Section 4.3
Solid Mechanics Part I
Kelly
111
4.3 One Dimensional Axial Deformations
In this section, a specific simple geometry is considered, that of a long and thin straight
component loaded in such a way that it deforms in the axial direction only.
The
x
-axis is
taken as the longitudinal axis, with the cross-section lying in the
y
x
−
plane, Fig. 4.3.1.
Figure 4.3.1: A slender straight component; (a) longitudinal axis, (b) cross-section
4.3.1
Basic relations for Axial Deformations
Any static analysis of a structural component involves the following three considerations:
(1)
constitutive response
(2)
kinematics
(3)
equilibrium
In this Chapter, it is taken for (1) that the material responds as an isotropic linear elastic
solid.
It is assumed that the only significant stresses and strains occur in the axial
direction, and so the stress-strain relations 4.2.8 reduce to the one-dimensional equation
xx
xx
E
ε
σ
=
or, dropping the subscripts,
ε
σ
E
=
(4.3.1)
Kinematics (2) is the study of deformation, the subject of §3.6-3.8.
In the theory
developed here, known as
axial deformation
, it is assumed that the axis of the
component remains straight and that cross-sections that are initially perpendicular to the
axis remain perpendicular after deformation. This implies that, although the strain might
vary along the axis, it remains
constant over any cross section
.
As defined in the
previous Chapter, the axial strain occurring over any section is given by
0
0
L
L
L
−
=
ε
(4.3.2)
This is illustrated in Fig. 4.3.2, which shows a (shaded) region undergoing a compressive
(negative) strain.
Note that individual particles/points undergo displacements whereas regions/line-
elements undergo strain.
In Fig. 4.3.2, the particle originally at
A
has undergone a
displacement
)
(
A
u
whereas the particle originally at
B
has undergone a displacement
)
(
B
u
.
From Fig. 4.3.2, another way of expressing the strain in the shaded region is
x
y
z

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