Page 240 - Finite Element Modeling and Simulations with ANSYS Workbench

Three-Dimensional Elasticity 225

• No separation: This contact type allows frictionless sliding along the contact faces,
but separation of faces in contact is not allowed.
• Frictionless: This contact model allows free sliding, assuming a zero coefficient of
friction. Gap can form in between regions in contact.
• Rough: This model assumes an infinite friction coefficient between the bodies in
contact. No sliding can occur.
• Frictional: This model allows bodies in contact to slide relative to each other, once
an equivalent shear stress up to a certain magnitude is exceeded.
In most cases, contact regions can be automatically detected and generated in the FEA
program. They can also be manually modified, if needed.

7.4 Formulation of Solid Elements
In this section, we will first summarize the FEA formulation for 3-D elasticity problems,
which are straightforward extensions of the FEA formulations for 1-D bar and 2-D elastic-
ity problems. We will then use an example of linear hexahedral (eight-node brick) element
to examine the element formulation in detail.

7.4.1 General Formulation
As in the FEA formulations for 1-D and 2-D problems, we first interpolate the displacement
fields within a 3-D element using shape functions N : i

N
u = ∑ N u i ,
i
i=1
N
∑
v = N v i , (7.8)
i
i=1 N
w = ∑ N w i
i
i=1

in which u, v , and w are nodal values of the displacement on the element, and N is the
i i
i
number of nodes on that element. In matrix form, we have:
 
u u 1
v 1
 
 u   N 1 0 0 N 2 0 0   
w 1
  (7.9)
 v  = 0 N 1 0 0 N 2 0  
u 2

 w   0 0 0 
 0
 (31 N 1 N 2 (33 N)  
v 2
× )
×
 
 w 2
×
3 ( N 1)