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

542
STRAIN TRANSFORMATIONS
y
y
y
γ xy dy
dy
O dx
ε x dx
FIGURE 13.2a
x
ε y dy
dy
O dx
FIGURE 13.2b
x
dy
O dx
FIGURE 13.2c
γ xy
x
The shear strain γ xy shown in Figure 13.2c is a measure of the change in angle between
the x and y axes, which are initially perpendicular to each other. Shear strains are considered
positive when the angle between axes decreases and negative when the angle increases.
Note that the sign conventions for strain are consistent with the stress sign conventions.
A positive normal stress (i.e., tensile normal stress) in the x direction causes a positive
normal strain ε x (i.e., elongation) (Figure 13.2a), a positive normal stress in the y direction
creates a positive normal strain ε y (Figure 13.2b), and a positive shear stress produces a
positive shear strain γ xy (Figure 13.2c).
13.3 Transformation Equations for plane Strain
y
t
dy
dn
θ
O dx
FIGURE 13.3
A
A'
x
n
The state of plane strain at point O is defined by three strain components: ε x , ε y , and γ xy .
Transformation equations provide the means to determine normal and shear strains at point
O for orthogonal axes rotated at any arbitrary angle θ.
Equations that transform normal and shear strains from the x–y axes to any arbitrary
orthogonal axes will be derived. To facilitate the derivation, the dimensions of the element
are chosen such that the diagonal OA of the element coincides with the n axis (Figure 13.3).
It is also convenient to assume that corner O is fixed and that the edge of the element along
the x axis does not rotate.
When all three strain components (ε x , ε y , and ε xy ) occur simultaneously (Figure 13.3),
corner A of the element is displaced to a new location denoted by A′. For clarity, the deformations
are shown greatly exaggerated.
Transformation Equation for Normal Strain
The displacement vector from A to A′ (shown in Figure 13.3) is isolated and enlarged in
Figure 13.4a. The horizontal component of vector AA′ is composed of the deformations
due to ε x (see Figure 13.2a) and γ xy (see Figure 13.2c). The vertical component of AA′ is
caused by ε y (see Figure 13.2b).
A′
n
A′
n
ε y dy cos θ – ( ε x dx + γ xy dy)
sin θ
t
θ
y
εy dy
t
θ
y
εy dy
( ε x dx + γ xy dy) cos θ+
ε y dy sin θ
A
θ
ε x dx + γxy
dy
x
A
θ
εx
dx+
γ xy dy
x
FIGURE 13.4a
FIGURE 13.4b