1.17 Axisymmetric Plane Strain
17
where
E
∗
=
E
1 − ν 2 , ν
∗
=
ν
1 − ν
.
(1.21)
Noting that
E ∗
1 + ν ∗ =
E
1 + ν
,
(1.22)
we come to the following conclusion:
the dependency between relative elongations and normal stresses in the case of plane strain
(ε z = 0) can be obtained from the respective dependencies in the case of a plane stressed
state (σ z = 0) by substituting E and ν through E ∗ and ν ∗ using formulas (1.21).
In the case of such substitution, the dependency between tangential stresses and
shears towards the ratios of (1.22) and (1.10) will not change.
1.16 Homogeneous Model of a Solid Body
During mathematical studies of a deformable solid body, we consider infinitely
small elements of the body that are attributed with mechanical properties found
in experiments with relatively large specimens. Thus, a real body is substituted by
its ideally homogeneous continuous model.
Movements of points of such a model are represented by functions of coordinates
of these points differentiated for the required number of times. In this case, the
stress-strain condition in sufficiently small volumes of a body can be deemed
homogeneous.
In some cases, let us assume to associate each atom of a solid body with a
geometric point of an ideal model. For specific loading, relative movements of any
two respective points of a real body and its continuous model must almost coincide
only if the distance between points is sufficiently long as compared to distances
between atoms.
This section suggests that strains and stresses of a continuous model of a solid
body are connected by Hooke’s law unless there are special exceptions. A real solid
body is substituted by the specified ideally homogeneous model.
1.17 Axisymmetric Plane Strain
Assume that during deformation, all points of a body move in the directions
perpendicular to a straight line being the symmetry axis, whereas the value of these
movements (u) is only a function of the distance (r) of the point from this axis.
17
where
E
∗
=
E
1 − ν 2 , ν
∗
=
ν
1 − ν
.
(1.21)
Noting that
E ∗
1 + ν ∗ =
E
1 + ν
,
(1.22)
we come to the following conclusion:
the dependency between relative elongations and normal stresses in the case of plane strain
(ε z = 0) can be obtained from the respective dependencies in the case of a plane stressed
state (σ z = 0) by substituting E and ν through E ∗ and ν ∗ using formulas (1.21).
In the case of such substitution, the dependency between tangential stresses and
shears towards the ratios of (1.22) and (1.10) will not change.
1.16 Homogeneous Model of a Solid Body
During mathematical studies of a deformable solid body, we consider infinitely
small elements of the body that are attributed with mechanical properties found
in experiments with relatively large specimens. Thus, a real body is substituted by
its ideally homogeneous continuous model.
Movements of points of such a model are represented by functions of coordinates
of these points differentiated for the required number of times. In this case, the
stress-strain condition in sufficiently small volumes of a body can be deemed
homogeneous.
In some cases, let us assume to associate each atom of a solid body with a
geometric point of an ideal model. For specific loading, relative movements of any
two respective points of a real body and its continuous model must almost coincide
only if the distance between points is sufficiently long as compared to distances
between atoms.
This section suggests that strains and stresses of a continuous model of a solid
body are connected by Hooke’s law unless there are special exceptions. A real solid
body is substituted by the specified ideally homogeneous model.
1.17 Axisymmetric Plane Strain
Assume that during deformation, all points of a body move in the directions
perpendicular to a straight line being the symmetry axis, whereas the value of these
movements (u) is only a function of the distance (r) of the point from this axis.