2.3 Definition of Movements
25
∂u
∂y
,
∂u
∂z
,
∂v
∂x
,
∂v
∂z
,
∂w
∂x
,
∂w
∂y
are small as compared to one. If the conditions are met, the angles between elements
parallel to the coordinate axes change as a result of deformation by the values
γ xy =
∂u
∂y
+
∂v
∂x
, γ yz =
∂v
∂z
+
∂w
∂y
, γ zx =
∂w
∂x
+
∂u
∂z
.
(2.3)
2.3 Definition of Movements
Let us associate a body with the system of rectangular coordinates Oxyz whose
origin is located in the point O inside the body. Assume that the body endures
elongation in three directions parallel to the coordinate directions. Let us use
(u, v, w) to designate the movement components of a point A(x, y, z) sufficiently
distanced from the point O. Using formulas (2.2), they can be represented as
u ≈ ε x x, v ≈ ε y y, w ≈ ε z z.
(2.4)
These formulas define movement due to deformation without rotation (no
rotations of areas that have no tangential stresses).
As shown in Fig. 1.5, the action of tangential stress, for example, τ xy , can be
represented as elongation and compression in the plane xOy at the angle ±45 ◦
with the coordinate axes. In this case, formula (1.9) defines the rotation angles of
sections that were parallel to the respective coordinate axes before deformation. If
we assume that there are rotations of square diagonals (Fig. 1.5), the linear element
parallel to the axis Ox will turn to the angle γ xy , and the linear element orthogonal
to it will turn in the opposite direction by the same value, e. g. by the angle −γ xy .
By summing movements due to these (small) shears, we can write
u ≈ yγ xy + zγ xz , v ≈ xγ yx + zγ yz , w ≈ xγ zx + yγ zy .
(2.5)
Apart from movements caused by linear (2.4) deformations and shears (2.5), the
body can turn around coordinate axes by the angles ω x , ω y , ω z as a rigid body. Let
us recall [3] that positive rotation occurs from the axis Ox to the axis Oy, from Oy
to Oz and from Oz to Ox. Movements due to these rotations will be
u ≈ zω y − yω z , v ≈ xω z − zω x , w ≈ yω x − xω y .
(2.6)
By summing the respective components of movement (2.4)–(2.6), we obtain the
movement components of sufficiently distanced points in the case of homogeneous
deformation:
25
∂u
∂y
,
∂u
∂z
,
∂v
∂x
,
∂v
∂z
,
∂w
∂x
,
∂w
∂y
are small as compared to one. If the conditions are met, the angles between elements
parallel to the coordinate axes change as a result of deformation by the values
γ xy =
∂u
∂y
+
∂v
∂x
, γ yz =
∂v
∂z
+
∂w
∂y
, γ zx =
∂w
∂x
+
∂u
∂z
.
(2.3)
2.3 Definition of Movements
Let us associate a body with the system of rectangular coordinates Oxyz whose
origin is located in the point O inside the body. Assume that the body endures
elongation in three directions parallel to the coordinate directions. Let us use
(u, v, w) to designate the movement components of a point A(x, y, z) sufficiently
distanced from the point O. Using formulas (2.2), they can be represented as
u ≈ ε x x, v ≈ ε y y, w ≈ ε z z.
(2.4)
These formulas define movement due to deformation without rotation (no
rotations of areas that have no tangential stresses).
As shown in Fig. 1.5, the action of tangential stress, for example, τ xy , can be
represented as elongation and compression in the plane xOy at the angle ±45 ◦
with the coordinate axes. In this case, formula (1.9) defines the rotation angles of
sections that were parallel to the respective coordinate axes before deformation. If
we assume that there are rotations of square diagonals (Fig. 1.5), the linear element
parallel to the axis Ox will turn to the angle γ xy , and the linear element orthogonal
to it will turn in the opposite direction by the same value, e. g. by the angle −γ xy .
By summing movements due to these (small) shears, we can write
u ≈ yγ xy + zγ xz , v ≈ xγ yx + zγ yz , w ≈ xγ zx + yγ zy .
(2.5)
Apart from movements caused by linear (2.4) deformations and shears (2.5), the
body can turn around coordinate axes by the angles ω x , ω y , ω z as a rigid body. Let
us recall [3] that positive rotation occurs from the axis Ox to the axis Oy, from Oy
to Oz and from Oz to Ox. Movements due to these rotations will be
u ≈ zω y − yω z , v ≈ xω z − zω x , w ≈ yω x − xω y .
(2.6)
By summing the respective components of movement (2.4)–(2.6), we obtain the
movement components of sufficiently distanced points in the case of homogeneous
deformation: