
D
Δy
n ∼ n
Δx
(x,y )
+
Fig. 32.14 Deformations of
the cut frame
y
D
C
x
B
ω
A 2
A 1