Page 195 - Finite Element Analysis with ANSYS Workbench

186 Chapter 10 Thermal Stress Analysis

10.1 Basic Equations

Since the differential equations and related equations
for heat transfer and stress analyses were presented in details in the
preceding chapters, this chapter will review essential equations and
show additional equations that relate the two disciplines together.

10.1.1 Differential Equations
The conservation of energy at any location in an isotro-
pic three-dimensional solid is represented by the differential
equation,

 T    T     T     T   
c     k  k      k   Q   0
t   x   x   y   y   z   z  
    
where  is the mass density, c is the specific heat, k is the ther-
mal conductivity coefficient, and Q is the internal heat generation
rate. In the above differential equation, T is the temperature that
varies with the x-, y-, z-coordinates and time t.
If the temperature change does not significantly alter
the solid strain rate, the quasi-static analysis may be used for ther-
mal stress solutions. The condition simplifies the analysis pro-
cedure and reduces overall computational time. The computed
temperature at a given time is input into the stress analysis to
determine the corresponding thermal stress solution. The thermal
stress solution is solved from the governing differential equations
of the solid,
  x +   xy +   xz  0
x  y  z 
  xy +   y +   yz 
x  y  z  0
  xz +   yz +   z  0
x  y  z 
where   y ,  are the normal stress components in the ,x ,y z
,
x
z
directions, respectively, and   xz ,  are the shearing stress
,
yz
xy
components.