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100 Chapter 6 Three-Dimensional Solid Analysis

6.1 Basic Equations

6.1.1 Differential Equations
The equilibrium conditions at any location of a three-
dimensional elastic solid in x-y-z coordinate system, with exclusion
of body forces, are governed by the three partial differential
equations,
    xy  
x   xz  0
x  y  z 
     
xy  y  yz  0
x  y  z 
     
xz  yz  z  0
x  y  z 
where    are the normal stress components in the ,x ,y
,
,
x
z
y
z coordinate directions, and   xz ,  are the shearing stress
,
xy
yz
components.
The three differential equations of the problem suggest
that there must be three basic unknowns. These unknowns are the
(, , ) in
displacement components (, , ),ux y z (, , )vx y z and wx y z
the x, y and z coordinate directions, respectively. Thus, the stress
components in the differential equations must be written in forms
of the three displacement components prior to solving them.

6.1.2 Related Equations
The six stress components can be written in forms of
the six strain components according to the Hooke’s law as,
      

C



(6 1) (6 6) (6 1)
T

where        xy  xz  
yz 
z
y
x
and   T      x y  z  xy  xz  yz 

C
The matrix  is the elasticity matrix which depends on the
Young’s modulus and Poisson’s ratio.