Introduction to Contact Problems 15.1 INTRODUCTION: THE GAP In many practical problems, the information required to develop a finite-element model, for example, the geometry of a member
Trang 1Introduction to Contact Problems
15.1 INTRODUCTION: THE GAP
In many practical problems, the information required to develop a finite-element model, for example, the geometry of a member and the properties of its constituent materials, can be determined with little uncertainty or ambiguity However, often the loads experienced by the member are not so clear This is especially true if loads are transmitted to the member along an interface with a second member This class
of problems is called contact problems, and they are arguably the most common boundary conditions encountered in practical problems The finite-element commu-nity has devoted, and continues to devote, a great deal of effort to this complex problem, leading to gap and interface elements for contact Here, we introduce gap elements
First, consider the three-spring configuration in Figure 15.1 All springs are of
(15.1)
From the viewpoint of the finite-element method, Figure 15.1 poses the following
FIGURE 15.1 Simple contact problem.
15
k
c= <
≥
2 3
δ
contactor P
δ
g
target
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below it on the target, these nodes are not initially connected, but are later connected
constraint whereby the middle spring does not move through the target If the nodes are considered unconnected in the finite-element model, there is nothing to enforce the nonpenetration constraint If, however, the nodes are considered connected, the stiffness is artificially high
This difficulty is overcome in an approximate sense by a bilinear contact element
(15.2)
(15.3)
Elementary algebra serves to demonstrate that
(15.4)
Consequently, the model with the contact is too stiff by 0.5% when the gap
is open, and too soft by 0.33% when the gap is closed (contact) One conclusion that can be drawn from this example is that the stiffness of the gap element should
be related to the stiffnesses of the contactor and target in the vicinity of the contact point
FIGURE 15.2 Spring representing contact element.
contactor
target
P
δ
kg
k
m
g
−
1
k
g= >
≤
/100
δ δ
k
c≈ + >
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15.2 POINT-TO-POINT CONTACT
Generally, it is not known what points will come into contact, and there is no guarantee that target nodes will come into contact with foundation nodes The gap elements can be used to account for the unknown contact area, as follows Figure 15.3
must contain all points for which there is a possibility of establishing contact
node on the contactor is connected to each node of the target by a spring with a bilinear stiffness (Clearly, this element may miss the edge of the contact zone when
it does not occur at a node.) It follows that each node of the target is connected by
a spring to each of the nodes on the contactor The angle between the spring and
(15.5)
FIGURE 15.3 Point-to-point contact.
k
ij ijlower ij ij
ijupper ij ij
≥
δ
candidate contactor contact surface
candidate target contact surface
dSc
dSt
c1
α 31
k(g31)
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As an example of how the spring stiffness might depend upon the gap, consider
the function
(15.6)
attain a narrow transition range from the lower- to the upper-shelf values In the
in Figure 15.4
The total normal force on a contactor node is the sum of the individual
contact-element forces, namely
(15.7)
Clearly, significant forces are exerted only by the contact elements that are
“closed.”
FIGURE 15.4 Illustration of a gap-stiffness function.
k g
ij ij ij
ij ij ij ij
−
−
δ
0
1 2
2
f tj k g ij ij ij ij ij i
N c
kij
k0/2
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Trang 5Introduction to Contact Problems 199
15.3 POINT-TO-SURFACE CONTACT
We now briefly consider point-to-surface contact, illustrated in Figure 15.5 using a triangular element Here, target node t3 is connected via a triangular element to
geometric part of the stiffness matrix of a triangular elastic element The stiffness matrix of the element can be made a function of both gaps Total force normal to the target node is the sum of the forces exerted by the contact elements on the candidate contactor nodes
In some finite-element codes, the foregoing scheme is used to approximate the tangential force in the case of friction Namely, an “elastic-friction” force is assumed
in which the tangential tractions are assumed proportional to the normal traction through a friction coefficient This model does not appear to consider sliding and can be considered a bonded contact Advanced models address sliding contact and incorporate friction laws not based on the Coulomb model
15.4 EXERCISES
1 Consider a finite-element model for a set of springs, illustrated in the following figure A load moves the plate on the left toward the fixed plate
on the right
What is the load-deflection curve of the configuration?
For a finite-element model, an additional bilinear spring is supplied, as shown What is the load-deflection curve of the finite-element model?
finite-element model is close to the actual configuration
Why is the new spring needed in the finite-element model?
FIGURE 15.5 Element for point-to-surface contact.
candidate target contact surface
candidate contactor contact surface
element connecting node t3 with nodes c1 and c2
dSt
dSc
t2 t1
t3
ˆ
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2 Suppose a contact element is added in the previous problem, in which the stiffness (spring rate) satisfies
F
H L
k
k
k k
k
k g
ij ij
ij ij ij ij
−
−
δ
0
1 2