Page 320 - Finite Element Modeling and Simulations with ANSYS Workbench

Thermal Analysis 305

9.2.2.2 2-D Cases
For plane stress, we have:

∆
 ε   α T 
x
   
∆
ε o =  ε  =  α T  (9.14)
y
   0 
 γ xy o   
For plane strain, we have:
 ε  (1 + να T 
∆
)
x
   
∆
)
ε o =  ε  =  (1 + να T  (9.15)
y
   0 
 γ xy o   
in which, ν is Poisson’s ratio.
9.2.2.3 3-D Case
 ε   α T∆ 
x
   α T∆ 
 ε y   
 ε    α T∆ 




z
ε o =   =   (9.16)
 γ xy   0 
γ  yz   0 
   

 γ  zx o     0 
Observation: Temperature changes do not yield shear strains.
In both 2-D and 3-D cases, the total strain can be given by the following vector equation:
ε = ε + ε o (9.17)
e
And the stress–strain relation is given by
σ = Eε = E(ε − ε ) (9.18)
e
o
9.2.2.4 Notes on FEA for Thermal Stress Analysis
Need to specify α for the structure and ΔT on the related elements (which experience the
temperature change).

• Note that for linear thermoelasticity, same temperature change will yield same
stresses, even if the structure is at two different temperature levels.
• Differences in the temperatures during the manufacturing and working environ-
ment are the main cause of thermal (residual) stresses.

A more comprehensive review of thermal problems, their governing equations and
boundary conditions can be found in the references, such as References [13,14].