Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

y
dx′
541
PLANE STRAIN
dx
θ
′ zx
dy′
θ′ xy
θ
′ yz
dz′
dy
dz
(b)
(a)
x
z
FIGURE 13.1
The Cartesian components of strain at the point can be expressed in terms of the
deformations by using the definitions of normal and shear strain presented in Section 2.2.
Thus, we have the following equations:
dx′ − dx
ε =
dx
dy′ − dy
ε =
dy
ε
π
= − θ′
2
π
= − θ′
2
x xy xy
y yz yz
dz′ − dz
=
dz
π
= − θ′
2
z zx zx
γ
γ
γ
(13.1)
In a similar manner, the normal strain component associated with a line oriented in an arbitrary
n direction and the shearing strain component associated with two arbitrary initially
orthogonal lines oriented in the n and t directions in the undeformed element are respectively
given by
ε
dn′ − dn
=
dn
and
n nt nt
γ
π
= − θ′ (13.2)
2
13.2 plane Strain
Considerable insight into the nature of strain can be gained by considering a state of strain
known as two-dimensional strain or plane strain. For this state, the x–y plane will be used
as the reference plane. The length dz shown in Figure 13.1 does not change, and the angles
θ′ yz and θ′ zx remain 90°. Thus, for the conditions of plane strain, ε z = γ xz = γ yz = 0.
If the only deformations are those in the x–y plane, then three strain components may
exist. Figure 13.2 shows an infinitesimal element of dimensions dx and dy that will be used
to illustrate the strains existing at point O. In Figure 13.2a, the element subjected to a
positive normal strain ε x will elongate by the amount ε x dx in the horizontal direction.
When subjected to a positive normal strain ε y , the element will elongate by the amount
ε y dy in the vertical direction (Figure 13.2b). Recall that positive normal strains create
elongations, and negative normal strains create contractions, in the material.