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Two-Dimensional Elasticity 119

(a) (b)
z

y
y
x

x

FIGURE 4.3
Plane stress condition: (a) structure with in-plane loading; and (b) 2-D model.

(a) (b)
y y
x
z x

FIGURE 4.4
Plane strain condition: (a) structure with uniform transverse load; and (b) 2-D model.

4.2.3 Stress–Strain (Constitutive) Equations
For linear elastic isotropic materials, we have the following strain–stress relation for the
plane stress case:

 ε   1/ E −ν E/ 0  σ x   ε x0 

x

      
/
 ε  = −ν E 1/ E 0   σ y  +  ε y0  (4.3)

y
   0 0 1/ G    
 τ xy
 γ xy      γ xy0 
or,
ε= E −1
σ+ ε 0
where ε is the initial strain (e.g., due to a temperature change), E Young’s modulus, ν
0
Poisson’s ratio, and G the shear modulus. Note that

E
G = (4.4)
21 ( + )ν

for homogeneous and isotropic materials.