18
1 Summary of Elasticity Theory: Basic Concepts
Movements u are deemed positive if the distance of the point from the symmetry
axis is increased. Otherwise u 0.
If the movement equals u in the point located at the distance of r from the
symmetry axis, the movement in the point r + dr can be represented as
u(r + dr) = u(r) +
du
dr
dr.
(1.23)
If two points located on the same perpendicular to the symmetry axis were
distanced before deformation from each other to the distance of l = dr, after
deformation this distance will be dr +
du
dr
dr. Consequently, the increment of the
distance between points will be l =
du
dr
dr, and relative elongation ((l/ l) in the
radial direction will be
ε 1 =
du
dr
.
(1.24)
Due to deformations, distances between body points in the direction perpendicular to the symmetry axis will change along with the lengths of coordinate
circumferences. The length a circumference with radius r with the center on the
symmetry axis will get an increment l = 2π(r + u) − 2πr. Relative elongation
l/ l, l = 2πr, ,l = 2πu) of this circumference will be equal to
ε 2 =
u
r
.
(1.25)
1.18 Lame Task
Let us consider deformation of a round hollow cylinder with walls of permanent
thickness that is subject to the action of evenly distributed internal (p a ) and external
(p b ) pressures (Fig. 1.8a). In the following, we will consider a round ring cut from
a cylinder by two planes perpendicular to its axis and located from each other at the
distance equal to one.
The symmetry condition follows that only normal stresses will act on the sides
of the element mnn 1 m 1 shown in Fig. 1.8b and limited by two coaxial surfaces and
the axial plane. Let us designate annular normal stresses on the sides mm 1 and nn 1
as σ t , and normal radial stresses on the sides mn and m 1 n 1 as σ r .
When the radius r is increased by dr, the radial stress will change for
dσ r
dr
dr.
Therefore, the normal radial stress on the side m 1 n 1 , (Fig. 1.8b) will be
σ r +
dσ r
dr
dr.
(1.26)
1 Summary of Elasticity Theory: Basic Concepts
Movements u are deemed positive if the distance of the point from the symmetry
axis is increased. Otherwise u 0.
If the movement equals u in the point located at the distance of r from the
symmetry axis, the movement in the point r + dr can be represented as
u(r + dr) = u(r) +
du
dr
dr.
(1.23)
If two points located on the same perpendicular to the symmetry axis were
distanced before deformation from each other to the distance of l = dr, after
deformation this distance will be dr +
du
dr
dr. Consequently, the increment of the
distance between points will be l =
du
dr
dr, and relative elongation ((l/ l) in the
radial direction will be
ε 1 =
du
dr
.
(1.24)
Due to deformations, distances between body points in the direction perpendicular to the symmetry axis will change along with the lengths of coordinate
circumferences. The length a circumference with radius r with the center on the
symmetry axis will get an increment l = 2π(r + u) − 2πr. Relative elongation
l/ l, l = 2πr, ,l = 2πu) of this circumference will be equal to
ε 2 =
u
r
.
(1.25)
1.18 Lame Task
Let us consider deformation of a round hollow cylinder with walls of permanent
thickness that is subject to the action of evenly distributed internal (p a ) and external
(p b ) pressures (Fig. 1.8a). In the following, we will consider a round ring cut from
a cylinder by two planes perpendicular to its axis and located from each other at the
distance equal to one.
The symmetry condition follows that only normal stresses will act on the sides
of the element mnn 1 m 1 shown in Fig. 1.8b and limited by two coaxial surfaces and
the axial plane. Let us designate annular normal stresses on the sides mm 1 and nn 1
as σ t , and normal radial stresses on the sides mn and m 1 n 1 as σ r .
When the radius r is increased by dr, the radial stress will change for
dσ r
dr
dr.
Therefore, the normal radial stress on the side m 1 n 1 , (Fig. 1.8b) will be
σ r +
dσ r
dr
dr.
(1.26)