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

Next, the displacement vector AA′ will be resolved into components in the n and t directions.
Unit vectors in the n and t directions are
n = cosθi + sinθ j and t =− sinθi + cosθ
j
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TRANSFORMATION EQuATIONS
FOR PLANE STRAIN
The displacement component in the n direction can be determined from the dot product:
AA′ i n = ( ε dx + γ dy)cosθ + ε dysinθ
(a)
x xy y
The displacement component in the t direction is
AA′ i t = ε dycos θ − ( ε dx + γ dy)sinθ
(b)
y x xy
The displacements in the n and t directions are shown in Figure 13.4b.
The displacement in the n direction represents the elongation of diagonal OA (see
Figure 13.3) due to the normal and shear strains ε x , ε y , and γ xy . The strain in the n direction
can be found by dividing the elongation given in Equation (a) by the initial length dn of the
diagonal:
ε
n
( εx dx + γ xydy)cosθ + εydysinθ
=
dn
⎛ dx dy ⎞ dy
= εx + γ xy cosθ εy
sinθ
⎝
⎜
dn dn⎠
⎟ +
dn
(c)
From Figure 13.3, dx/dn = cos θ and dy/dn = sin θ. By substituting these relationships into
Equation (c), the strain in the n direction can be expressed as
2 2
ε = ε cos θ + ε sin θ + γ sinθcosθ
(13.3)
n x y xy
Now recall the double-angle trigonometric identities:
2
1
cos θ = (1 + cos2 θ)
2
2
1
sin θ = (1 − cos2 θ)
2
2sinθcosθ = sin 2θ
From these identities, Equation (13.3) can also be expressed as
εx + εy εx − εy γ xy
εn
= + cos2θ
+
2 2
2 sin2 θ
(13.4)
Transformation Equation for Shear Strain
The component of the displacement vector AA′ in the t direction [Equation (b)] represents
an arc length through which the diagonal OA rotates. With this rotation angle denoted as α
(Figure 13.5a), the arc length associated with radius dn can be expressed as
αdn = ε dy cos θ − ( ε dx + γ dy)sinθ
y x xy
y
t
dy
dn
θ
dx
FIGURE 13.5a
α
α dn
A'
A
n
x